Solution of the population balance equation using constant-number Monte Carlo

We formulate a Monte Carlo simulation of the mean-field population balance equation by tracking a sample of the population whose size (number of particles in the sample) is kept constant throughout the simulation. This method amounts to expanding or contracting the physical volume represented by the simulation so as to continuously maintain a reaction volume that contains constant number of particles. We call this method constant-number Monte Carlo to distinguish it from the more common constant-volume method. In this work, we expand the formulation to include any mechanism of interest to population balances, whether the total mass of the system is conserved or not. The main problem is to establish connection between the sample of particles in the simulation box and the volume of the physical system it represents. Once this connection is established all concentrations of interest can be determined. We present two methods to accomplish this, one by requiring that the mass concentration remain unaffected by any volume changes, the second by applying the same requirement to the number concentration. We find that the method based on the mass concentration is superior. These ideas are demonstrated with simulations of coagulation in the presence of either breakup or nucleation.     

Analysis of four Monte Carlo methods for the solution of population balances in dispersed systems

Monte Carlo (MC) constitutes an important class of methods for the numerical solution of the general dynamic equation (GDE) in particulate systems. We compare four such methods in a series of seven test cases that cover typical particulate mechanisms. The four MC methods studied are: time-driven direct simulation Monte Carlo (DSMC), stepwise constant-volume Monte Carlo, constant number Monte Carlo, and multi-Monte Carlo (MMC) method. These MC's are introduced briefly and applied numerically to simulate pure coagulation, breakage, condensation/evaporation (surface growth/dissolution), nucleation, and settling (deposition). We find that when run with comparable number of particles, all methods compute the size distribution within comparable levels of error. Because each method uses different approaches for advancing time, a wider margin of error is observed in the time evolution of the number and mass concentration, with event-driven methods generally providing better accuracy than time-driven methods. The computational cost depends on algorithmic details but generally, event-driven methods perform faster than time-driven methods. Overall, very good accuracy can be achieved using reasonably small numbers of simulation particles, O(10³), requiring computational times of the order 10²−10³ s on a typical desktop computer.     

A population balance framework for nucleation, growth, and aggregation

Nucleation, growth, and aggregation for particulate systems are explored by distribution kinetics and population balances to build a new framework for understanding a range of natural and manufacturing phenomena. Nucleation is assumed to follow classical homogeneous theory or to be caused by heterogeneous nuclei added to the solution. Growth due to monomer addition from solution to clusters, and aggregation between clusters are both represented by integrals of the cluster distribution. When growth and aggregation rate coefficients are independent of cluster size, the population balance equations are readily solved by the moment method. Equations for steady-state well-mixed flow and unsteady-state closed (batch) vessels have relatively straightforward solutions. By incorporating solute (monomer) depletion, the results afford reasonable behavior for the cluster number and mass concentration. The monomer addition terms are shown to be consistent with (and a generalization of) conventional differential growth and growth dispersion expressions.     

Monte Carlo simulation of particle breakage process during grinding

The Monte Carlo method is quite useful in the modeling of particulate systems. It is used here to simulate the particle brekage process during grinding that can be represented by a population balance equation. The simulation technique is free from discretization of time or size. The results of simulation under restricted conditions of grinding compare very well with the available analytical solution of the population balance equation. The procedure is extended to simulate the grinding process in its entirety. This method provides an alternative to the modeling of the grinding process where the governing population balance equation cannot be readily solved.     

Method of moments with interpolative closure

The article summarizes the principal details of a method of moments with interpolative closure. This is a mathematically rigorous yet numerically economical approach to particle dynamics, describing time evolution of a particle ensemble undergoing simultaneous nucleation, coagulation, and surface growth. The method was introduced some time ago and since then has undergone further development as well as extensive testing in reactive flow simulations of practical systems. These results, scattered over quite diverse literature, are presented here in a unified form, focussing on logical development rather than on chronological order. In addition, the validity of the numerical approach is addressed on rigorous mathematical grounds. Also discussed are method shortcomings along with possible directions to their resolution.     

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.     

On the stochastic simulation of particulate systems

The stochastic chemical kinetics approach provides one method of formulating the stochastic crystallization population balance equation (PBE). In this formulation, crystal nucleation and growth are modelled as sequential additions of solubilized ions or molecules (units) to either other units or an assembly of any number of units. Monte Carlo methods provide one means of solving this problem. In this paper, we assess the limitations of such methods by both (1) simulating models for isothermal and nonisothermal size-independent nucleation, growth and agglomeration; and (2) performing parameter estimation using these models. We also derive the macroscopic (deterministic) PBE from the stochastic formulation, and compare the numerical solutions of the stochastic and deterministic PBEs. The results demonstrate that even as we approach the thermodynamic limit, in which the deterministic model becomes valid, stochastic simulation provides a general, flexible solution technique for examining many possible mechanisms. Thus the stochastic simulation permits the user to focus more on modelling issues as opposed to solution techniques.     

A model for turbulent binary breakup of dispersed fluid particles

Accurate predictions of particle size distributions, and therefore of the underlying processes of fluid particle breakup and coalescence are of vital importance in process design, but reliable procedures are still lacking. The present paper aims at developing a modular formulation for the turbulent particle breakup process. The model is to be included in a population balance model which is formulated such as to facilitate the direct future implementation into a full multifluid CFD model.The breakup process is described without introducing adjustable parameters. The current model is a further development of an existing model by Luo and Svendsen (AIChE J. 42 (5) (1996) 1225), which has been expanded and refined, and where an inherent weakness regarding the breakup rate for small particles and small daughter particle fragments are removed. A new criterion regarding the kinetic energy density of the colliding turbulent eddy causing breakup has been introduced. This new criterion is a novel concept describing the breakup process. The details are thoroughly discussed together with possible further modifications. The results from the new model are encouraging because the breakup rate is greatly reduced when the dispersed fluid particles are reduced in size. Further, the response to changes in system variables is reasonable and the distribution of daughter sizes vary in a reasonable way for the different collision possibilities.     

Identification of thermal zones and population balance modelling of fluidized bed spray granulation

Air temperature measurements in a fluidized bed of glass beads top sprayed with water showed that conditions for particles growth were fulfilled only in the cold wetting zone under the nozzle which size and shape depended on operating conditions (liquid spray rate, nozzle air pressure, air temperature, and particles load). Evolution of the particle size distribution during agglomeration was modelled using population balance and representing the fluidized bed as two perfectly mixed reactors exchanging particles with particle growth only in the one corresponding to the wetting zone. The model was applied to the agglomeration of non-soluble glass beads and soluble maltodextrin particles spraying respectively an acacia gum solution (binder) and water. Among the three adjustable parameters, identified from experimental particle size distributions evolution during glass beads agglomeration, only one describing the kinetics of the size distribution evolution depended on process variables. The model allowed satisfying simulation of the evolution of the particle size distribution during maltodextrin agglomeration.     

A new technique to determine rate constants for growth and agglomeration with size- and time-dependent nuclei formation

Fluidized bed spray agglomeration is a particle size enlargement process where particles stick together with the help of spraying binder. High impact forces between particles lead to attrition. Attrition may be modeled as poly-disperse nucleation. Furthermore, particulate event like over-spray leads to the formation of particles in a wide range of volume. A new technique for the determination of agglomeration, growth and nucleation parameters is presented in this work. The model is based on a previous approach which takes mono disperse nuclei formation in the smallest class into account. Frequently in crystallization processes, nucleation is assumed to be mono-disperse. The technique presented here incorporates nuclei formation in a certain range of volume. It is quite general and applicable to consider size- and time-dependent nuclei formation. For two particular cases of growth and agglomeration including size-dependent nuclei formation, simulation data was generated by continuous feeding of nuclei in a certain range to demonstrate the capability of parameter extraction of the model. Further, the new technique is applied to extract rate constants from experimental data measured during fluidized bed spray agglomeration. This technique is also useful for the prediction of bimodal behavior of particle size distributions (PSD).     

Direct simulation of Monte Carlo analysis of nano-floating effect on diamond-coated surface

It is well known that the diamond-coated sliding surface has good quality with respect to sliding movement. We have ensured that the partly polished diamond-coated sliding surface shows much lower coefficients of the kinetic friction than the teflon-coated sliding surface in our experiments. Also, we confirmed that the gas-layer lubrication effect appears due to the gas in the concavo–convex sliding surface when its relative counter speed reaches in the order of 1 m/s. Then, in this study, the numerical investigation was performed to simulate the gas-layer lubrication effect on the sliding surface. The microscopic flow conducted in this study has very small length scale which is in the order of the mean-free path of the air at the atmosphere pressure. Consequently, the Navier–Stokes equations are not applicable to analyze the present microscopic flow because the continuum flow is assumed. In addition, we observed many unpolished concave parts on the diamond-coated surface after polishing, and the vertical flow will appear induced by such complicated geometric boundary. For this reason, any analytical method that assumes quasi-parallel flow cannot be applied to the present flow problem. Herein, we used direct simulation of Monte Carlo (DSMC) method to handle the nano/micro flow, which cannot be assumed to be continuum. We clarified the feature of gas-layer lubrication on the sliding surface, which would be dominated by the surface roughness, the relative counter speed of surfaces, and the surface roughness including its surface geometry in the nano/micro scales.     

Population balance discretization for growth,attrition, aggregation,breakage and nucleation

This paper presents a new discretization method to solve one-dimensional population balance equations (PBE) for batch and unsteady/steady-state continuous perfectly mixed systems. The numerical technique is valid for any size change mechanism (i.e., growth, aggregation, attrition, breakage and nucleation occurring alone or in combination) and different discretization grids.The developed strategy is based on the moving pivot technique of Kumar and Ramkrishna and the cell-average method of Kumar et al. A novel contribution is proposed to numerically handle the growth and attrition terms, for which a new representation of the number density function within each size class is developed. This method allows describing the number particle fluxes through the class interfaces accurately by preserving two sectional population moments.By comparing the numerical particle size distributions with analytical solutions of one-dimensional PBEs (including different size change mechanisms and particle-size dependent kinetics), the accuracy of the proposed numerical method was proved.     

Computationally efficient solution of population balance models incorporating nucleation, growth and coagulation: application to emulsion polymerization

A computationally efficient solution technique is presented for population balance models accounting for nucleation, growth and coagulation (aggregation) (with extensions for breakage). In contrast to earlier techniques, this technique is not based on approximating the population balance equation, but is based on employing individual rates of nucleation, growth and coagulation to update the PSD in a hierarchical framework. The method is comprised of two steps. The first step is the calculation of the rates of nucleation, growth and coagulation by solving an appropriate system of equations. This information is then used in the second step to update the PSD. The method effectively decomposes the fast and the slow kinetics, thereby eliminating the stiffness in the solution. In solving the coagulation kernel, a semi-analytical solution strategy is adapted, which substantially reduces the computational requirement, but also ensures the consistency of properties such as the number and mass of particles.     

Constant number Monte Carlo simulation of population balances with multiple growth mechanisms

We present a complete simulation scheme for particulate processes based on the constant number Monte Carlo methodology. Specifically, the proposed scheme can be applied towards the solution of population balances that include nucleation, coagulation and surface deposition, coupled to chemical reactions. The synthesis of titania (TiO₂) by flame oxidation of TiCl₄ is employed as a comparison basis of the relative advantages and weaknesses of Monte Carlo against more classical numerical approaches. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Comparison of numerical methods for the simulation of dispersed phase systems

This contribution deals with a new numerical method for an accurate and efficient simulation of particulate processes. As an example for dispersed phase systems a detailed model for crystallization processes is considered. After the model derivation, which is based solely on physical principles, different techniques for the numerical simulation of the mathematical model are discussed. State-of-the-art finite volume schemes based on the ‘method of lines’ approach are compared to the recently published ‘Space-time conservation element and solution element method’. The presented simulation results show a strong dependence on the chosen numerical method. Guidelines for a proper selection of numerical methods for the treatment of population balance based models are given.     

Experimental characterization and population balance modelling of the dense silica suspensions aggregation process

Concentrated suspensions of nanoparticles subjected to transport or shear forces are commonly encountered in many processes where particles are likely to undergo processes of aggregation and fragmentation under physico-chemical interactions and hydrodynamic forces. This study is focused on the analysis of the behavior of colloidal silica in dense suspensions subjected to hydrodynamic forces in conditions of destabilization.A colloidal silica suspension of particles with an initial size of about 80 nm was used. The silica suspension concentration was varied between 3% and 20% of weight. The phenomenon of aggregation was observed in the absence of any other process such as precipitation and the destabilization of the colloidal suspensions was obtained by adding sodium chloride salt.The experiments were performed in a batch agitated vessel. The evolution of the particle size distributions versus time during the process of aggregation was particularly followed on-line by acoustic spectroscopy in dense conditions. Samples were also analyzed after an appropriate dilution by laser diffraction. The results show the different stages of the silica aggregation process whose kinetic rates depend either on physico-chemical parameters or on hydrodynamic conditions. Then, the study is completed by a numerical study based on the population balance approach. By the fixed pivot technique of Kumar and Ramkrishna [1996. On the solution of population balance equations by discretization—I. A fixed pivot technique. Chemical Engineering Science 51 (8), 1311-1332], the hypothesis on the mechanisms of the aggregation and breakage processes were justified. Finally, it allows a better understanding of the mechanisms of the aggregation process under flowing conditions.     

Two methods for sensitivity analysis of coagulation processes in population balances by a Monte Carlo method

We consider two stochastic simulation algorithms for the calculation of parametric derivatives of solutions of a population balance equation, namely, forward and adjoint sensitivity methods. The dispersed system is approximated by an N-particle stochastic weighted ensemble. The infinitesimal deviations of the solution are accounted for through infinitesimal deviation of the statistical weights that are recalculated at each coagulation. In the forward method these deviations of the statistical weights immediately give parametric derivatives of the solution. In the second method the deviations of the statistical weights are used to calculate a finite-mode approximation of the linearized version of the population balance equation. The linearized equation allows for the calculation of the eigenmodes and eigenvalues of the process, while the parametric derivatives of the solution are given by a Lagrange formalism.     

A moment methodology for coagulation and breakage problems: Part 3—generalized daughter distribution functions

Population balances for simultaneous coagulation and breakage (and their analogs, e.g., polymerization and depolymerization) are employed in describing many systems including aerosols, powders and polymers, and many unit operations including reactors, crystallizers, and size reduction/enlargement equipment. The birth term for the breakage process is usually formulated in terms of a distribution of breakage products known as the daughter distribution. There are many daughter distribution forms proposed in the literature in part because these distributions are notoriously difficult to determine experimentally. Here, a generalization of these forms is developed for multi-particle breakup which has the flexibility to represent a wide variety of distribution shapes. The simplicity of the generalized expression renders the population balance equations for simultaneous coagulation and breakage accessible to analytical attack, leading to an analytical expression for the fine end of the steady-state product size distribution. This expression has potential utility in both design and analysis of experiments aimed at measuring daughter distribution parameters.     

Part IV: Dynamic evolution of the particle size distribution in particulate processes. A comparative study between Monte Carlo and the generalized method of moments

The present work provides a comparative study on the numerical solution of the dynamic population balance equation (PBE) for batch particulate processes undergoing simultaneous particle aggregation, growth and nucleation. The general PBE was numerically solved using three different techniques namely, the Galerkin on finite elements method (GFEM), the generalized method of moments (GMOM) and the stochastic Monte Carlo (MC) method. Numerical simulations were carried out over a wide range of variation of particle aggregation and growth rate models. The performance of the selected techniques was assessed in terms of their numerical accuracy and computational requirements. The numerical results revealed that, in general, the GFEM provides more accurate predictions of the particle size distribution (PSD) than the other two methods, however, at the expense of more computational effort and time. On the other hand, the GMOM yields very accurate predictions of selected moments of the distribution and has minimal computational requirements. However, its main disadvantage is related to its inherent difficulty in reconstructing the original distribution using a finite set of calculated moments. Finally, stochastic MC simulations can provide very accurate predictions of both PSD and its corresponding moments while the MC computational requirements are, in general, lower than those required for the GFEM.