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.     

Population balance modeling. Promise for the future

Population balance modeling has received an unprecedented amount of attention during the past few years from both academic and industrial quarters because of its applicability to a wide variety of particulate processes. In this article, a fresh look is taken of the basic issues of the application of population balances towards strengthening the approach as well as widening the scope of their applications with regard to formulation, computational methods for solution, inverse problems, control of particle populations and stochastic modeling.     

Kinetic modelling of nickel powder precipitation by high-pressure hydrogen reduction

The active mechanisms in the precipitation of nickel powder by hydrogen reduction were investigated by means of mathematical models based on the moment form of the population balance equation (PBE). The objective of the work was to establish the mechanisms involved in powder formation and how these are affected by the presence of impurities. The effects of two major impurities were considered namely; iron, inherently present as ferrous sulphate, and a morphology modifier, a polyacrylic acid derivative used as an additive. Experiments were conducted on a laboratory and pilot-plant scale using a 0.5 and 75 L stainless steel autoclave, respectively. Nickel powder samples were collected from the autoclaves after each successive batch reduction (densification) and the particle size distribution (PSD) analysed using a laser diffraction technique. The PSD data was then transformed into moments and the experimental values were compared with those simulated using different models based on the moment form of the PBE. Five models were tested namely: (a) aggregation-only; (b) aggregation and growth; (c) nucleation and aggregation; (d) nucleation and growth and (e) aggregation and breakage. Under standard operating conditions, the process was best simulated by a size-independent aggregation and growth model in the early stages of the cycle, with breakage and growth becoming significant in later stages of the cycle when large particles have been formed. Crystallisation in the presence of Fe was characterised by a size-independent aggregation and growth model with varying degrees of nucleation depending on the Fe concentration and the available surface area. At modifier dosages of 0.25 and 5 vol% the process was best modelled by a size-independent aggregation and growth model coupled with a constant breakage frequency model. Based on the mathematical modelling results and evidence from scanning electron micrographs, the spherically shaped nickel powder particles were proposed to be formed through the formation of a pre-cursor by secondary aggregation followed by spherulitic growth. The degree of compactness of the spherulites was proposed to be determined by the number of active growth sites on the nickel particle surface. The morphology modifier was found to act as a growth inhibitor, decreasing the number of growth sites leading to more open spherulites. Iron was found to induce surface nucleation, thus, creating more growth sites on the particle surface and leading to more compact spherulites.     

Population balances for particulate processes—a volume approach

Population balance models have been used in chemical engineering since the 1960s and have evolved to become the most important tools for design and control of particulate processes. In this paper we show that the intrinsic particle parameter that determines changes in the process and should thus be included in the population balance is the particle volume. The basic population that is modeled should be the mass distribution, or the volume distribution if the density is constant. The population balance thus describes the change of the volume distribution of volume with time. Furthermore, we suggest that the “birth” and “death” terms that are often used to describe discrete events in particulate processes can almost always be replaced by a rate of change term.To design and control existing and future processes, a multi-dimensional population balance model is required. We propose a volume-based model in which the particle properties that are modeled are the volumes of solid, liquid, and air, respectively. In the most general case the model will consist of a properties vector and a distribution tensor. Depending on the complexity of the process, one or more of the properties may be omitted from the model. This is shown in three examples of increasing complexity: comminution, sintering, and granulation.     

A unified framework for detection, isolation and compensation of actuator faults in uncertain particulate processes

This paper presents a methodology for the robust detection, isolation and compensation of control actuator faults in particulate processes described by population balance models with control constraints and time-varying uncertain variables. The main idea is to shape the fault-free closed-loop process response via robust feedback control in a way that enables the derivation of performance-based fault detection and isolation (FDI) rules that are less sensitive to the uncertainty. Initially, an approximate finite-dimensional system that captures the dominant process dynamics is derived and decomposed into interconnected subsystems with each subsystem directly influenced by a single manipulated input. The decomposition is facilitated by the specific structure of the process input operator. A robustly stabilizing bounded feedback controller is then designed for each subsystem to enforce an arbitrary degree of asymptotic attenuation of the effect of the uncertainty in the absence of faults. The synthesis leads to (1) an explicit characterization of the fault-free behavior of each subsystem in terms of a time-varying bound on an appropriate Lyapunov function and (2) an explicit characterization of the robust stability region in terms of the control constraints and the size of the uncertainty. Using the fault-free Lyapunov dissipation bounds as thresholds for FDI in each subsystem, the detection and isolation of faults in a given actuator is accomplished by monitoring the evolution of the system within the stability region and declaring a fault if the threshold is breached. The thresholds are linked to the achievable degree of asymptotic uncertainty attenuation and can therefore be properly tuned by proper tuning of the controllers, thus making the FDI criteria less sensitive to the uncertainty. The robust FDI scheme is integrated with a robust stability-based controller reconfiguration strategy that preserves closed-loop stability following FDI. Finally, the implementation of the fault-tolerant control architecture on the particulate process is discussed and the proposed methodology is applied to the problem of robust fault-tolerant control of a continuous crystallizer with a fines trap.     

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.     

Techniques for the reconstruction of a distribution from a finite number of its moments

The reconstruction of a distribution knowing only a finite number of its moments is an extremely important but in practice still unsolved question for many fields of science (chemical and process engineering, electronic engineering, nuclear physics, image analysis, biotechnology…). Several methods have been proposed and corresponding mathematical formulations have been introduced in the literature during the last decades. Nevertheless, all these are generally limited to particular, often simple cases and require specific assumptions. It is indeed extremely difficult from a theoretical point of view (it is necessary, however, not sufficient, that all moments are available for a correct reconstruction) as well as from a practical point of view (ill-posed inverse problem) to find an accurate and relatively fast method which can be applied to all scientific areas. In the present paper, different possible methods (prescribed functions, discrete method, spline-based reconstruction) allowing such a reconstruction are explained, compared in terms of efficiency and accuracy, and validated for chemical engineering applications using examples with different degrees of difficulty.     

Integral approximation—an approach to reduced models for particulate processes

In this contribution, a model reduction technique for population balance systems describing particulate processes is presented. This technique is based on integral approximation and allows the derivation of highly accurate moment models. In contrast to other model reduction methods which can be found in literature, this integral approximation technique can be applied for arbitrarily complex phenomena specifications. The applicability of the presented method will be demonstrated for different example processes by comparing the dynamic behavior of the original population balance models with those of the derived reduced models of moments.     

Approximate ODE models for population balance systems

We propose an approximate polynomial method of moments for a class of first-order linear PDEs (partial differential equations) of hyperbolic type, involving a filtering term with applications to population balance systems with fines removal terms. The resulting closed system of ODEs (ordinary differential equations) represents an extension to a recently published method of moments which utilizes least-square approximations of factors of the PDE over orthogonal polynomial bases. An extensive numerical analysis has been carried out for proof-of-concept purposes. The proposed modeling scheme is generally of interest for control and optimization of processes with distributed parameters.     

The quadrature method of moments and its relationship with the method of weighted residuals

Population balance models are generally computationally intensive, so in many practical applications only a few moments of the density function are computed, minimizing the computational costs. Nevertheless, the moment formulation contains an excess of unknowns with respect to equations denoting a closure problem. One possible solution to this closure problem might be to apply a numerical quadrature approximation.In this work, the relationship between the quadrature approximation and the well-known method of weighted residuals (MWR) is discussed. An important result obtained in this work is that the problem of reconstruction of the density function is avoided using the MWR version of the quadrature approximation. Numerical experiments are performed in order to elucidate the advantages and disadvantages of the quadrature approximations.     

Operation of semi-batch emulsion polymerisation reactors: Modelling, validation and effect of operating conditions

A detailed dynamic model was developed for a styrene emulsion polymerisation semi-batch reactor to predict the evolution of the product particle size distribution (PSD) and molecular weight distribution (MWD) over the entire range of monomer conversion. A system exhibiting zero-one kinetics was employed, with the model comprising a set of rigorously developed population balance equations to predict monomer conversion, PSD and MWD. The modelling equations included diffusion-controlled kinetics at high monomer conversion where the transition from the zero-one regime to a pseudo-bulk regime occurs. The model predictions were found to be in good agreement with experimental results. Both particle growth and the PSD were found to be strongly affected by the monomer feedrate. Reactor temperature had a major influence on the MWD which was, however, insensitive to changes in the monomer feedrate. These findings were confirmed experimentally. As a result, it seems reasonable to propose that the use of the monomer feedrate to control the PSD and the reactor temperature to control the MWD are appropriate in practical situations. Consequently, an optimal monomer feed trajectory was developed off-line (using the validated reactor simulation) and verified experimentally by producing a polymer with specific PSD characteristics.     

Monitoring the particle size and shape in the crystallization of paracetamol from water

In this work, a technique capable of restoring bidimensional particle size distributions from images of the particles in suspension is applied to the seeded cooling crystallization of paracetamol from water. The effects of cooling rate and stirring rate on the final particle size and shape are studied and the average growth rates along different directions of particles are found to be strongly dependend on supersaturation. This observation is in line with previous studies, though in this work it has been established for the first time using populations of particles. The technique was capable of quantifying changes in particle size and shape, indicating particle sizes and shapes that correlated well with observations from electron microscopy images.     

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.     

Solution of bivariate population balance equations with high-order moment-conserving method of classes

In this work the high-order moment-conserving method of classes (HMMC) (Alopaeus et al., 2006) is extended to solve the bivariate Population Balance Equation (PBE). The method is capable of guaranteeing the internal consistency of the discretized equations for a generic moment set, including mixed-order moments of the distribution. The construction of the product tables in the case of aggregation, breakage and convection in internal coordinate space are discussed. Eventually, several test cases are considered to assess the accuracy of the method. The application to a realistic mass transfer problems in a liquid–liquid system is preliminarily discussed. The comparison with analytical solutions of pure aggregation problems shows that the proposed method is accurate with only a limited number of categories.     

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.     

Accurate size measurement of needle-shaped particles using digital holography

In this paper, a digital holography based method is proposed to accurately measure the length and orientation of the needle-shaped particles in solution. The method involves recording of the hologram and numerical reconstructions (focusing) of the hologram at several depths. An image analysis routine is then used to determine the length, location and orientation of the particles from the reconstructed images without any a priori information about the orientation of particles. The performance of the method is verified using a single fiber with known size and orientation. Subsequently, the proposed method is applied to a suspension of fibers, where the length measurements are found to be in good agreement with the true values. This proposed technique can overcome the shortcoming of existing 2D imaging tools, which can provide only the projected lengths of randomly oriented particles.     

Measurement of particle size and shape by FBRM and in situ microscopy

In this work a model is defined allowing for a rapid calculation of chord length distributions as well as the prediction of in situ microscopy data. Both calculations are done using the same underlying algorithm. The model assumes convex polyhedral particles that are defined by their vertices only, connected by straight lines, but imposes no further restrictions on particle geometry. Due to its speed, the model can easily be used for the prediction of experimental data from in situ monitoring tools based on whole particle populations, also with non-constant shape. The model has been verified using in situ microscopy to characterize a population of disc shaped particles.The applications of the model are focused on crystallization processes, but are not limited to these. Several relations between data measured by in situ instruments and the underlying multidimensional particle size distribution have been derived. The model is used extensively in a method that is presented allowing for the calculation of bidimensional growth rates from Focused Beam Reflectance Measurement or in situ microscopy measurements.     

Using the discrete element method to predict collision-scale behavior: A sensitivity analysis

Discrete element method (DEM) simulations have recently been used to investigate collision-scale measurements such as collision frequency and impact velocity distributions. These simulations are typically validated against particle velocity fields using experimental techniques such as particle image velocimetry or positron emission particle tracking. An important question that has not been addressed is whether validation of a macroscopic velocity field or solid fraction field also implies a validation of collision-scale measurements such as collision frequency. In this study, DEM measurements of solid fraction, shear rate, collision frequency, and impact velocity are made in a small region just beneath the free surface in a rotating drum. The effects of periodic drum length, particle stiffness, coefficient of restitution, and particle size are investigated. The solid fraction and shear rate do not vary with particle stiffness or coefficient of restitution over the range of values studied. However, the collision rate increases with increasing particle stiffness and coefficient of restitution. In addition, the average collision speed decreases as particles become stiffer or less elastic. The shear rate varies with particle size, but the average collision velocity remains constant. These findings indicate that validation against particle velocity and solid fraction fields does not necessarily imply validation of collision frequency and impact velocity. Indeed, the velocity and solid fraction fields were found to be relatively insensitive to a range of DEM contact stiffnesses and coefficients of restitution while the collision distributions were sensitive.     

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.