Population Balances: Theory and Applications to Particulate Systems in Engineering

Population Balances

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Population Balances Theory and Applications to Particulate Systems in Engineering

Doraiswami Ramkrishna Purdue University School of Chemical Engineering West Lafayette, Indiana

ACADEMIC PRESS A Harcourt Science and Technology Company

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This book is printed on acid-free paper. @ Copyright © 2000 by Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to the following address: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887-6777. ACADEMIC PRESS A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, CA 92101-4495 USA http://www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London, NWl 7BY, UK Library of Congress Catalog Card Number: 00-100463 ISBN: 0-12-576970-9 Printed in the United States of America 00 01 02 03 04 05 EB 9 8 7

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With Love to Geetha For Her Fealty, Fondness and Forbearance

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CONTENTS

Foreword Preface

Chapter 1

ix xiii

Introduction

References

Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12

1 5

The Framework of Population Balance

7

Particle State Vector The Continuous Phase Vector The Number Density Function The Rate of Change of Particle State Vector The Particle Space Continuum The Reynolds Transport Theorem The Population Balance Equation Population Balance Equation for Open Systems Equation for the Continuous Phase Vector Random Changes in Particle State Formulation of Population Balance Models Concluding Remarks References

8 10 11 12 13 14 15 22 24 26 29 45 45

VII

viii

Contents

Chapter 3 3.1 3.2 3.3

Birth and Death Functions

47

Birth and Death Rates at the Boundary Breakage Processes Aggregation Processes References

48 49 70 114

Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6

Existence of Solution The Method of Successive Approximations The Method of Laplace Transforms The Method of Moments and Weighted Residuals Discrete Formulations for Solution Monte Carlo Simulation Methods References

Chapter 5 5.1 5.2 5.3 5.4

6.3 6.4

Index

Inverse Problems in Population Balances

The Inverse Breakage Problem: Determination of Breakage Functions The Inverse Aggregation Problem: Determination of the Aggregation Frequency Determination of Nucleation and Growth Kinetics Other Inverse Problems References

Chapter 7 7.1 7.2 7.3 7.4 7.5

Similarity Behavior of Population Balance Equations

The Self-Similar Solution Similarity Analysis of Population Balance Equations Self-Similarity in Systems with Breakage and Aggregation Processes Self-Similarity in Systems with Growth References

Chapter 6 6.1 6.2

The Solution of Population Balance Equations

The Statistical Foundation of Population Balances

The Master Density Function The Master Density Equation for Particulate Processes Stochastic Equations of Population Balance On the Closure Problem Some Further Considerations of Correlated Behavior References

117 118 123 128 136 144 167 192 197 197 197 213 217 219 221 222 235 257 264 272 275 277 288 299 324 339 349

351

Foreword

A recent conference sponsored by the United Engineering Foundation, Inc. of New York brought together a group of about 40 engineers and scientists with remarkably diverse areas of expertise. The areas represented included comminution of ores and other solids; recovery and purification of solids by crystallization from melts and solutions; behavior of polymerization reactors; formation of monodisperse colloidal suspensions; formation of powders for use in paints, pigments, pharmaceuticals, etc.; deposition of proteinaceous material on the surfaces of dairy processing equipment; flocculation in water treatment processes; sedimentation; formation of smoke and soot during combustion of fuels; growth of microbial and cell populations; the nature of crystal growth in various geological situations; and numerical solution techniques for partial differential-integral equations. The conference extended over four and half days and one would think that it would have been difficult to get good attendance at all of the sessions especially since they were held in January—dismal in the northern latitudes where many of the attendees came from, but distractingly pleasant in the Kona District of the Big Island of Hawaii where the conference was held. Attendence was not a problem, however, and was essentially perfect. What was it that kept a group of people with such diverse professional backgrounds and interests coming back to the successive sessions of the conference? The common theme that attracted them was that everyone had to deal with a collection of objects—molecules, particles, cells, etc. — having a distribution of properties that changes in time and perhaps also in space.

X

Foreword

The conference attendees wanted to understand the natural laws that govern the evolution of the distributions that concern them: the engineers so that they could control the distributions produced by various processing operations, and the scientists not only for the same reason as the engineers, but also because these laws are of fundamental scientific importance. Of course, the atomic and molecular processes that are involved in, for example, crystal nucleation, growth, agglomeration, and breakage are entirely different from those that occur in the growth and reproduction of a population of microbes or animal cells. Such differences can, and have, kept knowledge compartmentalized. Nevertheless, certain general concepts, such as those embedded in so-called population balance equations, are appHcable to all of the processes dealt with at the Kona Conference. Anyone who desires a hohstic view of a situation that involves a collection of objects with an evolving distribution of properties will need to understand those concepts and the techniques of population balance modeling. Population balance equations are not new. Perhaps the famous Boltzmann equation of chemical physics was the first and is now more than a century old. I became aware of the importance and difficulties of these equations when I began my research into how to model the growth of microbial populations about 40 years ago, and the author of the present book got his start in the theory of population balance equations when he did his doctoral and postdoctoral work with me and my lamented colleague Professor Henry M. Tsuchiya. Many people from other fields also started using population balance equations around that time. In retrospect, it is clear to me that I did my work in isolation, not knowing and perhaps not even caring that many other people were being confronted with the same or similar conceptual and computational difficulties. Other workers may have similar confessions to make, but I shall not attempt to speak for them. The Kona Conference and many other events that I could cite show that it is now time to end this compartmentalization of knowledge, get our act together, and understand that there is a common body of concepts and techniques that apply to a large domain of very important processes and situations. Professor Doraiswami Ramkrishna has made a major contribution to the needed unification of theory and computational techniques of population balances with the preparation of his book Population Balances: Theory and Applications to Particulate Systems in Engineering. It should be, and I hope it will be, the source that workers from many diverse fields turn to when they seek to learn the concepts and techniques of population balance modeling of particulate systems.

Foreword

xi

Professor Ramkrishna has worked on many of the problems, not just on one class of problems, of modeling particulate processes for a long time and writes from a broad perspective of the field. He brings to it a breadth and depth of mathematical and statistical knowledge that is far beyond mine and, probably, beyond that of most of the field's practitioners. This is particularly evident in the later chapters of the book which deal with the stochastic aspects of particulate processes. I will say frankly that this book is not meant for people who demand a Sesame Street approach to learning. A book on population balance modehng using such an approach would have to be shallow and would not get to the bottom of things. The expenditure of some mental "blood and toil, tears and sweat" therefore will be required from those readers who want to get to the foundations of population balance modehng. But that is true of anything of large intellectual value, and I am sure that those who persevere, who are wilhng to make the effort, will be richly rewarded by their study of this book. A. G. Fredrickson University of Minnesota

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Preface

One review of the proposed contents of this book had wondered if the subject matter, in view of its expansiveness, would be better served by some collaboration. While this is indisputably true and a lone author is apt to produce the excuse of the inertia of a joint effort, I am eager to clarify that this book is the consequence of my association with several individuals including mentors, colleagues, and students. This acknowledgement arises from a deep sense of realization of the truth behind the dedication^ by an author, mentor, and friend who said, "to our students who have taught us much more than we have taught them." I will therefore endeavor to narrate how a substantial part of this treatment of the subject grew out of my students' efforts. Population balance may be regarded either as an old subject that has its origin in the Boltzmann equation more than a century ago, or as a relatively new one in light of the variety of applications in which engineers have more recently put it to use. The latter trend is, of course, associated with the realization that the methodology of population balances is indispensable for a rational treatment of dispersed phase processes in engineering. Yet, its recognition as a salient component of modeling in transport phenomena and reaction engineering has been relatively slow even to this R. Aris and N. R. Amundson, "Mathematical Methods in Chemical Engineering," Volume 2. First Order Partial Differential Equations and Applications, Prentice-Hall, Englewood Cliffs, New Jersey, 1973.

xiv

Preface

day. Transport and chemical reaction in a dispersed phase system occur in conjunction with the evolutionary processes producing the dispersion. It is the capacity of population balance to address the evolutionary aspects of a dispersion that affords its distinctive value to the analysis of dispersed phase systems. Because this book is a general treatment of population balance concepts, applications have been used to demonstrate a generic issue rather than to be comprehensive in any sense about the area of application. Thus, it differs from the books referenced in Chapter 3 — one by Randolph and Larson which focussed on crystallization, and another by Hidy and Brock that addressed aerocolloidal systems. Therefore, in addition to being a reference book on population balances, this book may also be suitable for use as a collateral text in a course on transport phenomena or chemical reaction engineering. The application of population balances has been growing enormously in recent times and the author sincerely regrets his limited coverage of this fast expanding literature. The author's first encounter with the subject occurred at the University of Minnesota during his graduate studies in the early 1960s while working with biological populations under the tutelage of Professors Arnie Fredrickson and Henry Tsuchiya. This introduction to the subject was most timely, for it was just then that chemical engineers began a formal association with the concepts of population balance. Notable among these efforts were those of Hulburt and Katz (1964), and Randolph and Larson (1964), who were the first to raise the issue of a generic population balance in chemical engineering. Although the publication of Fredrickson et ai, (1967), which followed soon after, addressed biological populations, its generality had considerable import for the study of other populations as well. That the natural relationship of population balance to the analysis of dispersed phase processes called for development of the tool on several fronts was one of the issues which consumed the author's early academic career. Fundamental understanding of the statistical background of population balances depended on the theory of stochastic point processes that had its roots in the physics literature. For one, who had acquired his background of stochastic processes by methods that were somewhat random, the able support of his good friend and colleague. Professor Jay Borwanker of the Department of Mathematics at the Indian Institute of Technology (IIT), Kanpur, was more than an ordinary scientific collaboration. The author fondly recalls his friendship and association, and regrets his untimely demise towards the end of the last year.

Preface

xv

An aspect of population balances that has often intimidated chemical engineers, who are famihar with differential equations, is the integrodifTerential nature of the equations describing population balance models. Establishing mathematical tractability of population balance models was therefore an attractive issue with which to begin. The author recalls G. Subramanian, his first ever graduate student, who solved the transient population balance model of a microbial culture by the method of weighted residuals in his Master's thesis in 1971. This problem is an order of magnitude more difficult than most that have served even in recent times to demonstrate the efficacy of current solution techniques. Soon after, Bharat Shah's arrival to work with Professor Borwanker and the author saw the development of Monte Carlo techniques for population balances that has remained the main source of inspiration for other papers on simulation techniques. P. N. Singh had found improved methods for the choice of trial functions for the method of weighted residuals, but this is an issue with an everlasting potential for refinement. Following the author's arrival at Purdue University in the Fall of 1976, Kendree Sampson had found ways of improving accuracy through fine tuning of collocation points by engineering trial functions for the method of orthogonal collocation. Sanjeev Kumar, who joined the group as a post-doctoral fellow several years later, revived computational efforts on population balance equations with new insights on discretization methods. He played a significant role in influencing our research group at Purdue and particularly T. Pirog's Ph.D. work on the application of population balance to destabilization of emulsions. To a long stretch of collaboration with G. Narsimhan both as a graduate student at IIT Kanpur and a post-doctoral fellow at Purdue, the author owes much to the early development of the inverse problem for drop breakage and the numerous experimental techniques for measuring drop size distributions. From a theoretical viewpoint, R. Muralidhar's role was significant in the adaptation of regularization methods for inverse problems and their extension to aggregating systems. This, together with his work on the use of stochastic methods on aggregation efficiencies, laid the groundwork for subsequent experimental work by Harold Wright and Tom Tobin on drop coalescence, and Arun Sathyagal's work on drop breakage. Alan Mahoney's recent solution of the inverse problem for nucleation and particle growth from dynamic particle size distributions is notably free from dependence on self-similarity and therefore represents a promising approach. The author owes a special sense of acknowledgement to his collaboration with Professors R. Kumar and K. S. Gandhi at the Indian Institute of

xvi

Preface

Science (IISc), Bangalore, where he has had a continuing interaction from the early 1980s. This collaboration began with modeling of coalescence efficiencies through P. K. Das's doctoral dissertation at IISc and his subsequent post-doctoral effort at Purdue. Subsequently, the collaboration led to stochastic applications of population balance through the doctoral theses of S. Manjunath and R. Bandyopadhyaya at IISc. In this connection, the author gratefully acknowledges financial support from the Indian Institute of Science, Bangalore, the Jawaharlal Nehru Centre (JNC) for Advanced Scientific Research at Bangalore, the TOKTEN program of the United Nations, the National Science Foundation, and Purdue University which made this collaboration possible. In particular, the author acknowledges Professor C. N. R. Rao who, as president of the JNC, provided special encouragement with a visiting professorship to the author on numerous occasions. There are several individuals to whom the author is indebted for helping with the preparation of the manuscript for this book. However, an especially profound acknowledgement is due to my wife whose forbearance and support have been invaluable. This book is dedicated to her. I am overwhelmingly grateful to my children for their consideration and understanding, and ever conscious of the inspiration of my parents and siblings. The author thanks Suzie Flavin for her assistance with numerous aspects of the preparation of this book's manuscript, Tanmay Lele for tracking down several references, and Alan Mahoney for help with the preparation of several figures.

CHAPTER 1

Introduction

Engineers encounter particles in an innumerable variety of systems. The particles are either naturally present in these systems or engineered into them. In either case, the particles often significantly affect the behavior of such systems. In many other situations, systems are associated with processes in which particles are formed either as the main product or as a by-product. We will refer to systems containing particles as dispersed phase systems or particulate systems regardless of the precise role of the particles in them. Analysis of a particulate system seeks to synthesize the behavior of the population of particles and its environment from the behavior of single particles in their local environments. The population is described by the density of a suitable extensive variable, usually the number of particles, but sometimes (with better reason) by other variables such as the mass or volume of particles. The usual transport equations expressing conservation laws for material systems apply to the behavior of single particles. Population balances are essential to scientists and engineers of widely varying disciplines. They are of interest to physicists (astrophysicists, highenergy physicists, geophysicists, meteorologists) and chemists (colloidal chemists, statistical mechanicians). Biophysicists concerned with populations of cells of various kinds, food scientists dealing with preparations of emulsions or sterilization of food all have an indispensable need for population balances.

2

1. Introduction

Among engineers, population balance concepts are of importance to aeronautical, chemical, civil (environmental), mechanical, and materials engineers. Chemical engineers have put population balances to the most diverse use. Applications have covered a wide range of dispersed phase systems, such as sohd-Hquid dispersions (although with incidental emphasis on crystallization systems), and gas-hquid, gas-solid, and Hquid-liquid dispersions. Analyses of separation equipment such as for liquid-liquid extraction, or sohd-Hquid leaching; and reactor equipment, such as bioreactors (microbial processes) fluidized bed reactors (catalytic reactions), and dispersed phase reactors (transfer across interface and reaction) all involve population balances. Although most of the foregoing applications are known, it is significant to cite more modern applications such as the preparation of ceramic mixtures and fine particles (nanoparticles) for a variety of applications, in which population balances play a critical role in the analysis, design and control of such processes. For example, the manufacture of superconducting ceramic mixtures requires very tight specifications on their composition on a fine scale of mixing. Coprecipitation of the oxide mixture from the liquid phase represents a promising process for the same. Thus, the use of microemulsions involving reverse micelles or vesicles for conducting precipitation in small systems must be guided by the use of stochastic population balance concepts. It will be the objective of this monograph to expound deterministic as well as stochastic population balances for numerous applications. Although the chief distinguishing feature of this monograph is its wide scope of population balance applications, it will be essential to impose some constraints on the topics to be covered. Most significantly, it will exclude the vast area of the fluid mechanics of dispersions, even though it falls within the scope of population balances, for this field has had a growth of its own mainly through the efforts of hydrodynamicists. Examples are the books of Happel and Brenner (1973) and of Kim and Karrila (1991). The treatment in these applications is deeply linked to the solution of the Navier-Stokes equations around one or more submerged bodies with the ultimate objective of calculating effective properties of dispersions. In the application of population balances, one is more interested in the distribution of particle populations and their effect on the system behavior. In this sense, other examples of multiphase flows in which substantial variation of the void fraction in the flow domain affects the flow behavior would seem more naturally within the scope of our

1. Introduction

3

treatment.^ Such applications will not be pursued in depth. Instead, we shall endeavor to treat population balance formulations in the context of particle coordinates more general than physical location such as those "internal" to the particle. Such an approach greatly widens the scope of applications, which constitutes the main thrust of this monograph.^ Another distinguishing feature of the systems of interest to this book is that they contain particles which are continually being created and destroyed by processes such as particle breakage and agglomeration. The phenomenological treatment of such breakage and aggregation processes is of focal interest into the population balance modehng of such systems. The particles of interest to us have both internal and external coordinates. The internal coordinates of the particle provide quantitative characterization of its distinguishing traits other than its location while the external coordinates merely denote the location of the particles in physical space. Thus, a particle is distinguished by its internal and external coordinates. We shall refer to the joint space of internal and external coordinates as the particle state space. One or more of either the internal and/or external coordinates may be discrete while the others may be continuous. Thus, the external coordinates may be discrete if particles can occupy only discrete sites in a lattice. There are several ways in which the internal coordinates may be discrete. A simple example is that of particle size in a population of particles, initially all of uniform size, undergoing pure aggregation, for in this case the particle size can only vary as integral multiples of the initial size. For a more exotic example, let the particle be an emulsion droplet (a liquid) in which a precipitation process is carried out producing a discrete number of precipitate "particles." Then the number of precipitate particles may serve to describe the discrete internal coordinate of the droplet, which is the main entity of population balance. Fundamental to the formulation of population balance is the assumption that there exists a number density of particles at every point in the particle For example, the treatment of the so-called "Boycott effect" by Acrivos and Herbolzheimer (1979) considers the flow of a dispersion in which particles segregate to create stratification between two fluids, one of which is clear and the other packed with particles. The first paper to appear in the chemical engineering literature on the general formulation of population balance is by Hulburt and Katz (1964), although slightly earlier a short communication appeared by Randolph and Larson (1964). At the same time, the author is personally aware also of an unpublished document by A. G. Fredrickson containing a general formulation at the University of Minnesota, which was subsequently pubhshed with special focus on microbial populations (Fredrickson et al, 1967).

4

1. Introduction

state space. The number of particles in any region of the state space is obtained by integrating the number density over the region desired. In a discrete region the integration amounts to simply summing over the discrete states in the region. The population balance equation is an equation in the foregoing number density and may be regarded as representing a number balance on particles of a particular state. The equation is often coupled with conservation equations for entities in the particles' environmental (or continuous) phase. The population balance equation basically accounts for various ways in which particles of a specific state can either form in or disappear from the system. When particle states are continuous, then processes, which cause their smooth variation with time, must contribute to the rates of formation and disappearance of specific particle types. Such processes may be viewed as convective processes since they result from convective motion in particle state space. They cause no change in the total number of particles in the system except when particles depart from the boundaries of the system. Other ways in which the number of particles of a particular type can change is by processes that create new particles ("birth" processes) and destroy existing particles ("death" processes). Birth of new particles can occur due to breakage or splitting processes, aggregation processes, nucleation processes and so on. Breakage and aggregation processes also contribute to death processes, for a particle type that either breaks (into other particles) or aggregates with another particle no longer exists as such following the event. The phenomenological content of population balance models lies in the convective processes as well as the birth-and-death processes. Consequently, an issue of considerable importance to this book is elucidation of the methodology for modehng of the above processes. These models pertain to the behavior of individual particles, singly (as, for example, in particle splitting due to forces arising in the environmental phase), in pairs (as in binary aggregation processes), and so on but with the important proviso that it be considered in the population setting in which it occurs. The number density function, along with the environmental phase variables, completely determines the evolution of all properties of the dispersed phase system. The population balance framework is thus an indispensable tool for deahng with dispersed phase systems. This book seeks to address the various aspects of the methodology of population balance necessary for its successful application. Thus Chapter 2 develops the mathematical framework leading to the population balance equation. It

References

5

goes into the factors required for the choice of the particle state space with various examples. Chapter 3 delves further into issues of formulation such as those of birth and death functions for breakage systems as well as aggregating systems. Chapter 4 deals with methods for the solution of population balance equations. It also probes into Monte Carlo simulation techniques. In Chapter 5, the self-similarity behavior of solutions to the population balance equations is considered with various examples. The subject of inverse problems for the identification of population balance models from experimental data on dynamic particle distributions is treated in Chapter 6. The exploitation of self-similar solutions in inverting experimental data is of particular interest. Chapter 7 is concerned with the statistical foundation of population balance models. The chapter deals with master density formulations leading to mean field equations for the average behavior of the system and fluctuations about average behavior. This represents the subject of stochastic population dynamics applicable to small systems the relevance of which to engineering is discussed. Departures of the mean field equations from population balance equations are demonstrated. The mean field equations so obtained suffer from lack of closure. Closure approximations are presented suggesting more complex mean field equations than population balance along with applications. Finally, Chapter 7 also presents some formulations of population balance models applicable to biological systems in which correlated or anticorrelated behavior between siblings and between parent and offspring can be accommodated. Examples of applications pervade throughout the different chapters in the book introduced primarily as an aid to understanding the different aspects of population balance modeling.

REFERENCES Acrivos, A., and E. Herbolzheimer, "Enhanced Sedimentation in Settling Tanks with Inclined Walls," J. Fluid Meek 92, 435-457 (1979). Fredrickson, A. G., D. Ramkrishna, and H. M. Tsuchiya, "Statistics and Dynamics of Procaryotic Cell Populations," Math. Biosei. 1, 327-374 (1967). Happel, J., and H. Brenner, Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media, Noordhoif International Publishing, Leyden, 1973.

6

1. Introduction

Hulburt, H. M., and S. L. Katz, "Some Problems in Particle Technology. A Statistical Mechanical Formulation," Chem. Eng. Sci. 19, 555-574 (1964). Kim, S., and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Butterworth-Heinemann, Boston, 1991. Randolph, A. D., and M. A. Larson, "A Population Balance for Countable Entities," Can. J. Chem. Eng. 42, 280-281 (1964).

CHAPTER 2

The Framework of Population Balance

We are concerned with systems consisting of particles dispersed in an environmental phase, which we shall refer to as the continuous phase. The particles may interact between themselves as well as with the continuous phase. Such behavior may vary from particle to particle depending upon a number of "properties" that may be associated with the particle. The variables representing such properties may be either discrete or continuous.^ The discreteness or continuity of the property pertains to its variation from particle to particle. There are several examples of discrete variables. First, a population consisting of particles of different materials may be distinguished by assigning a variable taking on discrete values each representing a particular material. Second, one may, merely for the sake of convenience, take a discrete view as is often done by engineers in characterizing a continuous spectrum of particle sizes by discrete mesh sizes (obtained by sieve analysis). Third, the particle may itself be distinguished by a discrete number of entities in it. An example is that of a liquid phase emulsion droplet in a precipitation process containing a limited number of precipitated particles. 1

From a more general mathematical viewpoint, it is not essential to distinguish between discrete and continuous variables if one is willing to admit the concept of generalized functions and derivatives. Since this is not common background among engineers, this route is not taken.

8

2. The Framework of Population Balance

Continuous variables may be encountered more frequently in population balance analysis. They often arise as a natural solution to dealing with indefinite or variable discreteness. For example, a particle-splitting process where the products of sphtting could conceivably have any size smaller than the parent particle is most naturally handled by assigning particle size as a continuous variable. The external coordinates denoting the position vector of (the centroid of) a particle describing continuous motion through space represent continuous variables. The temperature of a particle in a fluidized bed is another example of a continuous variable. In following the temporal evolution of the particulate system, we shall regard time as varying continuously and inquire into the rate of change of the particle state variables. It is more convenient to deal with continuous variables in this regard. A fundamental assumption here is that the rate of change of state of any particle is a function only of the state of the particle in question and the local continuous phase variables. Thus we exclude the possibility of direct interactions between particles, although indirect interaction between particles via the continuous phase is indeed accounted for because of the dependence of particle behavior on the "local" continuous phase variables. In order to enable such a local characterization of the continuous phase variables, it is necessary to assume that the particles are considerably smaller than the length scale in which the continuous phase quantities vary. The continuous phase variables may be assumed to satisfy the usual transport equations with due regard to interaction with the particulate phase. Thus, such transport equations will be coupled with the population balance equation.

2.1

PARTICLE STATE VECTOR

We shall be primarily concerned with particle phase variables that are continuous. The choice of the particle state variables depends on the application. For example, chemical engineers concerned with the modeling of crystallizers will be interested in predicting and controlling the size distribution of crystals in the product. The particles in this application are of course crystals, and the size of the crystal is the main particle state. A growing crystal changes its size at a rate often determined only by the size of the crystal (besides the prevailing supersaturation in the continuous phase with respect to the crystallizing solute). Thus, we need only specify the size of the crystal to predict its growth rate. The particle state is therefore

2.1. Particle State Vector

9

characterized in this example by a single quantity, viz., crystal size. Notice that the local supersaturation, although important to determining the growth rate of a crystal, is a continuous phase variable and hence does not enter the characterization of the particle state. Consider another example. Suppose we are interested in following the total number of cells in a population of bacteria that are multiplying by binary division. Assume that the cells do not divide until after a certain age has been reached. In this case, it becomes essential to define cell age as the particle state although it is not of explicit interest originally. Thus, the identification of age as the particle state in this case was dictated by its influence on the birth rate. In general we may conclude that the choice of the particle state is determined by the variables needed to specify (i) the rate of change of those of direct interest to the application, and (ii) the birth and death processes.^ The particle state may generally be characterized by a finite dimensional vector, although in some cases it may not be sufficient. For example, in a diffusive mass transfer process of a solute from a population of liquid droplets to a surrounding continuous phase (e.g., hquid-liquid extraction) one would require a concentration profile in the droplet to calculate the transport rate. In this case, the concentration profile would be an infinite dimensional vector. Although mathematical machinery is conceivable for dealing with infinite dimensional state vectors, it is often possible to use finite dimensional approximations such as a truncated Fourier series expansion. Thus it is adequate for most practical apphcations to assume that the particle state can be described by a finite dimensional vector.^

The dependence of particle processes (i) and (ii) on the current particle alone of course implies that we are neglecting memory effects. In other words, the choice of the particle state must be suitably made to support this assumption. The finite dimensional state vector can accommodate the description of particles with considerable internal structure. For example, consider a cell with m compartments. Each compartment may be considered as well mixed containing a total of n quantities. Suppose now the cell changes its state by interaction between its compartments and with the environment. The particle state can be described by a partitioned vector [ X i , X 2 , . . . , x j where x^ represents the vector of n components in the ith compartment. It is also interesting to observe that a finite dimensional vector is adequate to describe particles with spatial, internal morphology where several discrete components may be located anywhere within the particle relative to, say, the centroid of the particle. In this case, the elements in the partitioned vector above may be interpreted as position vectors of such components.

10

2. The Framework of Population Balance

It is convenient to distinguish between external coordinates r = (r^, r2, r^), which may be used to denote the position vector of the particle (as determined by that of its centroid), and internal coordinates x = (x^, X25 • • •' ^d) representing d different quantities associated with the particle. The particle state vector (x, r) accounts for both internal and external coordinates. We shall further let Q^ represent the domain of internal coordinates, and Q^ be the domain of external coordinates, which is the set of points in physical space in which the particles are present. These domains may be bounded or may have infinite boundaries. The particle population may be regarded as being randomly distributed in the particle state space, which includes both physical space and the space of internal coordinates. Our immediate concern, however, will be about large populations, which will display relatively deterministic behavior because the random behavior of individual particles will be averaged out. We conclude this section with the observation that many problems in population balance may feature particles distributed only according to their size or some other scalar variable. We shall exploit the simplicity of such problems to demonstrate concepts applicable to the more general problems of population balance.

2.2

THE CONTINUOUS PHASE VECTOR

The continuous phase variables, which affect the behavior of each particle, may be collated into a finite c-dimensional vector field. We thus define a continuous phase vector Y(r, t) = lY^{r, t), ¥2(1, t),..., Y^{r, t)], which is clearly a function only of the external coordinates r and time t. The evolution of this field in space and time is governed by the laws of transport and interaction with the particles. The actual governing equations must involve the number density of particles in the particulate phase, which must first be identified. In some applications, a continuous phase balance may not be necessary because interaction between the population and the continuous phase may not bring about any (or a substantial enough) change in the continuous phase. In such cases, analysis of the population involves only the population balance equation.

2.3. The Number Density Function

2.3

11

THE NUMBER DENSITY FUNCTION

We postulate that there exists an average number density function defined on the particle state space, Eln{x, r, 0] = /i(x, r, t),

xeQ,,

reQ„

(2.3.1)

the left-hand side denoting the expectation or the average of the actual number density n{x, r, t) while the right-hand side, displaying the average number density /^(x, r, t), anticipates a future notation arising from a more general stochastic theory in Chapter 7. Definition (2.3.1) implies that the average number of particles in the infinitesimal volume dV^dV^ (in particle state space) about the particle state (x, r) is /^(x, r, t)dV^dV^. However, we will save ourselves some verbiage by loosely referring to particles in the volume dV^dVj. about the particle state (x, r) merely as "particles of state (x, r)," although the latter statement is technically incorrect. The average number density /^(x, r, t) is assumed to be sufficiently smooth to allow differentiation with respect to any of its arguments as many times as may become necessary. The foregoing (average) number density allows one to calculate the (average) number of particles in any region of particle state space. Thus, the (average) total number of particles in the entire system is given by dVj,{x, r, 0 where dV^ and dV^ are infinitesimal volume measures in the spaces of internal and external coordinates respectively. The local (average) number density in physical space, i.e., the (average) total number of particles per unit volume of physical space, denoted N{r, t), is given by iV(r, t) =

dVJ,{x,rjy

(2.3.2)

If we desire the spatial number density of a selected class of particles belonging to some subset A^ of the space Q^, then the integration above must be over the subset A^. Other densities such as volume or mass density may also be defined for the particle population. Thus, if v(x) is the volume of the particle of internal state X, then the volume density may be defined as v(x)/i(x, r, t). The volume

12

2. The Framework of Population Balance

fraction density, (/)(x, r, t) of a particular state is defined by 0(x, r, 0 = — - v(x) /i(x, r, t),

0(r, t) =

rfKv(x)/i(x,r,0.

(2.3.3)

The denominator above represents the total volume fraction of all particles. Similarly, mass fractions can also be readily defined. For the case of scalar internal state using only particle size (volume) denoting the number density by /i(v, r, 0, the volume fraction density of particles of volume v becomes

*-"=trf' ^'•" =

v/,(v,r,Orfv.

(2.3.4)

A cumulative volume fraction that represents the fraction of particles with volume at most v, denoted F(v, r, t), is given by

where the denominator is given by Eq. (2.3.4). In contrast with the number density, the volume and mass densities are concerned with the amount of dispersed phase material, and consequently are often physically more relevant. The foregoing discussion and relations have been for continuous particle states. Discrete particle states are easily handled by replacing the integrals by summations.

2.4

THE RATE OF CHANGE OF PARTICLE STATE VECTOR

We observed earlier that particle states might vary in time. We are concerned here with smooth changes in particle state describable by some vector field defined over the particle state space of both internal and external coordinates. While change of external coordinates refers to motion through physical space, that of internal coordinates refers to motion through an abstract property space. For example, the growth of a crystal represents motion along the size coordinate, chemical reaction in a droplet may be viewed as motion through a multidimensional concentration space, and so on. We had collectively referred to these as convective processes for the reason that they might be likened to physical motion. It will prove convenient to define "velocities" R(x, r, Y, t) for internal coordinates and

2.5. The Particle Space Continuum

13

X(x, r, Y, t) for external coordinates separately. These functions are assumed to be as smooth as necessary. Generally, explicit dependence of X on external coordinates r is unnecessary, although this is not an assumption forbidding analysis. Clearly, in the foregoing discussion, the change of particle state has been viewed as a deterministic process. It is conceivable, however, that in some situations the change could be occurring randomly in time. In other words the velocities just defined may be random processes in space and time. It will be of interest for us to address problems of this kind. For the present, however, we postpone discussion of this issue until later in this chapter. Since velocities through both internal and external coordinate spaces are defined, it is now possible to identify particle (number) fluxes, i.e., the number of particles flowing per unit time per unit area normal to the direction of the velocity. Thus /^(x, r, t)R{x, r, Y, t) represents the particle flux through physical space and /^(x, r, t)X{x, r, Y, t) is the particle flux through internal coordinate space. Both fluxes are evaluated at time t and at the point (x, r) in particle state space. Indeed these fluxes are clearly important in the formulation of population balance equations.

2.5

THE PARTICLE SPACE CONTINUUM

Following earlier work of the author (Ramkrishna, 1985), it is convenient to define a particle space continuum that pervades the space of internal and external coordinates. For reasons to be clarified subsequently, we shaU deem the particles to be imbedded in this continuum. This continuum may be viewed as deforming in space and time in accordance with the field [X(x, r, Y, t), R(x, r, Y, 0 ] relative to fixed coordinates."^ Thus for any point on the continuum initially at ( x ^ , r j , its location at some subsequent time t may be described by coordinates [X(t; x^, r J, R(t; x^, r j ] which must satisfy the differential equations ^ = X ( X , R , Y , 4 ^ = R(X,R,Y,0; dt dt X(0; X,, O = x„ R(0; x,, r J - r,,

(2.5.1)

This continuum should not be confused with the fluid phase in which the particles are physically dispersed. They are the same only when there is no relative motion between the particles and the fluid phase.

14

2. The Framework of Population Balance

where we must regard the vector Y as a function of R and t. The solution field (x, r) = [X(r;x^,rJ, R(t;Xo,rJ] represents a time dependent transformation of (x^,rj coordinates to (x, r) coordinates. If a particle were initially imbedded at (Xo,rJ, it will change its location with time along a path in particle state space parametrically represented by the vector field [X(t; XQ, r j , R(t; x^, OJ.We shall refer to this as the particle path originating at(x„rj. When the particle state space is one-dimensional, this particle space continuum may be viewed as an elastic string deforming everywhere with the imbedded particles; the particle path must be along this single coordinate. For the example considered earlier of a population of crystals growing in a supersaturated medium in which the particle state is described by its size, the particle path just given is along the size coordinate traversed at a velocity equal to the growth rate. We are now in a position to derive the population balance equation for the one-dimensional case. The reader interested in this may directly proceed to Section 2.7 since the next section prepares for derivation of the population balance equation for the general vectorial particle state space.

2.6

THE REYNOLDS TRANSPORT THEOREM

The Reynolds transport theorem is a convenient device to derive conservation equations in continuum mechanics. Toward derivation of the general population balance equation, we envisage the application of this theorem to the deforming particle space continuum defined in the previous section. We assume that particles are embedded on this continuum at every point such that the distribution of particles is described by the continuous density function /^(x, r, t). Let (/^(x, r) be an extensive property associated with a single particle located at (x, r). Consider an arbitrarily selected domain A^ in the particle space continuum at some arbitrary reference time t = 0. Note that A^ consists of a part A^ in the space of internal coordinates (Q^), and a part A^ in the space of external coordinates (Q^). As time progresses, the domain A^ deforms continuously and is represented by the set A{t) consisting of A^{t) in Q^ and A^{t) in Q^. We focus on the total amount of the extensive property ij/ associated with all the particles in the domain A(r), denoted ^(t), and

15

2.7. The Population Balance Equation

given by

^w =

dV,il,ix, r)/i(x, r, 0. A,(l)

Using a readily established generalization of the Reynolds transport theorem in three-dimensional space ^ to general vector spaces we may write d_ dt

dF>/i =

dV^ AxW

A,(t)

dV^ A,(()

dV^

JA,(()

dt

^A+v.-xiA/i+V/R^A (2.6.1)

where the differentiation with respect to time is carried out holding the domain A^ fixed at all points. In the above equation, V/ represents the regular spatial divergence (for fixed internal coordinates) in any convenient spatial coordinate system. In particular, note that the spatial partial derivatives in V/ do not hold the environmental vector Y constant. This observation becomes important because of the spatial divergence in (2.6.1) operating on the Y-dependent function R. The partial divergence V^* acting on any dififerentiable vector field F in the particle state space is defined so as to imply that

V . - F 1== i1 ( ^ \(^^i/xkik^i),T,t

(2.6.2)

reflecting the choice of a Cartesian frame for the abstract internal coordinates. Of course, other choices of coordinate frames are sometimes more appropriate. Equation (2.6.1) is crucial to the development of the population balance equation for the general case, which is treated in the next section.

2.7

THE POPULATION BALANCE EQUATION

Although we are ready for the derivation of the general population balance equation, we shall begin for the sake of simplicity with the one-dimensional case. 5 For an elegant derivation of the Reynolds transport theorem, see Serrin (1959).

16

2. The Framework of Population Balance

2.7.1

The One-Dimensional Case

Consider a population of particles distributed according to their size x which we shall take to be the mass of the particle and allow it to vary between 0 and oo. The particles are uniformly distributed in space so that the number density is independent of external coordinates. Further, we assume for the present that the environment does not play any explicit role in particle behavior. Such a situation can be approximated, for example, in a crystallizer containing a highly supersaturated solution of the crystallizing solute. The process involves nucleation resulting in the formation of a rudimentary particle and its subsequent growth by transferring solute from the solution phase to the particle surface. In actuahty, in addition to mass transfer, heat transfer also occurs, serving to remove the heat of crystallization, although the latter is generally considered negligible.^ If the supersaturation is sufficiently high, the nucleation and growth rates may remain relatively unaffected as crystallization progresses. This unnecessarily restrictive assumption is made only for simplifying the preliminary derivation of the population balance equation. We let X{x, t) be the growth rate of the particle of size x. The particles may then be viewed as distributed along the size coordinate and embedded on a string deforming with velocity X(x, t). Choose an arbitrary region [a, b'] on the stationary size coordinate with respect to which the string with the embedded particles is deforming. We are interested in the rate of change of the number of particles in this size interval. As the string deforms, particles commute through the interval [a, fc] across the end points a and b, changing the number of particles in the interval. If we denote the number density by f^{x, t), the rate of change in the number of particles in [a, fc] caused by this traffic at a and b is given by X{a, t)Ma. t) - X{b, t)f,{b, t), the first term of which represents the "particle flux" in at a and the second the particle flux out at b. Assume for the present that there is no other way in which the number of particles in the interval [a, b'] can change. Then we may write for the number balance in the interval d {' /i(x, 0 dx = X(fl, 0/i(«, t) - X{b, t)Mb, tl 6 See, for example, Coulson and Richardson (1991), p. 663.

2.7. The Population Balance Equation

17

which may be rewritten as " ''^I^

+ !_(^x{x,c,t)/i(x,t)) t)Mx, t)) <\dx = 0

(2.7.1)

because all functions involved are assumed to be sufficiently smooth. Since the interval [a, 6] in Eq. (2.7.1) is arbitrary, the smoothness of the integrand implies that it vanishes altogether.^ Thus, we have the population balance equation

f l M + i_ (x(x, t)Mx, t)) = 0.

(2.7.2)

This equation must be supplemented with initial and boundary conditions. If we started with no particles we set /^(x, 0) = 0. For the boundary condition, we let the nucleation rate be h^ particles per unit time and assume that the newly formed particles have mass zero. This rate should be the same as the particle flux in at x = 0. Thus, X{0,t)f,(0,t)

= h„,

(2.7.3)

which is the required boundary condition.^ If Eq. (2.7.2) is integrated over all particle masses one obtains dN _ d dt dt

/i(x, 0 dx = x(0, o/i(o, t) - x((X), 0/i(^, 0 = K^

the equality on the extreme right arising from the fact that particles can increase in number in this process only by nucleation. From (2.7.3) and the equation just given, we conclude that X(cx), 0/i(c^, 0 = 0,

(2.7.4)

which is sometimes referred to as a regularity condition. It does not insist 7

Equation (2.7.1) can be sustained by allowing the integrand to vanish almost everywhere in the interval but not at some selected points in the interval, but this would contradict the smoothness requirement. Alternatively, if the integrand did not vanish at some point in the interval, then a smooth function must retain its sign in a small interval about this point. In such a case (2.7.1) cannot be sustained in this small interval which contradicts the arbitrariness of the interval [a, fe]. 8 More complicated boundary conditions can be envisaged in other applications. For example, the right-hand side of (2.7.3) may actually depend on particles in the entire size range, giving rise to an integral boundary condition. These will be encountered subsequently.

18

2. The Framework of Population Balance

that the number density itself vanish at infinite mass if the growth rate vanishes for large particles. If, however, the growth rate does not vanish for larger particles, Eq. (2.7.4) implies that the number density must vanish for arbitrarily large sizes. In the above derivation, we did not envisage the birth and death of particles in the interval [a, fc]. For example, crystals in a slurry may undergo breakage and/or aggregation contributing to the birth and death of particles in the interval of interest. To assess the rates of this contribution detailed modeling of breakage and aggregation processes will be needed. We defer these considerations to a later stage and instead propose that the net rate of generation of particles in the size range x to x -\- dx hQ described by h{x, t) dx where the identity of h{x, t) would depend on the models of breakage and aggregation. In this case, Eq. (2.7.1) must be replaced by ^ h ^

+ ^{X(x,

t)Mx, t)) - h{x, t) dx = 0

SO that the population balance equation becomes ^[l^

-f j - (Z(x, t)Ux, t)) = h{x, t).

(2.7.5)

As before, the preceding equation must be supplemented with initial and boundary conditions. Equation (2.7.3) continues to serve as the boundary condition. In view of the total number balance dN

h{x, t) dx, 0

the regularity condition (2.7.4) also holds. Suppose we relax the constraint that particle behavior above is independent of the environment. Consider the continuous phase to be described by a scalar quantity Y, which is assumed to be uniform in space. In a well-mixed crystallizer, Y may represent the supersaturation at the surface of the crystals. We introduce the following additional features: (1) The nucleation rate depends on Y, i.e., h^ = h^{Y). (2) The growth rate may also be assumed to depend on 1^ i.e., X = X{x, X t). (3) The growth process depletes the supersaturation at a rate proportional to the growth rate of the crystals, the proportionality being dependent on particle size, i.e., at the rate a(x)X(x, Y, t).

2.7. The Population Balance Equation

19

The net birth rate h may or may not depend on Y In this case, the process of derivation of the population balance equation used earlier is not influenced in any way, so that the proper substitute for (2.7.5) is given by

^ ^ ^ + ^ (x(x, Y, o/i(x, t)) = hix, X ty

(2.7.6)

The initial condition remains the same as before while the boundary condition recognizes the dependence of the nucleation and growth rates on ZThus (2.7.7)

X(0, Y, 0/i(0, t) = h,{Y).

Equation (2.7.6) must be coupled with a differential equation for Y accounting for its depletion because of the growth of all the particles in the population. This is easily found to be dY_ dt

(2.7.8)

cc{x)X{x, Y, t)f^{x, t) dx.

An initial condition for Y now completes the formulation of the problem. We now consider the derivation of the population balance equation for the general particle state space.

2.7.2

The General Case

We recall the domain Mf) in particle state space considered in Section 2.6, which is initially at A^ and continuously deforming in time and space. For the present, the particles are regarded as firmly embedded in the deforming particle state continuum described in Section 2.5. The only way in which the number of particles in Ait) can change is by birth and death processes. We assume that this occurs at the net birth rate of /2(x, r, Y, t) per unit volume of particle state space so that the number conservation may be written as dV

dt]^

\it)

dVJ, =

dV^h{x,r,Y,t).

dV A(t)

Kit)

Using the Reynolds transport theorem (2.6.1) with i^ = 1, for the left-hand side of the preceding equation, we obtain dV^ ^ / i + V . - X / , + V / R A -

dV, Ax(0

JA^W

= 0.

20

2. The Framework of Population Balance

The arbitrariness of the domain of integration above and the continuity of the integrand together imply that the integrand must vanish everywhere in particle state space, leading to the population balance equation |/i+V,-X/i+V,-R/i=/i.

(2.7.9)

The equation must be supplemented with initial and boundary conditions. The initial condition must clearly stipulate the distribution of particles in the particle state space, including internal and external coordinates. For the particular case in which the particles are all of the same internal state, say, x^, it is most convenient to use the Dirac delta function S{x — xj, which has the properties (i) (5(x - x j = 0, (ii) f

x^ x^

f{x)dix -x,)dV^

= f{x,)

where / is any function of x; indeed, when /(x) = 1, property (ii) implies that the integral of the function 8{x — xj is unity. Because of property (i), we may also infer that the property of the integral in (ii) is preserved even when the integration is considered over any volume containing the point x^. Note also that the density is technically infinity at x^.^ The initial condition for the population balance equation for this specific case is then written as Mx,T,0)

= N,g{r)S{x - xJ

where N^ is the initial, total population density and ^(r) their spatial distribution. The boundary condition requires more extended discussion and will be dealt with in the next section. 9

Such a function, which comes under the category of generalized functions, is generally understood as the limit of a suitable sequence of functions, say {(p„{x)} with (/?„(x) vanishing outside some domain AV„ (in Q J containing x^ and satisfying the property (p„ix) dV, = 1 where as n becomes large the domain AQ„ progressively contracts around the point x^. Clearly one such sequence is obtained by choosing „(x) = 1/AV^ for every x in AQ„ where AV„ is the volume of AQ„.

2.7. The Population Balance Equation

21

2.7.3. Boundary Conditions for the General Case

Suppose Eq. (2.7.9) is integrated over the entire particle state space. Denote the boundary of the particle state space by dQ^ for internal coordinates and 5Qr for external coordinates. These boundaries may be completely bounded or may have all or parts of them stretching to infinity. In either case, we may use the divergence theorem to write

s i fi. "-•

dVJ^ = \ dV^\ Q.

JQ.

dV,h - (b

X/i • dA^ - (h

R/i • dA

dQ^

JQ,

(2.7.10) where dA^ and dA^. are local, infinitesimal area vectors pointing out of the surfaces 5Q^ and SQ^, respectively. The left-hand side of this equation represents the net rate of change of the total number of particles in the entire system. The right-hand side represents the various ways in which the total number can change in the system. The volume integral, which is the first term, represents the total net birth rate for all particles in the entire space. The surface integral terms represent the rate at which particles are introduced into (or removed from) the particle state space across the bounding surfaces. An example of this was encountered in the one-dimensional case where nucleation contributed particles at size zero. Generally, boundaries at "infinity" represent no sources or sinks so that the particle fluxes vanish there. Thus, X/i^O,

llxll^o);

R/i-^0,

llrll^o)

(2.7.11)

which is a regularity condition similar to (2.7.4). Alternatively, we may assume that the spatial domain Q^ containing the particulate system is bounded. In what follows we will assume that the system is closed, by which it is implied that particle flux vanishes everywhere on dQ^. We now return to the issue of boundary conditions. Basically, this is a question of specifying the component of the particle flux normal to the boundary or (equivalently) the number density at each point on "appropriate" parts of the boundary. We shah presently see what these appropriate parts are. Note that the population balance equation (2.7.9) features a first-order partial differential operator on the left-hand side. Although the nature of the complete equation is governed by the dependence of the right-hand side on the number density function, the solution to Eq. (2.7.9) may be viewed as evolving along characteristic curves which (are the same

22

2. The Framework of Population Balance

as the particle paths identified in Section 2.5 and) originate at the boundary (where particles enter the system). Such characteristic curves may terminate at other boundaries (where the particles leave the system). In specifying the boundary condition we are concerned with the boundary at which characteristics originate and not on that on which they terminate. If we denote the part of dQ^ at which characteristics originate by dQl and the local outer normal vector to dQ.^ by n^, the boundary condition becomes - ( X -nJ/.Cx, r, 0 = h,,

xEdQl

(2.7.12)

where h^ is to be specified from physical models. The boundary condition just given is a generalization of the one-dimensional version (2.7.7). The physical models for h^ may include its possible dependence on the particles in the entire particle state space. For example, we may say that h^ depends on particles in the space of internal coordinates alone as represented by h,=

\

di;,X(x,x',r)/,(x',r,0,

xeSQ^,

(2.7.13)

where X(x, x', r) measures the local dependence of the birth process on the particle state. The combination of (2.7.12) and (2.7.13) yields an integral boundary condition. Examples of this type of boundary condition are generally encountered in biological systems. The foregoing discussion was for closed systems in which the particle flux vanished on all of dQ^. For open systems, we may distinguish a portion 5Qr,in of 5Qr through which the entering particle flux may be specified. If particles leave the system through another portion 5Qr,out of dQ^, then the flux may be allowed to vanish on ^Q^ barring ^Q^jn (and of course ^^rout)- No boundary condition is needed on 5Qr,out where the number density can be calculated by integrating along appropriate characteristics. The next section deals with applications of population balance to open systems which are obtained by a macroscopic balance and are of special importance to engineers.

2.8

POPULATION BALANCE EQUATION FOR OPEN SYSTEMS

We deal directly with the general case. Consider an open system defined by the domain of external coordinates, Q^ with boundary dQ^ consisting of an inlet domain 50^ j^, which admits the particle-fluid mixture into the system, and an outlet domain 50^ ^ut through which the mixture exits the system. We

2.8. Population Balance Equation for Open Systems

23

integrate the population balance equation (2.7.9) over the domain fi^ and use the divergence theorem to obtain the result 8_

It where we have used the fact that parts of the boundary SQ^ other than SQ^jn and 5Qr out ^^e impervious to the transport of particles. Of particular interest to this section is the case of a population uniformly distributed in space throughout Q^ so that the number density does not depend on external coordinates. Many of the applications fall into this category. Letting F, be the volume of Q^, the preceding equation becomes |(^r/l) + KV,-XA+/i,

R-dA, = V,h dii, „,„

(2.8.1) where we have defined spatial averages 1

""'v.

dvX

h = - \ dV,h.

Generally, the functions X and h inherit the spatial uniformity of /^ so that X = X and h = h. The surface integral over the inlet domain is the volumetric flow rate of the particle space continuum (and not necessarily the continuous phase!) entering the domain Q^, whereas that over the outlet domain is the flow out. If it is assumed that there is no relative motion between the continuous phase and particles, then the volumetric flow rates above are also those of the particle-fluid mixture. For steady, incompressible flow into a filled vessel, the flow rate in is the same as that out. Denote this flow rate by q. A second assumption at this stage is that the vigorous fluid mixing provides for a uniform number density of the particles in the region consisting of the following components: (i) The "interior" of Q^ excluding a small region (of negligible volume) near 5Qr,in ^^ which the number density changes continuously from the uniform value /^ ^^ the interior to the boundary value /^ j^ entering from the outside at dQ^^^^. (ii) The exit domain 5Qr,ouf In fact it is assumption (i) that justifies the use of the divergence theorem to

24

2. The Framework of Population Balance

obtain Eq. (2.8.1). Assumption (ii) implies that /^ = /i,out so that Eq. (2.8.1) becomes

I (KA) + KV, • X/i - qfuir. + qfi = Kh,

(2.8.2)

the negative sign associated with the inlet term of course arising from the outer normal being directed opposite to that of the flow in. Equation (2.8.2) is the population balance of interest. When V^. is constant (2.8.2) assumes the more famihar form ^

+ V,• X/i = ^ ( / , , , „ - / i ) +/I

(2.8.3)

where 6 = VJq is the residence time. This equation is useful for many diverse applications to particulate processes carried out in continuous reactors. It needs to be supplemented with an initial condition, as well as boundary conditions such as the ones described in the previous section. In some applications, the open system may feature a flowing population, which is transversely weU mixed entering at a cross-section under known conditions and exiting at another. The required population balance is obtained by integrating Eq. (2.7.9) over any cross-section along the flow and obtaining an equation in terms of cross-section averaged quantities. Section 3.3.4 in Chapter 3 considers an example of such a system, so that no further details are included here. We now return to the consideration of the continuous phase equation for the general case.

2.9

EQUATION FOR THE CONTINUOUS PHASE VECTOR

Recall that the continuous phase variables were described by the vector field Y(r, t). In general, the components of this vector field should encompass all continuous phase quantities that affect the behavior of single particles. These could include all dynamic quantities connected with the motion of the continuous phase, the local thermodynamic state variables such as pressure and temperature, concentrations of various chemical constituents, and so on. Clearly, this general setting is too enormously complex for fruitful applications so that it is necessary to suitably constrain our domain of interest. In this connection, the reader may recall our exclusion of the fluid mechanics of dispersions, so that we shall not be interested in the equation

2.9. Equation for the Continuous Phase Vector

25

of motion for the continuous phase, except perhaps under some very special circumstances. We shall assume the velocity field v(r, t) of the continuous phase to be known for most purposes. A similar statement may also be made of the pressure field p(r, t). The continuous phase vector Y(r, t) will include temperature (under nonisothermal situations), and concentrations of various chemical components that may be involved in transport between the continuous phase and the particles, and in chemical reactions in either phase. Liquid-liquid dispersed phase reactors are a common feature of the chemical process industry where the preceding processes are encountered. For the present we shall assume isothermal conditions and consider only concentration components in Y(r, t). Alternatively, this strategy would be appropriate even for nonisothermal situations if temperature were to be isolated as another variable to be dealt with through an energy transport equation. In writing the balance for Y(r, t), we recognize the following: (i) The total mass flux vY(r, t) + Jy in the continuous phase, the first term in the sum representing the convectiveflux,and the second (Jy) the diffusive mass flux relative to the mass averaged velocity v(r, t). (ii) The transfer rates of continuous phase entities towards each particle located at (x, r), denoted by jy(x, r, Y, 0^^ displaying its dependence on location in the particle state space as well as the local continuous phase vector Y(r, t). (iii) A volumetric source a as, for example, due to chemical reaction in the continuous phase. The particles are viewed as "point" sources because of the assumption made earlier that the variation of Y(r, t) occurs over length scales considerably larger than that of particle size. The transport equation for Y(r, t) may now be written as - Y ( r , f ) + V / [ v Y + J^] +

dVJ,ix,r,t)i^

= <7.

(2.9.1)

Except for the third term due to the presence of particles on the left-hand side, Eq. (2.9.1) is a famihar transport equation (e.g.. Bird et a/., 1960). The population balance equation (2.7.9) must in general be considered together with the above continuous phase equation. 10 Note that jy is a vector whose elements are scalars while Jy is a vector whose elements are spatial vectors.

26

2. The Framework of Population Balance

We now consider the well-stirred open system of Section 2.8 with the continuous phase vector represented by the spatially uniform Y{t) in the domain Q„ and Yj^ at the entrance region 50^ jn- Integrating Eq. (2.9.1) over the region Q^ and recognizing that the diffusive flux Jy must vanish everywhere, we obtain the equation

j ^ Y(0 + [ dVJ,{x, r, Oiy = I (Yi„ - Y) + (7,

(2.9.2)

where we have assumed the same residence time as that for the particulate phase based on neglecting all relative motion between the continuous phase and the particles. If relative motion of the particles cannot be neglected, different residence times must be used for the particle phase and the continuous phase. In Section 2.4, we had raised the possibility that particle state could change in a random manner. Since the deliberations until this stage have taken a deterministic view of the rate of change of particle state, we shall address this issue at some length in the next section. For this section to be comprehensible, the reader must be familiar with Ito's stochastic calculus and elementary aspects of the theory of stochastic differential equations.

2.10

RANDOM CHANGES IN PARTICLE STATE

The population balance equations considered so far were for systems in which particles changed their states deterministically. Thus specification of the state of the particle and its environment was sufficient to determine the rate of change of state of that particle. Applications may, however, be encountered where the particle state may change randomly as determined, for example, by a set of stochastic differential equations. Since, however, the population balance equation is a deterministic equation, our desire is to seek the expected displacement of particles moving randomly in particle state space during an infinitesimal interval dt. Although it is possible to address this situation for the general particle state vector including internal and external coordinates, we shall take the route of establishing the results for the one-dimensional case and proceed to infer the generalization for the vectorial case without elaborate derivation. Consider again a population of particles distributed according to a scalar state variable x, which we shall take to vary over the entire real, line and let /i(x, 0 be the number density. The scalar state x is presumed to vary in

2.10. Random Changes in Particle State

27

accord with the Ito stochastic differential equation dx = X(x, t) dt + ^2D{x,t)dW,

(2.10.1)

where dW^ represents the increment of a standard Wiener process (during the time interval dt) and its coefficient in the equation has been assigned a form in anticipation of a convenient future definition. Continuous phase influence has been neglected for the present (although this is not a forbidding assumption) to keep other details to a minimum. Consider any property of the population calculated by summing that associated with each particle in the population. Denote the property associated with a single particle of state x by g{x). Then the property associated with the entire population is given by g(x) f^{x, t) dx.

G{t) =

Suppose we are interested in how the property G{t) is changing with respect to time. Then we may write dGjt) dt

9{^) ^ fM. t) dx.

The foregoing rate of change must include contributions from the following: (i) The net birth rate of particles given by j ! ! ^ gix)h{x, t) dx. (ii) The expected change caused only by the random movement of the particles as determined by the stochastic differential equation (2.10.1); we may write this contribution as " ' ^dg{x{t)) /i(x, 0 dx dt In order to calculate (ii) we make use of Ito's formula/^ which gives dg{x{t)) dW , = g'i^) X(x, 0 + ^ V 21>(x, t) + g"(x)D{x, t\ dt dt whose expectation is given by JW, ^dg(x(t)) E—^J2D{x,t) + g'\x)D{x, t). E — ^ = g(x) X{x,t) + dt 11 See, for example, Gardiner (1997), p. 95.

2. The Framework of Population Balance

28

Since the expectation within the square brackets on the right-hand side of the above equation is zero by definition of the Wiener process, we have dg{x{t)) . E—-— = g {x)X{x, t) + g {x)D{x, t). at The contribution of (ii) is given by ) lg'(x)X(x, t) + g"{x)D{x, t)']f,{x, t)dx, oo

SO that we may now write for the balance of entity G{t)

,

^W ^ /i(^, 0 dx

=r

(JL

{g{x)h{x, t) + lgXx)Xix, t) + g'Xx)D{x, 0]/i(^, 0} dx.

Integrating the terms in square brackets by parts, we obtain ^W j ^ fiix, t) - h{x, t)+—

{X{x, t)f,{x, t)) -—{D{x,t)Mx,t)) dx^

dx = 0.

Because of the arbitrariness of g{x), we conclude that the expression within the square brackets in the integrand above must vanish yielding j^fiix,

t) +-^iX{x,

0/i(x, 0) = 1 ^ W ^ ' 0/i(^, 0) + h{x, t), (2.10.2)

which is the desired population balance equation. We note in passing that if the stochastic differential equation (2.10.1) were to have been interpreted in terms of what is known as the Stratonovich integral, the population balance equation just given would have been written as P

P

P\

/

P\

\

Jt ^^^^' ""^^Tx^^^^' ^^^^^^' ^^^ ^ Tx (^^^' ^^ Tx ^'^^' ^V ^ ^^^' ^^' ^^'^^'^^ A proper explanation of Stratonovich integration is somewhat outside the scope of this book. The generalizations of (2.10.2) and (2.10.3) for the general vector case including continuous phase dependence are identified as follows. Let the rate of change of particle state be given by stochastic differential equations of the

2.11. Formulation of Population Balance Models

29

form dX = X(X, R, Y, 0 dt + 7 2 D J X , R, Y, 0 dW^,,

(2.10.4)

dR = R(X, R, Y, t) dt + y2D,(X, R, Y, 0 dW^,,

(2.10.5)

where dW,,^ and dW^^ are vector (standard) Wiener processes in the spaces of internal and external coordinates respectively each with uncorrelated components. D^ and D^ are square matrices of orders d and 3, respectively, whose coefficients depend on internal coordinates, external coordinates, and the continuous phase vector. The population balance equation for the Ito case can be shown to be ^ / i + V, • Xf, + Vr • R/i = V,V,: BJ,

+ V X 'Afi+h

(2.10.6)

where D^ = D^Djand D^ = 0^0/^.^^ If (2.10.4) and (2.10.5) are interpreted in the Stratonovich sense, then the population balance equation becomes

I /i + V,-XA + V,-RA = V , - ( D , V , D J A ) + W,-{BX^rfi)

+ h. (2.10.7)

Either of equations (2.10.6) and (2.10.7) must be considered simultaneously with the continuous phase equation (2.9.1).

2.11

FORMULATION OF POPULATION BALANCE MODELS

In the foregoing sections of this chapter, our concern has been of how a particle population redistributes itself within the particle state space as time progresses. The death of existing particles and the birth of new ones were recognized only through a net birth rate without concern for the detailed phenomenology of such processes. The birth and death processes are extremely important in population balance models; in fact, they represent the crux of such models as they view the very process of how particles appear or disappear in the system and not merely how they are redistributed. Although Chapter 3 will address the problem of modeling the 12 The notation of the double dot inner product used in Eq. (2.10.6) is consistent with that generally used in transport phenomena. For example,

30

2. The Framework of Population Balance

birth-and-death processes in detail, the formulations of this chapter are indeed useful for a class of applications that we shall presently demonstrate. The applications are selected so as to emphasize the factors leading to the choice of particle state in each case.

2.11.1

Dissolution Kinetics

Consider a well-stirred vessel initially containing a given mass (MJ of a solid present as a population of polydispersed particles in a hquid in which it is soluble. Assume that mass transfer controls the dissolution of each particle and that the heat of dissolution is negligible. The particles may all be assumed to be spherical and distributed according to their mass x. The rate of change of mass of a particle of mass x by dissolution, X{x, Y), can be described by X{x,Y)=

r 3x T'^

-47^ —

K{X)(Y*-Y)

where 7* is the solubility of the solute at the prevailing temperature, Y is the uniform concentration of the solute in the well-stirred bulk, K{X) is the mass transfer coefficient, and the remaining term can be seen to be the surface area of the sphere by recognizing the particle density p. The mass dependence of the mass transfer coefficient may be assumed to be negligible and we may set K' = 4n{3/4np)^^^K to be a new constant for the convenience of expressing X{x, Y) = —K'X^^^{Y* — Y). The only particle property on which the dissolution rate depends is the particle size. The well-stirred nature of the system obviates the need for external coordinates. Thus, the choice of particle size as the only particle state variable is justified. The domain of particle size is the positive real interval [0, oo). Alternatively, we state that Q^ = [0, oo). The population balance equation in the number density function /^(x, t) is given by (2.7.6) with the right-hand side set equal to zero. Thus, ^-^^

+ ^[-K'x''HY*~Y)Mx,m=0,

(2.11.1)

which must be coupled to the continuous phase equation

f = K'(y*-y)

x^^^Mx,t}dx,

(2.11.2)

2.11. Formulation of Population Balance Models

31

which accounts for the accumulation of solute in the continuous phase by dissolution. Equations (2.11.1) and (2.11.2) must be subject to initial conditions. Let the initial size distribution of particles be given by g{x). Then the initial number density function is given by

<x> =

Mx,0)=^gix),

xg{x) dx,

where <X> is the average mass of the initial population of particles. If the continuous phase contained no initial solute, we obviously have 7(0) = 0. The model is now complete. We seek to analyze this model from both physical and mathematical viewpoints. First we shall restrict ourselves to the situation where the volume of the continuous phase is large relative to the number of particles. This implies that the number densities are small enough to permit the assumption dY/dt ^ 0, so that Y will remain approximately at its initial value of zero. Thus, the population balance equation becomes ^

^ + ±l-K'x''^Y*Mx, t)-] = 0, (2.11.3) dx dt which may be readily solved, but our interest for the present is not in the solution. If we integrate the preceding equation over all particle masses, then using the regularity condition, at

x^o

It would appear from the above that the right-hand side will be zero if the number density is bounded at zero. This would imply that the total number density of particles could not change in the system at any time. However, it will be readily apparent that the time taken by a particle of finite mass x to completely dissolve and disappear from the system is finite, thus reducing the number of particles as the dissolution is in progress. This "paradox" of course disappears at once when we recognize that the number density is not bounded at zero. This does not mean that there is an infinite number of particles, as is often interpreted by the novice who fails to recognize the difference between the number density in particle size coordinate {and space) and the total number density in space. The total number density in space, which is obtained by integrating the not necessarily bounded number density in particle state space over the internal coordinates (i.e., Eq. (2.3.2)), is generally bounded.

32

2. The Framework of Population Balance

The number density /^(x, 0 satisfying Eq. (2.11.3) is readily obtained by the method of characteristics. Since our goal is to understand the nature of the solution we refrain from the diversion of demonstrating the technique of solution and directly state the result as /i(^, t)

{x"^ + K'Yn/3fgl{x''^

+ K'YH/3fl

(2.11.4)

the singularity of which at x = 0 is clearly reflected at all times t > 0. The evolution of the particle size distribution is shown in Figure 2.11.1 for an initial particle size distribution of ^(x) = x^e'""/!. Of course if we had picked particle radius as the size coordinate this problem of the singularity would not have arisen at all! It will then transpire that the number density would have a finite, nonzero value at zero radius, which we leave for the student to discover. Although particle radius was obviously a better choice in this example, such inspiration is not always available in more subtle cases. Furthermore, particle mass is a more convenient variable in situations

0.5

1

Time, t

0

FIGURE 2.11.1 Singular behavior of the number density function near the origin for the dissolution process in Section 2.11.1.

2.11. Formulation of Population Balance Models

33

where, for example, particle agglomeration or breakage occurs in conjunction with dissolution. This exercise shows that we must be sensitive to the possibility of singularities in the number density function. When the particle densities are not small, the coupled equations (2.11.1) and (2.11.2) must be solved simultaneously. Generally, such solutions can only be obtained numerically. The solution of population balance equations is of concern in Chapter 4. EXERCISE 2.11.1 Crystallization is the exact opposite of the dissolution process described above. Consider a well-mixed continuous crystallizer unit in which the supersaturated solution is cooled by a cooling jacket. The crystallizer slurry may be assumed to be uniformly at a temperature Y^ and solute concentration Y2 so that the supersaturation of the solution is given by a = Y2 — ^2*(^i) where Y^iY^) represents the solubiHty of the crystalHzing material at temperature Y^. The crystallizer is fed with a solution free of solids at a temperature Y^j and concentration Y2J, while simultaneously withdrawing the well-stirred slurry at a volumetric flow rate q identical to that of the feed. The population of crystals is distributed according to mass X. Crystals of mass x grow at the rate X{x, a). Identify the steady-state population balance equation for the crystal phase in terms of the number density /i(x) where x is crystal mass. If the specific heats of the solution and the crystals are Cpi and Cp^ respectively, identify the steady-state energy balance for the system. State the required boundary conditions for the population balance equation given that crystals of mass "zero" are formed by nucleation at rate h^{(j). 2.11.2

Synchronous Growth of Ceil Population

In this example we consider a cell population in a batch stirred reactor where the cells are distributed according to their age, denoted T, ranging between 0 and 00. The main purpose of this example is to demonstrate the boundary condition that arises. Each cell beyond a certain age T^ has a constant "rate" of division, say, k. The division of the cell of age T results in the loss of that cell, but also in the gain of two new cells of age zero each. The only particle state variable is the cell age T. N O external coordinates are needed because the population is in a well-stirred reactor. The continuous phase is assumed to have no explicit influence on the cells, presumably because the necessary nutrients are present in saturating proportions. We again have the particle state domain Q^ = [0, 00).

34

2. The Framework of Population Balance

Initially, the cells of total population density N^ are assumed to be all of age zero. The number density function /^(T, t) must satisfy the population balance equation | / i ( T , 0 + ^ / i ( T , t ) = /i(T,0.

(2.11.5)

where the velocity of cells along the age coordinate is properly reflected as unity. The net birth rate of cells of age T, /I(T, t), should only involve the disappearance of cells of age greater than T^. The total rate of cells disappearing by division is given by -h{T,t)=kH{T

-T,)MT,t)

where I 1, -^ > -^o is the Heaviside step function. Integrating Eq. (2.11.5) over all ages, we obtain

f-/,.o,0

MT,t)dT,

(2.11.6)

where we have used the regularity condition at infinite age. We may now derive the boundary condition at age zero in either of the following ways. First, we may recognize that the total number of cells will increase, and the rate at which this occurs is given by

dN

r°^

— = ^|

MT,t)dT.

(2.11.7)

Combining (2.11.6) and (2.11.7) we obtain the boundary condition A(0,0 = 2/cf"/,(T,r)dT, (2.11.8) which could have been directly obtained as follows. The left-hand side may be interpreted as the "flux" of cells of zero age, since the number density is multiplied by the unit velocity along the age coordinate. The right-hand side, which is the rate of formation of newborn cells (from all dividing cells of age larger than i J , reflects the fact that each dividing cell contributes two cells of age zero.

2.11. Formulation of Population Balance Models

35

The initial condition for the number density is given by /,(T,0)=iV,(5(T). This problem has been solved by Laplace transforms^^ to obtain the solution for the total number density as Nit)

^^/^"^

- ^ = 1 + X 2-~'f{k(t - mxX m) ^^ o

m=l

(2.11.9)

where [x] is an integer-valued function of the largest integer smaller than its argument x. The function f{y, m) is defined by

f{y. ^) =

(m - 1)! J

e-'^x'^-^dx.

The interesting attribute of the solution (2.11.9) is that it shows the gradual loss of the initial synchrony in the population.

EXERCISE 2.11.2 Consider a sexually reproducing population distributed according to age in which the male population density /I,M('^' 0 raust be distinguished from the female density /i p.(T, t). Assume random mating between males of age T and females of age x' with a frequency ca(T, T') resulting in new live births of equal likelihood for male and female offspring. The death rates of males and females of age x may be assumed to be kj^{x) and kp{x\ respectively. Identify the population balance equations and boundary conditions for the two densities and their boundary conditions, assuming that the likelihood of multiplets may be negligible. Debate the form of the frequency function CO(T, T') if you were to apply the model to a human population.

2.11.3

Budding of Yeast Population

This example is selected with a view to show how discrete particle states can arise rather than develop a very realistic model of a yeast population. Also, it gives us an opportunity to discuss differences in the boundary condition from that used in the previous example. 13 See Tsuchiya, Fredrickson, and Aris (1966).

36

2. The Framework of Population Balance

A yeast cell multiplies by forming a "bud" which gradually develops into a daughter cell. Eventually the daughter cell separates off as an independent cell leaving a scar on the mother cell. The mother cell continues to form new buds forming daughter cells, which also form buds and continue the process of reproduction and growth. As a cell accumulates more scars, its ability to bud is progressively impaired, thus eventually arresting its abihty to reproduce. In formulating a simple population balance model, the formation of a new cell from a mother cell can be viewed in much the same way as we did in the previous example. However, the number of scars on the mother cell, which is a discrete particle state, becomes important in determining the rate of forming a new cell. We shall again use cell age T as the continuous particle state as in the previous example. We denote the number density of cells of age T with i scars by /i,i(T, 0- The rate of division of one such cell is allowed to be kiH{T — T^), where we have not used the option of letting T^ depend on i. We shall set no limit on the number of scars since we let /c^ -^ 0 as i -> 00, reflecting the progressive loss of reproductive power with increasing number of scars. The net birth rate for cells of age i with i scars is given by /I,(T,

t) = k,_,H{T -

TJ/I,,_I(T,

t) - k,H{T -

TJ/I,,(T,

0,

f = 1,2,..., (2.11.10)

which shows how cells with / scars and age T are mother cells of i — 1 scars with age T left over at the instant of giving birth to new daughter cells (with no scars and age zero). Equation (2.11.10) also shows how cells with i scars and age i are lost by division because of an additional scar on each mother cell left over after birth. For / = 0, we have MT,f)=-/c„/f(T-T„)/,,„(T,t).

(2.11.11)

The population balance equation for / i I(T, t) is given by I / i , , ( T , t)+~

f,/T,

t) = /i,.(T, t),

i = 0,1,2,...,

(2.11.12)

where the right-hand side is specified by (2.11.10) and (2.11.11). The boundary condition for the birth of daughter cells of age zero and zero scars must account for the fact that all scarred mothers must contribute to the

2.11. Formulation of Population Balance Models

37

formation of such cells. Thus, the boundary condition is given by /i,o(o,

t)=Y.^i

A,{T,t)dT.

(2.11.13)

For completing the specification of the problem we must state the initial age distribution of cells for each /. If there are no cells with scars originally, then the number densities must vanish identically for all i > 1. The reader is referred to Hjortso and Bailey (1982, 1983) for a detailed treatment of yeast cell populations with population balance models. 2.11.4

Gas Holdup in a Stirred Tank

We shall now develop an application of the concepts of Section 2.10 in which a random change of particle state was considered. The problem is to calculate the holdup of gas bubbles in a stirred reactor (of uniform cross-section) containing a hquid through which a gas is being bubbled continuously by introducing it at the bottom of the reactor through a sparger. ^"^ The bubbles enter at the bottom of the vessel and are immediately split into a distribution of sizes. Thereafter, because of their buoyancy, the bubbles climb through the vessel but are randomly jostled around by the turbulence in the continuous phase. Eventually, the bubbles leave the vessel by forcing themselves out through the liquid film at the free surface at the top. We let the vertical height of the liquid level from the plane of entry of the bubbles be denoted by H. It should be recognized that H is itself dependent on the holdup of bubbles, which is the main object of the calculation, so that in this sense H is itself unknown. It will prove useful to let H^ be the height of the hquid level when no gas is sparged through the vessel. We shall assume that the gas is saturated with the liquid vapor as it enters the vessel and that evaporation of the hquid into the gas is negligible. It is clearly of importance to find the population of bubbles in the entire vessel. We make the following assumptions: (i) The system is operating under steady-state conditions. (ii) The bubbles do not coalesce or break up further as they climb through the vessel. 14 The hold-up is an important quantity for the calculation of transfer and/or reaction rates in gas-hquid dispersions.

38

2. The Framework of Population Balance

(iii) The change in hydrostatic pressure in the vessel is not significant enough to change the size of the bubbles. (iv) The bubbles, on an average, climb at the terminal velocity, which is determined by their size. Large population densities will affect the single bubble velocity but this effect will be neglected for this demonstration. The local turbulence also causes the bubbles to display random diffusive displacement in accord with the stochastic differential equation of the type (2.10.1). We shall present this after the particle state variables have been defined. (v) Near the liquid surface at the top, surface tension forces will significantly reduce the velocity of the bubbles, requiring more detailed dynamic analysis. We shall, however, assume that the velocity at the top surface is nearly zero and allow the bubble to escape across the liquid surface by a special boundary condition. A somewhat similar situation occurs at the sparger where the bubbles enter. The bubbles detach from the sparger holes at velocities different from their terminal velocities. We assume that it is possible to specify the size and velocity of the bubbles leaving the sparger. Consider the particle state for the model. Clearly, external coordinates are needed because the vertical position of the bubbles is needed to recognize their exit from the vessel. However, transverse coordinates are not important because they do not affect their vertical climb. Because the rise velocity depends on the size of the bubble, we let bubble volume represent its internal coordinate. Hence, the particle state must therefore consist of volume X as its internal coordinate and vertical location z as its external coordinate. Clearly, Q^ = [0? Qo) and Q, = [0, H]. We let the number density of bubbles be represented by /^(x, z, t). Since the volume of a bubble is assumed not to change as it climbs through the vessel, we have X{x) = 0. The vertical motion of the bubble is modeled as dz = Z,{x) dt + ^2D{x,z)dW,

(2.11.14)

where Z,(x) is the terminal velocity of the bubble is (of the form ax^), and D(x, z) is the bubble diffusion coefficient with dependence on space as well as its volume. Further, we assume Stratonovich integration in the stochastic differential equation (2.11.14) above. The steady-state population balance equation may now be immediately written from Eq. (2.10.7) with /i = 0 by

2.11. Formulation of Population Balance Models

39

virtue of assumption (ii) above, as (2.11.15)

i)(x,z)|A

where / j = /^(x, z) is the steady-state population density in the vessel. Since the differential equation above is second-order with respect to z, two boundary conditions are required. At z = 0, we use the continuity of total fluxes between those at the sparger and those into the bed to write NX(^)gix)

= Z,ix)Mx,

0) - D(x, 0) | / i ( x , z )

,

(2.11.16)

Jz = 0

where Z^{x) is the velocity of the bubble of volume x detaching from the sparger, ^(x) is the size distribution leaving the sparger, and it is assumed that the terminal velocity of the bubble has been reached at the vessel entrance itself. The total number density N^ is obtained by a volume balance with the gas supply rate. We let the superficial velocity of the gas (under the conditions in the vessel) be v^.^^ Thus, (2.11.17)

Z^{x)g{x) dx = V,.

N.

When we review assumption (v) further, the boundary condition at z = H becomes -D{x,H)

dz

= X(x)/i(x, HI

Z{z, H) ^ 0. (2.11.18)

z=H

The problem is now completely stated. An analytical solution is easily obtained for Eq. (2.11.15) satisfying boundary conditions (2.11.16) and (2.11.18). The result is exp f,{x,z)=NX{x)g{x)

f" Z,{x dC Jz Dix,l K{x)

exp

+

.D{x,i:') Dix, 0

dC (2.11.19)

Since the height of the gas-liquid dispersion is still to be obtained, we 15 The superficial velocity is defined as the volumetric flow rate divided by the entire cross-sectional area of the vessel, whether or not all of the area is available for flow.

40

2. The Framework of Population Balance

conserve the liquid volume to write H-H

= 0

dxx

dzMx,z),

(2.11.20)

0

which yields a nonlinear algebraic equation in H when (2.11.19) is substituted into (2.11.20). The holdup of the gas per unit cross-sectional area is then obtained as the left-hand side of (2.11.20), once we have solved for if. In the foregoing demonstration, we had limited ourselves to include only the kinematic aspects of bubble motion. A dynamic model including force balances on bubble motion would have called for adding the bubble velocity also as a particle state variable. Such a model could also have been considered allowing for bubble velocity to be a random process satisfying a stochastic differential equation of the type (2.11.14). The basic objective of this example has been to demonstrate applications in which particle state can be a random process. The next and the last example in this chapter considers a similar application, but with a distinction that can help address an entirely different class of problems.

2.11.5

Modeling of Cells with Dynamic Morphology

We are interested here in living cells with morphology as determined by the location of specific organelles within the cell. The motivation for this consideration arises because the behavior of the cell may depend critically on such morphology. For example, cell division may depend upon the time it takes for a divided nucleus in the interior of the cell to migrate to the cell's periphery. Since the purpose of our example is to demonstrate formulations, we shall consider a relatively simple situation. The model assumptions are as follows: (i) We envisage a population of cells described by two particle state variables, viz., the mass of the cell m, and the location of one of its nuclei following nuclear division, which migrates to the periphery of the cell. The position of the migrating nucleus is described by a radial coordinate, say x, measured from the center of the cell, which is assumed to be spherical. Note that the spatial coordinate x is still an internal coordinate. The particle state is given by [m, x] with a

2.11. Formulation of Population Balance Models

41

domain

r

r /3mY'^'

Q, = | m 6 [ 0 , o o ) ; x e ^ 0 , ( ^ ^ j

where p is the density of the cell and the maximum value of x is shown to be the radius of the cell. Although the foregoing particle state domain may seem like a two-dimensional one, since the "spherical" cell must itself be viewed as three-dimensional, the domain Q^ is four-dimensional and defies pictorial representation. In terms of this domain, the total population density N(t) is given by 1*00

n3m/7rp)l/3

N{t) = \ dm \ Jo Jo

A%x^f^{m, x, t) dx.

The particle state just defined clearly refers to cells with divided nuclei. We could refer to them as "pregnant" cells and distinguish them from those that have single nuclei with only cell mass as the assigned particle state. While this formulation is quite practicable, we will, for the sake of simplicity, regard all cells to be pregnant and hence described by the particle state [m, x]. Further, each cell will be assumed to have its migrating nucleus at its center at the instant of its birth. (ii) The continuous phase is assumed not to limit cell growth so that the growth rate, M, is regarded as solely dependent on the cell mass m. (iii) Assume that the migration of the nucleus occurs by two mechanisms. One involves migration caused by the growing cell, and the other that caused by diffusion. The distance to the periphery depends on the mass of the cell, which is simultaneously increasing. We represent the migration of the nucleus by the stochastic differential equation similar to (2.10.4) dx = ^ ^ ( ^ I 4np \3mJ

dt + j2D{m,x)dW,, ^

(2.11.21)

where the first term on the right-hand side represents the displacement due to growth calculated by attributing the cell's peripheral radial velocity of the cell to the nucleus as well; the second term denotes diffusion migration where the dependence of the "diffusion coefficient" on the cell mass could be a device to account for slower diffusion due to the nucleus acquiring a proportionate increase in

42

2. The Framework of Population Balance mass. Again, as in the previous example, we assume Stratonovich integration in (2.11.21). (iv) A cell of mass, say, m' whose second nucleus has reached the periphery is assumed to immediately undergo cell division and give rise to two new daughter cells of masses, say, m and (m' — m), respectively, with a probabihty density function, say, p{m \ m'). Both cells are assumed to continue the process of reproduction by cell division in the manner just recounted. Of course both cells have only single nuclei and therefore are devoid of a migrating nucleus, but assumption (i) deems these cells to become immediately pregnant; thus, both instantaneously acquire a second nucleus located, as assumed in (i), at the center of the cell.

The entire process is depicted diagrammatically in Figure 2.11.2. In writing the population balance equation for the number density function /i(m, x, t), we invoke the general form (2.10.7), remembering that x originates from the spherical coordinate system. |/,(m,x,0+^[M(m)/,(m,x,0]-f^|^

M(m)f4npY"

\__d_ x^D(m,x) — / ,fi(m, ( m , xx,0 , 0 |,, x^ dx

' (2.11.22)

which is written for cells in the interior of the domain of the particle state space Q^ above, i.e., for cells with 0 < x < {3m/4np)^^^ where the cell mass m can be any positive value. Equation (2.11.22) displays no net birth rate term because "new" cells that have x = 0 will appear through the boundary condition at x = 0. To obtain this boundary condition, we first recognize that cells that give rise to newborns are those that arrive at the curve X = {3m/4npy^^ belonging to the boundary dQ^ of ^x- Second, to represent the certainty with which a cell at this boundary immediately undergoes cell division, we may use the "absorption" boundary condition characteristic of diffusion problems, lim

/i(m,x,r) = 0.

(2.11.23)

Thus, the particle flux at the boundary x = {3m/4np)^^^ will only consist of the diffusive flux. A new cell of mass m and x = 0 will come about in either of two mutually exclusive ways. First, it may be the daughter cell that is born of the preceding process of cell division in which the mother cell has

2.11. Formulation of Population Balance Models

(viewed as instantaneous) New-bom cell

43

V^V'

(^^^^ growth & migration of Pregnant" second nucleus) ^^^^

Two daughter cells —_ ^

mass m FIGURE 2.11.2

Schematic of model of cell growth and division.

any mass larger than m. The other is that it is the leftover mother cell after a daughter cell (of any mass) has separated. In order to identify the mathematical description of this boundary condition, it is convenient to define the particle fluxes FM

M(m) fAno\^'^ ^, t) ^ - ^ i-j^j fM

d ^. t) - D{m, X) — Mm, ^, t) (2.11.24)

F^(m, X, t) = M{m)f^(m, x, t)

(2.11.25)

and the normal vector n = [n^, n j to the curve x = {3m/47ipy^^ facing away from the region Q^, where n^ and n^ are the respective components along the m and x axes, given by

njm) =

l/3m-^\^/^ l/3m-^\ "3V471P ) l/3m-^V/3^

'^'A-^)

n^{m) =

1

'1 + U3m-^V

(2.11.26)

9 \ 4np

so that the particle flux normal to x = {3m/4np)'^'^ is given by f„(m, X, t) = n„F„{m, x, t) + n^F^{m, x, t).

(2.11.27)

44

2. The Framework of Population Balance

The boundary condition for the population balance equation at x = {3m/4npY^^ can now be expressed compactly in terms of (2.11.23) to (2.11.26). lim {47ix^F^{m, x, 0} = 2

dm'p{m I m')

lim

{A7ix^F^{m\ x, t)]. (2.11.28)

The left-hand side represents the newborn cells with their second nuclei at their centers. The right-hand side represents the total number of daughter cells of mass m produced per unit time by division of all cells larger than m following migration of their nuclei. In view of the boundary condition (2.11.23), the flux of cells at the boundary involves only diffusive flux. The formulation of the foregoing problem is complete when the initial condition is specified for the bivariate number density function and we take explicit cognizance of the boundary condition at x = 0, lim x'D{m, X) ^^'^"^^""^^^ = o,

(2.11.29)

^^

x^0+

which limits the singularity of the radial gradient at the center of the cell. The model can be readily improved by relaxing some of the assumptions made. For example, distinction is easily made between nonpregnant and pregnant cells by allowing the former to be distributed according to their mass alone and undergoing transition to the latter in the course of their growth. The restriction to spherically shaped cells (of a single characteristic length, viz., the radius x) can be relaxed to accommodate more complex shapes by adding more characteristic lengths. The value of this example lies in showing how the population balance framework, viewed in suitably abstract terms can accommodate even the detail of spatial morphology of the particles. The author is not aware of such models in the literature.

EXERCISE 2.11.1 From the population balance (2.11.22), and the boundary conditions, show that the total numbqr balance is given by dN ~dt

^M{m) —^

\3m J

f^{m, 0, t) dm.

References

2.12

45

CONCLUDING REMARKS

We have in this chapter developed the various features of formulation of population balance. Section 2.11 discussed several examples in which the different features were demonstrated. However, in most of the examples, the net birth term could be dealt with through the boundary conditions. In the next chapter it will be our concern to investigate closely the nature of the birth and death terms in population balance due to breakage and aggregation processes

REFERENCES Bird, R. B., W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, John Wiley, New York, 1960. Coulson, J. M. and J. F. Richardson with J. R. Backhurst and J. H. Harker, Chemical Engineering, Vol. 2, Pergamon Press, New York (4th edition), 1991. Gardiner, C. W., Handbook of Stochastic Methods, Springer-Verlag, Berlin, 1997. Hjortso, M. A. and J. E. Bailey, "Steady-State Growth of Budding Yeast Populations in Well-Mixed Continuous-Flow Microbial Reactors," Math. Biosci. 60, 235-263 (1982). Hjortso, M. A. and J. E. Bailey, "Transient Responses of Budding Yeast Populations," Math. Biosci. 63, 121-148 (1983). Ramkrishna, D. "The Status of Populations Balances," Revs. Chem. Eng., 3, 49-95 (1985). Serrin, J., "Mathematical Principles of Classical Fluid Mechanics," in Handbuch der Physik, Bd. VIII/1 S. Flugge and C. Truesdell, Eds., Springer-Verlag, Berlin, 1959. Tsuchiya, H. M., A. G. Fredrickson, and R. Aris, "Dynamics of Microbial Cell Populations," Adv. Chem. Eng., 6, 125-206 (1966).

This Page Intentionally Left Blank

CHAPTER 3

Birth and Death Functions

In Chapter 2, we had considered systems in which the number of particles changed because of processes that could be accommodated through the boundary conditions of population balance equations particularly with respect to internal coordinates. In other words, new particles appeared or disappeared only at some boundary of the internal particle coordinate space. The example on dissolution kinetics in Section 2.11.1 featured particles, that disappeared at zero particle size. Similarly, in appHcations to crystallization processes, the formation of nuclei of "zero" size by nucleation processes is a birth process that occurs at the boundary of particle size. Although this chapter shall further dilate on such birth and death processes, its major concern is the modeling of processes in which particles may appear or disappear at any point in the particle state space. Birth and death events of the latter type are generally a consequence of particle breakage and/or aggregation processes. Thus we will at first be concerned with the nature of the birth and death rates of particles of any specific particle state for both breakage and aggregation processes which will display the broad phenomenological implements of these processes. However, applications can be served only by modehng considerations more specific to the system under investigation. Such modeling, being often restricted to addressing particle behavior away from the population setting in which the particles are actually present, constitutes a thorny issue of population balance. The reader will undoubtedly recognize that this curse of the many-body problem is not peculiar to population balance! 47

48

3.1

3. Birth and Death Functions

BIRTH AND DEATH RATES AT THE BOUNDARY

We begin with revisiting the boundary condition (2.7.12), which represents the crucial boundary condition representing the birth of new particles at the boundary, which subsequently migrate to the interior of the particle state space. If the birth of new particles represented by the boundary condition (2.7.12) occurs at the expense of existing particles, then the right-hand side of the population balance equation (2.7.9) must include a corresponding sink term. Boundary conditions of the type (2.7.12) are important in crystallization where secondary nucleation, as pointed out by Randolph and Larson (1988), may be governed by the growth rate of existing particles. For example, consider a well-mixed crystallizer where the number density is only a function of the sole internal coordinate selected as particle size x as represented by a characteristic length, which should satisfy a population balance equation of the type (2.7.6). Randolph and Larson discuss a variety of nucleation mechanisms and conclude that contact nucleation is the most significant form of nucleation. Thus, the mechanical aspects of the crystallizer equipment which provide contact surfaces contribute to increased nucleation rate. When growing crystals, containing adsorbed solute on their surfaces, come into contact with other solid surfaces, nucleation is induced. The boundary condition for the formation of new nuclei in a real crystallizer is therefore considerably more complicated than that implied by (2.7.7). Instead, the boundary condition must read as 1(0, X 0/i(0, t) = h,lX{x, X t%Mx, t)

(3.1.1)

where Y here refers to the supersaturation. The right-hand side of (3.1.1) denotes a nonlinear functional of the number density function of crystals and their growth rates at each instant of time and represents the nucleation model. ^ The dependence on the growth rate is inspired by the observation that only growing crystals at any time contribute to contact nucleation. Randolph and Larson (1988) eminently discuss the issues with several references on the subject. They also present a model for the case of size-independent growth rate of crystals for which it need only be a functional of the number density function, since its dependence on the 1 By a functional is meant a mapping of a function into a number. The mapping generally depends on the values of the function defined over the entire domain of its arguments. In the present context, we are concerned about a functional, that depends on the entire size distribution at a fixed time. Note that the definition does not extend here to its time history.

3.2. Breakage Processes

49

growth rate is that of an ordinary function. This model uses a power law function of the growth rate and postulates the functional as the mass density of crystals in the slurry. Mathematically we may write MUx,tlX{Y)']=kp,X{Yy

0

x^f,{x,t)dx

(3.1.2)

where /c is a constant that depends on the shape of crystals, p^ is their density, X{Y) is the size-independent growth rate for this specific context, and i is the power law exponent for growth rate dependence of the nucleation rate. The boundary condition is therefore obtained by combining the model (3.1.2) with Equation (3.1.1). More complicated functional can be envisaged which would of course complicate the solution of the problem. The reader is invited to revisit the examples in Section 2.11 to develop a proper appreciation for the birth and death rates in population balance equations that appear through the boundary conditions. In this regard, the example in Section 2.11.5 presents the boundary condition (2.11.23), which is a particularly interesting example of a birth process occurring at a boundary. We shall now turn our attention to the birth and death functions associated with breakage and aggregation processes.

3.2

BREAKAGE PROCESSES

Although the term "breakage" would seem to connote mechanical fracture of particles, the discussion of this section will apply not only to systems in which particles undergo random breakage, but also to those in which new particles arise from existing particles by other mechanisms. Thus, cell division by asexual means constitutes an example of such a process. The underlying theme stresses independent behavior of individual particles with respect to breakage. Consequently, it would seem that breakage of a particle resulting from collision with other particles would not qualify for discussion here. However, comminution operations in which particles are in intimate contact with each other have been traditionally modeled using the methodology to be outlined here. The justification for this may come from viewing the environment of each particle as an average medium of particles transmitting forces leading to particle breakup. Consider the problem in the general setting of the vector particle state space of Section 2.1 in an environment with a continuous phase vector as

50

3. Birth and Death Functions

described in Section 2.2. Thus, we let the net birth rate be /i(x, r, Y, t) which appears on the right-hand side of the population balance equation (2.7.9). We assume that /i(x, r, Y, t) may be expressed as the difference between a "source" term /z"^(x, r, Y, 0 and a "sink" term /i~(x, r, Y, 0, both due to breakage processes. Our objective is to provide a characterization of the breakage process so as to calculate the foregoing source and sink terms.

3.2.1

The Breakage Functions

If breakup of particles occurs independently of each other, we let b{x, r, Y, t) be the specific "breakage rate" of particles of state (x, r) at time t in an environment described by Y. It represents the fi-action of particles of state (x, r) breaking per unit time. Then we have /z-(x, r, Y, 0 = 5(x, r, Y, 0/i(x, r, 0,

(3.2.1)

the average number of particles of state (x, r) "lost" by breakage per unit time. In order to characterize the source term we should be concerned about the following quantities: v(x', r', Y, t)\

The average number of particles formed from the breakup of a single particle of state (x', r') in an environment of state Y at time t.

P(x, r|x', r', Y, t)\

Probability density function for particles from the breakup of a particle of state (x', r') in an environment of state Y at time t that have state (x, r). This is a continuously distributed fraction over particle state space.

The preceding functions must be obtained by physical models. Collectively, the functions fo(x, r, Y, t), v(x', r', Y, t) and P(x, r|x', r', Y, t) may be referred to as the breakage functions. We have been liberal with the choice of arguments for them in order to stress all their potential dependencies, but several ad hoc simplifications will guide applications. In particular the usefulness of phenomenological models of this kind lies in their being free of temporal dependence. However, the inclusion of time will serve as a remainder of the need for the assumption to be made consciously. The function fo(x, r, Y, t) has the dimensions of reciprocal time and is often called the breakage firequency. It is also sometimes referred to as the

3.2. Breakage Processes

51

transition probability function for breakage. Implicit in the definition of this function is the assumption that breakage, when it occurs, is an instantaneous process. By "instantaneous" we imply a time scale small compared with that in which the particle state varies or (in the cases where particle state does not vary with time) with the time scale used for observing the population. The modeling of the breakage frequency must proceed by examining the events on the time scale in which they occur before either leading to breakage or leaving the particle intact. Because the processes of interest are often random, the modeling will generally proceed using probabilistic arguments. We shall consider examples of such modeling at a later stage but for the present merely make the following broad observation. The analysis generally affords an average time of survival of breakage whose reciprocal may be viewed as the breakage frequency. It is of interest to point out that the breakage frequency is not necessarily a bounded function of particle state. In particular, if particles approaching some hypersurface in particle state space are certain to break, this situation can be described by allowing the breakage frequency to become unbounded as the hypersurface in question is approached. The average number of particles v(x', r', Y, t) formed by breakage of a particle of state (x', r') at time t in an environment of state Y is frequently known. It has a minimum value of 2 but, being an average number, is not restricted to being an integer. In the analysis of microbial populations the property of binary division by cells causes v to be identically 2. In a multiple-splitting process, however, detailed modeling of the breakage process is essential to arrive at the value of v. Its determination from experiments also represents a potential alternative. The function P(x, r|x', r', Y, t), which represents the distribution of particle states for the fragments from breakage, is also a quantity to be determined from experimental observation or by detailed modeling of the breakage process. The function inherits certain properties from conservation laws which must constrain the breakage process. First, it must satisfy the normalization condition P{x,r\x\r\Yj)dV^ = 1.

(3.2.2)

Suppose m(x) represents the mass of a particle of internal state x. Then conservation of mass requires that P(x, r|x', r', Y, 0 = 0,

m(x) ^ m(x').

(3.2.3)

52

3. Birth and Death Functions

Further, we must have m(x') ^ v(x', T\ Y, 0

mix) P(x, r|x', r\ Y, t)dV,,

(3.2.4)

the equahty holding if there were no loss of mass during breakage. Equation (3.2.4) represents the obvious principle that the mass of all fragments within the system formed from breakage of a parent particle must be no more than the mass of the parent. A more subtle inequality arising from the conservation of mass will be discussed at a later stage. Returning to the issue of calculating the source term for particles of state (x, r) originating from breakup, we may write /z+(x,r,Y,0 = dV,.v{x\ r', Y, t)b{x\ r', Y, t)P{x, r\x\ r', Y, 0/i(x', r', 0,

dV, Qr

(3.2.5)

nx

which reflects the production of particles of state (x, r) by breakage of particles of all particle states, internal and external. The integrand on the right-hand side of (3.2.5), which represents the rate of formation of particles of state (x, r) formed by breakage of particles of state, (x', r') is arrived at as follows. The number of particles of state (x', r') that break per unit time is b{x\ r\ Y, t)f^{x\ r', t)dV^,dV^, thereby producing new particles numbering v(x', r\ Y, t)b{x\ r', Y, t)f^{x\ r', t)dV^dV^, of which a fraction P(x, r|x', r', Y, t)dV^dV^ represents particles of state (x, r). The net birth rate of particles of state (x, r) is given by /i(x, r, Y, t) = h^{x, r, Y, t) — h~{x, r, Y, t) the right-hand side of which is given by Equations (3.2.1) and (3.2.5). We have now the complete population balance equation for a breakage process from Equation (2.7.9) whose right-hand side has just been identified. In the following sections, we shall consider some applications of the breakage process. It is of interest to consider the breakage process for a particle population distributed according to mass alone in several applications. 3.2.2

Breakage Process: Mass Distribution of Particles

Particles distributed according to their mass (or volume) are frequently encountered in applications. The size reduction of solid materials is an example of such a breakage process. The evolution of drop size distributions

3.2. Breakage Processes

53

in a stirred liquid-liquid dispersion in which the dispersed phase fraction is small occurs mainly by drop breakage, since coalescence effects will be negligibly small at least at the initial stages. The growth of a bacterial population in which reproduction occurs by binary cell division provides another example of such a "breakage" process. Regardless of the application, we shall consider the breakage process for a population of particles distributed according to their mass denoted x. No continuous phase variables will be considered. The breakage functions consist of a breakage frequency b{x), a mean number of particles on breakage of a particle of mass x' denoted v(x'), and a size distribution for the fragments broken from a particle of mass x' given by P{x\x'), all of which are assumed to be time-independent.^ The conditions corresponding to (3.2.2), (3.2.3), and (3.2.4) respectively imply the following constraints for the function P{x\x'): P(xIx') = 0,

P{x\x')dx=l

x> x\

x'

^y{x')

xP(x\x')dx.

0

(3.2.6) The inequahty to the extreme right becomes an equality if there is no loss of mass during breakage. We shall assume this to be the case. If the breakage is binary we have v{x') = 2. Also for this case, P{x\x') must satisfy the symmetry condition P{x' — x\x') = P{x\x') because a fragment of mass x formed from a parent of mass x' (undergoing binary breakage) automatically implies that the other has mass x' — x so that their probabilities must be the same. For breakage involving more than two particles a somewhat more subtle inequality is imposed by the conservation of mass (see McGrady and Ziff, 1987). This inequality is stated as follows. Let z < x!II. Then

0

xP(x|x')dx ^

xP(x' — x|x')dx,

which may be understood in the following light. Consider the fragments from breakage (of a particle of mass x') in the range [0, z] of particle mass. It appears that Valentas and Amundson (1966) were the first to consider a population balance analysis of breakage and coalescence processes in dispersed phase systems in the chemical engineering literature. While at the University of Minnesota, the author recalls, in particular, Oleg Bilous' significant contribution to the foregoing effort at the initial stages, although he became uninvolved in subsequent development of the work.

3. Birth and Death Functions

54

While there could be several fragments in the foregoing range, there can be at most one particle in the complementary range of mass [x' — z, x']. If we assign the number distribution of the complementary range to the smaller range [0, z], this total "hypothetical" mass contained herein cannot exceed the actual mass in this range. If the function P{x\x') were monotone decreasing, then for X G [ 0 , z ] we clearly have P{x\x') ^ P{x' — x|x'), from which the preceding inequahty obviously holds. However, monotonicity is not often reasonable so that the inequahty constraint must be treated as it is. For binary breakage the number distribution in the two ranges being exactly equal, the inequality above becomes an equality. The population balance equation for the breakage process just discussed becomes

v{x')b{x')P{x\x')Mx\

t)dx - b{x)Mx, ty

(3.2.7)

Suppose further that growth processes are absent. Then the population balance becomes dMx. t) dt

v{x')b{x')P{x\x')f^{x\

t)dx - b(x)f,{x,

t).

(3.2.8)

In this process, if mass is conserved during breakage, then the total mass in the system must remain constant. It is of interest to examine the "first moment" of the number density function /i^ defined by /ii

=

x/i(x, t)dx,

which represents the mass density of particles in the system at any time. If mass is conserved during breakage, we expect this quantity not to vary with time. We shall investigate whether Equation (3.2.8) is consistent with this requirement. Multiplying the equation by x and integrating over the semi-infinite interval with respect to x, we obtain dt

xdx

v{x')b{x')P{x\x')f^{x\

t)dx' -

xb(x)/^(x, t)dx.

On the right-hand side, the integration region in the (x, x') plane implied by the first integral, viz. {x < x' < oo; 0 < x < oo}, can be also written as

3.2. Breakage Processes

55

{0 < X < x'; 0 < x' < 00}. Hence, the preceding equation becomes dx'b{x')f^(x'

dt

t)v(x') 0

xP{x\x')dx

—

Jo

xb{x)f^{x,t)dx.

Using the extreme right of (3.2.6) as an equahty because of assumed exact conservation of mass, we obtain from above that d^ijdt = 0, the resuk sought. Thus, the mass density of particles fi^ is time-invariant.^ In what follows we seek a characterization of the preceding breakage process in terms of the cumulative mass fraction of particles with mass in the range [0, x] defined by F,{x, t) =

1

x'/i(x', t)dx'.

(3.2.9)

/^i

Notice that this function was defined earlier for particle volume (Equation (2.3.5)). The function is clearly a cumulative distribution function because it is monotone increasing and approaches unity at infinite particle size, as it should. For continuous number density /^(x, t) we may write SF,{x, t)

(3.2.10)

/^i — - ^ — = •x:/i(:x;, t).

It is of interest to identify the equation in ^^(x, t) satisfied by the breakage process. Replacing x by ^ in Equation (3.2.8), multiplying by (^, and integrating with respect to ^ over the interval [0, x], we obtain fi,

dF (x t) C^ g^' = ^^^1

P{^\x')b{x')v(x')Mx',t)dx'

mi)M^, t)di

If we convert the domain of integration in the double integral above, { 0 < ( ^ < x ; ( ^ < x ' < o o } into the equivalent domain {0 < (^ < x'; 0 < x' < x} u {0 < (^ < x; X < x' < oo}, the preceding equation becomes 8F,{x, t) dt

+ 3

dx'b{x')f^{x\

dx'b{x')f^{x\

t)v(x')

t)v(x')

d^mi\xi

d^m^W)

^b{^)m.

t)di (3.2.11)

A subtlety in regard to the conservation of mass when the breakage frequency increases rapidly as particle size vanishes is discussed by Ernst and Szamel (1992). The issue, however, had arisen in a paper of Filippov (1961) referenced in Chapter 5. In this situation mass is "lost" to particles of mass zero.

56

3. Birth and Death Functions

Using the extreme right as an equahty in (3.2.6), the first term on the righthand side of the preceding equation can be seen to cancel with the third (and last). Furthermore, we define the function G{x\x') =

v(x') ' 0

(3.2.12)

d^iP(i\x'),

which is readily interpreted as the volume fraction of broken fragments (from breakage of parent of mass x') that have mass less than x. The function is clearly a cumulative distribution function satisfying the properties, G(0|x') = 0 and G{x'\x') = 1. Equation (3.2.11) can then be condensed into the following equation: Ml

dF,{x, t) dt

dx'h{x')x'f^{x\

t)G{x\x').

In view of (3.2.10) the right-hand side of this equation may be conveniently represented in terms of a Stieltjes integral"^ converting the equation into the following evolution equation for breakage processes: dF (x t) r°° g/ =1 b(x')G(x\x')8,.F^{x',t).

(3.2.13)

This equation does not show a sink term because particles with mass less than or equal to x cannot disappear by breakage; they can be formed by breakage of larger particles. We shall have occasion to use this equation in dealing with various aspects of breakage processes. Metallurgical engineers have used Equation (3.2.8) for the analysis of comminution operations. In particular, the reader is referred to the works of Kapur, referenced in Chapter 5. In the following sections, we consider further applications of breakage processes. These applications demonstrate additional issues not formally covered in the theoretical treatment because they are peculiar to the area of interest. 3.2.3

Evolution of Drop Size Distributions in Stirred Lean Liquid-Liquid Dispersions

Liquid-hquid dispersions are of considerable interest to appHcations in a diverse variety of industries. Such dispersions involve two almost immiscible 4

The definition of the Stieltjes integral can be found in any treatment of integral calculus. See, for example, Taylor (1955), p. 532.

3.2. Breakage Processes

57

liquids, with one dispersed in the form of fine droplets in the other continuous hquid phase. The dispersion is generally accomplished by a mechanical stirrer in a vessel provided with baffles at the periphery to avoid vortex formation. The drops break near the impeller zone and circulate through the vessel with almost neghgible breakage occurring in the recirculation zone. However, coalescence between droplets may occur in the recirculation zone. We restrict consideration to lean dispersions to emphasize low dispersed phase fraction, which will allow coalescence processes to occur at negligibly small rates during the initial stages of evolution of the drop size distribution. It is usual to assume that the stirring provides for a uniform population density in the vessel in spite of recognition of the fact that breakage rates of drops vary sharply with location of the drops in the vessel.^ Furthermore, the breakage functions are also assumed to be independent of the droplet's position, which is in even greater defiance of the observations made earher about drop breakage. As a consequence of these assumptions, the model for evolution of drop size in a batch vessel (closed to mass exchange with the external environment) is described by either Equation (3.2.8) or (3.2.13). The objective of this section is to discuss the circumstances under which the breakage process model as described by (3.2.8) or (3.2.13) may be a reasonable description of the drop size evolution in a stirred vessel. We propose to examine the assumptions of the model from a slightly more general point of view, which is represented in the following assumptions: (i) The breakage functions are a function of position but independent of time. Letting x be the drop volume or mass, and r the position vector of the drop, the breakage frequency is given by b{x, r), and the mean number of fragments from breakage of drop of mass x is denoted v(x, r). (ii) Assume that breakage is local in that the drop that breaks shares the same location as the fragments from breakage immediately after the event. Thus, the function relating to the sizes of the fragments is given by P(x, r|x'). The population balance equation for the number density function, denoted /i(x, r, t) is identified as follows. The left-hand side of the population balance equation to be satisfied is given by that of Equation (2.7.9) (replacing the vector x by the scalar x and 5 See Shah and Ramkrishna (1973).

58

3. Birth and Death Functions

dropping the term X{x) since drop size may be assumed to remain constant); the right-hand side is obtained as follows. First, we modify the arguments in Equation (3.2.1) and Equation (3.2.5) by replacing x by x and eliminating the vector Y. Next, we modify Equation (3.2.5) to include integration only with respect to size and not with respect to space, since breakage is confined to the local neighborhood. The resulting equation is

8t

b(x\ r)v{x\ r)P{x, r\x')f^{x', r, t)dx

/i(x, r, 0 + V , - R / i ( x , r, 0 =

- b{x, r)/i(x, r, t).

(3.2.14)

The task at hand is one of starting from Equation (3.2.14) and deriving Equation (3.2.8) by defining the volume-averaged number density function /i(x, r, t)dV, (with due apologies for using the same notation for the number densities on both sides) where Q^ is the domain of volume V^ in the vessel containing the dispersion. Integrating Equation (3.2.14) over Q^ we obtain

dt

VJ,{x, t) =

b{x\ r)v(x', r)P(x, r\x')f^{x\

dV,

r, t)dx

Or

dV^b{x, r)/i(x, r, 0. We now assume that the traversal of the drop population through the recirculation zone occurs so rapidly that the population density is the same all over the vessel. Also V^ may be assumed to be constant with respect to time so that the preceding equation becomes j/Ax,

t) =

dx'fiix', t)

-fM,

t) j ^

r J

1

b{x', i)v{x', r)P(x, r\x')dV,

dV^b(x, r).

(3.2.15)

3.2. Breakage Processes

59

When we set b(x)^

1 Kj

n.

dV,b{x, r),

' ^ ^

v(x')

' '

VMx') JQr

b{x\ r)v{x\ r)dVj.

Equation (3.2.15) becomes identical to (3.2.8). Thus the apphcabihty of Equation (3.2.8) by using volume-averaged breakage functions for describing the evolution of drop size distributions in a stirred vessel depends upon the rapid circulation of the drop population through the recirculation zone. This discussion also points to the inadequacy of Equation (3.2.8) in describing the process in large stirred vessels where the assumption of uniform population density in the vessel may not be borne out.

3.2.4

Mass Transfer in a Lean Liquid-Liquid Dispersion

We shall consider here an application that captures the essence of population balance modehng because it addresses a physical process occurring in a dispersed phase system in the course of formation of the dispersed phase. Consider a continuous flow device to which is fed the continuous phase as well as the dispersed phase in the form of drops of uniform size (i.e., monodisperse feed) while the well-stirred dispersion is being simultaneously withdrawn (see Figure 3.2.1). The dispersed phase fraction is maintained at a low value so that drop coalescence may be regarded as negligible and the drop population evolves by breakup alone. A solute contained in the dispersed phase droplets is being extracted into the continuous phase as the drops pass through this "continuous extractor." The objective of the exercise is to predict the total amount of solute removed from the dispersed phase as it flows through the vessel under steady state conditions. This problem, excerpted from the work of Shah and Ramkrishna (1973), is based on the following assumptions: (i) Drops can break only above a certain size. When a drop breaks, it does so into two equal halves. (ii) Mass transfer of the solute occurs by pure diffusion in a spherically symmetric manner. External resistance to transfer at the drop surface is neglected, although this is not a restrictive assumption.

60

3. Birth and Death Functions

Continuous Phase feed

Dispersed Phase feed

Outflow of dispersion

FIGURE 3.2.1 Continuous extractor of Section 3.2.4 fed with drops of fixed size and solute concentration. (iii) When a drop breaks all concentration gradients of the solute within the droplets established by diffusion are destroyed, so that the daughter droplets have spatially uniform solute concentrations at the instant of their birth. (iv) The solute concentration does not influence the breakage of the droplets. (v) Mass transfer does not significantly change the droplet size. (vi) The dispersed phase fraction being small, the concentration of the solute in the continuous phase is not affected substantially. We shall briefly deliberate the choice of the particle state. Since the quantity of interest is the mass transfer rate from the droplets, particle state must be chosen to yield from it the instantaneous mass flux from the droplet. The mass flux by diffusion requires the concentration profile near the surface. Indeed, drop size (say radius) is clearly important; if we choose the average solute concentration in the droplet as another variable, the two together cannot yield the surface mass flux. But since the concentration at birth is uniform, specification of drop age, the time elapsed since its birth, can be used as a third particle state variable. Shah and Ramkrishna (1973) provide the details of the calculation of how drop size x, the average solute concentration c, and drop age T together help to calculate the mass flux at the drop surface. The mass flux will directly provide the rate of change of average concentration C(x, c, T) SO that the rate of change of particle state required for the population balance model is also completely identified.

3.2. Breakage Processes

61

The breakage frequency of the drops was assumed to be \k{x - xj,

X > X,

Assumption (i) imphes that the value of v is identically 2, and since the division is exactly into two halves, the daughter drop size distribution can be described by the Dirac delta function

In identifying the steady-state population balance equation for the number density function/^(x, c, i), we appeal to the general form (2.8.3) and drop the time derivative. Also we take note of the fact that drops which appear in the vessel either by entering with the feed or by breakage of larger droplets must necessarily be of age zero so that they are accounted for in the boundary condition at age zero. Thus, the population balance equation becomes — [C(x, c, T)/I(X, C, T)] + ^ / I ( ^ ' C' '^) = ~ 0-^i(^' ^' '^^ ~ ^ W / i ( ^ . ^. ^l

(3.2.17) where 9 is the average residence time in the vessel. If we denote the feed distribution in terms of drop size and concentration by fij{x, c), the boundary condition at age zero may be written as /i(x, c, 0) = ^ / i , / ( x , c) + 2

b(x')P(x|x')/i(x', c, T)dx'

(3.2.18)

where the fact that the left-hand side represents the "flux" of particles of age zero will become more apparent by recognizing that particle velocity along the age coordinate is identically unity. The first term on the right-hand side of (3.2.18) represents the entering drops of age zero while the second denotes those that are born in the vessel by breakage of larger drops. Since the feed drops all have the same size with radius, say Xj-, then letting the dispersed phase fraction be we may write f^j{x,

c) = ^—3 d{x- Xj-)S{c - CfX

62

3. Birth and Death Functions

which displays the assumption of spherical drops, and uniform solute concentration in all the feed drops. Shah and Ramkrishna (1973) solve the population balance equation (uncoupled from the continuous phase equation because of assumption (vi)) analytically to be able to compute the total mass transfer rate M into the continuous phase is given by /•oo

M = —

/"oo

dx \

Jo

Jo

dc

dTC{x, c, T)/I(X, C, T), 0

where the limits of integration on size and concentration are in fact bounded from above by Xj- and Cf respectively because of the very nature of the process. It is of interest to peruse some of the results obtained from the model above because it brings out the special capabilities of the population balance model. Since the breakage frequency assumed forbids breakage below a size x^, the drops in the extractor must consist of a finite number of "generations" of successively decreasing sizes {xj-/2^; /c = 1, 2,..., X} where K is the smallest integer such that Xf/2^ ^ x^. Notice in particular that the classical concept of the residence time does not apply to any of the emerging droplets of the /cth generation (except when /c = 0) because the droplets could not have entered the vessel at all! Rather they were formed by breakup of larger droplets within the vessel. Because of the random nature of the breakage process, the solute concentration must be distributed in each generation of droplets. It will be interesting to see how this concentration distribution varies among the different generation of droplets. The total population density in the extraction vessel at steady state, denoted N, is given by f*Xf

(*Cf

f* ao

N = \ dx \ dc \ dif^ix, c, T) Jo Jo Jo so that the trivariate distribution of size, solute concentration, and age, denoted/;^cr(^' ^» ^\ is given by 7 JXCT\^^

_ 7i(x, c, ^y ^} —

j^

T) •>

from which the size distribution of droplets, /^(x) is obtained as /xW =

dc

0

3.2. Breakage Processes

63

The conditional density 7c|;f(c, x) is then calculated as Jc\x\^^ ^) ~

AW *

Letting x^ = Xj-/2' be the radius of the kth generation of droplets, we denote the concentration distribution in this generation of droplets by fi{c) = fc\xi^^ x j . It is of interest to examine this concentration distribution for each generation of droplets. Shah and Ramkrishna (1973) have calculated these distribution functions from the complete solution to the population balance equation (see Exercise 4.2.2 in Chapter 4). Their results are reproduced in Figure 3.2.2. For the sake of comparison, the concentration

0-2

0-3

0-^

as

0-6

07

10

Dimensionless Concentration

FIGURE 3.2.2 Concetration distributions,/;(c), predicted by the population balance, model of Section 3.2.4 for different drop sizes (continuous lines) compared with predictions by model based on instantaneous breakage and exponential residence time distribution (dotted lines). Reprinted from Shah and Ramkrishna (1973) with permission from Elsevier Science.

64

3. Birth and Death Functions

distributions in the respective generations are also obtained by assuming the breakage to be infinitely fast, i.e., the entire drop size distribution is instantaneously attained on entry of the feed droplets into the mixer. Such a perspective has appeared in the hterature (Gal-Or and Padmanabhan, 1968). The concentration distributions for the instantaneous breakage model, which evolve directly because of an exponential residence time for each generation, are represented by dotted lines in Figure 3.2.2. The zeroth generation has a concentration distribution shifted considerably to the right for the population balance model (relative to that for the instantaneous breakage model) because drops with lower concentration must have higher life spans which breakage renders unlikely. This finding is also true of succeeding generations but with the differences, however, gradually decreasing for the progressively smaller drops. The foregoing example is interesting because it shows population balance models can account for the occurrence of physicochemical processes in dispersed phase systems simultaneously with the dispersion process itself. Shah and Ramkrishna (1973) also show how the predicted mass transfer rates vary significantly from those obtained by neglecting the dynamics of drop breakage. The model's deficiencies (such as equal binary breakage) are deliberate simplifications because its purpose had been to demonstrate the importance of the dynamics of dispersion processes in the calculation of mass transfer rates rather than to be precise about the details of drop breakup.

EXERCISE 3.2 By absorbing drops of age zero into the population balance equation with the aid of the Dirac delta function, the equation can also be written as ^ [C(X, C, T)/,(X, C, T)] + ^ A(X, C,T)=^

[ / i , / x , C)S{T) - /^(x, C, T)]

-/7(x)/i(x, c, T) +2(5(T)

^(x')P(x|x')/i(x', c, x)dx'.

From the foregoing population balance equation, establish the boundary condition (3.2.18).

3.2. Breakage Processes 3.2.5

65

Modeling of Microbial Populations

Fredrickson, Ramkrishna, and Tsuchiya (1967)^ have developed a very general population balance framework for investigating the dynamics of microbial populations by defining the particle state vector as a physiological state vector. This may be considered as an example of a breakage process because cells are assumed to reproduce by binary division. The equations are somewhat similar to those discussed in Section 3.2.1 and will therefore not be recalled in this section. Fredrickson et al (1967) addressed their analysis specifically to what are known as "prokaryotic" populations in which internal organization and morphology are not pronounced. We shall outline here how a more general theory can be constructed for "eukaryotic" organisms characterized by an advanced state of internal organization with detailed morphology. The physiological state vector can recognize the amounts of various biochemical entities in different cellular compartments by means of a partitioned vector as pointed out in footnote 3 in Section 2.1. The component vectors would then represent the amounts of different entities (assumed to be uniform) in each compartment. The rate of change of each component vector could then accommodate interaction between different compartments through its dependence on all the other component vectors. One may also be able to accommodate morphological features that may be prerequisite to cell division by using developments along the same lines as in Section 2.11.5. For example, more complex shapes and their dynamics can be described by employing a finite dimensional vector of characteristic dimensions. Also, stochastic features that may be associated with the smallness of the system as modeled by stochastic differential equations of the type (2.10.4) are accommodated in the manner demonstrated in Section 2.10. Thus a considerably more general theory of microbial populations is within the scope of the population balance framework as expounded in this book. The heart of the population balance model for breakage processes lies in the breakage functions described in Section 3.2.1. The breakage functions must be obtained either directly from experiments or by modeling considerations related to the processes causing the breakage. This is the subject of the next section. 6

See Chapter 1 for reference.

66

3. Birth and Death Functions

3.2.6

Modeling of Breakage Functions

We shall at first be concerned with the breakage frequency function b{x, r, t) in which we have dispensed with the continuous phase vector Y, although the arguments to follow are not particularly dependent on this assumption. The existence of such a breakage frequency, which is essentially a probability per unit time, reflects the Markoffian nature of the assumed breakage process, viz., the breakage in the infinitesimal time interval (t to t + dt) to follow is not dependent on the "past history" of the particle. Yet another way of stating this assumption is that processes leading to the breakage of a particle occur on a time scale, say T, considerably smaller than the time scale t. The processes responsible for breakage of a particle may be viewed as occurring in cycles, each cycle occurring over a period very much smaller than the time scale in which the population changes observably. During each cycle, the particle state vector at the beginning of the cycle (say (x, r) as defined in Section 2.1) does not change perceptibly. Modeling of the breakage process during the cycle must be designed to calculate the breakage probability per unit time conditional on the particle state (x, r) at time t. Thus the model involves calculating a probability associated with breakage and a characteristic time in which the breakage is completed. The ratio of the two provides the breakage frequency. Often a dynamic probabilistic model of the process over the time scale of the cycle is possible which could be based on some additional internal coordinates (denoted by, say, z) not included in the vector x, yielding a probabihty function p{z, T|X, r, t). For example, the model would postulate regions of z-space constituting breakage of the particle so that the rate of probability flow into the boundaries of these regions will yield the breakage frequency using arguments along the following lines. By integrating the foregoing rate of probability flow with respect to t, one obtains the probability that breakup occurs over the cycle time scale. In order to calculate the breakage frequency, this probability must be divided by a characteristic breakage time. This characteristic time may be assumed to be the average time of breakup computed from the distribution of breakage times. The distribution of breakage times is obtained by dividing the rate of probability flow (into the regions of z-space characterizing breakup) by its integral with respect to time. Of course other choices of characteristic times may also be possible. We now provide some examples of formulation of breakage frequency models.

3.2. Breakage Processes

3.2.6.7

67

Drop Breakage

There are a few examples in the Hterature of calculation of the breakage frequency of hquid drops in the turbulent flow (such as that prevaihng in a stirred vessel) of a second immiscible, hquid phase. Coulaloglou and Tavlarides (1977) and Narsimhan et al (1979) have presented models based on somewhat different physical arguments for drop breakage. The former consider breakage to be binary and instantaneous, but to be reinforced by successful separation of the two droplet fragments before drainage of the intervening film. The probability calculated is that of the contact time between the drops being less than the required drainage time. Narsimhan et a/., on the other hand, consider breakup by bombardment of the drop by eddies (smaller than the drop), with at least, as much energy required to create the minimum amount of new interface. The breakage frequency is calculated as the ratio of the probability that an eddy of the appropriate amount of energy is incident upon the drop surface, to the average arrival time of the eddies. Thus, in this model the temporal element lies in the waiting period for the appropriate eddy to arrive, but upon its arrival breakup occurs instantly. We consider an interesting approach by Lagisetty et al (1986) to drop breakage, which although intended by the authors for calculating what is known as the maximum stable drop size, is also amenable to calculation of the breakage frequency along the general lines indicated earlier. In other words, we address the dynamics of the breakage process on a short time scale using additional internal coordinates to characterize breakage. Assume that the drop population is described in terms of their volume or mass distribution and that we seek the breakage frequency as a function of drop volume, say x. Lagisetty et al, describe the deformation of the droplet by a scalar strain measure z that we shall deem to be the additional internal coordinate. The physical process involves the drop being entrained in a turbulent eddy in which the drop is subject to deformation by viscous forces and restoration by surface tension forces.^ Their analysis shows that z, starting from zero, is always positive and that breakage occurs if and when it reaches the value of unity. The uncertainty of breakage arises from two sources. First, the deforming viscous force (available in the eddy) may only deform the droplet 7

There are more subtle issues here which are left for the reader to obtain from Lagisetty et al (1986).

68

3. Birth and Death Functions

to an extent short of breakage, i.e., z reaches a steady-state value smaller than unity, in which case regardless of the life time of the eddy, the droplet cannot break. Second, the viscous force in the eddy may be sufficiently strong to deform the droplet to break it in finite time, provided, however, that the eddy remains intact for this period of time. Thus, the breakage frequency may be calculated as follows. We identify the joint probability that the drop is entrained in an eddy of the appropriate size (capable of breaking the droplet) and that its lifetime is large enough to break the drop. This is the probabihty that the drop will undergo breakage, which must be divided by the characteristic breakage time in order to obtain the breakage frequency. The characteristic time is readily calculated by solving the difTerential equation for z as has been done by Lagisetty et al. (1986). Although the foregoing procedure is evident, we now outline an alternative approach for two reasons. First, the approach demonstrates how dynamic probabilistic modeling may be performed, which is of general pedagogical interest to this book. Second, a drop in a turbulent flow is subject to random pressure forces, which is more appropriately modeled as a dynamic stochastic process. Thus, the differential equation for the drop deformation given by Lagisetty et al. may be modified to include a stochastic perturbation over the average value of the shear stress. Confining ourselves to a Newtonian liquid drop, we write the equation in z as

^T^^

dz =

dx + ^dW^,

z(0) = 0,

(3.2.19)

where i is the time scale in the small, a is the mean stress, and a governs the size of the fluctuation of the turbulent stress about its mean. This stochastic differential equation is of the type encountered in Section 2.10; if o is regarded as independent of z, the partial differential equation in the probabihty density p(z, T|0; X) for the solution process, conditional on its initial value of zero and the drop being of size x, is given by 5p dx

d_ dz

1

2

2- +«

= a-i.

dz

(3.2.20)

At the boundary z = 1, the drop is deemed broken. Since z can take on only positive values, the process must remain in the interval, which implies that the total probability flux at z = 0 must vanish. Thus, the boundary condi-

3.2. Breakage Processes

69

tions for Equation (3.2.20) are given by p{i, T|0; X) = 0,

_^^p(Q> j l Q ; ^) ^L_^

jjp(^^ ^\^'^ ^) = ^-

(3.2.21)

The initial condition simply reinforces the fact that z must have the value zero so that p(z, 0|0; x) = 3{zl

(3.2.22)

which completes the specification of the stochastic deformation process. What remains now is the calculation of the breakage frequency. To this end, we recognize that the probabihty that the drop breaks during the interval T to T + dr is given by the total rate of probability flow out at z = 1, i.e., 5p(l,

(

T | 0 ; X)

-^ ^ ^J

\\

,,

,^

,

^ + U + -jp(l, T|0; X).

This probability must be viewed as being conditional on the survival of the eddy during the period from 0 to T. Since, for the drop to break, the eddy in which it is trapped must be "alive" for at least until T, we let the distribution of life spans of the eddy be denoted by I/^(T) and infer the probability of the lifespan of the eddy to exceed T as l^\l/{T')df. Thus the probability density in terms of time T, say JS(T|X), for the drop to break in the eddy in question is given by

P(T\X)

r

=

Jo

^{x')dx'

— 1(

dp{\, T|0; X) ( 1\ ^' — + l a + - l p ( l , T | 0 ; X)

"oo

dx'

—

dp{l,

T | 0 ; X)

8-,

f

+ [a +

1 \ ,,

,n

-jpil,x\0;

X)

X

where the denominator displays the probability that the drop will break if given "infinite" time in the eddy. Note, of course, that if the eddy in which the drop is entrapped is not capable of breaking the drop even in infinite time, the question of breakage in finite time does not arise at all. In fact, the analysis should incorporate the probability that entrapment occurs in an eddy, which is capable of breaking the drop at least in infinite time. The reader surprised by the infinite upper limit of integration should recognize this as a mathematical abstraction since it is still to be regarded as a short time scale. The average breakage time, denoted , may be calculated as

TjS(T|x)(iT.

= 0

70

3. Birth and Death Functions

In accord with the development given earher, the breakage frequency, denoted b(x), is given by poo

b{x) = ^lo

rao

\l/{x')dx'

di

JX

+

-a

a+ -

p(l,

T | 0 ; X)

dx



(3.2.23)

in which the short time scale does not appear. The actual calculation clearly depends on solving the boundary value problem comprising Eqs. (3.2.20)(3.2.22).^ More rigorous analysis of drop deformation and breakage can be attempted by considering the equations of motion. With small drops, the flow is likely to satisfy the approximation of creeping flow even if the continuous phase is in turbulent flow, as long as the density difference between the fluids is sufficiently small. We have thus seen how the breakage frequency can be modeled. The other breakage functions are considerably more difficult to model without a more detailed statistical formulation of the breakage process in which randomness in the discrete number of fragments formed from breakage has been addressed (see, for example, Derrida and Flyvbjerg, 1987; Davis, 1989; HiU and Ng, 1997). There have also been models for the breakage functions including the breakage frequency and the size distribution of broken fragments such as those of Nambiar et al. (1992,1994) and Tsouris and Tavlarides (1994) based on the assumption that particle breakage is binary. Direct experimental verification of whether breakage is exclusively binary is, however, quite difficult. A problem of great interest is the identification of breakage functions from experimental data on particle size distributions in breakage systems. We shall not dilate on this subject here as it is treated in detail in Chapter 6. Finally, there are situations in which breakage may occur by continuous erosion of the particles, which can be dealt with via the convective term X. Such processes do not require special discussion, although the identification of the erosion rate is an interesting problem. 3.3

AGGREGATION PROCESSES

We are concerned here with particulate events in which two or more particles may be involved. Aggregation processes occur commonly in nature 8

Boundary value problems of this type can be solved by using standard techniques. See, for example, Ramkrishna and Amundson (1985).

3.3. Aggregation Processes

71

and in engineering processes. For example, the formation of rain from a cloud of very fine droplets occurs by coalescence of the droplets (due to relative Brownian motion) to form larger drops that fall under the action of gravity. Coalescence occurs between bubbles or droplets in a diverse variety of dispersed phase systems arising in industrial processes. Aggregation between cells in biological processes, between particles in the manufacture of fine powders, between sites on a catalyst surface (called sintering), and numerous others forms a large class of applications. Aggregation must occur at least between two particles, although in very crowded systems, it is conceivable that several adjacent particles could simultaneously aggregate. Our concern will be restricted to systems that are sufficiently dilute to make only binary aggregation significant. Also, aggregation covers a variety of processes ranging from coalescence, in which two particles completely merge along with their interiors, to coagulation, which features a "floe" of particles loosely held by surface forces without involving physical contact. In intermediate situations particles may be in physical contact with each other without merger of their interiors. The framework of interest to us is somewhat insensitive to these details. However, the morphological changes in particles with progressive aggregation (which could significantly change their subsequent aggregation rates) can be accommodated, in principle, by a properly chosen set of internal coordinates.

3.3.1

The Aggregation Frequency

The chief phenomenological instrument of the population balance model of an aggregation process is the aggregation frequency. It represents the probability per unit time of a pair of particles of specified states aggregating. Alternatively, it represents the fraction of particle pairs of specified states aggregating per unit time. This interpretation must, however, be modified for the aggregation frequency commonly used in population balance models in which the population is regarded as well-mixed and external coordinates do not appear explicitly in the population density. We will subsequently derive this modified frequency from the quantity that we have just defined. We assume that the population density is so small that during a time interval dt, the probability of more than two particles aggregating simultaneously to form a single particle is only of order 0{dt^) while that of two

72

3. Birth and Death Functions

particles aggregating is of order 0{dt).^ Let the particles be described by the state vector (x, r) in Section 2.1 in a continuous phase of state Y. We define the probability that a particle of state (x, r) and another particle of state (x', r'), both present at time t in a continuous phase with state locally at Y, will aggregate in the time interval t to t -{- dt to be given by a{x, r; x\ r'; Y, t)dt.

(3.3.1)

Alternatively, we recognize that a(x, r; x', r'; Y, t) is the fraction of particle pairs of states (x, r) and (x', r') aggregating per unit time. The aggregation frequency is defined for an ordered pair of particles, although from a physical viewpoint the ordering of particle pairs should not alter the value of the frequency. In other words, a(x, r; x', r'; Y, t) satisfies the symmetry property a(x, r; x', r'; Y, t) = a{x\ r'; x, r; Y, t). It is essential to consider only one of the above order for a given pair of particles. The explicit time dependence in (3.3.1) in the aggregation frequency is generally not a desirable feature in models and is eliminated in the remaining treatment. In order to formulate the source and sink terms in population balance, it is essential to identify the state of the particle formed by aggregation. To this end, we define the internal and external coordinates of a particle formed by aggregation of the pair [(x, r), (x', r')] as ^(x, r; x', r') and p(x, r; x', r'), respectively. These transformations must obviously obey physical conservation principles. We shall come across them in applications and will not attempt to be general in this regard. Furthermore, we assume that it is possible to solve for the particle state of one of the aggregating pair given those of the other and the new particle. Thus, given the state (x, r) of the new particle, and state (x', r') of one of the two aggregating particles, the state of the other aggregating particle is denoted by [x(x, r|x', r'), r(x, r|x', r')]. Next, it is necessary to define the average number of pairs of particles at each instant t with specified states. Accordingly, we define /2(x, r; x', r', t) to represent the average number of distinct pairs of particles at time t per unit volumes in state space located about (x, r) and (x', r'), respectively. The source term for the rate of production of particles in volume (x, r) of state A

9

A.

•

J .

u

r

J

^/j„xfi-

A term is said to be of order 0{at") if lim '^'^o

0{dt")

fO,

^ i^J dt'' l
k
k=n

73

3.3. Aggregation Processes

(x, r), denoted h^{x, r, Y, t\ must account for the fact that the density with respect to coordinates [x(x, r|x', r'), r(x, r|x', r')] must be transformed into one in terms of (x, r) by using the appropriate Jacobian of the transformation and hence may be written as h^(x, r, Y, 0 =

dV,, ^r

d{x, r) 1 :dV^M{x, r; x^ r', Y)/2(x, r; x^ r\ t) 5(x, r)' (3.3.2)

where (x', r') is held constant in the definition of the Jacobian. ^^ In the above equation, the function a(x, r; x', r', Y) must vanish for states (x', r') for which (x, r) assume impossible values, i.e., not admitted by the particle state space; 5 represents the number of times identical pairs have been considered in the interval of integration in the right-hand side of (3.3.2) so that 1/(5 corrects for the redundancy. The sink term h~{x, r, Y, t) is more readily found to be h-{x, r, Y, 0 =

dV^M{x\ r'; x, r, Y)/2(x', r'; x, r, t).

dV,,

(3.3.3)

Since the birth and death functions are now identified the complete population balance equation for an aggregating population is given by substituting for h in the right-hand side of (2.7.9) h^ — h~ SiS calculated from (3.3.2) and (3.3.3).

10 The symbol

'-— represents the determinant d{x, r) dx^ dx^

dx-^ dx^ dx„

dx^

5XjL

dr2

^^3

dx^ dx^

dx dx„ dx„ dr.

dx„

dx„

sh

^

5^ dr.

drs dr,

dr2

dr. df2

dxi

¥7

dx„

dx^

dx^

dx^

dx„

en

Sr,

dr2

8h dr2

This factor is missing in an earher treatment by the author (Ramkrishna, 1985) referenced in Chapter 2.

74

3. Birth and Death Functions

The population balance equation so obtained is not closed because the right-hand side involves a fresh unknown in the pair density function /2. Even without a detailed consideration, it should be obvious that an equation for /2 would involve f^, and so on. Thus, an infinite hierarchy of equations is obtained, and unless some form of "closure" approximation is made the population density cannot be obtained. In population balance analysis, one makes the approximation /2(x', r'; X, r, t) =/i(x', r', 0/i(x, r, t), which represents the coarsest form of closure hypothesis. This assumption implies that there is no statistical correlation between particles of states (x', r') and (x, r) at any instant t. We are not in a position to comment on whether or not this is a reasonable assumption for it at this stage, for it is the subject of Chapter 7. However, we remark here in passing that the assumption becomes increasingly plausible for larger populations.

3.3.2

Aggregation Process. Mass Distribution of Particles

As in the breakage process in Section 3.2.2, we are concerned here with a population of particles distributed according to their mass, denoted x. The population is uniformly distributed in space so that no external coordinates are involved. Environmental effects as well as particle growth are also omitted in the considerations here. The aggregation frequency for particle pairs of masses x and x' is denoted by a{x, x'). This aggregation frequency, which depends on the form of relative motion between particles, can be obtained from that (containing spatial dependence) defined in Section 3.3.1 as follows. We recognize that the population density in particle size can be obtained by integrating the position-dependent density in Section 3.3.1 with respect to the spatial coordinate r over the domain Q^ and dividing by the volume V{Q^) of Q^. On using the spatial uniformity of the population to shed the spatial coordinates in the population density, we may define a(x, x') as dV^a{x, r; x\ r')

dV, a{x, x') =

«r

n^r)

(3.3.4)

where the same symbol has been used for the frequencies on either side of the above equation to check proliferation of notation. Note, of course, that

3.3. Aggregation Processes

75

since particle size is the only internal coordinate, the vectors x and x' are replaced by the scalars x and x\ respectively, in the aggregation frequency on the right-hand side. Equation (3.3.4) shows how the size-dependent aggregation frequency, a{x, x') has the dimensions of spatial volume per unit time while the frequency a{x, r; x\ r') has only the dimension of reciprocal time. The form of a{x, x') determines different apphcations of aggregation processes, and its derivation using (3.3.4) together with models for relative motion is considered in Section 3.3.5. The source function, which represents the rate of formation of particles of mass X by aggregation of smaller particles, is computed as follows. From conservation of mass, we have particles of mass x — x' { = x in terms of the notation used in the general considerations leading to (3.3.2)) aggregating with particles of mass x' to produce particles of mass x. Clearly as x' varies between 0 and X, so also does x — x' so that each pair in the set {[x — x', x ' ] ; 0 < x' < x} is considered twice (i.e., ^ = 2 in (3.3.2)). Thus the source term becomes^^ /i^(x, 0 = ^

a{x — x\ x')/^(x — x\ t)f^{x\

t)dx'.

0

This source term could also have been written alternatively as rx/2

/i+(x, t) =

a{x — x', x')/i(x — x\ t)f^{x\

t)dx'

where the range of integration has eliminated the double counting of pairs in the previous version and hence the factor of 1/2 which appeared there. The sink term represents h (x, t) = / i ( x , t)

a(x, x')/i(x', t)dx'.

The number density function /^(x, t) must then satisfy the population balance equation

g/ife 0 ^ 1 dt

2

a{x — x', x')/i(x — x', t)f^(x\ t)dx' —/i(x, t) 0

a(x, x')/i(x', t)dx'.

(3.3.5)

0

11 In deriving this source term from (3.3.2) we must recognize the transformation of coordinates from {x, r) to (x, f) thanks to the Jacobian present in the integral. The derivation of the sink term, on the other hand, is more straightforward.

76

3. Birth and Death Functions

As in the breakage process of Section 3.3.2, the total mass in the system must be conserved, since each aggregation event conserves mass. Following the development in Section 3.3.2, we obtain the following equation in fi^, the first moment of the number density function dx'a{x — x\ x')f^{x — x\ t)f^{x\ t)

xdx

dt

xf^{x, t)dx

(3.3.6)

dx'a{x, x')fi{x', t).

The domain of integration in the first term on the right-hand side of the foregoing equation, {0 < x < oo; 0 < x' < x}, may be rewritten as {0 < x' < o o ; x ' < x < o o } s o that the term in question becomes 1 2

dx'f^{x\ t)

xdxa{x — x', x')f^{x — x\ t) = 1

"oo

dx' d^ix' + Oa{x', OMx', t)Mi, t), Jo 2^0 J the right-hand side of which is obtained by making the substitution X — x' = ^. The integrand on the right is symmetric in the domain of integration so that it becomes equivalent to the second term on the right-hand side of (3.3.6). Thus, we obtain dfi^ldt = 0, which represents the conservation of total mass in the system. ^^ It is also of interest to identify the equation in the cumulative mass fraction F^{x, t) defined by (3.2.9) which calls for integrating Equation (3.3.5) over the interval [0, x], /^i

dF^ix, t) _ 1 ~dt "2

^d^

dx'a{^ - x\ x')M^

^m. t)d^

- x\ t)Mx\

dx'a{^, x')f^{x\ t).

t) (3.3.7)

The domain of integration in the first double integral on the right-hand side of Equation (3.3.7), {0 < x' < ^; 0 < ^ < x}, may also be represented by 12

For sufficiently strong size dependence of the aggregation rate on particle size, the increase in particle size can jeopardize the vanishing of the integrals on the right-hand side of (3.3.6) and thus endanger the implication of conservation of mass. Such processes involve phase transition (gelation). See, for example, Ernst et ah (1984). Conservation of mass in such cases will involve the particles and the gel together.

77

3.3. Aggregation Processes

{x' < (^ < x; 0 < x' < x} so that Equation (3.3.7) becomes - Ml

aFi(x, t)

1

dt

dx'

ma{^

- x\ x')f^{^ - x', t)f,{x\

UiiL t)di

t)

dx'a{^, x')/i(x', t).

Setting ^ — x' = u, the above equation may be further transformed to 5Fi(x, 0 /^i

1

dt

du{x' + u)a{u, x')f^{u, t)f^{x\ t)

dx' ^/i((^, t)d^

dua{^, u)f^(u, tl

\:

(3.3.8)

where we have replaced the dummy variable x' in the second integral by u for convenience. The domain of integration in the first double integral on the right-hand side of Equation (3.3.8) is depicted geometrically in Fig. 3.3.1. Since this domain and the integrand are both symmetric about u = x\ Equation (3.3.8) can be rewritten as i"i-

3Fi(x, 0 dt

x'dx'

dua{u, x')f^{u, t)f^{x', t).

iAii. t)di

dua{^, w)/i(w, 0-

Replacing the dummy variable x' by S, in the first double integral on the

FIGURE 3.3.1. Region of integration in the first term on the right-hand side of (3.3.8) appears shaded.

78

3. Birth and Death Functions

right side above and recognizing the symmetry of the aggregation frequency, we obtain 5f i(x, t)

duaii, W)/I(M, 00

Writing the above equation entirely in terms of F^{x, t) (see Equation (3.2.9)), we obtain the equation dF,{x, t)

8F,{i, t) 0

"^ ait u) -^^^^-^ dF,{u, 0,

(3.3.9)

which displays Stieltjes integrals on the right-hand side in the same way as Equation (3.2.13). Notice in particular that this equation contains only a sink term because particles with mass less than or equal to x can only disappear by aggregation and cannot be produced afresh. We shall have occasion to use this equation at a subsequent stage. In the following sections, we consider some examples in the application of aggregation processes.

3.3.3

Aerosol Dynamics

Aerosols comprise the dispersion of solid or liquid particles in a gas continuous phase. Atmospheric aerosols contain particles covering a wide range of size from the order of angstroms to microns. The size distribution of these particles evolves as a result of aggregation between particles and growth by accretion of vapor molecules. For a general discussion of the dynamics of aerosols, the reader is referred to the works of Hidy and Brock (1970), Seinfeld (1980), and Pandis and Seinfeld (1998). The issue of specific interest here is the formulation of population balance for an aggregating system in which the particle state has both discrete and continuous regions. This choice of particle state is often convenient for accurate description of numbers in the discrete range. As in the previous example, we assume particle volume x to represent the particle state. Further, the particle volume is distributed discretely between x^ = 0 and x^ in uniform increments of Xj, and continuously in the range x„ < X < 00. In the discrete range, we define the number (per unit volume) of particles of volume x^ i=ix^) by fij{t). The number density in the continuous range is described by /^(x, t). Note that the dimension of/^^(t) differs from that of f^{x, t) by a particle volume because the latter is a

3.3. Aggregation Processes

79

density in particle volume. The total number N{t) of particles per unit volume covering both discrete and continuous ranges is given by

m = E A,(o +

Mx, t)dx.

(3.3.10)

The following additional assumptions pertain to the behavior of the aerosol particles: (i) Each particle of volume x^ in the discrete range can "evaporate" to lose a unit entity x^ at a rate e{x^)io yield a particle of volume x,_ i. Similarly, a particle in the continuous range of volume x can evaporate at a rate e{x) to produce a particle of volume x — x^. However, in the continuous range, the difference between x and X — x^ may be regarded as infinitesimal, suggesting the adaptation of a continuous evaporation rate to be derived subsequently, (ii) The aggregation frequency of any particle pair of volume x^ and x^ in the discrete range is given by a^- (symmetric with respect to the indices i and j ) . The aggregation frequency of any two particles in the continuous range of volumes x and x' is denoted a(x, x'). For aggregation between a particle of volume x^ in the discrete range and X in the continuous range, the aggregation frequency is given by (iii) It is usual in this application to include source terms for homogeneous generation and removal of particles in the discrete as well as in the continuous ranges. In the discrete range, for a particle of state x^, the generation rate is denoted by S^(t), and the removal rate, assumed to depend on the population density, is denoted S[{f^i, t). In the continuous range, for particles of state x, the generation and removal rates are S"^(x, t) and S~{f^{x, t), x, t), respectively. The generation and removal rates in the continuous range are different from those in the discrete range in that the former are densities in particle volumes. In deriving the population balance equation we make the following observations. Since no continuous growth processes have been assumed, there will be no convective terms in the population balance. The birth and 13 The aggregation frequencies in the discrete and the continuous ranges are generally related by a^j = a(x-, Xj). The distinctive notations for particles in the discrete and continuous ranges serve as a reminder as to whence the aggregating pair arises.

80

3. Birth and Death Functions

death terms must account for aggregation between particles in the discrete and continuous ranges. Particles in the discrete range can aggregate to form particles in the continuous range, although necessarily of volume smaller than 2x„. Thus, a population balance of particles in the continuous range below 2x„ must entertain a source term entirely involving the discrete range, while that for particles larger than 2x„ need not. We first identify the population balance equation for the discrete range. Since the smallest particles of volume x^ do not possess a source term by aggregation, their number density will satisfy an equation slightly different from those of volume x^. Thus,/i^ will satisfy ^

(3.3.11)

= ht{t)-h;{t)

where the birth term has no contribution from aggregation because particles smaller than volume x^ do not exist. Thus, the only contribution to the birth term comes from the evaporation process in (i) and generation process in (iii) so that poo

n

hlit) = X e(xj)f,j(l

+ Sj^,) +

J=2

e{x)Ux,

t)dx + StU).

(3.3.12)

Jxn

The first term on the right-hand side above, which represents evaporation of particles in the discrete range, also accounts for the double contribution by evaporation of a particle of volume X2 ( = 2xj) through the use of the Kronecker delta Sj 2- The second term represents the evaporative contribution from the continuous range. The third term is the generation rate. The death term must account for aggregation as well as removal rates so that hiit)

= / i , i <^ X ^ i j / i j + U=i

«a,{x)f,{x, i W / i ( ^ , t)dx\ t)dx\ + 5 r ( / i , i , 0 (3.3.13) Jxn J

where the first term on the right-hand side in curly braces represents the loss of the smallest particles by aggregation with larger particles (in both the discrete and continuous ranges), and the second term is the removal rate. For particles of volume x^, we have the equation ^

= Kit) - hiit),

i = 2,...,n,

(3.3.14)

81

3.3. Aggregation Processes

where the birth term h^{t) is given by

Kit) =

1 '-^ 9 Z ^i-jjfl,i-jflj

I+

^(^r+l)/l,/+l + ( 1

e{x)f^{x, t)dx + St{t).

-^i,n)

(3.3.15)

In this equation, the first term on the right-hand side in square braces represents the formation of particles by aggregation of smaller particles that must necessarily be in the discrete range. The second term in square braces represents the contribution by evaporation of particles in the discrete range and the small part of the continuous range indicated, which arises only for the largest discrete size. The last term is the generation rate. Kit) =fui\

Z «ij/ij +

^iW/i(^. t)dx\ + e{Xi)f^^i + Si {f^j, t) (3.3.16)

The first term in curly braces on the right-hand side above represents loss by aggregation with all particles. The second term represents loss by evaporation and the third by removal. The balance of particles in the continuous range remains. The population balance equation for the continuous range may be written as dt

= h^{x, t) — h (x, t),

x„ < X < 00,

(3.3.17)

in which the birth and death terms, however, require more astute considerations. The continuous domain x„ < x < oo has discrete contributions by aggregation of particles in the discrete range. Because the number of particles in the continuous range is described by a number density in particle volume, the discrete contributions arising from the discrete range must be measured as those made to the density. For example, a particle of volume x^ in the discrete range may aggregate with another particle Xj in the discrete range such that i +j > n, so that the resultant particle is in the continuous range. This discrete addition of a single particle of volume Xi+j must be viewed as a contribution to the number density at x,+^ equal to 1/x^. This is equivalent to distributing the particle number (1 in this case), concentrated at Xi^j, uniformly in the interval Xj.+^- < x ^ x^+j + x^. Mathematically, this

82

3. Birth and Death Functions

distribution may be represented by i- -[ rHw( rxv _- vx,.^,)A -_ H(x m v _- vx , , , ^ i )^i] = = J^/-^i' ^' " ^-^i+j ' " ' < ^ <-^i+j+i ^ '^ '

(3.3.18)

where fO, [1,

X^ 0 X> 0

is the Heaviside step function. Thus, discrete contributions to the continuous density must be scaled by x^ and redistributed as in (3.3.18). It must be clear that the discrete contributions by aggregation to the continuous range can only cover the subrange x„+ ^ < x ^ ^m+i- This feature must be reflected in the source term h^{x, t). Particles in the continuous range x„ < x < oo can also be formed by aggregation of pairs with one in the discrete range and the other in the continuous range. It is useful to make a distinction between such contributions and the discrete contributions involving aggregation between both particles in the discrete range. Consider for a particle of volume x in the continuous range the function [ x / x j , which represents the largest integer smaller than x/xj. In the continuous subrange x„+i < x ^ X2„+i, we must have 1 ^ [ x / x j — n ^n. The number density at x must inherit the discrete contribution to the number density at x^^/^^] since the contribution is uniformly distributed between x^^/^jj and x^^/^^^^-^. Since the discrete contributions by aggregation to the continuous range can only cover the subrange x^^.^ < x =^ ^2n+i> the birth term in this part of the continuous spectrum will display terms not present in that for particles outside this range, i.e., X2„+i < x < oo:

r 1 (^'

^) ^

"

^ ^ L L^-^1 j = [x/xi]-n

^[x/x,]-jjfl,[x/xi]-jfuj

X [ i / ( x - Xf^/^^j) - H{X - Xj^/^^3+i)] min{[x/xi] — n,n}

X [ l - / / ( x - X 2 j ] +H(x-x„+i) X aj{x - Xj)f^jf^(x

xfiix

- Xj, t) +

2

X a{x\ X — x')/i(x', t)

— x', t) + e{x + x^)f^{x + x^, t) + S^{x, t), x„ < x < oo. (3.3.19)

3.3. Aggregation Processes

83

The first term on the right-hand side represents the discrete contributions to the continuous range. It vanishes identically for particle volumes above x 2n since no purely discrete contributions are possible in that range. The second term represents the contribution by aggregation between particles, one from the discrete range and the other from the continuous range, which can arise only for particles larger than x„+1. The third term refers to evaporation and the fourth to the generation rate. The death function h~{xi,t) is given by h (x, t) = /i(x, t)

a(x, x')/i(x', t)dx' .7=1

+ e{x)f,{x, t) + S-{Mx,

t), X, t).

(3.3.20)

The first term on the right-hand side above represents loss by aggregation with all particles (in the discrete and continuous ranges), the second loss by evaporation, and the third loss by removal. The population balance equation is now identified for the discrete and continuous ranges. The equations must be considered subject to initial conditions on the discrete as well as in the continuous ranges. Thus, /i,i(0) = v^, f = 1, 2,..., n;

/^(x, 0) = v(x), x„ < x < oo,

where the right-hand sides clearly are specified. The formulation of the population balance model is now complete. Before concluding this example, it is of interest to examine the population balance equation for the continuous particle volume range x„ < x < oo in regard to the birth and death terms arising from evaporation. The combination may be expanded in terms of Taylor series to obtain 3 x^ d^ e{x + Xi)/i(x + Xi, t) - ^(x)/i(x, t) =-^ le{x)f^(x, 0] - y ^ x[e(x)A(x, t ) ] + 0 ( x ? ) where 0{xl) represents terms of cubic order in x^^ and above. Seinfeld (1980) points out that for aerosols even the second derivative above is generally so small that the population balance equation acquires only the first derivative term above. The reader will recognize it as the regular convective term characteristic of a continuous change in particle volume occurring due to evaporation. 14 The multiplicative factor [1 — if (x — X2n)] is redundant when the discrete range is initially as specified and no generation is possible outside it.

84

3. Birth and Death Functions

In aerosol dynamics, it is possible to have more internal coordinates representing concentrations of reacting species requiring a multidimensional particle state of internal coordinates. Although we do not include this here, we next discuss a bubble aggregation process encountered in fluidized bed reactors, which involves similar features.

3.3.4

A Bubbling Fluidized Bed Reactor

We consider here an apphcation to a fluidized bed catalytic chemical reactor whose performance is affected by coalescence between gas bubbles. The problem has been considered in detail by Sweet et al. (1987) (based on the earlier work of Shah et al, 1977, which addressed the bubbling process without chemical reaction) and we shall discuss here the formulation aspects of the model. The process of interest consists in blowing a gas containing a reactant A through a bed of catalyst particles at a velocity in excess of the minimum fluidization velocity. The excess gas forms bubbles at the bottom of the bed, which ascend by virtue of their buoyancy up the bed of a "dense" phase of catalyst particles, and eventually escape from the top surface. The catalyst particles in the dense phase are in vigorous circulatory motion through the bed. The reactant in the gas bubbles has inadequate contact with the catalyst particles so that no reaction takes place. However, the gas in the dense phase does undergo reaction to products. The formation of gas bubbles bypasses reactant away from contact with the catalyst phase, although this effect is alleviated to some extent by exchange of reactant via mass transfer as well as bulk flow across the bubble surface. The bubbles coalesce to large sizes in their ascent through the bed. Our interest is in calculating the conversion in the reactor, which is assumed to occur under isothermal conditions. The following considerations are extremely important in the formulation of the population balance model. Since the reaction occurs in the dense phase, the balance of reactant in the dense phase is essential to the calculation of conversion in the reactor. The volumetric rate of gas phase which enters the dense phase is only that which corresponds to the minimum fluidization velocity so that the flow rate in and, hence, out of the reactor (based on assumptions of incompressibility) are both known. Because reactant can pass between the bubbles and the dense phase at rates depending on the sizes and reactant concentrations of individual bubbles, the number distribution of bubbles accounting for size as well as reactant concentration wifl be required. Thus, a population

3.3. Aggregation Processes

85

balance of the gas bubbles is indispensable for calculation of conversion. Also, the population balance equation must be coupled to the reactant balance in the dense phase. The particle state space must include external coordinates because the exit of bubbles from the reactor can only be recognized by their location at the vertical end. In this regard, this consideration is identical to that in the example treated in Section 2.11.4. Also, bubble location is important in determining coalescence rate of bubbles. In addition, we must also entertain internal coordinates, including (1) Bubble size (say volume, denoted x) because it determines its rise velocity through the bed, exchange rate of reactant with the dense phase, and the rate of coalescence between bubbles (2) Reactant concentration (denoted c) because it is required for calculating the rate of reactant transfer to the continuous phase We make the following more specific model assumptions: (i) The dense phase is assumed to be perfectly stirred so that the reactant concentration, denoted c^, is the same everywhere.^^ The reaction is first order and occurs only in the dense phase at the rate kc^ per unit volume of dense phase. (ii) Bubbles enter the reactor at a uniform size x^ with concentration c^ and at a velocity determined by the formula in (ii). Thus, the total number density N^ of bubbles entering the reactor is given by

where U is the actual superficial gas velocity (volumetric flow rate per unit cross-section of bed) and U^j- is the superficial velocity at minimum fluidization. It should be clear that the concentration of reactant in the bubble must always be less than c^ because of chemical reaction in the dense phase. (iii) The bubbles describe random transverse motion while ascending through the bed with a deterministic vertical velocity. We deem only the vertical coordinate z to be necessary for the model. The vertical rise velocity for a bubble of volume x is given by an empirical 15 For more realistic assumptions in this regard see Muralidhar et al. (1987).

86

3. Birth and Death Functions

formula: Z(x) = ocx^. Other formulas for the bubble velocity accounting for the presence of neighboring bubbles exist. ^^ (iv) Bubble breakup may be considered negligible. Only coalescence between bubbles can occur. This assumption implies that the bubble size can only increase above that of the entering bubbles. Furthermore, since reaction must deplete the reactant, its concentration in the bubbles can never increase above the initial concentration of the entering bubbles. We are thus in a position to identify the domain of particle state space exactly as Q^ = {x >

XQI

0 < c <

CQ};

{0 <

Q^ =

Z

<

H}

where H is the height of the fluidized bed. Note that the domain has been identified separately for internal as well as external coordinates. Also, although not an essential part of the formulation, it is possible to identify an upper bound x^^^ for the bubble volume by estimating a lower bound N^^^ for the number density of bubbles exiting the bed. The maximum bubble volume x^^^ is then calculated by requiring that gas outflow rate as bubbles must equal that which enters in excess of the fluidization velocity, ^maxA^min^(^max) = ^

-

^mf'

The estimation of N^^^, however, requires some analysis outside the scope of the present stage. ^"^ Hence, we shall assume for this formulation that bubble volume can vary between zero and infinity. (v) Coalescence occurs only between bubbles whose vertical coordinates differ by a distance d determined by summing the radii of the bubble pair corrected for departures from spherical shape. More precisely. d{x, x') = -

K/

+ •

X

1/3-

(3.3.21)

\K

where K represents the correction factor.

16

/6x\ 1/6

For example, Davidson et al. (1977) suggest Z(x) = 0.71 I — J

+U - U^j-.

17 Such estimates can be made by using a uniform upper bound for the aggregation frequency and solving the population balance equation analytically for the density at the exit of the bed.

3.3. Aggregation Processes

87

(vi) Bubbles that can coalesce (i.e., satisfying condition (v)) coalesce in accord with a coalescence frequency 0, a{x, z; x\ z') = { ^^^ ^ ^ Kd{x, x'f,

\z — z'\> d{x, x') ,_ _,, ^ ^ . J ,,„ \z - z'\ ^d(x, x')

(3.3.22)

where X is a dimensional constant. Note that the coalescence frequency is assumed to be proportional to the projected area of an equivalent bubble whose radius is the sum of the radii of the pair considered for coalescence.^^ (vii) Coalescence between two bubbles of respective particle states [x, c, z] and [x', c\ z'] results in a new bubble with volume x + x'; concentration (xc + x'c')/(x + x'), which implies complete mixing in the bubble during coalescence; and vertical location given by the mean location (z + z')/2 (other possibilities in which the location of the coalesced bubble depends on the volumes of the coalescing bubbles can of course be entertained also). Moreover, if [x', c\ z'~\ is the state of one of the coalescing bubbles to form a new bubble of state [x, c, z], the state of the other coalescing bubble, denoted [x, c, z] is calculated as X = X — x';

c = {xc — x'c')l{x — x')\

z = 2z-— z'.

In these equations, in order for the state for [x, c, z], to be in the particle state space [x', c\ z'~\ it must satisfy the following constraints: 0<x'<x;

0<

xc — x! d 7-
Iz-U


(3.3.23)

The middle inequality above implies that

1-4(1


X

xc <—-. X

However, in view of the fact that 0 < c' < c^, d must lie at the intersection of the two sets, which is readily shown to be max jo,c„ 1 - ^ 1 - X \

.

Co

18 See Argyriou et al (1971) for arguments concerning this frequency.

(3.3.24)

88

3. Birth and Death Functions

Similarly, in addition to the inequality on the extreme right in (3.3.23) for z', we must also have 0 < z' < if, so that z' must be contained within their intersection or, equivalently, max{0, 2z - H}
+ pjp^g

where p^ is the bubble pressure at the top of the bed, p^ is the density of the dense phase, and g is the acceleration due to gravity. Note that the contribution of mass transfer of the reactant to the change in the bubble volume is considered negligible. (ix) The concentration of reactant in the bubble changes due to convective exchange of gas with the dense phase and mass transfer. A simple mass balance of reactant leads to C(x, c) =

-^

(c^ - c),

where K^a is the mass transfer coefficient times the interfacial area of the bubble, and q is the exchange rate obtained by a simple hydrodynamic analysis due to Davidson et a\. (1917) as ^ = 3^ I 471/ T:)

^mf

The size dependence of the rate of change of reactant concentration in the bubble arises from both q and K^a. The particle state vector and its rate of change have now been identified. To write the population balance equation, the birth and death terms due to bubble coalescence remain to be identified. In what follows, we denote

3.3. Aggregation Processes

89

the steady-state population density by f^{x, c, z). The source term h^{x, c, z) requires more detailed considerations than the sink term h~{x, c, z)}^ To determine h^{x, c, z), we seek the set of bubble states [x', c\ z'], which on pairing for coalescence with bubble states [3c, c, z] yield bubbles of state [x, c, z]. In regard to this, we carefully review the dehberations in (vii) to arrive at distinct bubble-pairs, which coalesce to form bubbles of state [x, c, z], to identify the source function as 'jc/2

h'-{x,c,z)=K

dx'

'min[Co,xc/x']

dd

X /i(x', c\ z')

dz' max[0, z — d{x — x',x')]

'••'['--A'-TJI X d{x — x\ x'Y f^{x

'min[//, z + d{x — x',x')]

— x\ {xc — x'c')l{x — x'\ Iz — z')

Ix X — X

where the limits of integration with respect to z originate from the last inequality in (vii) showing that the bubbles in each pair with sizes and concentrations coalescing to form the bubble of state [x, c, z] can be (symmetrically) placed on either side of z. The limits of integration with respect to bubble volume x' are designed to eliminate double counting of pairs because the bubble of volume x — x' is always larger than that of volume x'. Particular attention is also called to the fact that the Jacobian of the transformation referred to in Equation (3.3.2) on page 51 is included in the source term just shown; the Jacobian here is easily evaluated to be 2x/(x — x').^^ In what follows, it is shown that more detailed considerations can fix the limits of integration with respect to concentration.

19 The paper by Sweet et al. (1987) contains errors in the population balance equation. However, since their computations involved direct simulation of the system, the errors did not affect their results. 20 The Jacobian of concern here is

dx dx dx dx dc dz d{x, c, z) d{x, c, z)

dc dx dz dx

dc Jc dz Jc

1

dc = c/{x — x') Jz dz 0 Jz

0

0

x/{x — x')

0

0

2

2x (x — x')

90

3. Birth and Death Functions

For c > |co, we may write f

rm\n{H,z +

d(x-x',x')]

dc'

"^ dx'

dz' max[0,z-d(x-x',x')]

h^{x, c, z) = I

+

'min[H,z4-d(x-x',x')]

d^'

dd

I x( 1 - (cjco))

•^ c J 1 - (x/x')( 1 - {cjc^))^

max[0,z-d(x-x',x')]

Kd{x — x\ x'Yf^{x — x\ {xc — x'c')/{x — x'% 2z — z')/^(x', c\ z')

dz' 2x

X — X

where the curly brackets enclose an integral operator operating on the integrand that appears outside the brackets. For c <^c^ the source function may be written as f*m\n[H,z +

'x(c/c)

dx'

d(x-x',x')]

dc'

dz' max[0,z - d{x - x',x')]

h^{x, c,z) = l rxn

/'min[H,z-t-d(x-x',x')]

*xc/x'

dc'

dx' 0

xic/c„)

dz'\ J max[0,z — d(x — x',x')]

Kd(x — x', x')^/i(x — x', (xc — x'c')/(x — x'), Iz — z')f^{x', c', z')

2x

X — X

It is of course possible to resolve the integration with respect to z further, but we forego this detail. The sink term for bubbles of state [x, c, z] can be obtained more easily as f z + d(x,x')

h {x, c, z) = / i ( x , z, c)

dx'

dc'Kdix, x'ff,{x',

dz'

z', c'

J z — d(x,x')

regardless of the value of c. If the upper bound x^^^ is available from (iv), the upper limit of integration with respect to x' may be replaced by x^^^. Thus, the population balance equation is now completely identified as -

P

p

[7i(x, c, z)X{x, z)] + -

P

llix,

c, z)Z{x)\ + -

[7i(x, c, z)C{x, c)] =

/z"^(x, c, z) — /i~(x, c, z). The equation must be subject to the boundary condition 7i(x, c, 0) = N,b{x - x^)b{c - c j

(3.3.25)

91

3.3. Aggregation Processes

where iV^, the total number density of bubbles in the feed, is obtained by a volume balance 00

_

.

U — U

xMx, c, 0)Z{x)dx = N,x„Z{xJ

= U- t/„^^iVo = - ^ — ^ .

The population balance equation (3.3.25) contains the reactant concentration c^ through the term C(x, c) (identified in (ix)) in the dense phase, which is unknown. Thus, the complete specification of the population balance model requires an equation for c^. This is obtained by a mass balance for the reactant in the dense phase. If we remember that the dense phase is perfectly mixed, a mass balance for the reactant leads to

Umfi^o - Q ) = ^<^d^mf +

^^

dxxC(x, c)

dcf^{x, c, z).

(3.3.26)

In the above equation, the reaction term features the bed height H^f at minimum fluidization, which gives the volume of the dense phase per unit cross-section. The mutually coupled Eqs. (3.3.25) and (3.3.26) represent the mathematical formulation of the population balance model. Since the reaction rate is hnear it is possible in this case to solve explicitly for Q to obtain (*H

UmfCo + CA

dxx{q + KQO)

dz

dcf^ix, c, z)

= •

UmfCo + feQ^m/ +

dz

dxx{q + KQO)

dcf^ix, c, z)

Indeed, nonlinear reaction rates will not permit such explicit solutions. The height H of the fluidized bed is in fact unknown and must be calculated by equating the sum of the volumes of the dense phase and the bubbles to the volume of the bed. Based on unit cross-section one obtains H - H^f =

dxx

dz

dcf^{x, c, z)

which is an implicit equation in H. The number density function can be used to calculate all quantities associated with the fluidized bed reactor. For example, the average concentration of reactant in the bubble gas phase

92

3. Birth and Death Functions

exiting the bed, denoted c^^^, is given by dccf^{x, c, H)

dxx 0

dcf^ix, c, H)

dxx Xo

.

0

The reader is referred to Sweet et a/., (1987) for further details. The problem was solved by Monte Carlo simulation, which is the subject of discussion in Chapter 4. Calculations show that with progressive increase in coalescence rate, the bubble bypassing effect causes more reactant to escape in the outgoing gas. Of course, the advantage of bubbling lies in the agitation of the particle phase and minimizing the effect of transport processes. Population balance models built to account for such effects can guide the engineer in assessing the extent to which bubbling must be maintained in the reactor. Bubble bypassing may also significantly influence selectivity in multireaction systems, which is another issue that can be addressed by population balance models of the type just discussed.

3.3.5

Modeling of Aggregation Frequencies

Aggregation between particles requires first their proximity to one another, which comes about by relative motion between particles. Thus, modeling of the aggregation frequency must be concerned with relative motion between particles. Suppose for the present that physical contact between two particles is necessary and sufficient to cause aggregation. If the full spatial (external) coordinates of the particle were involved in the particle state, the relative location of the particles is clearly contained in the number density. Furthermore, the velocity of the particle would automatically enter the dynamic description. If the velocity were deterministic, then the probability that a given pair of particles at time t would collide with each other in the time interval between t and t + dt would be that the relative velocity of one with another has the proper direction and magnitude to bridge the gap between the particles with due regard to their sizes. Of course, the analysis of relative motion during the time interval t to t -{- dt must account for forces that may act on or between the particle pair.^^ However, it is not usual to attack the problem in this generality, for the full external coordinates of the 21

Considerations of this sort are precisely those in treatments of the kinetic theory of matter.

3.3. Aggregation Processes

93

particle are often not included in the particle state in favor of the assumption of well-mixedness in space either fully or partially. In what follows, we shall therefore assume that the sizes of the particle in the pair considered for aggregation are specified at time t, and we are interested in the probability that in the time t to t -\- dt the pair aggregates to form a single particle. Whether or not external coordinates are involved in the population balance model of the aggregation process, it is clear that the aggregation frequency must be modeled by analyzing relative motion on a time scale considerably smaller than that in which the population evolves by aggregation. In fact. Equation (3.3.4) shows how the aggregation frequency a(x, x') may be obtained from more detailed considerations of relative displacement of particles. Thus, a model must be formulated for the relative location between the particles. In the treatment to follow, we assume the particles to be spherical and distributed according to their volumes denoted by variable X. Accordingly, we consider the evolution of relative motion between the particles of volume x and x' following instant t when they are at locations r and r\ respectively. It is convenient to locate the origin at the centroid r of the particle of volume x and trace the motion of the particle of volume x' relative to the first particle. The location of the particle of volume x' relative to the new origin is clearly r' — r. Equation (3.3.4) may then be rewritten in terms of the chosen relative coordinates as (IVQ

dV^>-^a(x, 0; x\ r'— r)

a{x, x') = The first integration can then be immediately performed to obtain a(x, x') =

dV,._M^, 0; x\ r' - r).

(3.3.27)

The quantity a{x, 0; x\ r' — r)dt represents the probability that the particle of volume x' at relative location r' — r encounters the other particle of volume X located at the origin during the next time interval dt and must be obtained in each case by a suitable model. We consider a few cases next. 3.3.5.1

Aggregation by Relative Deterministic

Motion

Consider the case where the particles move with deterministic velocities and no net force exists between the particles. Assume that the particles of

94

3. Birth and Death Functions

volumes x (radius R = (3x/47r)^''^) and x' (radius R' = {3x'/4nY'^) have respective velocities \{x) and v(x') (which may be calculable from some prior considerations of a dynamic nature neglecting transient effects. For the calculation of the aggregation frequency a{x, x') using Eq. (3.3.27) it is necessary to obtain the function a{x, 0; x\ r' — r) based on probabiUstic arguments. The velocity of the particle of volume x' relative to that of volume X is clearly v^ei = v(x') — v(x). We proceed as follows. First, we mount a coordinate frame of reference on the particle of volume X with its origin at the centroid of the particle and follow the location of the particle of volume x' relative to this frame by denoting its variable position vector as r". The unit vector along the radial direction will be denoted 5^.. We define p(r", r + T|r' — r) as the probability density for the location of the particle of volume x' (relative to the new frame moving with the particle of volume x) at a subsequent time t + T conditional on its initial location (measured at instant t) being at r' — r. This density function can be shown to satisfy the partial differential equation ^ p(r", r + TI r' - r) + v,,, • Vp(r", t + 11 r' - r) = 0

(3.3.28)

where the gradient is with respect to r". Equation (3.3.28) must be solved subject to the initial condition p(r", r | r ' - r ) = ^ [ r " - ( r ' - r ) ] .

(3.3.29)

Equation (3.3.29), by using the Dirac delta function on its right-hand side, merely stipulates the initial location in a self-consistent manner. The solution to (3.3.28) is immediate and may be expressed as pir'\ r + T|r' - r) = 5[r" - V,,,T - (r' - r)].

(3.3.30)

The function a(x, 0; x\ r' — r) can now be obtained from the probability flux from the exterior of the particle toward the surface r" = {R -\- R')5^. since this implies contact between the two particles upon which aggregation is presumed immediately. Thus, we write a{x, 0; x\ r' - r) = p{{R + R')d,., t + dt\r' - r)(-^,.-v,„). Substituting into this equation from the solution (3.3.30), we obtain a(x, 0; x\ r-T)

= S{(R + R')S,. - v,e,T - (r' -

r)}{-d,ry,J,

3.3. Aggregation Processes

95

which, on substitution into (3.3.27), gives for the aggregation frequency a(x, X') =

dV,_,d{(R

+ R')3,. - y,^,dt - (r' - r)}{-5,ry^^,).

(3.3.31)

In terms of spherical coordinates using p = (r' — r)- d^. as the radial distance from the origin we may write for the axisymmetric situation dV^_^ = 2np^ sin 9d9. Because the integration is over all initial orientations of the particle of volume x\ we must make sure that the particle does not overlap with that at the origin. Thus, we must have p > {R -\- R'). As the integration in (3.3.31) will only allow p = (R -\- R') ~ dt\^^i'3^>,, we must arrange for 9 to vary over the region in which v^ei * ^r" is negative, v^ei being the magnitude of Vj-ei- Thus, (3.3.31) may be rewritten as a(x, x') = 2n{R + R')

sin 9d9(-5,.'\,,,).

(3.3.32)

We shall now apply the relation (3.3.32) to the case of gravitational aggregation caused by differential sedimentation so that \{x) = - k

^

,

pp > p,

(3.3.33)

where Pp is the density of the particle, /i and p are respectively the viscosity and density of the surrounding fluid, and g is the acceleration due to gravity. Thus, the relative velocity vector becomes

Since k • 3^> is cos 9, the region of integration shown in (3.3.32) turns out to be cos 9 > 0 or 0 < 9 < n/2. On substituting for v^ei from above into (3.3.32), we obtain for the aggregation frequency >.i^^:ipp-p)Q\,2

a(x, x') = 2n{R + R')

9,

•7r/2

d9 sin 9 cos 0, 0

which yields the final form a(x, x')=^^^\

9}ji

^^\l^J^IAfl^x'i^ ^

+ x'"y\x'^

~ x \

(3.3.34)

This frequency is applicable to the study of differential sedimentation of dilute suspensions. If the population density is high enough for the effects of hindered settling to be important, then, following the approach in hy-

96

3. Birth and Death Functions

drodynamics (Batchelor, 1982), the velocities may be allowed to depend on the dispersed phase volume fraction. The resulting population balance equation for the aggregating population would then feature a higher degree of nonlinearity since the aggregation frequency will itself depend on the first moment of the population density.

3.3.5.2

Aggregation by Random Relative Motion

Suppose particles describe random motion through the suspending medium, as for example, by Brownian motion. Smoluchowski^^ was the first to address the problem of calculating the collision rate of particles describing Brownian motion. We shall present here a somewhat different derivation of his result because it naturally accommodates some generalizations of interest to us. As before we assume the population to be uniformly distributed in space at all times. Each particle is assumed to describe random Brownian motion independently of all the other particles. The problem at hand is the calculation of the aggregation frequency (or the pair-specific aggregation rate) a(x, x') between two particles of volumes x and x\ Our objective in what follows is to obtain the function a{x, 0; x\ r' — r) from the analysis of relative motion and calculate the aggregation frequency via Equation (3.3.27). The Brownian motion of the particle of volume X may be described by the stochastic differential equation

where dW^ is the differential, standard Wiener process increment with the properties already stated in Chapter 2. The diffusion coefficient D{x) is obtained by invoking its roots to the concept of osmotic pressure and the friction coefficient arising from hydrodynamic theory. ^-^ A similar differential equation may be written for the particle of volume x' inserting the standard Wiener process dWj, which will be regarded as independent of the Wiener

22 23

Smoluchowski (1914). See also S. Chandrasekhar (1943). See, for example, Einstein (1956). The osmotic pressure may be written as p = vkT where V is the number density of particles, k the Boltzmann constant, and T is the absolute temperature. The net osmotic force per unit volume is given by — Vp = —/cTVv = fv\ where vv is the diffusive flux and / is the friction coefficient that arises from hydrodynamic analysis. The diffusion coefficient must therefore be given by D _ , = kT/f.

3.3. Aggregation Processes

97

process dW^. In order to calculate the probability that the particle of volume x' will aggregate by relative diffusive motion with the particle of volume x, we shall be interested in the random displacement of the particle of volume x' relative to that of volume x. This relative displacement vector R^'_^ = R^' — R^ must satisfy the equation dR^._^ = ^2D{x')dW,

- ^2D{x)dW,

= ^2lD{x)

+ D(x')]dW,, (3.3.35)

the expression to the extreme right arising from the properties of dW^ and dW[ and their independence. Assume that a particle of volume x' is located at some radial position r' — r > R + R\ We exploit the spherical symmetry inherent in this situation to consider relative displacement between the two particles only along the radial coordinate. Accordingly, we let r" be the radial displacement of the particle of volume x' relative to that of volume x and p{r'\ t + T\r' — r) be the probabihty density for the spatial distribution of the particle of volume x' given that their initial separation is r' — r. This density function must satisfy the Fokker-Planck equation^"*^ associated with the stochastic differential equation (3.3.35), viz..

-pir'U^.W-r)=-^-

r^dr^ pi^"^ ^ + '^k' - ^) T > 0,

r'>R-\-

R\

(3.3.36)

where we have set the relative diffusion coefficient D^>_^ = D{x) + D{x'). The initial condition stipulates that the particle of volume x' is at r' — r so that p{r\tW-r)

=

5{/' — r' -\- r)

(3.3.37)

where the term in the denominator is inherited from integration in spherical coordinates. The boundary conditions are formulated as follows. Aggregation of the particle of volume x', which occurs at r" = R -\- R\ will terminate the diffusion process so that the surface /' = R + R' becomes an "absorbing" barrier. In other words, p{R +R\t

+ T\r' -r)

=0

24 See, for example, Gardiner (1997), referenced in Chapter 2.

(3.3.38)

98

3. Birth and Death Functions

while the boundary condition at infinity is given by Hm p{r\ t +

T\/

(3.3.39)

- r) = 0.

We are now in a position to calculate the function a{x, 0; x\ r' — r) introduced in Equation (3.3.27) since it is the "probability flow" at r" = J^ + R' toward the surface from the exterior. Thus we have a{x, 0; x\ r' - r) - 4n{R +

R'fD^._^

dp{r\ t + T | r ' - r ) dr"

The aggregation frequency a{x, x') can now be computed from (3.3.27) as follows. We let p = r' — r so that from spherical symmetry we have dV,._, = Anp^dp. Then (3.3.27) yields a{x, x') =^An{R +

R'fD^,. R + R'

dp{r'\ t + T\P) dr"

Anp^'dp. (3.3.40) r =R + R'

If we set P{r\ t +

T)

p{r\ t + T\p)4np^dp,

(3.3.41)

the aggregation frequency may be written as a{x, x') =4n{R + R'YD^,_

dP{/\

t+ dr"

T)

(3.3.42)

Thus, the determination of the aggregation frequency depends on that of the function P[r\ t + T). It would appear from (3.3.42) that the aggregation frequency is dependent on the time elapsed since the instant t. We shall return to this issue presently. It follows from using (3.3.36) and (3.3.37) that _d_ ^ ' ^

^

/''

dr" r - - P ( r V + T)

T > 0,

r" > R-\- R' (3.3.43)

and

p(r'\ t + T) = 1.

(3.3.44)

The boundary condition of absorption at r" = R + R' is inherited from (3.3.38) so that P{R + R',t + T) = 0.

(3.3.45)

3.3. Aggregation Processes

99

The boundary condition at /' = co needs careful consideration as (3.3.39) may lead one to hastily conclude that it also holds for P{r'\ t + T). The constraint hes in the bar against taking the limit inside the integral in (3.3.41). On the other hand, since the diffusion process does not ever affect at infinity the initial uniform state of the population throughout the region, we may conclude that lim P{r", t + T) = L

r"-^oo

(3.3.46)

The solution of the partial differential equation (3.3.43) written for P{r'\ t -\- T) subject to the initial condition (3.3.44) and boundary conditions (3.3.45) and (3.3.46) is readily obtained. However, since the time scale of random relative motion may be viewed as being considerably smaller than that of aggregation, the preceding diffusion process may be construed to have reached "steady state." The steady-state version of (3.3.43) written for P{r'') = lim P{r\ t + T) has the solution

r The aggregation frequency from Equation (3.3.42) may be obtained as a(x, x') = 4n{R + R'fD^^^,

dP\ ,,

r = R + R'

We have thus obtained the well-known Brownian coalescence frequency. It displays a singularity at x = 0 (or x' = 0) reflecting the very high rate of agglomeration between particles of disparate sizes because of the vigorous diffusion of the smaller particle toward its sluggish larger partner. Mechanisms of relative motion between particles other than by Brownian diffusion have also been of interest in obtaining aggregation frequencies for appropriate situations. In obtaining the Brownian coalescence frequency, we had assumed that particles move independently of one another even when they are in the immediate proximity of each other. Thus, the foregoing analysis does not account for any correlation between the movement of particles as a result of interparticle forces and/or viscous forces in the intervening fluid. We next outline the manner in which such effects may be included in the derivation of the aggregation frequency.

100

3. Birth and Death Functions

3.3.5.3

Aggregation from Correlated Random Movement

Consider first the effect of interparticle forces. We return to the view of relative motion of the particle of volume x' from the center of the particle of volume x. The interactive force F is generally described as the gradient of an interactive potential, say V{T), where r is the position vector of the (center of the) particle of volume x' relative to the particle of volume x. Thus, F = — VK Neglecting acceleration effects, we may relate the above force to a steady velocity of the particle of volume x' by dividing the force by the "friction coefficient" / = 6Kfi{3x'/4ny^^. Note in particular that this procedure continues to neglect the viscous interaction between the two drops. The stochastic differential equation for the relative displacement of particle of volume x' should now be written as VF ^ R . ' - . = - y ^f + V 2 ^ . ' - x r f W ,

(3.3.47)

The first term on the right-hand side of this stochastic differential equation (arising from the interparticle force), referred to as the "drift" term, impHes a correlation between relative displacements in different time intervals and hence correlation between the movements of the two particles. The relative diffusion coefficient D^>_^ is determined in the present context from (3.3.36) as the sum of the individual diffusion coefficients. The Fokker-Planck equation for the stochastic differential equation (3.3.47) will differ from Equation (3.3.36) because of the drift term. However, since the calculation of the aggregation frequency depends on the function P{r", t + T) as defined in Section 3.3.5.2, we will directly proceed to the differential equation in P{r'\ t + T). Recognizing spherical symmetry, we have dP{r\ t + T) _ 1 _ ^ ^ „ , . X

—X

dP{r\ t + dr"

T)

, P{r\ t + T)dV f dr' (3.3.38)

The boundary conditions are the same as those in Section 3.3.5.2. As before, only the steady-state version of the differential equation is of interest. The

3.3. Aggregation Processes

101

solution is given by C'"

Pin

1

exp

JR + R' '

_~inn-- V{r)} 1

dr

~v{ry dr _kT _

R+R

where we have replaced the term fD^..^ by fcT because of the definition of the diffusion coefficient (see footnote 23). To obtain the aggregation frequency for this case, however, we cannot use (3.3.42) since it does not include the probability flux at particle surface in toto. The required modification is dP

a{x, x') = 4n{R + R'f

P dV' r" = R + R'

1

^47rD,,_, R + R>

r

exp

Vir)' dr kT

-1

(3.3.49)

which recovers, as it should, the aggregation frequency obtained earher for the case of no interparticle force when the potential is allowed to vanish identically. Although the foregoing frequency accounts for correlation between the displacements of the two particles, the random (diffusive) components of their motions occur independently of each other. Speilman (1970) accounted for the correlation between their diffusive motions by invoking the relationship between the relative diffusion coefficient and the friction coefficient / calculated from a hydrodynamic analysis of the interaction between the two spherical particles (Batchelor, 1982).^^ With the foregoing friction coefficient used in calculating D^>_^, the aggregation frequency for this case is given by Equation (3.3.49).

3.3.5.4

Aggregation by Multiple Mechanisms of Relative Motion

In applications, one more frequently encounters particles aggregating by more than one mechanism. For example, Brownian motion and differential settling in a gravitational field could simultaneously contribute to relative motion between particles and consequent aggregation. The calculation of 25

For a detailed treatment of this problem for spherical Hquid drops accounting also for hydrodynamic interaction effects, the reader is referred to Zhang and Davis (1991).

102

3. Birth and Death Functions

the aggregation frequency in such cases should proceed by analysis of relative motion by combining the mechanisms. Splicing aggregation frequencies for the individual mechanisms, such as by simple addition or otherwise, does not constitute a rational procedure, although it may occasionally provide satisfactory answers. We now provide an illustration. Consider relative motion by Brownian motion as well as by gravitational settling. We neglect interparticle force and correlated random motion, although their inclusion in the manner in which it was done in Section 3.3.5.3 is straightforward. In fact, the issue of interest in Section 3.3.5.3 does not differ from that in this section. The relative motion of particles of volume x' as viewed from a particle of volume x is described by the stochastic differential equation

where k is the unit vector directed vertically upward, g is the acceleration due to gravity, v^ is the settHng velocity in gravitation, and D^_^, as in Section 3.3.5.2, is the sum of the diffusion coefficients of the individual particles. Since the vertically downward settling velocity of particles destroys the spherical symmetry, the function P{/\ 6, 0, t) must satisfy the convective diffusion equation in spherical coordinates given by /).-

dr'\

dry

sin^a^

»-l

+ Kcos^|^or

Ksin e dP dP r" ~de~^' (3.3.50)

The boundary conditions are the same as in Section 3.3.5.2. This problem has been addressed in detail by Simons et ai (1986). The resulting aggregation frequency shows sizable deviations from that obtained by splicing the Brownian and gravitational coagulation frequencies.

3.3.5.5

Modeling of Aggregation Efficiency

In the preceding discussion, collision between two particles was assumed to be sufficient for aggregation. In many situations, collision could result in the particles bouncing off in different directions. In other situations, a thin film of the continuous phase may exist between the particles even following a "collision" and a further force may be required to squeeze out the continu-

3.3. Aggregation Processes

103

ous-phase film before aggregation can occur. Thus, it is necessary to associate an efficiency of aggregation for a complete characterization of the aggregation frequency. In other words, we set Aggregation frequency = ColHsion frequency x Aggregation efficiency. The methods in Sections 3.3.5.1 through 3.3.5.4, which were used for modehng the aggregation frequency, could instead be used to model the collision frequency. Thus, the calculation of the aggregation frequency will depend on modeling of both the colhsion frequency and the aggregation efficiency. In what follows we shall introduce deliberations on modeling of the aggregation efficiency. The aggregation efficiency so defined may also be interpreted as the probabihty that, given that two particles have entered into a collision, they will aggregate to form a single particle. Thus, while the colhsion and aggregation frequencies represent rates, the aggregation frequency is not a rate function. We consider two spheres of fixed radii to have collided to form a film between them of thickness, say, h^ (less than some value h^ to be defined presently). We further assume a random force such as that arising due to turbulent pressure fluctuations that produces a random film drainage process. A positive force is assumed to drain the film while a negative force causes it to thicken by inflow. Although the process is strictly threedimensional, we shall assume a one-dimensional model, letting the force be always normal to the film. Further, we stipulate that if the film drains to some critical thickness, say h^, the film snaps to allow aggregation between the particles. We shall see later how such a model can be formulated mathematically.^^ The instantaneous film thickness H will serve to describe the position of one of the particles relative to the other. Based on a quasi-static assumption, we adopt Taylor's equation for the drainage rate of a film between two spheres of volumes x and x' under the action of a constant force by replacing the constant force by a timedependent F{t): dH _ dt

2HF{t) 3nfi

3x\-^/^

/3x;\-^/^'

471/

\47r

(3.3.51)

The implication of the quasi-static assumption is that changes in the squeezing force occur at a rate considerably lower than that at which 26 The foregoing model was formulated by Das et al. (1987), The reader is also referred to Murahdhar et al. (1986) for a more comprehensive analysis of this problem.

104

3. Birth and Death Functions

steady-state film drainage is reached. The squeezing force is modeled as a stochastic process described by the differential equation fF -F\ dF= -(-—^

5 dt + -j= dW,

(3.3.52)

where F represents the mean force (we assume F > 0 so that drainage occurs on the average) and d is the standard deviation; Tf represents the "autocorrelation" time, i.e., the average period over which the values of the fluctuating force are statistically correlated. Such a process is often referred to as band-limited noise.^^ The coupled stochastic differential equations (3.3.51) and (3.3.52) represent the relative motion between the two particles that have collided, the separation between the particles being described by the film thickness. If at any instant the film thickness drops to the critical thickness h^ with the particle pair experiencing a squeezing (positive) force, then the film drainage process ends with the aggregation of the particle pair. Suppose we now assume that, following collision, if the particle pair at film thickness h^ experiences a separating (i.e., negative) force, the pair is no longer under collision. Then the film drainage process may be said to have terminated into an unsuccessful aggregation for the particle pair. The model for the aggregation efficiency may now be regarded as complete. Let the probability density for the stochastic film drainage process be represented by p{h, / ; t|/i^, f^) where h^ and fi are the initial values of H and F, respectively. This density function p = p{Kf; ^|/i,,X) satisfies the Fokker-Planck equation dp

d_

Tt^'dh

d

^y.

(3.3.53)

where R and R' are the particle radii. Note that the "convection" terms in this equation arise from film drainage along the h coordinate and the "drift" of the stochastic force towards its mean along the / coordinate. Diffusion 27

A stochastic process is also characterized by its "spectral density," the Fourier transform of its autocorrelation function. The autocorrelation function of a (stationary stochastic process) measures the correlation of the process at different time intervals while the spectral density measures the amplitudes of the component waves of different frequencies. A "white noise" process has a constant spectral density (i.e., the same amplitude for all frequencies) and the "band-limited noise" has a frequency band over which the spectral density is nearly constant.

3.3. Aggregation Processes

105

occurs only along the / coordinate with a diffusion coefficient that depends on the standard deviation of the fluctuation of the force about its mean. The initial condition is given by p{h,f;0\h,J,)-

5ih-h,)d{f-f^

(3.3.54)

while the boundary conditions are

p{KJ;t\h,j,) = o, / < o ,

p{KJ;t\h,j^ = o, / > o . (3.3.55)

The boundary conditions in (3.3.55) arise from the fact that at either boundary the physical process cannot exist across it. In other words, the film thickness cannot grow from below the critical value or drainage cannot occur from a value larger than the maximum stipulated for colhsion. The entire process is conveniently visualized as the convective diffusion of a hypothetical particle (only abstractly connected to the real particle pair in question) on the /z — / plane shown in Figure 3.3.2. The particle moves to the left on the upper half of the plane (where the force is positive and causes film drainage reducing the film thickness) and

sample path terminating in aggregation

/

^T\ SANK^ S^ / ^

h =h

sample path terminating in separation

h=K

FIGURE 3.3.2 Film drainage under the action of a stochastic force. Sample paths of the process, one leading to aggregation and the other to separation.

106

3. Birth and Death Functions

moves to the right on the lower half of the plane until arriving either on the upper portion of h = h^ where aggregation occurs or on the lower portion of h = h^ where separation occurs. The solution of the boundary value problem^ ^ will yield the probability density p{h,f; t|/i^,/^), which is conditional on the initial film thickness and force at the instant of colhsion. If we represent the unconditional probability density by p{h, / ; t), its calculation from the conditional density p{h, / ; t\hi, fi) is straightforward if we assume a distribution for the initial film thickness and the force. For example, assuming that the initial film thickness is uniformly distributed between h^ and h^ and the force is Gaussian (with mean F and standard deviation <5) and further that the two random variables are uncorrelated we obtain

p{hj;t)

=

Cho

A-

ih„-K)

f"

df,

}-^^5

exp

(ft - FY

p{Kf',t\h,j,). (3.3.56)

The efficiency of aggregation of the particle pair denoted r]{x, x'), being the probability of escape of the hypothetical particle on the positive half of the line h = h^, is computed from the probability flux along the positive / axis. r]{x, x') =

2h f dt I df^{R-'+R^-'fp{KJ;t).

(3.3.57)

That the integration with respect to time in (3.3.57) is carried through to infinity is a reflection of the assumption that the time scale of this process is considerably smaller than that in which the population of particles is changing. The actual calculation of such an efficiency is accomplished by methods demonstrated by Muralidhar ei al. (1988), referenced in footnote 28. The objective of the discussion herein has been to provide broad guidelines for modehng of the aggregation efficiency, since mechanisms and (hence details) in individual applications can vary rather diversely. Before concluding this discussion, we refer to one further aspect of modeling that can produce valuable simplifications in the foregoing treatment. Such simplifications arise from a consideration of times scales as in the treatment of Muralidhar and Ramkrishna (1986). 28 See Muralidhar et al. (1988) for a solution of the boundary value problem (3.3.53)-(3.3.55).

3.3. Aggregation Processes

107

Suppose the time scale of film drainage (for a constant force) is large compared with the autocorrelation time of the fluctuating force. Then it may be possible to assume that the left-hand side of Equation (3.3.52) may be equated to zero so that Fdt = Fdt + S^TfdW,,

(3.3.58)

which states that the force fluctuation about its mean is a "white noise" process. The white noise process has the property that its values at any two instants are completely uncorrelated. In the present context, we observe that the "slow" film drainage process does not perceive the correlation of the force fluctuation assumed to occur over smaller time scales. The advantage of the foregoing assumption is that it converts the pair of stochastic differential equations (3.3.50) and (3.3.52) in the previous model into a single stochastic differential equation in the film thickness given by 2H a = ^ ( i ^ - ^ + R'-'f.

dH= -(x{Fdt + 3JTfdWX

(3.3.59)

Associated with this stochastic equation is the Fokker-Planck equation in the probability density p{h, t|/z^) given by

dpJK t\h,)

d

1

-

+ -:^l-oiF^=:^oi^S^T.

d^K

^\

t\K)

' '\

(3.3.60)

The film drainage process is now to be viewed from the different perspective of both diffusion and convection (of the hypothetical particle) occurring along the h coordinate alone. Moreover, there is no / coordinate, since the drainage is occurring at the mean force F about which fluctuations occur as described by Equation (3.3.58). The initial condition is given by p{K 0\hi) = d{h-hi).

(3.3.61)

The boundary conditions at /i = h^ and h = h^ must respectively reflect the definite aggregation and separation of the particle pair and are obtained by setting (3.3.62)

p{K,t\h,)=p{K,t)^0.

If, as in the previous model, the initial film thickness h- is distributed uniformly between h^ and h^ then the unconditional probability density of the film thickness is given by 'ho

p{K t) =

dh

(K - K)

piK t\h,y

108

3. Birth and Death Functions

The efficiency of aggregation is obtained from integrating the probabiHty flux dXh = h, over time. Thus ^(x, x')=^a^5^T^

dt ^^^^' ^^ dh

(3.3.63)

The simphcity of this model allows an analytical solution to the boundary value problem (3.3.60)-(3.3.62) so that an explicit expression is possible for the aggregation efficiency. For details of such analytical solutions, the reader is referred to Das et al (1988). Another example of a simple model for aggregation (coalescence) efficiency is that due to Coulaloglou and Tavlarides (1977). From the perspective presented here, their model is obtained by setting identically F = F so that drainage occurs under a constant force in a deterministic manner, although the particle pair could be dislodged randomly at any instant with a constant transition probabihty, thus ending the drainage process without aggregation. If, on the other hand, the particle pair survives the dislodging process until drainage occurs through to the critical film thickness, the pair will have aggregated. The aggregation efficiency in this case is obtained from the probability that the dislodging time exceeds the required time for drainage to the critical film thickness.

3.3.6

Simultaneous Aggregation and Breakage

When both aggregation and breakage processes occur together, the birth and death functions are generally accounted for in the population balance equation by algebraically summing them. This procedure cannot be valid unless aggregation and breakage processes occur independently of each other. Such independence can occur only under special circumstances, however. In modeling aggregation or breakage, it was necessary to address in some manner the processes leading to either of them. Thus, agglomeration required the description of relative motion between particles while breakage involved considerations such as deformation of individual particles preparatory to breakage. For example, if relative motion was affected by deformation, then it should be clear that the processes of aggregation and breakage interfere with each other. In such cases, one is forced to consider modeling of the associated frequency functions in which both processes are considered simultaneously in much the same manner as aggregation of particles by more than one mechanism was analyzed in Section 3.3.5.

3.3. Aggregation Processes

109

In some situations, the particle population may undergo circulation between different spatial domains in which only one of the processes of breakage and aggregation occurs. Thus, the two processes occur independently of each other and their contributions to the population balance (with due regard to spatial inhomogeneity of the population density and the frequency functions) become additive. This situation may be envisaged in a liquid-liquid dispersion in a mixing vessel provided with a mechanical impeller whose rotation may be viewed to broadly estabhsh two regions in the vessel. One is a zone close to the impeller of very high shear in which breakage of droplets occurs with high frequency with almost no coalescence, and the other a recirculating zone where very httle droplet breakup occurs. If the droplets commute between the two zones sufficiently rapidly, the volume-averaged population balance equation for the entire mixing vessel will feature coalescence and breakage terms in an additive manner. This may be demonstrated more rigorously with equations using arguments very similar to those in Section 3.2.3. Before we conclude this section it is of interest to discuss population balance models in which both coalescence and redispersion events are lumped into a single step. It has been of some utility in modeling both dispersed phase mixing (Curl, 1963) and mixing in homogenous phases (Rao and Dunn, 1970).

3.3.6.1

Coalescence-Redispersion

Models

We consider a liquid-liquid dispersion in a continuous reactor in a steady state with dispersed phase droplets and the continuous phase entering and leaving with a constant holdup for both phases. Further: (i) The droplets are all assumed to be of the same (average size) but containing a chemical reactant of concentration c which may vary from one droplet to another. A chemical reaction of the type A -^ Reactants takes place with intrinsic reaction rate r{c). The only internal coordinate is the reactant concentration in the droplet. For simphcity no mass transport is assumed between the continuous phase and the drops, although it should be clear that such effects are easily incorporated into the model without much additional complexity.

110

3. Birth and Death Functions

The rate of change of concentration can then be described by C(c) = -r{c). (ii) The average residence time of the droplets as well as the continuous phase is given by 6. The volume fraction of the dispersed phase is denoted (/>. The size of the droplets is then given by 0/N. All the droplets enter with the same concentration c^. Since reaction depletes the concentration of the reactant c, this concentration has the range 0 < c < c^. (iii) The droplets interact with some frequency a^, which is proportional to the total population density iV, which remains constant since there is no change caused by coalescence-redispersion processes. Two drops having difTerent concentrations c^ and C2, respectively, will coalesce and redisperse into two different droplets, each of concentration (ci + 02)12. The proper formulation of the source term for drops of concentration c in the population balance equation requires careful consideration of the type that led to (3.3.2). Let the population density in the reactor at steady state be /i(c). Suppose we let one of the drop pair that combines to form two drops of concentration c have concentration c^ and denote the concentration in the other by C2. Then we must have c^ + C2 = 2c, and the source term for drops of concentration between c and c + dc is given by 2ajc2

dcj^{c^)f^{c2\ J c i +C2 = 2c

the factor of 2 arising because of the immediate redispersion of the coalesced pair into two separate drops of the same mean concentration c. The foregoing term is more conveniently represented by variables c and c' by using the transformation c^ = c — c\

C2 = c -\- d.

Using the Jacobian of the transformation d{c^, C2)ld{c, d) = 2, we have dc.dcj = -z-^—^dcdd ' ' 3(c, d)

= Idcdd,

which converts the preceding source term into h'^{c)dc = Aa^dc

ddf^{c - d)f^{c + d),

3.3. Aggregation Processes

111

where we have left the hmits of integration unspecified because of their dependence on the range of c.^^ More precisely, we may write the source term as dc'Mc - c')f,{c + c'l

4a„

h^ic) =

0 < c < cJ2

0

(3.3.64) dc'Mc - c')f^(c + c'\

4a„

c^>c>

cjl

Jo

In writing the foregoing, we have accounted for the fact that the concentration c cannot exceed c^. Thus, for a balance on drops of concentration less than c^/2, the maximum concentration 2c in the coalescing drop is indeed less than c^. On the other hand, for a balance on drops of concentration c exceeding c^/2, the upper limit of integration is chosen to prevent the maximum concentration from exceeding c^ and the minimum concentration from becoming negative. Our goal is This equation conversion in considerations to be dc

to formulate the steady-state population balance equation. could then be used to study the effect of mixing on the the reactor. Since the sink term is easily identified, from in (iii) the population balance equation for the reactor is seen

- ric)Uc)-] = ~ lN5ic - c j -lie)-]

- aM{c)

+ /z^(c),

(3.3.65)

where h^{c) is given by (3.3.64). It is desirable to check the formulation by making an overall balance on the number of drops by integrating Equation (3.3.64) over the interval [0, c^]. It is left for the reader to verify that

-ricJMcJ=-a„N^

+ 4a„ dc

'co/2

dc

dc'Mc + c')/i {c - c') + 2a,

dc'f^ic + c')7i(c - c').

(3.3.66)

Co/2

29

Special attention is called to the factor of 4, which is not at all obvious and is likely to be replaced by 2 in "physically intuitive" derivations! The proper use of the Jacobian is emphasized here.

112

3. Birth and Death Functions

The region over which integration is performed on the right-hand side of the foregoing equation is indicated in Fig. 3.3.3. If we transform to coordinates w, v defined by u = c — c\

V = c + c\

c = (w + v)/2,

c' = (v — u)/2,

the Jacobian of the transformation is readily seen to be 1/2 so that the integral in question is given by 2a^

dudvf^{u)f^{v)=a^ v>u

f^{u)du Jo

Mv)dv = a,N'

c (u = v)

FIGURE 3.3.3 Integration zone (shown in the shaded region) for calculating total number of droplets in coalescence-redispersion process. Transformation to u, v coordinates by u = c — c\ v = c + c\

3.3. Aggregation Processes

113

in which the symmetry of the integrand is exploited to define the integral over the entire square (0 ^ w, v ^ cj. Substituting into the overall number balance (3.3.66), one obtains the result thsitf^{c^) = 0. This is understandably so because, following the entry of the droplets with the concentration c^, reaction would instantly begin to deplete the reactant and reduce the number density to zero at c^. We have thus verified the overall number balance. The attribute of the foregoing model is its remarkable simplicity and ability to assess the effect of drop mixing on conversion. Of course a drop population can have a broad spectrum of sizes. There have been attempts to improve this feature by incorporating a size distribution as measured experimentally and to view the coalescence of a pair of unequally sized droplets to result in the same pair of droplets except for the mixing of their contents! It cannot be said that this viewpoint is an improvement, for the assumption of such memory in redispersion is less realistic than that of uniform size. However, it is of interest to see whether a uniformly distributed redispersion event can predict a size distribution that is anything like what is observed. We discuss this as another example in the formulation of population balances.

3.3.6.2

Coalescence and Uniform Redispersion

We consider the evolution of drop size in a well-stirred dispersion in a batch mixer. The population is distributed according to drop volume x through a number density /^(x, t). The frequency of interaction is assumed to be a^ as in the previous section, independently of drop volume. The uniformly distributed redispersion of a drop (formed by coalescence at any instant) of size y into two other drops imphes that the probabiHty density for the size of either of the newly formed pair is 1/y. Thus, the source term for drops of volume x is given by dy

h'^{x, t) = la^

'y

y J0

fi(y -z.

t)f^{z, t)dz.

Since the sink term is readily identified the population balance becomes dt

= 2a

dy^

y

y J0

My - z, t)Mz, t)dz - a„Nf,{x, t)

(3.3.67)

At equilibrium the left-hand side vanishes and the resulting integral equation can be solved analytically to get an exponential distribution for drop

114

3. Birth and Death Functions

size that compares rather favorably with experimental data from numerous sources (Bajpai et a/., 1976) .

EXERCISE 3.3.1 Consider the effect of mixing on the reaction considered in Section 3.3.6.1 within the setting of the model in Section 3.3.6.2. Identify the population balance equation for the bivariate size, concentration distribution.^^

REFERENCES Argyriou, D. T., H. L. List, and R. Shinnar, "Bubble Growth by Coalescence in Gas Fluidized Beds," A.LCh.E. JL, 17, 122-130 (1971). Bajpai, R. K., D. Ramkrishna and A. Prokop, "A Coalsecence Redispersion Model for Drop-Size Distributions in an Agitated Vessel," Chem. Eng. Sci., 31, 913-920, (1976). Batchelor, G. K., "Sedimentation in Dilute Polydisperse System of Interacting Spheres. Part 1. General Theory," J. Fluid Meek 119, 379-408 (1982). Chandrasekhar, S., "Stochastic Problems in Physics and Astronomy," 15, 1-49, (1943). The latter article is reproduced in Seleeted Papers on Noise and Stochastic Processes. (N. Wax, Ed.), Dover, 1954. Coulaloglou, C. A. and L. Tavlarides, "Description of Interaction Processes in Agitated Liquid-Liquid Dispersions," Chem. Eng. Sci., 32, 1289-1297 (1977). Curl, R. L., "Dispersed Phase Mixing: 1. Theory and Effects in Simple Reactors," A.LCh.E. Jl. 9, 175-181 (1963). Curl, R. L., "Dispersed Phase Mixing Effects on Second Moments in Dominantly First Order Back Mix Reactors," Chem. Eng. Sci. 22, 353-358 (1967). Das, P. K., R. Kumar, and D. Ramkrishna, "Coalescence of Drops in Stirred Dispersion. A White Noise Model," Chem. Eng. Sci. 42, 213-220 (1987). Davidson, J. F., D. Harrison, R. C. Darton, and R. D. Lanauze, "The Two-Phase Theory of Fluidization and its Application to Chemical Reactors," Chapter 10 in Chemical Reactor Theory. A Review. (L. Lapidus and N. R. Amundson, Eds.) pp. 583-685. Prentice-Hall, NJ, 1977. Davis, H. T., "On the statistics of randomly broken objects," Chem. Eng. Sci., 44, 1799-1805 (1989).

30 For an application of this model to phase transfer catalytic reactions, see Hibbard and Ramkrishna (1981).

References

115

Derrida, B. and H. Flyvbjerg, "Statistical Properties of Randomly Broken Objects and of Multivalley Structures in Disordered Systems," J. Phys. A: Math. Gen. 20, 5273-5288 (1987). Einstein, A., Investigations on the Theory of the Brownian Movement. Dover, 1956. Ernst, M. H., R. M. Ziff, and E. M. Hendriks, "Coagulation Processes with a Phase Transition," J. Coll. & Interface Sci. 97, 266-277 (1984). Ernst, M. H. and G. Szamel, "Fragmentation Kinetics," J. Phys. A: Math. Gen. 26, 6085-6091 (1992). Friedlander, S. K., Smoke, Dust and Haze, John Wiley & Sons, 1977. Gal-Or, B., and L. Padmanabhan, "Coupled Energy and Multicomponent Mass Transfer in Dispersions and Suspensions with Residence Time and Size Distribution," A.I.Ch.E. Jl. 14, 709-714 (1968). Hibbard, J. L. and D. Ramkrishna, "Analysis of Phase Transfer Catalytic Reactions in Liquid-Liquid Systems," in Process and Fundamental Considerations of Selected Hydrometallurgical Systems, (M. C. Kuhn, Ed.) pp. 281-289. Society of Mining Engineers of American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc., New York, 281-289, 1981. Hidy, G. M. and J. R. Brock, The Dynamics of Aerocolloidal Systems. Pergamon, Oxford, 1970. Hill, P. J. and K. M. Ng, "Statistics of multiple particle breakage," AIChE Jl. 42, 1600-1611 (1997). Lagisetty, J. S., P. K. Das, and R. Kumar, "Breakage of Viscous and Non-Newtonian Drops in Stirred Dispersions," Chem. Eng. Sci., 41, 65-72 (1986). McGrady, E. D. and R. M. Ziff, "The Shattering Transition in Fragmentation," Phys. Rev. Lett. 58, 892-895 (9)(1987). Muralidhar, R. and D. Ramkrishna, "Analysis of Droplet Coalescence in Turbulent Liquid-Liquid Dispersions," Ind. Eng. Chem. Fundls. 25, 554-560 (1986). Muralidhar R., Gustafson, S., and D. Ramkrishna, "Population Balance Modeling of Bubbhng Fluidized Bed Reactors-II. Axially Dispersed Dense Phase," Sadhana, 10, 69-86 (1987). Murahdhar, R., D. Ramkrishna, P. K. Das and R. Kumar, "Coalescence of Rigid Droplets in a Stirred Dispersion-II. Band-Limited Force Fluctuations," Chem. Eng. Sci. 43, 1559-1568 (1988). Nambiar, D. K., R. Kumar, T. R. Das, and K. S. Gandhi, "A New Model for the Breakage Frequency of Drops in Tubulent Stirred Dispersions," Chem. Eng. Sci. 47, 2989-3002 (1992). Nambiar, D. K., R. Kumar, T. R. Das, and K. S. Gandhi, "A Two-Zone Model for the Breakage Frequency of Drops in Tubulent Stirred Dispersions," Chem. Eng. Sci. 49, 2194-2198 (1994). Narsimhan, G., J. P. Gupta and D. Ramkrishna, "A Model for Transitional Breakage Probabihty of Droplets in Agitated Lean Liquid-Liquid Dispersions," Chem. Eng. Sci. 34, 257-265 (1979).

116

3. Birth and Death Functions

Pandis, S. N. and J. H. Seinfeld, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change. Wiley, New York, 1998. Ramkrishna, D. and N. R. Amundson, Linear Operator Methods in Chemical Engineering. Prentice-Hall, Englewood Cliffs, NJ, 1985. Randolph, A. D., and M. A. Larson, The Theory of Particulate Processes, Academic Press, New York, 1971. Rao, D. P. and I. J. Dunn, "A Monte Carlo Coalescence Model for Reaction with Dispersion in a Tubular Reactor," Chem. Eng. Sci., 25, 1275-1282 (1970). Seinfeld, J. H., "Dynamics of Aerosols," in Dynamics and Modeling of Reactive Systems. (W. E. Stewart, W. H. Ray, and C. G. Conley, Eds.), pp. 225-258. Academic Press, New York, 1980. Shah, B. H. and D. Ramkrishna, "A Population Balance Model for Mass Transfer in Lean Liquid-Liquid Dispersions," Chem. Eng. Sci. 28, 389-399 (1973). Shah, B. H., D. Ramkrishna, and J. D. Borwanker, "A Simulative Analysis of Agglomerating Bubble Populations in a Fluidized Bed," Chem. Eng. Sci. 32, 1419-1425 (1977). Simons, S., M. M. R. Wilhams, and J. S. Cassell, "A Kernel for Combined Brownian and Gravitational Coagulation," J. Aerosol Sci. 17, 789-793 (1986). Smoluchowski, M. v., "Studies of Molecular Statistics of Emulsions and their Connection with Brownian Motion," Sitzungsberichte. Abt. 2a, Mathematik, Astronomic, Physik, Meteorologie und Mechanik. 123, 2381-2405, 1914. Spielman, L. A., "Viscous Interactions in Brownian coagulation," J. Coll. and Interface Sci. 33, 562-571 (1970). Sweet, L R., Gustafson, S. and D. Ramkrishna, "Population Balance Modeling of Bubbling Fluidized Bed Reactors-I. Well-Stirred Dense Phase," Chem. Eng. Sci. 42, 341-351 (1987). Taylor, A. E., Advanced Calculus. Ginn, Boston, 1955. Tsouris, C. and L. L. Tavlarides, "Breakage and Coalescence Models for Drops in Turbulent Dispersion," AIChE Jl. 40, 395-406 (1994). Valentas, K. J., O. Bilous, and N. R. Amundson, "Analysis of Breakage in Dispersed Phase Systems," I & E. C. Fundls. 5, 271-279 (1966). Valentas, K. J., and N. R. Amundson, "Breakage and Coalescence in Dispersed Phase Systems," liScE.C. Fundls. 5, 533-542 (1966). Valentas, K. J., and N. R. Amundson, "Influence of Droplet, Size-Age Distribution in Dispersed Phase Systems," liScE.C. Fundls. 7, 66-72 (1968). Zhang, X., and R. H. Davis, "The Rate of Collisions due to Brownian or Gravitational Motion of SmaU Drops," J. Fluid Mech. 230, 479-504, 1991.

CHAPTER 4

The Solution of Population Balance Equations

This chapter will be concerned with solution methods for population balance equations. It is a source of reassurance before attempting a solution to be aware that a solution exists so that we shall begin with this step. Furthermore, this existence exercise also leads to an approximate method of solution of the population balance equation. However, we shall desist a protracted mathematical treatment of issues generally unfamiliar to the engineer, in favor of the following approach. We show that the population balance equation, by proper recasting, is equivalent to a Fredholm or Volterra integral equation of the second kind whose existence of solution is a standard subject of treatment in mathematical texts (Petrovsky, 1957). Thus the existence of solution to a population balance equation will depend on the model hypotheses satisfying mathematical existence criteria. We shall then cover some cases of analytical solutions of population balance equations where our objective will be to focus on the structure that leads to such solutions rather than be exhaustive in our coverage of the literature. Often, engineering calculations seek only moments of the number density function and it is sometimes possible to obtain moment equations directly from the population balance equation. We shall explore the domain of this procedure and seek remedies where it leads to difficulties. This naturally leads to a discussion of approximate methods for the solution of population balance equations based on weighted residual and orthogonal collocation methods. 117

118

4. The Solution of Population Balance Equations

A particularly attractive approach that has evolved more recently is that of discretizing population balance equations and solving the discrete equations numerically. The effectiveness of this technique lies in rapid solutions of selected properties of the population that may be of interest to a specific application. Lastly, we shall explore Monte Carlo simulation techniques, which, with an abundance of computing power, have an incomparable degree of omnipotence.

4.1

EXISTENCE OF SOLUTION

We shall consider the vectorial case but with no distinction between external and internal coordinates, since it is of not of any consequence to this discussion. Thus we let z = (x, r) and Z(z, t) = [X(x, r, t), R(x, r, 0 ] and rewrite the population balance equation (2.7.9) as | / . + V , - Z / , =Mz, 0

(4.1.1)

where/i = /^(z, t). Clearly, Eq. (4.1.1) neglects any influence of the continuous phase. This constraint, although not necessary, is imposed only to simplify the initial discussion. Also, we have automatically excluded from our consideration, the case of random changes in particle states developed in section 2.10. However, Eq. (4.1.1), from the point of view of applications, is of sufficient generality to warrant the discussion that follows. The net source function, h{z, t), on the right-hand side may display any of the following characteristics. (1) It is independent of the number density, i.e., a specified function of z and t. This occurs obviously in systems in which the particles may appear or disappear because of events occurring in the continuous phase. An example of this would be homogeneous nucleation in crystallization or precipitation processes. (2) It is a linear function of / i , which occurs in systems where particles are lost spontaneously such as by "death" in biological populations. (3) It is a functional of the number density function, which requires the specification of the number density over a range of particle states. Such population balance equations are integro-partial differential in nature. Many of the examples from Sections 3.2 and 3.3 yield

4.1. Existence of Solution

119

equations in this category. The breakage processes of Section 3.2 give rise to /z(z, t) that are linear functionals of/^ whereas the aggregation processes of Section 3.3 are examples h{z, t) being nonlinear functionals of Z^. Following the discussion in Section 2.7.3 we impose the following initial and boundary conditions on Eq. (4.1.1): Mx,0)=g{zl

ZGQ,;

Mz,t)=

^^l'/\,,

n^-Z(z, t)

z e dQl

(4.1.2)

where F(z, t) is the (not necessarily continuous) specified normal particle flux on dQl which is a part of the total boundary dQ^. The conditions (4.1.2) are fairly general.

4.1.1

Existence for Cases 1 and 2

Cases 1 and 2 are in fact the same because, in the latter, the right-hand side could be subsumed into the left by transposition to obtain an equation similar to that in case 1. We shall thus focus on Eq. (4.1.1) regarding the right-hand side as known so that it is a first-order partial differential equation for which the existence of solution is well known. ^ It is generally solved by what is known as the method of characteristics.^ The characteristic curves (which, we shall show here, are the same as the particle paths defined in Section 2.5) originating from the hyper surfaces t = 0 and dQl in (z — t) space (containing no characteristic curves) on which "initial" data are specified for the number density. These "initial" data include both the specification of the number density at t = 0 (which corresponds to the actual initial state of the population) as well as the boundary conditions in (4.1.2). The specification of the number density at t = 0 or on 5Q^ provides the initial conditions required for integrating along the characteristic curves in order to compute the number density at any point (z, t) in Q^. Of course the identity of the initial point on the characteristic curve for a fixed (z, t) at which the number density is desired can only be obtained by backward integration along the characteristic until it meets either 5QJ or the surface t = 0. 1 See, for example, Courant and Hilbert (1956), pp. 39-55. 2 See Rhee et al (1986).

120

4. The Solution of Population Balance Equations

In order to show that the characteristic curves are the particle paths we rewrite Eq. (4.1.1) as -f,^t-VJ,

= [/i(z, t) - V,• Z ] f,.

(4.1.3)

The characteristics are then given by the solution of the differential equations ^ = Z(z, 0,

1 = 1,

z(0)=z„,

m

= t„,

(4.1.4)

where 5 represents the parameter along the characteristic curve and is assumed to vanish at the beginning of the characteristic curve. The second of the differential equations (4.1.4) is readily solved to obtain t = t^ + 5. Indeed Eqs. (4.1.4) can be seen to be the same as those defining the particle path in Section 2.5. We denote the solution of the first of differential equations (4.1.4) by Z(s, z^). The solution of Eq. (4.1.3) is obtained by solving the following differential equation for /^ along the characteristic curve: ^

= /,(z, f ) - ( V , - Z ) / „

fm=fuo-

(4.1.5)^

where the initial condition on / j , in consonance with (4.1.2), will be given by

,

P"^

.

'" = '

(4.1.6)

ln.-Z(z„,g' '''^''In (4.1.5), we have used the square brackets to represent the population density purely as a function of the characteristic parameter s. Thus the solution for the population density may be written as / i M =/i,oexp

+

[-V,-Z(Z(5', z j , r, + 5')]^5'

ds'h{Z{s\ zj, t^ + s') exp

[-V,-Z(Z(s",z„),f„ + s")]is"|,

0

(4.1.7) 3 The term V^-Z in the differential equation (4.1.5) represents the rate of strain of an infinitesimal volume of the particle phase continuum of Section 2.5 in an evolving population.

4.1. Existence of Solution

121

where f^ls'] = /i(Z(5, z j , t^ + s) = /i(Z(t - t^, z j , t). Of course, for a given (z, t), it is essential to determine (z^, t^) by setting ^{t

- to, 1o) = Z.

t^ =

t-s,

As observed earlier, the solution of the preceding equations is equivalent to integrating the differential equations (4.1.4) backwards until either t drops to zero or the particle state intersects 5Q^ (at some z^) for some positive t^. Once (z^,tj has been identified, /^^ is obtained from Eq. (4.1.6). We may then rewrite the solution (4.1.7) as /i(z, 0 = / i , , e x p

+

i-v,'Z{z{f -t,,zj, n:\df

dt'h{Z{f — t^, z^), f) exp

[-V,-Z(Z(t"-f„,zJ,

f")]df"|. (4.1.8)

The foregoing solution is computationally well defined. It can however be more succinctly represented by defining the function K{z, t; f) SLS X(z, t; f) = exp

l-V^'Z{Z{f-t,,z,),n-]df

(4.1.9)

so that the solution (4.1.8) may be rewritten as /i(z. t) = /i,,K(z, t; t,) +

K{z,t;nhiZ(f

-t,,z^),ndt,

(4.1.10)

where it must be remembered that (z^, t^) are both functions of z and t. This solution is contingent on the evaluation of the function K{z, t; f) via (4.1.9), which may be either analytical or computational depending on the complexity of the model. From a computational point of view, we may consider the evaluation of the solution in a somewhat different hght. The entire solution surface /^(z, t) may be generated from (4.1.10) by discretizing the coordinates (z^, tj and performing only forward integrations. Thus the solution (4.1.10) is computationally defined in the foregoing sense by A(Z(t - t„ z j , t) = h,M^{t

- t„ z j , t; Q

+ I K{Z{t - t„ z j , t\ t')h{Z{t' - t,, z j , t')dt', (4.1.11)

122

4. The Solution of Population Balance Equations

which gives the number density at Z(t — t^, z^) at time t. The existence of solutions for cases 1 and 2 is thus estabhshed through that of a system of ordinary differential equations (4.1.4) and (4.1.5) the stipulations for which are assumed to be satisfied by the models for Z(z, t) and /i(z, t)^ The actual solution is contained in (4.1.11). We shall next address case 3.

4.1.2

Existence for Case 3

We begin with assuming that /i(z, t) = H [ { / J ; z, r],

(4.1.12)

which is a functional of the number density /^ and a function of (z, t). This functional is, quite frequently, an integral involving the population density. The several examples of breakage and agglomeration systems discussed in Chapter 3 satisfy the form (4.1.12) and involve integrals of the population density over some region of the particle state space. Consequently, the population balance equation (4.1.1), in the light of (4.1.12), becomes | / i + V , - Z / , = / / [ { / , } ; z,(],

(4.1.13)

which is an integro-partial differential equation. We also assume the boundary conditions (4.1.2) with the modification that the function F(z, t) is of the form F(z, 0 - = O [ { / i } ; z ] ,

(4.1.14)

the right-hand side being a functional of the population density / i , which subsumes boundary conditions of the type (2.7.13), encountered in Chapter 2. If on the right-hand sides of (4.1.13) and (4.1.14) we pretend that the number density f^ is known, then we can use the methods of the previous section to write down the "solution" to the population density on the leftThus the functions Z(z, t) and h{z, t) must be globally Lipschitzian by which is meant that for any two vectors z and z' in the particle state space there are constants K^ and K^ such that ||Z(z, t) - Z ( z ' , r ) | | ^ X 2 : l | z - z ' | | , and |/z(z, t) - h{z', r)| ^ K J | z - z'||. For existence of solutions to ordinary differential equations, see Chapter 1 of Coddington and Levinson (1955).

4.2. The Method of Successive Approximations

123

hand side of (4.1.13) using the form (4.1.10), viz.,

/i(z, 0 = fi,oK{z, t- tj +

K(z, t;f)Hl{f,};Z{t'

- r „ z j , fW-

(4.1.15)

Equation (4.1.15) is an integral equation in the population density since, on its right-hand side, the functional H usually involves an integral of the population density with respect to the particle state. The above integral equation is analyzed using fixed-point methods, which also generate criteria for existence of the solution.^ The contraction mapping theorem used for establishing existence also assures uniqueness of the solution.

EXERCISE 4.1.1 Identify the functional H for the breakage process in Section 3.2.2 described by the Eq. (3.2.8). Determine the integral equation that must be satisfied by the population density.

4.2

THE METHOD OF SUCCESSIVE APPROXIMATIONS

The existence of solution to Eq. (4.1.15) is generally established by a procedure that guarantees the convergence of the method of successive approximations (also called Picard's iteration). This method consists in substituting into the right-hand side of (4.1.15), the nth approximation for the population density denoted by //"^ in order to calculate the {n + l)st approximant. Thus we have

fr%,

0 = /i,„-K(z, t; tj + r X(z, t; t')Hl{ffy,

Z{t'-

t„, zj,

t'W

J to

(4.2.1) The actual calculation has already been covered in the preceding section. If 5 See Ramkrishna and Amundson (1985), pp. 95, 139, for a demonstration of existence of solution for a one-dimensional linear integro-differential equation. For a nonlinear integrodifferential equation, the reader is referred to Naylor and Sell (1971).

124

4. The Solution of Population Balance Equations

the criteria for existence of a unique solution of (4.1.15) are satisfied, the convergence of the iteration imphed by (4.2.1) is guaranteed. The following simple example will serve to illustrate the ideas. Consider the cell population of Section 2.11.2, which multiplies by binary division for the more general case of an arbitrary cell division rate r(T, t) with an initial age distribution of ^(T). The population density /^(T, t) satisfies the equation, g/l(T, t)

g/,(T, t)

(4.2.2)

and the following initial and boundary conditions: A(T, 0) = gix),

A(0, 0 = 2

r(T', t)/i(T', t)dt'

(4.2.3)

Although from the point of view of Eq. (4.2.2), the problem at hand is an example of case 2, the boundary condition in (4.2.3) converts it to be one of case 3. To solve (4.2.2) by the method of characteristics, we identify the characteristic curves by the differential equations dt = h ds

dx = 1, ds

ds = - r ( T , o / i

(4.2.4)

subject to initial values t^, ^o^fi,o respectively, at s = 0. Integrating these differential equations gives T = T, + 5,

t = t^-hS,

riT^ +

/ i [ 5 ] = / i , , exp

s\t^^s')ds'y (4.2.5)

where again, as before, we have used the square brackets to represent the functionality of /j purely in terms of the characteristic parameter s. The population density may also be written from (4.2.5) as /i(^,

0 = /I(TO' O

exp

r{T\t,

+ T'-T^)dT'}.

(4.2.6)

In the region i > r of the T — r plane, the characteristics trace back to the line t = 0, whereas in the region T < t they trace back to the line i = 0 (see Fig. 4.2.1). 1

+ b{x, y) /i(^, 0

4.2. The Method of Successive Approximations

125

characteristic

characteristic

FIGURE 4.2.1 The characteristic curves for the solution of the partial differential equation (4.2.2).

Thus (4.2.6) becomes /I(T,

t) = g{T ~ t) exp

/I(T, t) = A(0,

r(T', t -\-

t-T)QXp{-

T'

—

T) dz'K

r(T', t + T' - T)dT'y

T^ t t > T.

(4.2.7a) (4.2.7b)

If we let h{t) = /i(0, t), substitute (4.2.7a) and (4.2.7b) into the boundary condition (4.2.3), we obtain the Volterra integral equation h{t) =

h{t - T)iC(t, T)dT + G(0

(4.2.8)

where K{t, G{t) = 2

T)

= 2r(T, t) exp

g{T — t)r{T, t) exp

r(T', t + x'

-T)df

r(T', f + T' - T)dT' V di.

(4.2.9) (4.2.10)

Indeed, G(t) is a known function since the initial age distribution ^(t) is specified. Thus the solution of the original population balance problem is

126

4. The Solution of Population Balance Equations

now equivalent to the solution of the integral equation (4.2.8). It is this integral equation that can be solved by the method of successive approximations. Following (4.2.1), the (n + l)st approximation is obtained from the Mth approximation as + 1)M = h^'^^'Xt)

/2^"^(t - T)K{t, T)dT + G{ty

(4.2.11)

which could be initiated, for example, by setting h^''\t) = G{t). Such a procedure may also be referred to as the method of successive substitution.

EXERCISE 4.2.1 For the special case considered in Section 2.11.2, solve the integral equation (4.2.8) to obtain the solution for the total population density as given in (2.11.9). EXERCISE 4.2.2 For the example considered in section 3.2.4, apply the method of successive approximations to obtain an analytic expression to the number density and the solute concentration distributions of the different generations of droplets (see Shah and Ramkrishna (1973) in Chapter 3. EXERCISE 4.2.3 Apply the method of successive approximations to solve the aggregation problem for the constant frequency. Assume that the population balance equation is given by Eq. (4.3.4) in terms of the dimensionless variables of Section 4.3 with a(x, x') = 1 and that the initial distribution is given by d{x — 1).

4.2.1

The Method of Successive Generations

Liou, Srienc, and Fredrickson (1997) have developed an interesting approach to solving population balance equations based on what they refer to as the method of successive generations. It can be conveniently demonstrated on an equation of the type (4.2.2) although it is applicable to more general situations. Although it is computationally equivalent to the method of

4.2. The Method of Successive Approximations

127

successive approximations, it has an interesting conceptual attribute not present in the latter. It is based on resolving the overall population of a particular state as the sum of contributions from all succeeding generations. Considering Eq. (4.2.2) for a population of cells, we let /^(T, t) be

M^, t)=t

k= 0

fl'V t)

where fl^\T, t) is the number density of cells of age T from the /cth generation, the initial population being defined to be the zeroth generation. No cells of the zeroth generation can exist of age zero at t > 0 so that we have the boundary condition //>, 0=0,

(4.2.12)

(t>Oy

Since the fcth generation arises from the (fe — l)st, Eq. (4.2.3) implies that

/r'(o, t) = 2

r(T', t)/i

V , f)

(t > 0),

dr', k = l,2,...

(4.2.13)

whereas the initial condition yields / r ( T , 0) = 0(T),

/ f V , 0) = 0, k=l,2,...

(4.2.14)

Equation (4.2.2) must be satisfied by /I''*(T, t) for each integer k so that the integral equation (4.2.8) using (4.2.12) and (4.2.14) becomes /i<°»(f) = Git),

h^^Xt) =

h<''-'>{t-T)K{t,x)dT,

k=l,2,...,

(4.2.15)

where h^''\t) = /^^(O, t). Thus, using (4.2.7a) and (4.2.7b), the solution for t) is determined as

/I'''(T,

r ( T ' , t -\- T' — T)dT'>,

fi%,t)=0, //"'(T, 0 = 0 , fl%,

T^ t

k=l,2,... t >T

t) = /i<*>(t - T) exp

T(T\

t + T' -T)dT'y

k= 1,2,... .

(4.2.16) The successive iteration of h^^\t) using (4.2.15) defines the solution (4.2.16) for fl^\T, t) completely.

128

4. The Solution of Population Balance Equations

EXERCISE 4.2.4 Resolve Exercise 4.2.1 by the method of successive generations.

4.3

THE METHOD OF LAPLACE TRANSFORMS

Laplace transforms are particularly suitable for obtaining analytical solutions for certain forms of population balance equations. In aggregating systems, the population balance equation in particle mass (or volume) features a convolution integral in the source term which makes it amenable to solution by Laplace transforms. We shall illustrate the solution of the aggregation problem represented by Eq. (3.3.5), for suitably selected aggregation frequencies. We recall the population balance equation (3.3.5) as

St

-

'

2 ;o

a(x - x\ x')/i(^ - x\ t)f^{x\ t)dx' - / i ( x , t) 0

a{x, x')/i(x', t)dx'

(4.3.1)

subject to the initial condition /i(x, 0) = NM^)

(4.3.2)

where N^ is the initial total population density, and g{x) is the initial size distribution. We assume that particle mass x has been nondimensionalized with respect to the average particle size at r = 0, so that xg{x)dx=

1.

(4.3.3)

It is convenient to nondimensionalize the population balance equation using dimensionless variables. Further, we introduce the dimensionless quantities T = a^Nj,

fix,

T)

= — /i(x, t),

a(x, x') =

^^ o

V(T)

=

Jo

/(-x, x)dx =

'

,

^o

-—-

^^ o

where a^ is some characteristic value of the aggregation frequency. The

129

4.3. The Method of Laplace Transforms

dimensionless population balance equation may now be written as df{x,

T)

1

dT

oc{x — x')f{x a(x, x')f(x\

— x\ T)f{x\ x)dx' — f{x, T) T)dx\

(4.3.4)

which must be subject to the initial condition /(x, 0) = g{x).

(4.3.5)

Because of (4.3.3) and the conservation of mass, it readily emerges that (4.3.6)

xf(x, T)dx = 1.

We shall develop the solution for (4.3.4) and (4.3.5) by defining the Laplace transform with respect to the size variable /(s, T) =

/ ( x , x)e '""dx.

Note in particular that /(O, i) = V(T). We consider below two different aggregation frequencies, viz., the constant frequency given by a(x, x') = 1, and the sum frequency, a(x, x') = x + x'. 4.3.1

The Constant Aggregation Frequency

Taking the Laplace transform of (4.3.4) for the case a(x, x') = 1, we obtain a/(5, T) dx

1

dxe~ f{x\

f{x -x\

x)f{x\

x)dx'

dxe ^V(x,

T)

x)dx'.

By modifying the limits of integration in the first term on the right-hand side of the preceding equation as done just below Eq. (3.3.6) in Section 3.3, we obtain dfjs, T) ^ 1 dT 2

dx'f(x',

x)e'

f{x', T)dx'

e-'"f{u, x)du

dxe '"'fix, T) (4.3.7)

130

4. The Solution of Population Balance Equations

where we have set x — x' = w as a new integration variable. Equation (4.3.7) becomes ^ ^

= -f{s,

(4.3.8)

TY - V(T)/(5, T).

Taking Laplace transform of the initial condition (4.3.5) we have the initial condition for (4.3.8) given by (4.3.9)

f{s, 0) = ^(5).

By letting s = 0 in Eq. (4.3.7), and noting that /(O, T) = V(T), we obtain the following ordinary differential equation in V(T): v(0) = 1 whose solution is given by V(T) =

(4.3.10)

(2 + T)

By dividing (4.3.8) by/^, we may rewrite it as the following linear differential equation in l/J: 1

+ V(T)

This is readily solved to give f(s, T) =

gis) 2+T

1

2+T

(4.3.11) g(s)

If the initial particle size distribution is monodisperse with dimensionless size unity, then ^(5) = e~^ and the transform (4.3.11) may be inverted by using the expansion of (1 — y)~^ in powers of y, which converges for y less than unity. Thus lis, T) =

-s{n+

1)

which is readily inverted to give the solution 2 + ty „% \2 + T

(4.3.12)

4.3. The Method of Laplace Transforms

131

This above solution could also have been obtained by the method of successive approximations discussed in Section 4.2 (see Exercise 4.2.3). Seinfeld and co-workers have made extension of such Laplace transform solutions to multidimensional problems in the coagulation of aerosols.^

EXERCISE 4.3.1 Allow for particle growth with X{x) = KX in the population balance Eq. (4.3.4) and solve by the method of Laplace transforms for the monodisperse initial condition.

EXERCISE 4.3.2 Solve by the method of Laplace transforms the population balance equation given in Section 3.3.6.2 under equilibrium conditions. And show that the solution is an exponential distribution. Next, we illustrate the solution of the aggregation problem for the case of the dimensionless aggregation kernel given by x + x\

4.3.2

The Sum Frequency

The dimensionless population balance equation may be written as dfix, T) dz

1 x'f{x 2 Jo

— x\ T)f{x\ T)dx' — f{x, T)

{x + x')f{x\

x)dx'. (4.3.13)

Taking the Laplace transform of (4.3.13), we obtain df{s, T) ST

1 dxe~ 2 Jo

xf{x — x\ T)f{x\

X

x)dx'

dxe

^""fix, T)

0

(x + x')f{x\ T)dx' Jo By modifying the limits of integration in the first term on the right-hand 6 See Gelbard and Seinfeld (1978a,b).

132

4. The Solution of Population Balance Equations

side of the preceding equation as done in the previous section, we obtain dfis,T)

1

dx'f{x\

dT

x)e

dxe-'Y{x,

0

T)

(X + x')/(x', T)dx\

(4.3.14)

If we recognize that dfjs, T) 35

xf{x, x)e ^^dx

and Eq. (4.3.6), we may write (4.3.14) as df(s, i) dx

df{s, i) [/(S, T) - V(T)] -f(s, ds

T),

(4.3.15)

which is subject to the initial condition (4.3.9). By letting s = 0 in Eq. (4.3.14), we obtain the following ordinary differential equation in V(T) dv{x) ~d^

= -V(T),

V(0) = 1.

whose solution is given by V(T) = e~\ Equation (4.3.15) may be solved by the method of characteristics. The characteristics in the three dimensional space of (i, 5, / ) coordinates are given by

^ = (7-v),^=-/,

^[0] = 5„

7m=g{So\

(4.3.16)

The second of these initial conditions is clearly obtained from (4.3.9). Note that the first of the preceding differential equations shows that on the (5, T) plane, the line 5 = 0 is a characteristic, which follows from the fact that /(O, i) = V(T). The differential equation for / is readily solved to obtain its value along the characteristic starting at s = s^ and i = 0:

/ M = ^(0^"^

(4.3.17)

In order to obtain /(s, r), it is of interest to calculate s^ as a function of s and T by backward integration along the characteristic from the point (5, i). Using the solution for V(T), we solve the differential equation for 5 to obtain ds dx

ms„) - l ] e -

s-s„ = ( l - e - ) [ 0 ( s j - l ] ,

(4.3.18)

4.3. The Method of Laplace Transforms

133

which is difficult to solve analytically for s^ in terms of 5 and T and therefore thwarts this approach to a solution. However, this situation is also suggestive of a remedial transformation of / designed to render it invariant along the characteristic. The actual procedure will provide a better clarification. Define 7(5, T) = ^ ( 5 , T)IA(T)

(4.3.19)

where I/^(T) will be chosen so that the partial differential equation in ^(5, T) arising from (4.3.15) displays only derivatives of ^(s, T) with respect to T and s (i.e., not ^(5, T) by itself). Substituting (4.3.19) in (4.3.15), one obtains d(p{s, T) d(p{s, T) W — - ^ — = -^ — ^ ^ — [^(s, ^)W -

V(T)]

- cp{s,

T)

In order to eliminate the term ^(5, T), we must set its coefficient in the preceding equal to zero, which gives I/^(T) = e~'' interestingly the same as V(T). If we use this result, the partial differential equation in ^(s, T) becomes 5^(5, T) — - ^ — = -^

d(p{s, T) — ^ ^ — [^(s, T) - 1].

(4.3.20)

It is now convenient to define an alternative independent variable T by dT = \l/{T)dT, T= 1

-e-\

the second of which follows from the first by arbitrarily setting T = 0 at T = 0. Then (4.3.20) may be written as

^ = _ ^ [ * „ r , - i :

(43.2.)

where we have set $(s, T) = (p{s, T). The initial condition for $(s, T) is given by 0(s, 0) = gis).

(4.3.22)

The characteristic equations are then given by

The above are readily solved to obtain

from which we obtain the implicitly expressed solution for 0(s, T)(= $ [ r ] )

4. The Solution of Population Balance Equations

134

given by 0(5, T) = -g{s + (1 - 0(5, T)T)l

(4.3.23)

Note further that hne s = 0 is a characteristic on the (s, T) plane along which $(s, T) must remain constant at the value of unity in order that /(O, T) = II/{T) = e~\ Expression (4.3.23), although implicit in 0(5, T), can generate all the moments of the population density at every instant, using the property that ^fc

_

I

poo

- ^ [ 0 ( 5 , T)] ^^

x^(D(x, T)dx

={-lf |s = 0

Jo

SO that v, = (-l)''(l

(4.3.24)

r ) ^ [ 0 ( 5 , r)],=o,

where v^ is the /cth moment of the dimensionless population density f{x, T). As an example, we shall demonstrate the calculation of the second moment. Thus, differentiating (4.3.23) once. aO(0, T) ds

g'iO) 1 + Tg'iO)

(4.3.25)

Since the average initial particle mass has been assumed to be unity we have ^'(0) = — 1, so that (4.3.24) implies that Vj = 1 which merely rediscovers the known result from conservation of mass. Differentiating (4.3.23) twice, we obtain V2=(l

^'(0)

T ) ^ [ 0 ( 5 , T)-], = , = r{0) 1

1 + Tg'iO)

Since ^"(0) represents the initial value of the second moment, denoted, by, say, V2,o? the foregoing expression becomes = V 2,0

'1 + T^^

The route to the higher moments is now established. The expression (4.3.23) can also be considered for direct inversion of the Laplace transform of the function 0(s, T) by expanding it in Taylor series about T=0 °c T^ 5^0(s k\

dT"

T)

(4.3.26)

4.3. The Method of Laplace Transforms

135

where we have used the initial condition (4.3.22) and convergence may be presumed in the entire interval 0 < T < 1. The derivatives of $(s, T) with respect to T at r = 0 is obtained after some tedious algebra from (4.3.23) as a'0(5, T) dT^

1 L^-gis)f (k + 1) ds"

r=o which may be rewritten using the binomial expansion as d^^is,

T)

Qjk

1 'y(_i). (^ + 1)' r\{k + \-r)\ ih + 1) r=i

^=0

<^'9^')' d^

(4.3.27)

Substituting (4.3.27) into (4.3.26), one obtains ^(s,T)=g{s)-

00

k+1

X

Z

k=l

(-ir

r=l

d'-gisf r!(fe + l - r ) ! ds"

(4.3.28)

By interchanging the order of summation with respect to k and r, we obtain from (4.3.28) 00

$(s, T)= X r=l

00

Z (-!/"' k=r-l

r\{k + 1 - r)! ds"

If we redefine the summation index within the inner sum on the right-hand side of the equation as ; = fe + 1 — r, the foregoing expression becomes (4.3.29) It is now convenient to invert the preceding Laplace transform within the inner sum as follows. The inverse Laplace transform of the derivative appearing in the inner sum of (4.3.29) is given by L"

'9(sr = {-xr^^-'L-'{g{sn

d^' dsJ^"--^

which, on substitution into the Laplace inverse of (4.3.29), gives
T)] = Z r=l

^^^^L-\g(sy) ^'

Z 7= 0

^ ^ . J-

On recognizing the inner sum as the exponential function, we finally have 0(x, T) = e - ^

l^^^JJ^L-HgisY). r!

(4.3.30)

If we now assume that the initial particle size distribution is the Dirac delta

136

4. The Solution of Population Balance Equations

distribution d{x — 1), then g{s) = e~^ and the foregoing expression becomes a)(x, T) = ^-^^ X ^'

1— ^(^ - '*)•

(4.3.31)

Thus the dimensionless number density f{x, T) for the aggregation problem with the sum frequency is obtained as fix, T) = e-[T+x(i - . - ) ] X

^

<5(x - r).

Solutions for other initial conditions can of course be obtained by inverting afresh the Laplace transform (4.3.30). Similarly, solutions for the product frequency can also be obtained by the method of Laplace transforms (Scott, 1968). For an integrated treatment of the constant, sum and product aggregation frequencies, the reader is referred to Hidy and Brock. ^

4.4

THE METHOD OF MOMENTS AND WEIGHTED RESIDUALS

Not infrequently, practical needs can be fulfilled by calculating the (generally integral) moments of the number density function. The calculation of such moments can occasionally be accomplished by directly taking moments of the population balance equation producing a set of moment equations. Consider, for example, a breakage process without growth terms for which the population balance equation is given by (3.2.8). Further, assume that breakage is binary so that v(x') = 2. Multiplying (3.2.8) by x" and integrating over the semi-infinite interval together with switching integration ranges as shown in Section 3.2.2, we obtain by setting £, = x/x' 1

dt

0

d^i"

P(x'i\x')b(x')Mx', 0

t)dx' -

x"b{x)fi{x,

t)dx.

0

(4.4.1) If we regard the function P{x\x') as a function purely of x/x\ then we may write P{x\x') = p{^) and define moments r/7(0rf^,

n = 0,1,2,...

0

7

See Hidy and Brock (1970), Chapter 6, pp. 305-315, 1970, referenced in Chapter 3.

4.4. The Method of Moments and Weighted Residuals

137

all of which can be evaluated. Indeed n^= 1, and n^ = 1/2 from conservation of mass. Equation (4.4.1) now becomes x"fo(x)/i(x, t)dx.

(4.4.2)

0

In order for the moment equation to generate only terms containing moments, the breakage rate b{x) must be a polynomial function of x. However, even this requirement leads to a set of unclosed moment equations because the differential equation in any moment involves higher moments. Thus the only way to get an exactly closed set of moment equations is to require that b{x) = b^, a. constant which yields from (4.4.2) the equation

^

= nM2n„ - 1),

which is readily solved to obtain //„(r) =//„(0)^^^^"~^^^''^ from which it follows that the total population density /i^(t) = fi^{0) e^°^ and the total mass fi^ is invariant with time. If we allow particle growth, it is interesting to note that closed moment equations could be obtained for linear growth rate. Recalling (3.2.7) for breakage processes with X{x, t) = kx, where /c is a constant, and taking moments one obtains ^

= [/cn + b„(27r„-l)]^„,

which has the solution /x„(t) = /i„(0)^^''"'^^''"~^^^^^ that remains the same for li^(t) without growth but allows the exponential solution fi^{t) = jUi(O) e^^ for the first moment, as one would expect. One encounters similar constraints with aggregation processes to generate closed integral moment equations, i.e., a constant aggregation rate and at most a linear growth rate. In order to demonstrate the moment equations for this case, we recall the population balance equation (3.3.5) for the constant aggregation rate, a{x, x') = a^, incorporate a linear growth term X{x, t) = kx, and take moments. The result is

du

1

r* 0

0

dx'Ux', t),

duf^{u, t) — a^

Jo

^"/i(^? 0 d^

(4.4.3)

138

4. The Solution of Population Balance Equations

where we have switched integration variables as done in Section 3.3. Using the binomial expansion in the first integral on the right-hand side of the foregoing equation, one obtains the set of moment equations

1

" fn\ X

= knfl„+-a^

(

1/^n-r/^r-^oMoMn.

(4.4.4)

which are clearly closed. The zeroth moment satisfies the equation

from which we obtain the solution /^o(O)

f^oit) =

l+-fi,{0)aj Since for n = 1, the differential equation contains only the first term on the right-hand side of (4.4.4), the first moment has the solution in^(t) = iij^{0)e^\ For higher moments, the moment equations (4.4.4) can be solved analytically by rewriting as

whose solution is readily found to be /i„(0=/^„(0)^^- + ^ a .

\^nit-t'/^ 0

fA^^_^^t^)^^^f)dt\ r=l

n^l,

(4.4.5)

\^/

Thus, not surprisingly, all the moments are analytically accessible for this case. More often, however, the constraints under which moment equations can be obtained directly are violated in examples of applications. Trouble frequently arises due either to the moment equations being unclosed or to the emergence of nonintegral moments. Hulburt and Akiyama (1969) sought to cure this problem by expanding the number density function in terms of a finite number (say M) of Laguerre polynomials, L„(x) as M

1

/i(x, t) = e-- X -««Wi^„W,

(4.4.6)

4.4. The Method of Moments and Weighted Residuals

139

where L,{x)^e^^{e-^xn= dx

t ynX. ,%

n = 0,1,2,...,

satisfy the orthogonality conditions^ •oo

e'^L„ix)L„{x)dx

= in\yS„„,

(4.4.7)

0

from which we identify the coefficients of expansion in (4.4.6) as 1

1 " Ux,t)L„(x)dx=-Yynrf^r''. =0

(4-4.8)

Thus (4.4.8) shows how any term in the population balance equation involving the number density can be expressed purely in terms of the first M integral moments. While the use of generalized Laguerre polynomials with additional parameters provided some flexibility for the method of moments, Hulburt and Akiyama have reported difficulties with this method. The foregoing approach was shown by the author (Ramkrishna, 1971) to be an application of the well-known method of weighted residuals (Finlayson, 1972) in which the solution to an equation is expanded in terms of a finite set of functions. This set of functions is generaUy obtained by truncating an orthonormal family (complete in the space of functions to which the solution is supposed to belong^). The coefficients of expansion are obtained by orthogonalizing the residual with either the same set of functions used to expand the solution (as in Galerkin's technique) or another suitable set of what are known as weighting functions. This recognition has led to an application of the method of weighted residuals with its attendant variations for the solution of population balance equations. We now discuss some of the early efforts in this direction although since then the literature on this subject has ever been on the rise. Subramanian and Ramkrishna (1971)^° were the earliest to use the method of weighted residuals on population balance equations. They considered a microbial population distributed according to their mass x, growing by assimilating substrate at concentration y from the surrounding 8 9

See, for example, Courant and Hilbert (1956), p. 95. In this case, the functions of interest are those whose squares can be integrated over the semi-infinite interval. We shall denote them by L2IO, 00). 10 See also Subramanian et al. (1970).

140

4. The Solution of Population Balance Equations

abiotic phase, and reproducing by binary fission. The model equations for a continuous bioreactor with the feed containing substrate at concentration yf and no microbes were given by^^

^

+ |^[X(x,,)/,(x,0]=2

b{x', y)P{x, x')/i(x', t)dx' -Q + b{x, y) Mx, t)

dt

6^^^

i?(x, y)Ux,

^'

(4.4.9) (4.4.10)

t)dx.

The foregoing equations are subject to the initial conditions Mx,0)

= g{x),

y(0)=y„.

The number density /^(x, t) was expanded in terms of Laguerre functions (/)„(x) =e-^L„(x)/n! as M

X C„(0„M.

Ux,t)=g{x)+

(4.4.11)

The initial condition is accommodated by allowing all the coefficients C„'s to vanish at time t = 0. The residual, on substitution of the trial solution (4.4.11) into Eqs. (4.4.9) and (4.4.10), may be represented by dt

~UL',y)^nfuy)

(4.4.12)

where 3 i denotes the operator (accounting for all the terms in the popula-

11 The forms of the various functions, the growth rate X{x, y), the partitioning function P{x, x'X the cell division rate b{x, y) and the specific substrate consumption rate p{x, y) in (4.4.9) and (4.4.10) were Xix, y)

jxxy_ k+y

/I'x,

2exp 8^71 erfc

P{x, x') = 30x2(x' - xflx'\

s

/

• •Xix,y),

b{x, y)

yxy p{x,y)=-j^, k+y

in which /x, fi', k, y, £ are model parameters and x* is a "critical" cell mass at or about which cell division becomes highly probable.

4.4. The Method of Moments and Weighted Residuals

141

tion balance equation (4.4.9) excluding the time derivative) which maps functions in L2IO, 00) into other functions in the same space. Since the term associated with the number density in Eq. (4.4.10) is not a function of x, substitution of the trial solution (4.4.11) into (4.4.10) produces no residual in L2IO, GO) that has to be orthogonalized. The orthogonalization of the residual ^{fi; y) is accomphshed by using a set of weighting functions {il/j{x)}; that is

r 9iif,;y)il^j{x)dx^i%iPjy

= 0.

From (4.4.12), we obtain A — - BC = b, C(0) = O, (4.4.13) at where the matrix A has for its ijth coefficient, A^j = <(^j-, i/^^.>, matrix B has Bij = <30t? ^j}, and the vector b has bj = <3i^, ^j}- In the current application the weighting functions were given by il/j{x) =x^e~^''. The convergence of the solutions with 10 or 11 basis functions is displayed in Fig. 4.4.1. For a more complete demonstration of the accomphshments of the method, the reader is referred to Subramanian and Ramkrishna (1971). There are numerous variations of the method of weighted residuals relating to the choice of the trial basis functions as well as the weight functions for orthogonalizing the residual. ^^ In particular, the method of orthogonal collocation depends upon equating the residual to zero at selected interpolation points in the interval in which the function is defined.

4.4.1

Problem-Specific Trial Functions

Trial functions such as the Laguerre functions (4.4.6) may also be interpreted as arising from Gram-Schmidt orthogonalization^^ of the basis set {x"} using the inner product

<0, ^> =

12

^(j){x)\lj{x)dx

For several of the references in the 1980's and earher in this regard, the reader is referred to Ramkrishna (1985), referenced in Chapter 2. 13 See, for example, pp. 161 of Ramkrishna and Amundson (1985).

142

4. The Solution of Population Balance Equations

X, Cell mass (gm)x10

FIGURE 4.4.1 Transient solution of Eqs. (4.4.9) and (4.4.10) for the number density in a continuous bioreactor by the method of weighted residuals as described in Section 4.4. (From Subramanian and Ramkrishna (1971). Reproduced with permission from Elsevier Science.)

where 0 and i// are real-valued functions. This inner product can be changed by varying the (positive) weight function w(x) = e'"" to generate trial functions that may be specifically appropriate for a particular application (in the sense of requiring only a small number of trial functions). ^"^ Thus, solutions can be obtained for dynamic population balance problems by choosing the weight function w such that the trial functions adjust to both coordinates x and t. Several examples are cited in footnote 14. The dynamics of Brownian aggregation considered by Singh and Ramkrishna (1975) use time-dependent polynomials generated as follows. The Brownian aggregation frequency ^^ is almost constant for particles of nearly the same size. For 14 See Ramkrishna (1973). However, the idea of using such problem-specific polynomials had been exploited earlier by Golovin (1963). See also Enukashivili (1964). The methods used for solving population balance equations were somewhat ad hoc and resticted to calculating moments. For a more systematic application of the method of weighted residuals including orthogonal collocation procedures, using problem-specific trial functions, the reader is referred to Singh and Ramkrishna (1975, 1977).

4.4. The Method of Moments and Weighted Residuals

143

suitable initial conditions, the population balance equation for aggregation with a constant frequency can be solved analytically (see Section 4.3). The temporally varying number density so obtained may then be used as the weight function in the inner product of the Gram-Schmidt orthogonalizing process to generate polynomials with time-dependent coefficients. When such problem-specific polynomials are used in the trial solution such as (4.4.11) the number of functions required to correct for the nonconstant aggregation process is presumed to be small in this approach. Singh and Ramkrishna (1975) show the advantage of the method, using orthogonal collocation, although their choice of the exponential distribution for the initial condition led to a curiously erroneous solution near the origin. Since aggregation occurs at infinite rates between particles of size "zero" and those of nonzero size, the number density immediately drops precipitously to zero near the origin, which the approximate solution was unable to capture. We have dwelt at length on this issue because of its value to the discussion on the method of orthogonal collocation that follows.

4.4.2

The Method of Orthogonal Collocation

In this method, the residual is exactly equated to zero at a finite number of points Xj, normally the zeroes of the polynomial of the smallest degree not included in the expansion. This procedure produces a number of equations equal to the number of expansion coefficients to be estimated in the unknown trial solution. ^^ The quality of the solution depends on the location of the collocation points. Returning to the example of Brownian aggregation considered by Singh and Ramkrishna (1975) with respect to the error near the origin, Sampson and Ramkrishna (1985) have shown how the location of the collocation points may be manipulated by choice of the trial function. Thus by locating a collocation point as near the origin as desired, they showed that increasingly accurate solutions could be produced for the Brownian aggregation problem. Gelbard and Seinfeld (1978c) have investigated the solution of a general class of coagulation problems including particle growth using a finite element method in combination with collocation procedures. This method appears particularly useful for aggregation problems in which the aggregation kernel shows strong dependence on 15 See Chapter 3, Section 3.3.5.1. 16 See Michaelsen and Villadsen (1978).

144

4. The Solution of Population Balance Equations

particle size. More generally, sectionally sensitive methods are desirable for the solution of population balance equations in which particle behavior is strongly dependent on particle state. Other variations in the approach have been reviewed by the author (Ramkrishna, 1985 referenced in Chapter 2). In recent times, however, there has been a spurt of interest in the approximate solution of population balance equations and the number of pubhcations has grown substantially. Although a review of these contributions would have been desirable, we turn our attention to a somewhat different approach that has attracted considerable attention in recent times. While all approximate methods of solving continuous equations (including the method of weighted residuals discussed earlier) are based on discretizing the equations, a different form of discretization has led to interesting and computationally efficient methods for solving population balance equations.

4.5

DISCRETE FORMULATIONS FOR SOLUTION

The methods of Section 4.4 accomplished discretization of the population balance equations by expressing the number density of particles as being contained in a finite dimensional space spanned by a finite set of trial basis functions. Thus the number density was described by a finite dimensional vector whose components were the coefficients of expansion of the continuous number density in terms of the basis set. A direct solution of the population balance equation by finite difference methods, on the other hand, would require a natural discretization of particle state space to represent derivatives and integrals, which appear in the equation. The fineness of discretization of particle size (or more generally particle state space) in this case would necessarily be dictated by numerical considerations in approximating the derivative or the integral. However, we shall be concerned in this section with discretization of a considerably coarser nature so that the population balances under consideration would essentially be "macroscopic" balances in particle state space in which the phenomenological implements of the model may display significant variations. Viewed from this perspective, one must sense that an accurate estimate of the number density as a source for calculation of all properties of the population by such methods would be more fortuitous in any given application than one of general consequence. Nevertheless, the methods have been notably success-

4.5. Discrete Formulations for Solution

145

ful in a variety of applications in that they have greatly reduced computation times, an attractive attribute for repetitive calculations in optimization and control of particulate systems. In applications, certain properties of the particle population may be more significant than others either because they control product quahty or because they are easier to measure for the purposes of control. Thus the perspective with which we begin our discussion of these discretization methods is that we design our calculations for certain selected properties of the system rather than for an estimate of the number density accurate enough for estimating all properties of the population. Before developing this perspective, however, we concern ourselves with the literature in regard to how the methods originated. It is convenient to first set the proper scenario for this discussion by integrating the one-dimensional population balance equation in particle size (which we shall represent by particle volume v) over subintervals in a partition P^ = {0 = v^, v^,...,Vj^, ^M+i}' I^ vi^w of our interest in the solution of population balance equations for both breakage and aggregation processes we shall entertain source and sink terms for both processes. However, it will be convenient to neglect growth processes at first in the sequel. We thus begin with the population balance equation dAJv, t) = H[{A}; V, q dt

(4.5.1)

where the functional Hl{f^}; v, Q is SL functional that maps the number density function into a function of v and t. Since both breakage and agglomeration processes are envisaged, the functional Hl{f^}; v, Q is given by a{v - v', v')fi(v - v', t)f^(v', t)dv' -fi{v,

H [ { / i } ; v,q =

t)

a{v, v')fi{v', t)dv'

+

v{v')b{v')PivW)Mv',

t)dv' - b(v)Mv, t). (4.5.2)

We denote the interval [i;-, i^j+i) by /^ and obtain the macroscopic balance

146

4. The Solution of Population Balance Equations

of the particles in the interval /^ by integrating Eq. (4.5.2) over I^ d dt

i f [ { / J ; i ; , r]di;,

f^{v,t)dv=\

i - 0,1,2,...,M.

(4.5.3)

If we set the total number of particles in I^ as N,{t)^

r^'Mv,t)dv

(4.5.4)

and substitute it along with the functional in (4.5.2), the macroscopic balances (4.5.3) become dNi dt ~

dv - X

Jvi

/

><s M

^i^-

v\ v')f,{v - v\ t)f,{v\ t)dv' -f,{v,

t)

j = 0 Jvj

rt

a{v, v')f^{v\ t)dv'

j= 0 J

+ X

v{v')b{v')P(v\v')Uv',

t)dv' - b(v)f,(v, t)

(4.5.5)

j = i Jvj

in which the integral with respect to v' inherited from (4.5.2) is expressed as the sum of integrals over subintervals. We recognize the M + 1 equations (4.5.5) as being unclosed in the variables { N j , i.e., reflecting a lack of autonomy. The use of the discrete methods of this section depends on the restoration of autonomy, which lies in our ability to express the right-hand side of Eq. (4.5.5) entirely in terms of the dependent variables {Nj. Toward this end, we observe that the right-hand side of (4.5.5) involves double integrals over I^ and Ij. Consider, for example, one such term dva{v, v')

dv'f,{v, t)f,{v\ t).

(4.5.6)

The various attempts in the literature to cast the above expression entirely in terms of N^ and N^ belong to either of the following two categories. ^^ (1) We use the mean value theorem^^ to express (4.5.6) as Vj+l

a{x^, x^)

dv

dv'fM

t)Aiv\ t) = a(x„ Xj)N,Nj

17 This discussion originates from a paper of Kumar and Ramkrishna (1996a). 18 See, for example, pp. 51-52 of Taylor (1955), referenced in Chapter 3.

(4.5.7)

4.5. Discrete Formulations for Solution

147

where x^ and Xj may be described as pivotal points or simply pivots in li and Ij respectively. The pivot concentrates the particles in the interval at a single representative point. Thus we may write the number density /^ {v, t) as being given by M

Mv, t) = E N,5iv - X,)

(4.5.8)

i= 0

Substitution of (4.5.8) into (4.5.6) will yield the right-hand side of (4.5.7). The pivots x^- and Xj must depend on the frequency function a{v, v') as well as the number densities in the two intervals so that they must strictly be regarded as time dependent in a dynamic problem. Furthermore, the pivots would not remain the same for all the terms in the functional Hl{f^}; v, t]. These observations provide further corroboration of the point made earlier that an exact calculation of all properties of the population by such methods would not be a reasonable expectation. (2) The second category seeks to evaluate (4.5.6) by substituting the mean value of the population density in each interval in each integrand. Thus in I^, we take the population density to be identical to NJ i^i+i ~~ ^i) ^iid withdraw it from the integral. Hence (4.5.6) is written as N^N, '

^

\Vi+^-Vi){Vj+^-Vj)

dv

dv'a{v, v').

(4.5.9)

Some latitude exists here in the choice of the average value withdrawn from the integral so that shghtly different forms of (4.5.9) can also be envisaged. In this approach, one is then required to calculate the integral of the aggregation frequency at each step. In either of the preceding categories, since the integrand contains the unknown number density, the mathematically rigorous choice for the pivot, which is consistent with the mean value theorem is of course not accessible. The finer the interval, the less crucial would be the location of the pivot in I^. The fineness required would depend on the extent to which the phenomenological functions of the population balance model such as the aggregation and breakage functions vary in the interval.

148

4. The Solution of Population Balance Equations

In further quest of the restoration of autonomy to (4.5.5), we substitute (4.5.8) into the former to obtain dN-

1 '~^

^

+ X

Njv{xj)b{xj)

P{v\xj)dv-b{x,)N,, i = 0,l,...,M,

(4.5.10) 1 9

where it is understood that terms involving indices other than i = 0,1,..., M are automatically set to zero. Thus on the right-hand side above, the first term is zero for / = 0, while the third is zero for i = M. The birth (source) terms in Eq. (4.5.10) on the right-hand side, the first due to aggregation and the third due to breakup, contribute particles only to the size interval /^ so that the number balance in each subinterval of the partition is exactly upheld. This also implies that the total number balance in the entire interval (or alternatively the zeroth moment) is satisfied. It is convenient to define the rth sectional moment of the population denoted /i^'^ by l^it)

^

v%{v, t)dv = x\Ni,

r - 0,1,2,...,

(4.5.11)

the equality to the extreme right arising from the discretization (4.5.8). The rth moment of the population over the entire size interval, denoted fi^, may then be expressed as the sum of the rth sectional moments, viz., M

/x,(0 = E /i*'>(t),

r = 0,1,2,....

i=0

It is of interest to examine the differential equation satisfied by /i^. Such a differential equation may be obtained in either of two ways. First, it may be recovered directly from the population balance equation by multiplying Eq. (4.5.1) by v"" and integrating over the subinterval I^ to get the sectional moment equation in ju^'\ = ^ 1 Nj X N,a{x,, xj){xj + xJ dt ~ 2 j = 0 {Xj + xi,)eli 19

- ^

X ^A^i^

^j)

7= 0

Since particles in the smallest interval IQ can only disappear by aggregation, the equation for i = 0 will not feature the first term on the right-hand side in Eq. (4.5.10).

4.5. Discrete Formulations for Solution M

+

Y^(^j)Hxj)Nj

149

(4.5.12)

v''P{v\xj)dv-b{Xi)fil'\

followed by summation over all / to get da \ ^ -17 = ^1.

^ I. Nj

M

Z

(Xj + Xk)eli

Nj,a{xj,, Xj){xj + xj

M

^ -lirY.

7= 0

f*Vi+i

+ 1 1

Hxj)bixj)Nj

i=0 j=i

Nja(Xi, Xj)

M

X /7(x,)/i«

t/Piv\xj)dv-

(4.5.13)

1=0

J Vi

On the other hand, if Eq. (4.5.10) is merely multipUed by x^ we again obtain a differential equation for the sectional moment //^'\ ^

=^ 1 "^

E

Nj

^j=0

E iV.a(x,, x,)

N,a{x,, xj)x^ - i^

(^,+^k)6/f

j=0

+ X v(x,)fo(x,)iV,x^

P(t;|x^.)^t;-&(xO/i. ,

(4.5.14)

which is not the same as Eq. (4.5.12) because of the source terms for aggregation and breakup. Summing (4.5.14) over i, one has dii

\ ^

-77 = :^ Z at

^

Z ^j

Z i= Q j = Q M

+ Z

Z

^

N^a{x^,Xj)x\-ii^

ixj + xk)eli

v{xj)b{Xj)Njx[

^A^i^^j)

j=0

M

I

X

Piv\xj)dv-

M

X biXi)fi\,(i) (4.5.15)

Equation (4.5.15) is different from the differential equation (4.5.13), which is clearly attributable to the discrepancy between (4.5.12) and (4.5.14). Kumar and Ramkrishna (1996a) describe this situation as being internally inconsistent. If consistency must be restored, the partition Pj^ must satisfy two criteria, viz., (i) Xj + x^, = x^, which will eliminate the inconsistency due to the source term for aggregation (and has the advantage of being exact), and (ii) the pivot x^ must satisfy the property v''P{v\Xj)dv = x^i

P{v\Xj)dv,

Xi > Xj,

which can only be satisfied exactly for r = 0. It can be satisfied approximately for low order moments for sufficiently fine partitions. Internal consistency does not of course guarantee a number density sufficiently accurate for the

150

4. The Solution of Population Balance Equations

calculation of all associated properties of the population unless the partition of the size interval is suitably fine. Thus, fineness of the partition is indeed an underwriting requirement for accurate calculations of the number density even when condition (i) for aggregation is satisfied. For purely aggregating systems, condition (i) alone is required for internal consistency. This condition is satisfied, for example, by choosing the partition P^^ as follows: Xj = Vi = ih where /i is a constant. Clearly then Xfc + Xj = {k -\-j)h = Xj^+j = Xi the last of which applies to the inner summation of the aggregation source term in Eqs. (4.5.10), (4.5.12) and (4.5.13). The foregoing formulation of the aggregation problem, implying the choice of a linear grid for the partition, is due to Hidy and Brock (1970).^^ Accurate calculations require substantial computation times, however. For a sufficiently fine hnear grid as chosen by Hidy and Brock (1970), we may write the discrete equation for the population density from Eq. (4.5.10) for a process including aggregation and breakage as dN-

1 '"^

^

^

ai

^ j=o

j=o

j=i

i = 0,l,...,M,

(4.5.16)

where we have used the following abbreviations: a^j = a{Xi, Xj), bj = b{xjl

Vj = v{xj),

P{Xi\xj) = P^^j. (4.5.17)

Since the moments of the population for the entire size interval are often of interest in applications, we now present them now:

Ul

^ i=o

j=0

'A\j-^ (4.5.18)

This equation above shows again that for r = 1, the moment is timeinvariant reflecting mass conservation while, for other integral values of r, the preceding equation relates the dynamics of the moment to its discrepancy following aggregation and breakup events throughout the range of particle states. In arriving at Eq. (4.5.12), we were guided entirely by (4.5.7) (representing category 1) in describing terms of the type (4.5.6) in the balance equation 20

Referenced in C h a p t e r 3.

4.5. Discrete Formulations for Solution

151

(4.5.5) for the population density. We could also have been guided by (4.5.9), as in category 2, to arrive at an alternative differential equation for the sectional moment /x^'\ For a process involving aggregation alone, one obtains

^=t ^^

t ^u>^^^'^^-^^ E hi^'

7 = 0fe= 0

7= 0

(4.5.19)

where the coefficients a^j^ and P^^ are constants which are obtained by integrals of the type in (4.5.9). The regions of integration are dependent on the size interval and require careful geometric deliberations for their identification. We shall not labor to present the details here for two reasons: For one, the issue has been discussed in detail by Kumar and Ramkrishna (1996a) and for another, the calculation of integrals of the type (4.5.9) renders the method computationally inefficient at least where numerical integration becomes necessary.^^

4.5.1

The Geometric Grid

As pointed out earlier, the spirit of the discretized methods of this section lies in the coarseness of discretization for which the method imphed in the use of Eq. (4.5.14) will be unsuitable. In this connection, Bleck (1970) has suggested a geometric grid in which particle size increases in the partition in multiples of 2. Although no rationale, heuristic or otherwise, is discussed in the literature for the choice of a geometric grid, some arguments suggest themselves. In aggregation processes (described by particle mass), for example, aggregation between particles of a specific size would double the particle size. Thus it would be meaningless to dissect the size any more finely than between the original size and twice its value. Similar arguments can also be made for breakage events based on mass conservation. Hence, mass conservation presumably plays a role in the effectiveness of a geometric grid. In any case, overly coarse geometric grids can be further refined by letting i;j + i = l^v^ where s must consequently be in the interval (0,1]. 21

See Gelbard et al (1980) for details of the development as well as computational demonstrations. For an extension of the technique of Gelbard et al. (1980), the reader is referred to Landgrebe and Pratsinis (1990). A relatively recent review of the methods of solving population balance equations for aggregation processes is contained in Kostoglou and Karabelas (1994). Kumar and Ramkrishna (1996a) also provide a more recent and critical review of the literature on discretization methods.

152

4. The Solution of Population Balance Equations

As observed earlier, the choice of a coarse geometric grid makes it impossible for the discrete equations to be internally consistent for all moments. However, internal consistency may be sought for selected moments of the population. The early efforts were somewhat ad hoc in this regard in that they did not consciously address the problem just enunciated. Thus Batterham et al (1981), using a coarse geometric grid, redistributed particles formed by aggregation in the region between pivots among the pivots so that mass was conserved. The procedure led to a discrete set of equations that conserved mass but erred with respect to particle numbers (although this was partly due to double counting of particles of the same size in aggregation). Similarly, Bleck (1970), to whose work reference was made earlier, conserved particle mass but not numbers. To Hounslow and his co-workers (1988) must go the credit, however, for attending to particle numbers as well as particle mass which was accomphshed through the instrument of a correction factor. The factor did not depend on the aggregation frequency, however. In the next section, we provide a general perspective from the work of Kumar and Ramkrishna (1996a, b) that not only affords a framework for conveniently interpreting the work reported in the literature on discrete methods but also provides a natural route for improvements.

4.5.2

Moment-Specific Internal Consistency for Coarse Grids

The issue that confronts coarse discretization is one of finding effective ways of restoring autonomy to Eqs. (4.5.10) so that accurate calculation of selected properties (e.g., moments) of the population are made possible. Let us recall that the basic idea of discretization is that particles in a size range (say li) are assigned a pivotal size x,-. Processes such as aggregation and breakage, however, produce particles that are between such pivotal sizes (except in the case of a uniform linear grid) and must be reassigned to the pivots. It is then this reassignment that must be done with due care to preserve accurate calculation of selected moments. For example, particles formed between pivots Xj- and x^^^ could be assigned partly to x^ and partly to Xj + i and we are concerned with the rationale for this redistribution. The rationale adopted by Kumar and Ramkrishna (1996a) is that of internal consistency of the moment equations (4.5.13) with (4.5.15) for specific choices of r. Since no such rationale was used in arriving at (4.5.13), they were consequently inconsistent with (4.5.15). We therefore seek to incorpor-

4.5. Discrete Formulations for Solution

153

ate the rationality of internal consistency into the process of arriving at the discrete set (4.5.10) from the continuous version (4.5.1). Since the issue has to do with the birth terms for aggregation and breakage, we focus our attention on these. Denoting the total source term in (4.5.1) as H^l{f^}; v, Q, we may identify it as a(v — v\ v')f^{v — v\ t)f^{v\ t)dv'

H+[{/i};t;,f]=^

0

+

v{v')b{v')P{v\v')f^{v\

t)dv'.

It will be of interest to see how particles of pivotal size x^ arise from the foregoing terms. For x^^v < x^ + i, let the fraction of a particle of size v assigned to size x^ be denoted by yf{v\ where the superscript signifies the interval of origin of the particle and the subscript signifies the assignment, and a fraction 7-+i(i^) be assigned to size x^^^. Then the particles of size x^ must arise from new particles in [Xj.^, x^) as well as in [x^, x^ + i). Thus, the source term for particles of size x^ becomes

H*l{h};v,t-]yr»dv

+

H^l{h};v,t-]yT{v)dv.

(4.5.20)

We shall state the rationale for reassignment of the particle of size v as that which does not alter the rth moment of the population for some selected r. Because there are two variables yf{v) and yf^i{v) we may select two moments corresponding to r = r^ and r = r2. Since the contribution of the particle of size v to the rth moment before reassignment is v"", the reassignment will preserve the total rth moment provided •iPiv) x\ + 7fi M xU , = v-,

r = r„ r,.

(4.5.21)

These two equations above yield a unique solution for the quantity y^l\v) given by

yiV)=

,/W2

_

-r,

,,r2^ri

' : \ -/ilM-r.'

_

' ,

x,^v<x,,„ (4.5.22)

which provides, as it should, an immediate corroboration of the reassignment occurring entirely to x^ when v = Xi and none to x^ when i; = x^ +1. We shall now show that the property of internal consistency is displayed by the differential equations for the moments of order r^ and r2. To obtain the

4. The Solution of Population Balance Equations

154

substitute for Eq. (4.5.10), we first make use of (4.5.8) in (4.5.20) for the new source terms to become 'Xi

i-1

M

k=0

j=0

^

dvyf{v)a{x,,xj)

+ •

N,d{v - v'- x,) ^

+

yf~'\v)dv

+

yr{v)dv

I

Nj5W-Xj)

J= 0

k= 0

X

v{v')b{v')P{vW)

NjS{v'-Xj)dv'

7= 0 M

v{v')b{v')P{vW)

X

NjS{v'~Xj)dv'.

j=o

Further processing of the Dirac deltas in the foregoing expression leads to j^k Xi-i^{Xj

1 V

+ Xh)<Xi

^

z

Xi ^ (Xj + Xk) < Xi M

+ E

Njv{xj)b{xj)

yr

'(v)P{v\xj)dv+

„('»/ f''{v)P(v\xj)dv

(4.5.23) where in the aggregation terms, the sums (of the symmetric summand) over bothj and k running from 0 to M have been altered to cover only the range j ^ k in favor of eliminating the coefficient 1/2 except for j = k. Thus, the number balance of particles in /,-, after absorbing the source terms in (4.5.23) and retaining the same sink terms as before in (4.5.10) becomes dNj

IT

Z' Xi-^^{Xj

+ Xu)<Xi

f» -\^i^ \

Wr'\x,

+ x,)a{x„ x,)N,iVJ

^

j^k Xi^iXj

+ Xk)<Xi + M

M

- JV, X a{x„ Xj)Nj + X Njvixj)b{xj)no,j j=0

j=i

- Hx,)N„

(4.5.24)

4.5. Discrete Formulations for Solution

1 55

^ and yi are given by (4.5.22) and

where y^

vyr

{^)Piv\^j)dv +

vyy{v)P{v\xj)dv,

in which the subscript r is in anticipation of a similar derivation for the rth sectional moment to get Z

-^= dt

Xi-i^{Xj

1 -^Sj,

]l{xj + xjyf-'^ixj

+ x,)a{x,,

Xj)NjN,-]

+ Xk)<Xi

+ Xi^(Xj

+ Xk)<Xi+i

\

^

M

/ M

- iif X a(x„ x,)iV, + X Arjv(x,)fc(x,.)^Mj - M^a^r'*

(4.5.25)

Summing the above equation over i, one obtains for the rth moment the equation du

^

J^^

(

1

\ M

j

Z A^,-v(x,-)fc(^,-) Z ^ M J

+ yf\xj + x,)}a{x,,Xj)NjN,-]+

7= 0

/^.

(4.5.26)

Z a{Xi,Xj)Nj + b{Xi)

Lj = o

If we now set r^ = 0, then (4.5.21) implies that the expression in the curly brackets is unity and that v'P{vW)dv, i= 0

which converts (4.5.26) into du

^

J^^

dt

i= 0

(

1

Xl^(xj-\-x^c)<Xi^ M

+ Z iV,.v(x^.)ft(^,)'A.(^,)-/^J j=o

r M

Z a{Xi,Xj)Nj + b{Xi)

\_j=o

(4.5.27)

1 56

4. The Solution of Population Balance Equations

For internal consistency with respect to another moment of order r (say = r2) we need to show that (4.5.27) can also be obtained by multiplying (4.5.24) by x^ and summing over i. The procedure leads to M

j^k

/

i == 00 Xi ^ (Xj ++ Xk) < Xi H Xi^iXj Xk)<Xi +

1

\

^

/

:^ \

M

a{xj,,Xj)NjNj^']+

j

Z A^,v(x^.)^(^,-) Z x'Tio^ij j=0

i=0

' M -i".

Z a{Xi,Xj)Nj + b{Xi)

(4.5.28)

It is readily seen that (4.5.28) is identical to (4.5.24) in lieu of (4.5.21), provided r = r2. Thus the strategy of Kumar and Ramkrishna (1996a) provides for internal consistency of the discretized algorithm (4.5.24) with respect to two moments, one of order r = 0 and the other of order r = r2. If we set r2 = 1, the resulting algorithm would provide for conservation of numbers as well as mass, but it should be evident that this choice of the two moments is arbitrary.^^ As Kumar and Ramkrishna (1996a) have also pointed out, the development of the foregoing discrete algorithm need not be constrained by the choice of specific moments. Instead, the algorithm may accommodate any two extensive properties of the population derived by summing those of the individual particles. Thus, consider two properties G^it) and 62(1) defined by GM =

gMMv,

t)dv=Y.

N,it)g,ix^

r = 1,2,

(4.5.29)

where g^^iv) and g2{v) are the respective size-dependent particle-specific properties from which G^{t) and 62(0 are derived. The algorithm (4.5.24) can then be used by modifying the functions y^i\v) and y^'liiv) as below: (i).^ _

y^iv) = .0 . . _

giiv)g2iXi+-L) gi{Xi)g2{Xi+i)

-

g2iv)giixi+i) g2{Xi)gi{Xi+iy

g2iv)g.iXi)-gMg2iXi) gi{Xi)g2iXi+i)

, <, < ,

^^'^'^^^

- ^2(^i)^i(^/+i)

22 This idea was presented by Kumar and Ramkrishna (1995) in November 1995 at the Annual A.I.Ch.E. Meeting in Miami Beach.

4.5. Discrete Formulations for Solution

1 57

For the specific choice of r^ = 0 and r2 = 1, the algorithm (4.5.24) can be shown to yield the equations of Hounslow et al. (1988) whose ad hoc arguments call for a fresh derivation when, for example, the grid dimensions are altered or a different choice of moments is made for internal consistency. The algorithm here naturally accommodates selective refinement of the grid over particle size ranges where higher accuracy of the number density may be required. In fact this feature endows it with the capacity to deal with formulations of population balance in which particle size is discrete over the small ranges and continuous over the large ones. It also recovers the algorithm of Hidy and Brock (1970) on employing a uniform grid with internal consistency for moments of all order; the algorithm (4.5.24) possesses a feature of universal flexibihty that is clearly distinctive. The incorporation of internal consistency with respect to two chosen moments or properties represents a significant advance in the use of discrete methods over the earlier efforts of Bleck (1970), Gelbard et al. (1980), and Landgrebe and Pratsinis (1990). It is also transparent that one could demand internal consistency with respect to more than two moments by allowing for redistribution of particles formed in I^ among pivots in addition to x^ and x^^ ^ on either side of them. This natural extension is rather readily made without further discussion of the framework just presented. For extensive numerical demonstrations of the effectiveness of (4.5.24) using several examples encompassing pure breakage or pure aggregation as well as those that feature both aggregation and breakage, the reader is referred to Kumar and Ramkrishna (1996a). We shall therefore be brief in regard to the presentation of numerical results here. Comparisons of numerical solutions (with analytical ones when available and) with progressive refinement of the grid have both borne testimony to the veracity of the algorithm. Of the examples usually considered in the population balance literature, solutions of pure breakup problems have yielded better solutions than those of pure aggregation systems. For pure aggregation with frequencies generally increasing with particle size, the discrete solutions compare well with the analytical solution for small particle sizes away from the large size region in which the number density drops precipitously. In the latter region, the number density is overpredicted the extent of over prediction increasing with the strength of particle size dependence of the aggregation frequency. This trend is of course a natural consequence of discretization, which ignores the increase in the aggregation rate with particle size in the interval.

158

4. The Solution of Population Balance Equations

We address a problem in polymerization that was considered by Blatz and Tobolsky (1945). Since depolymerization occurred simultaneously with polymerization, the population balance equation featured both aggregation and breakage terms. The population density in polymer size evolved in time with a rate constant that was assumed to be independent of size for the polymerization step, while depolymerization varied linearly with the polymer volume (in excess of that of the monomer) uniformly into smaller polymers.^^ Algorithm (4.5.24) could be formulated for a linear grid for the smaller polymers (up to 15-mers) beyond which a geometric grid covered the larger polymer range. The results of Kumar and Ramkrishna (1996a) are reproduced in Fig. 4.5.1 alongside the analytical solution of Blatz and Tobolsky (1945). These computations show the especially high quality of the solution for smaller polymers while deviations are observed for large polymers where the number density drops steeply. It can be obviously improved by further refinement of the grid at a higher computational cost. Thus, selective refinement of the grid is desirable by focusing on size ranges in which steep variations of the number density are detected. These and other issues have been discussed in significant depth by Kumar and Ramkrishna (1996a). We shall turn our attention to another approach to discretization suggested by Kumar and Ramkrishna (1996b) to account for steeply varying number densities. This is based on the idea that the pivot in each size interval may be chosen to secure a better representation of the variation displayed by the number density in the interval. Since the number density in a given size interval can change with time in a dynamic situation, the pivot too must undergo dynamic variation to provide for a better representation in the interval. Thus, the pivot x^ in the interval /, must be correspondingly construed to vary with time.^"^ We consider the details in the next section.

23

For this example, the aggregation and breakage functions are as follows: a{v, v') = kj.; b{v) = k,{v - 1); v(i;) = 2 ; P{v\v') =

24

S(v — iv^,) , i = 1,2,.... v' — \

v^ is the volume of the monomer unit. Clearly, these functions forbid a monomer from splitting, while the Dirac delta in P accounts for the essentially discrete nature of polymer length, especially for small polymers. The mean value theorem as applied to terms such as (4.5.6) also implied that the pivots must move with time, which was ignored by the fixed pivot approach.

159

4 . 5 . Discrete Formulations f o r Solution 1

10000

100

V

E

0.01

1

1

1

1

\ \^^y>%,

I

1 c o

1

SIMULTANEOUS BREAKUP AND AGGREGATION i.e.— t=1e-3, Anal. soln. -— i ;k,.. numerical soln. o t=1e-2, Anal. soln. " 11 '^ "inumerical soln. + t=0.1,Anal.soln. - - numerical soln. • ' ' ^ t=:1,Anal.soln. ' ;i- x-'tx-VieewSk^ numerical soln. x s.s. anal. soln. — i x^ numerical soln. A ;

^

a

\

\

^

^ ^ '^ ^^ V \

-

E C

0.0001

^ ^ 1

1e-06

\

io.in

^

>

V

i

\

10

\

1

\

V

^ 1

\

'.X

\

1 + \

le-08

\x

\,

1

"i ^ •^ •

^B

•

•,a

J 100

n-mers

\f^ Steadystate H \ ^^^distribution U

*^ ^ \

LJ

1e3

\

1

U i

1 1e4

FIGURE 4.5.1 Comparison of the number density calculations of Kumar and Ramkrishna (1996a) using the discredization algorithm (4.5.24) with the analytical solution of Blatz and Tobolsky (1945) for a polymerization-depolymerization process (simulated as simultaneous breakup and aggregation). (Reproduced with permission from Elsevier Science.)

4.5.3

Moving Pivots

We return to the problem (4.5.1), entertain both breakage and aggregation processes so that (4.5.2) is valid, and seek the discretization (4.5.9). Further, we seek internal consistency with respect to two arbitrarily chosen properties G^(t) and G2(0 as appearing in (4.5.29). Such internal consistency will, however, be accomplished by allowing the pivot to move in a consistent manner. Thus, no redistribution need be made of particles arising from

160

4. The Solution of Population Balance Equations

breakage or aggregation among contiguous pivots as suggested in (4.5.20). Therefore, instead of the discretization algorithm (4.5.24) used for the fixed pivots, we restart from Eq. (4.5.1). Furthermore, we directly proceed to derive an algorithm for preserving the two chosen properties G^{t) and G2(0- Accordingly from (4.5.1), we must derive equations for the sectional variables "'" gMfM

Gl"{t) =

t)dv = gAXi)N^,

r = 1,2,

(4.5.31)

which must satisfy equations dGl'* iG*'*

^^ >^^

(1.

11

\

M

- G « X a(x,,Xj)Nj+

M

X

Njv(xj)b{xj)n,,j-b{xdG';\ r=l,2,

(4.5.32)

where ^r,ij

^

g^{v)P{v\Xj)dv,

r = 1,2.

The numerical scheme now consists of Eqs. (4.5.32) considered for, say, r = 1, which may be written purely in terms of either {Gf{t)} or (ATj using (4.5.31). The second set of equations is obtained from (4.5.32) by setting r = 2 and ensuring internal consistency in the property 62(1) by converting them as equations in the pivots {xi{t)}. More precisely, we define a function h{v) such that

so that at

at

at

The differential equations for the pivots are now obtained by substituting in the left-hand side above from (4.5.32) written for r = 2, and substituting in the right-hand side from (4.5.32) for r = 1. Of course, a choice of the initial location of the pivots must be made based on the initial population density as initial condition for the differential equation in x^(t).

4.5. Discrete Formulations for Solution

161

Following Kumar and Ramkrishna (1996b), the specific strategy is to solve for the number density using ^^(i;) = 1 and g2{v) — v, which ensures conservation of numbers as well as mass. /

j^k

1

\

^

dt M

+ I Njv{xj)b{xj)n,,j - b{x,)N„

(4.5.33)

j= i

where Tl^jj is obtained from (4.5.32) by setting g^(v) = 1. The differential equations for the pivots become dx.

^^^ {Xj -r XicfGli

( \

1

\ /

M

+ Z Njv{xj)b{xj){n,,j - x,n,,J

(4.5.34)

where 112,/^ is obtained from (4.5.32) by setting g^{v) = v. Kumar and Ramkrishna (1996b) consider several numerical examples from breakage and aggregation processes demonstrating the value of the foregoing algorithm in significantly reducing computational errors in particle size regions where the number density changes steeply. We shall present here only one example from their work toward making such a demonstration. This example pertains to an aggregation process with the sum kernel as the aggregation frequency. The problem has an analytical solution (Scott, 1968; see also Section 4.3.2) which is available for comparison of the numerical solution. A geometric grid is employed (for both the fixed and moving pivot algorithms), whose fineness could be adjusted by controlling the parameter p in the partition generated by The discrete algorithm (4.5.24) for fixed pivots, and the moving pivot algorithm (4.5.32) with internal consistency for the zeroth moment (conserving numbers, i.e., g^{v) = 1) and the first moment (conserving mass, i.e., g^iv) =vso that h{v) = v) are both compared with the analytical solution in Figure 4.5.2. A geometric grid is employed (for both the fixed and moving pivot algorithms), whose fineness could be adjusted by controlling the parameter p in the partition generated by t;^+i = pf^. Figure 4.5.2 shows how the moving pivot for a fixed geometric grid improves the overprediction over the fixed pivot technique and that progressive refinement of the grid improves both techniques with the moving pivot retaining the edge over the fixed pivot algorithm under all circumstances.

4. The Solution of Population Balance Equations

162

1e-05

u t OS

a 1e-10

1e-15

10 100 particie volume (v)

10000

FIGURE 4.5.2 Comparison of the fixed and moving pivot techniques of Kumar and Ramkrishna (1996b) on the solution of the aggregation problem for the sum kernel with its analytic solution. Reproduced with permission from Elsevier Science. Before we conclude our discussion on discretization, we consider the solution of population balance equations with growth terms, which were excluded from prior discussions. 4.5.4

Discretization in the Presence of Nucleation and Growth

As seen in Chapter 2, several applications in population balances involve the formation of new particles by nucleation and particle growth. Accordingly, the population balance equation of interest may be written as

^ ^ ^ + l^imAiv,

t)-] = //[{/J; V, q + s„{vi(4.5.35)

4.5. Discrete Formulations for Solution

163

where V{vX the growth rate of particles of volume v, is assumed for the present to be independent of continuous phase variables, and S„{v) is the nucleation rate contributing particles of size v. The functional if [ { / J ; v, t], given by (4.5.2), represents as before the sources and sink terms for aggregation and breakage processes. Integrating Eq. (4.3.35) over /• one obtains .

dN-

.

f'"'^^

^dt - vivdMvt, t) - v{v,^,)Mv,^,, t) + I +

Hi{f,Y

SMdv (4.5.36)

V, tl

The discretization of the last term on the right-hand side has already been elucidated in Sections 4.5.2 and 4.5.3. The nucleation term, the penultimate term on the right-hand side, is readily determined by recognizing that nucleation produces the smallest particles, all of which may be included in the size bin with i = 0.^^ Thus we may write SMdv

= Kd,^,

(4.5.37)

where h^ is the total nucleation rate including particles of all sizes by nucleation. We are thus left with the growth terms represented by the first and second terms on the right-hand side of (4.5.36). Since they involve the number densities evaluated at the end points of the interval /^, the discretization procedure would seek to relate n{vi,t)io N^. There have been several attempts in this regard, which are listed below: n{v,, t)=\

f^ ^ 2\^v^_^ All

n{v:, t) = ^^^^

+ ^ \ AvJ Av

(Marchal et al, 1990)

At)

--^—i^, 1 + ^'^

(1 H-^)V^. + i -^i-ij

(David et al, 1991)

(4.5.39)

Ai;^ (Hounslow et. aU 1988)

25

(4.5.38)

(4.5.40)

If this assumption is not valid as, for instance, the spectrum of sizes produced by nucleation may extend beyond the smallest bin, then the nucleation source may be included for larger discrete sizes also.

164

4. The Solution of Population Balance Equations

Clearly, each of the foregoing expressions represents an effort to interpolate for the number density n{Vi, t) between the average number densities in the intervals /j_i and I^ on either side of i;,. If ^ = 1 so that the intervals Av^_;^ and Ai?. are equal, then all of the expressions presented are the same. However, none of the discretized algorithms listed is insured against the possibility of negative solutions for the discretized number density, a contingency which Hounslow et al. (1988) sought to correct by setting the negative number density to zero. Kumar and Ramkrishna (1997) provide a critique of the foregoing discretization algorithms by evaluating them on the same examples as done by Hounslow et al. (1988). The problems inherent in these algorithms manifest in the simplest example of a nucleation process beginning with the particle size distribution starting from, say, an initial exponential distribution and evolving by a size-independent growth process. Since nucleation is assumed to produce particles of a single fixed size, the preceding scenario necessarily provides for a discontinuity in the solution beginning at the size of the nucleus and moving to the right with time because of particle growth. Such a discontinuity in solution, which is an attribute of the hyperbolicity of the governing population balance equation, is not accommodated by the "diffusive" behavior promoted by finite difference approximations characterizing the foregoing algorithms. Consequently, the solutions display oscillations with unreasonable excursions into negative number densities. The oscillations could be squelched by setting the negative number densities to zero. Furthermore, the algorithms also err on predicting accurately the location of the discontinuity. The reader is referred to Kumar and Ramkrishna (1997) for a more thorough discussion of these issues. The handling of particle growth terms must therefore depend on an alternative approach such as that due to Kumar and Ramkrishna (1997) on which we now dilate. We begin with observing that the effect of particle growth is to "deform" particle sizes in accord with the growth rate. The partition P ^ becomes a dynamic entity. Thus, the interval /^ at any instant may be regarded as a "material" interval lVi{t), i^,+ i(0)j which will vary in accord with the solution of the differential equation ^ = V{vl

v{0) = v,„,

(4.5.41)

where i;^^ is the ith site of the partition at some time t = 0. Denoting the solution of this differential equation by V{t; i;^^) which reflects its dependence on the initial condition i;^-^, we recognize that v^it) = V{t; v^J as the

4.5. Discrete Formulations for Solution

165

temporally varying image of i;^-^ caused by particle growth. Every element of the partition Pj^ varies in a similar manner, distorting the entire partition. The macroscopic balance over the material interval /^(t) = [i^i(fX ^i + M), obtained through the use of the transport theorem (2.6.1), is given by dNi _ d dt dt

•i^i + i W

Hl{f^};v,t']dv,

/i(i;, t)dv = Viit)

(4.5.42)

i>0.

Vi(t)

This equation shows no nucleation source, since it arises only for the smallest bin of size t;^, which corresponds to / = 0. As each discrete size v^ in the partition (including v^) moves as a result of particle growth, there is renewed addition of the smallest bin through the process of nucleation. Thus, the number (M + 1) of sizes in the partition also increases with time in such an algorithm. The right-hand side of (4.5.42) now may be replaced, for example, by the right-hand side of (4.5.24), which was derived by Kumar and Ramkrishna (1996a) for a fixed pivot. The expression on the right-hand side of (4.5.4) is applicable here since at any time the arguments behind its derivation are valid for the instantaneous partition. Consequently, we have the algorithm dNi_

^^

it) + Xkit)]<Xiit) d^ Xi-,{t)^[xj{t)+xk{m<xi(t)

/.

1

\

^

J

x,{t))N,N,-] +

'Z U-\ xmMxj(t)+xi,(t)\<Xi^^{t) \

^j^ bfixjit) ^ /

+ xM

a{xM x^{t))N,N,-\

M(t)

- N , Z ^i^M j=o

Mit)

^jit))Nj+

E

j=i

Njv{xj{t))b{xj{t))no,j-b{x,{t))Nt (4.5.43)

where the time-dependent pivots pervade the right-hand side. The ith pivot Xi{t) may be obtained by solving the differential equation ^=F(x,).

26

(4.5.44)2^

It is of interest here to show that Xi{t) satisfying differential equation (4.5.44) retains at all times its status as a pivot in the interval I^ i.e., v^ ^ x^ < f j + ^ with Vi and v^+j^ also satisfying the same differential equation. Generally the function V{v) is a monotone non-decreasing function of v so that the foregoing requirement is satisfied.

166

4. The Solution of Population Balance Equations

The general algorithm of Kumar and Ramkrishna (1997) now consists of differential equations (4.5.41) written for both v^ and v^ + y^, (4.5.43) and (4.5.44). The continuous addition of new particles to the population by nucleation is handled by adding new bins to the partition at regular discrete intervals. The authors in the reference cited have discussed these and several other details of implementation of the algorithm. The purpose of this discussion has been to point out that the technique here exploits the method of characteristics which, unlike the finite difference approximations, retains the hyperbolicity of the equations. Obviously, it depends upon the ease with which the characteristics defined by (4.5.41) are calculated. In several applications of population balance that have been covered in the hterature, the characteristics are even analytically obtainable. Figure 4.5.3 presents calculations with the foregoing algorithm by Kumar and Ramkrishna (1997) for the case of simultaneous growth (at constant rate) and aggregation (with constant aggregation frequency) for which analytical results are available for comparison. Several other cases demonstrating the efficacy of the algorithm that have been considered by the authors are not included herein. To summarize, the main advantage of the discretization methods of Section 4.5 is the coarseness, which they tolerate allowing the rapid computation of key quantities of interest that is essential for control applications. However, properties of the particle population for which internal consistency is not ensured may not be predicted with accuracy by such methods. Because the literature on approximate methods for the solution of population balance equations has grown substantially over the years, an exhaustive coverage of the same has been difficult. It must also be said, however, that not infrequently papers display different techniques to solve the same set of problems rather than generalizations expanding the scope of problems solved. In the next section, we are concerned with Monte Carlo simulation techniques whose scope is limited only by the available computing power. The approximate solutions considered in the prior sections of this chapter were designed to solve problems of limited dimension; in other words, the particle state was characterized often by a single variable whereas in applications there may be several. Consequently, it behooves one to be familiar with the simulation techniques in the next section.

4.6. Monte Carlo Simulation Methods

167

Simultaneous Growth and Aggregation 1—'—'-n—'—"-n

•'• ' n

p

r-TT

1

r-TT

1

-^^^T—

100 / ^ ^ .

^—' n—'—'-r I.e. - ^ analytical numerical analytical numerical

time = 0.01

f V\

\

S 0.01 L o

\

-— -+-— i -& -

J

V

(D D

E c 0.0001

^

H

W

\'\

1 1+ le-06 h

J

!\ • '\

•;\

1 I

1

le-08

1 '•

1 <

lis

''

\\

time = 0.1

H

'• \

1e-10 . . 1e-05

i

.i._..i

0.0001

i

ii_.i

0.001

t.-ii

0.01

j

1

1

0.1 particle volume

It 1

\k

\ i 1 li 1 10

100

1000

FIGURE 4.5.3 Comparison of the calculations of Kumar and Ramkrishna (1997) using the discredization algorithm (4.5.43) and (4.5.44) for a growth and aggregation process with its analytical solution for the case of constant growth rate and constant aggregation frequency. Reproduced with permission from Elsevier Science.

4.6

MONTE CARLO SIMULATION METHODS

Broadly, Monte Carlo methods are based on artificial realization of the system behavior. Our primary concern will be the simulation of systems in which the particle state changes deterministically. Thus, the systems encountered in Section 2.10 with random changes in particle state will be excluded for the present from our discussion in favor of a sketchy treatment of this case at a later stage.

168

4. The Solution of Population Balance Equations

We envisage a population of particles in a prescribed spatial domain with initial states that are known in a statistical sense; i.e., the initial distribution of particle states is known. The processes that cause the disappearance of existing particles and appearance of new particles in the population domain are generally random with specified probabihties. We therefore broadly expect that a "sample path" of the process can be created by artificially generating random variables that satisfy the specified probability laws of change. By generating numerous such sample paths, the expected or mean behavior of the system can be calculated by averaging all of the sample paths. Although this is a very broad sketch of what Monte Carlo simulations are about, it is of interest to develop a somewhat better perspective of how the solution of population balance equations and Monte Carlo simulations accomplish the same calculations starting from the model hypotheses of the system behavior. Since the appropriate theoretical framework is yet to be established for accomplishing the foregoing objective in detail, we shall be content for the present with the schematic representation shown in Figure 4.6.1 of the two routes of predicting system behavior from

Hypotheses about Single Particle Behavior

Master Density Equation

Direct Averaging

Solution

Computationally Complex Solution

Simulation Averaging and Elimination of Low Probability Events

FIGURE 4.6.1 The connection between computations of average population behavior via the population balance equation and Monte Carlo simulations.

4.6. Monte Carlo Simulation Methods

169

the model hypotheses.^^ Although the methodology for arriving at the population balance equation from single particle models was the subject of Chapter 2, the formulation of the Master Density equation from stipulations about single particle behavior has not been encountered yet, as it will form the subject matter of Chapter 7. What is of interest to this section is the procedure by which average information about population behavior is obtained by simulation, an artificial realization process that automatically acts as a filter for the sample paths of relatively low probability. The crux of the simulation process is then one of translating the given deterministic and probabiHstic information about how single particles change with time into a systematic procedure for realizing the dynamic behavior of the system.

4.6.1

Simulation Techniques Based on Discrete Deterministic Steps

We first briefly look into some early simulations of particulate systems that were based on discretizing time (or any other evolutionary coordinate) along which the system undergoes changes. These include continuous, deterministic changes in particle states, specified by ordinary differential equations, and discrete, random changes because of processes such as breakage or aggregation of particles resulting in the birth of new ones and in the death of existing ones.^^ We shall assume that, at any intermediate instant t of reckoning, the population is completely specified in number as well as the states of each individual particle. In the absence of random changes during the period following t, the deterministic changes in each particle state with time can be readily tracked by solving the differential equations for each particle. It is convenient to denote any period during which no random changes occur in the system as a period of "quiescence".^^ During this period of quiescence, no existing particle in the population is destroyed or new particle created, and the simulation of the system proceeds

27 A detailed treatment of this issue is deferred to a subsequent stage. It can also be found in Ramkrishna (1981). 28 The discontinuous changes in the number of particles due to birth and death of particles are strictly a feature of individual sample paths but not of the average behavior of the system, for otherwise the average number density would not be differentiable with respect to time. 29 The term "quiescence" is borrowed from Kendall (1950).

1 70

4. The Solution of Population Balance Equations

like that of any deterministic system. Clearly, then, the central issue facing the simulation of particulate systems is the random interruption of the period of quiescence by events causing discontinuous changes as a result of particle birth or death. From the preceding discussion, it is evident that the interval of quiescence is a random variable (presumably depending on the state of the system at the instant of reckoning). Returning to the simulations of particulate systems that were based on discretization of the evolutionary coordinate, they may be viewed as distributing the random events at suitably chosen discrete intervals. In other words, the random interval of quiescence is assigned fixed (although not necessarily its statistically exact) average values. At the discrete times chosen, the random events are identified by generating the values of random variables with probabilities for different outcomes (that depend on the length of the interval of quiescence as well as the state of the system at the beginning of the interval). Following the identification of the random event, the system is appropriately updated for further continuation of the simulation beyond the current quiescence interval. The procedure is now defined for continuation upto a specified time or point along the evolutionary coordinate. Spielman and Levenspiel (1965)^^ appear to have been the earliest to propose a Monte Carlo technique, which comes under the purview of this section, for the simulation of a population balance model. They simulated the model due to Curl on the effect of drop mixing on chemical reaction conversion in a liquid-liquid dispersion that is discussed in Section 3.3.6. The drops, all of identical size and distributed with respect to reactant concentration, coalesce in pairs and instantly redisperse into the original pairs (after mixing of their contents) within the domain of a perfectly stirred continuous reactor. Feed droplets enter the reactor at a constant rate and concentration density, while the resident drops wash out at the same constant rate. Reaction occurs in individual droplets in accord with ^zthorder kinetics. In the foregoing example, the deterministic event is the chemical reaction in the drops while the random events are those of drop entry into and exit from the reactor, and coalescence-redispersion within the reactor. The interval of quiescence, therefore, represents the period in which none of the following processes occur: (i) addition of drops with the feed, (ii) loss of 30 A somewhat similar approach was used in homogeneous phase mixing by Rao and Dunn (1970) referenced in Chapter 3.

4.6. Monte Carlo Simulation Methods

1 71

drops with the exiting stream, (iii) coalescence between any pair of droplets in the reactor. Thus, only chemical reaction takes place in the different droplets during this time interval. In this section, we are concerned with simulation algorithms based on the approach of a fixed discrete time step for the quiescence interval. The concentrations in each drop can be updated using their initial values and reaction rates without interruption by drop entry, exit, or coalescence-redispersion. At the end of the interval, however, by generating a suitably calculated random number (to be presented subsequently), the process which disturbed the quiescence may be identified. The state of the population is now readily updated for further continuation of the simulation. There are several further simplifications that can be made to produce a speedier algorithm. For example, Spielman and Levenspiel (1965) assumed that drop entry and exit could be perfectly synchronized at regular discrete intervals during which a fixed number of coalescence-redispersion events occur. Thus, the only random numbers to be generated are those associated with identification of the drop pairs involved in coalescence and redispersion. Generally, the mean behavior is obtained by averaging several sample paths. However, in steady-state simulations such as the one under discussion, the mean behavior can be more conveniently calculated by averaging of a single simulation at different times. We have avoided mathematical details in our description of the foregoing simulation approach in view of the more complete discussion to follow on what we shall refer to as statistically exact simulation approaches. Before entering that discussion, however, it is worth briefly reviewing some of the other detailed simulations that have appeared in the literature. Collins and Knudsen (1970) simulated drop breakup and coalescence in turbulent pipe flow of a liquid-liquid dispersion emerging from a feed nozzle with a specified distribution. The evolutionary coordinate in this work is the axial distance along the flow so that the simulation was carried out along discrete lengths along the flow direction chosen somewhat arbitrarily. One of the most detailed discrete simulations in the Hterature is that of Zeitlin and Tavlarides (1972a,b,c), which addressed the evolution of drop size distributions in mechanically agitated hquid-liquid distributions produced in batch, semibatch, and continuous flow vessels. In these simulations, the flow vessel was divided into distinct spatial regions among which individual droplets commuted with specified velocities. Thus, both internal and external coordinates characterized the individual particle state. Drop

1 72

4. The Solution of Population Balance Equations

breakup and coalescence rates were allowed to vary in the different regions of the flow. An initial distribution of the droplets in the reactor was used to start such a simulation, which proceeded by discretizing the time interval thereafter. The simulations that were discussed in this section depended on several simplifications of the random events in the actual process. Such simplifications may indeed be necessary to enable efficient computation of the system behavior. They were arrived at, however, by not entirely rational methods so that time-consuming verification procedures could offset the advantage of the facilitated computation. Furthermore, these simulation techniques cannot be used in situations in which fluctuations about mean behavior are of interest to calculations. Such would be the case with a small population of particles for which the randomness of individual particles cannot average out to a deterministic behavior of the overall system. In the following section, we discuss a statistically exact procedure for simulating particulate systems that is useful for calculating the average as well as the fluctuations about average behavior, and can also provide the basis for rational simplification to more efficient algorithms when only average information is required.

4.6.2

Statistically Exact Simulation of Particulate Systems

The development in this section arises from the generalization of an idea due to Kendall (1950) for a simple birth-and-death process. This generalization, accomplished by Shah et al. (1977)^ \ was significant in that it provides a route to the simulation of a particulate system of arbitrary complexity and is limited only by the amount of computational power available. It is statistically exact in that the random numbers to be generated satisfy exactly calculated distribution functions from the model for particle behavior, thus allowing not only the calculation of the average system behavior but also the fluctuations about it. Furthermore, we shall see that it is free from arbitrary discretizations of time (or any other governing evolutionary coordinate) that were characteristic of the simulations of Section 4.6.1. 31 This work was completed in Shah's (1974) doctoral effort and was not submitted for publication until the author's move to Purdue University in 1976. The authors were unaware of the publication of the very same idea that was specific to the aggregation process by Gillespie (1975) in the geophysical literature.

4.6. Monte Carlo Simulation Methods

1 73

Kendall's idea is based on the concept of an interval of quiescence (reckoned from some arbitrary instant t at which the state of the population is known). This interval of quiescence was defined earlier as the time during which there occurs no discontinuous change in the population due to either the birth of a new individual or the disappearance of an existing one. Since birth and death events are assumed to be random, the interval of quiescence is clearly a random variable whose distribution function is of interest to the technique of simulation. The particles envisaged by Kendall were indistinguishable so that the transition probabilities concerning the birth and death of individuals were constants. The generalization of Shah et al. (1977) considered particles that were distinguished by a scalar state x, and the processes of birth and death were biased by this state variable. We shall consider here a population of particles distinguished from one another by a finite dimensional vector x of internal coordinates and distributed uniformly in space.^^ Further, we shall be concerned with the open system of Section 2.8 whose behavior is dictated by the population balance equation (2.8.3). Thus the number density in the feed,/i in(x), may be assumed to be Nj-f{x) where Nj- is the total number density in the feed stream and /(x) is probability density of particle states in it. It will also be assumed that the continuous phase plays no role in the behavior of the system. Relaxing this assumption does not add to any conceptual difficulty, although it may increase the computational burden of the resulting simulation procedure. In expounding the technique of simulation, our strategy will be first to stipulate the state of the system (i.e., in the domain Q^ of volume ]Q at any instant t and then to show how to simulate the change at a later instant (an interval of quiescence away). Clearly, once such a strategy is available, it is only necessary to show how the initial state of the system is to be specified. If the initial state is known exactly, the issue is immediately resolved. If it is known probabilistically, as by specification of its distribution function, then a particular realization of the initial state can be created by the technique of random number generation that is indicated in footnote 33. The random state of the entire system at any instant t is given by the states of all the particles that exist at that time. Hence, we let N^'^\t) represent the total number of particles in Q^ (distinguish this from the total number density N{t) = N^'^\t)/VJ, and X,(0 the state of, say, the ith particle, 32 Of course one may also include spatial nonuniformity for a more general treatment the methodology for which is not, in principle, difficult to obtain from the treatment here.

1 74

4. The Solution of Population Balance Equations

i varying from 1 to N^'^\t). It is convenient to introduce the notation A, ^ {N^^\t) = v; X,{t) = X,; i = 1,2,..., v} where the symbol on the left is an abbreviation of the assertion on the right that there are v particles in the system at time t with states x^, X2,...,x^. We now assume that the particles may undergo binary breakup with breakage functions b{x) and P(x|x'), the former representing the breakage frequency and the latter the distribution for the state of the fragment of breakage. Further, we assume binary aggregation with frequency a(x, x'); the processes disturbing quiescence are particle entry and exit, breakage, and aggregation. The foregoing random processes are assumed to occur independently of one another so that their simultaneous occurrence during an infinitesimal interval (i, T + dx) is of order dz^. During quiescence, the particles are assumed to change their states continuously according to the differential equations --i = X{XX / = 1,2,...,V, dt

t'>t

(4.6.1)

X.(r) = X, where we have used t' during the period of quiescence. We now precisely define the interval of quiescence T such that quiescence is interrupted immediately following f = t -h T by any one of the random processes of particle entry or exit, breakage, or aggregation. Clearly 7 is a random variable, which depends on the state of the population A^ at time t; its distribution function Fj(T\t) is defined by F^(T|0

= Pr{T ^

TI^J-

= 1 - Pr{T\t); P^(T|f) = P r { r >

T\A,}.

Fj{T\t) is a quantity of prime interest to the simulation since it generates the quiescence time after t. We shall presently show that it is easier to calculate the function Pj^{T\t). Clearly we may write PJ{T -\- dT\t) = P/(T|r) X Pr [None of the processes disturbing quiescence occur during the interval T to i + J i ] (4.6.2) Since particle entry, exit, breakup, or aggregation disturbs quiescence, we may write for the probability that no quiescence disturbing events occur during the interval

4.6. Monte Carlo Simulation Methods

1 75

1 — Pr [Any one of the processes of particle entry, particle exit, breakup of one of the particles, or aggregation between any one pair of particles occurs during the time interval T to T + di] (4.6.3) The individual probabilities for the different events that interrupt quiescence are readily identified. The probabihty of particle entry occurring during (T, T + (IT) is obtained directly on multiplying by dz the constant number feed rate, yielding Nj-qdx. The probabihty of each (of the v particles) exiting during (T, T + dx) is given by qdx/V^ = dx/O. Pr [Particle entry in T to T + dx] = Nj-qdx: Pr [Particle exit in x to X + dx'] =-^dx u

Pr[One out of the particles present at time t breaks in the time interval V

T to T + di] = ^ biXiit + x))dx Pr[One pair out of the particles present at time t aggregates in the time interval V— 1

V

Z

X to X + dx']= YJ

^O^iit + ^l ^ji^ + ^)) dx

Adding the individual probabilities above, substituting into (4.6.3) and subsequently into (4.6.2), one obtains Pjix +dx\t) = Pj.{x\t)

+y

l-lNfq i

+ ^-^t^b{X,{t + x))

a(X,(t +T),X/t+T))jdT

from which a suitable transposition of terms, followed by division by dx and letting it tend to zero yields the differential equation dPj : dx

^ L.

V

= - pjNfQ +l+i fc(X,(f + T)) + l '

t

«(Xi(t + T), X^.(f + T)) I (4.6.4)

for which the initial condition is given by p^(0|t) = 0.

(4.6.5)

176

4. The Solution of Population Balance Equations

Equation (4.6.5) reflects the fact that the interval of quiescence time is strictly greater than 0. Equation (4.6.4) can be readily solved subject to the initial condition (4.6.5) to yield the cumulative distribution function for the interval of quiescence as Fj{T\t) == 1 - e x p

Z biXiit + TO)

- M ^ f ^ + nh +

(4.6.6)

dx' i = l j=i + l

The cumulative distribution function for the quiescence interval is thus known from knowledge of the state of the population at time t. The random number for the interval of quiescence must be generated so that the distribution function (4.6.6) is satisfied.^^ If the particles in the population did not grow so that the states identified at time t remained the same with the passage of time, expression (4.6.6) becomes Fj{T\t) — 1 — exp

^/^ + n + Z bi^i) + Z "

1=1

Z «(Xp X.) \ T

i = l 7 = i+l

(4.6.7) for which it is considerably easier to compute the quiescence interval since it involves equating the right-hand side of (4.6.7) to the generated uniform random variable and inverting it for i (see footnote 51). The average quiescence interval conditional on the state of the population at time t, denoted <(T|^^>, is given by

ix\A,) = \N,q + ^ + Z Mx,) + Z' Z «(^.' ^j)] '' L

^

i=i

i=i j=i+i

J

(4.6.8)

which applies, of course, to the simplified distribution function (4.6.7). If any single term dominates in the sum above, it implies that the event it represents occurs (and hence will interrupt quiescence) most frequently 33 The generation on the computer of a continuous random variable X with cumulative distribution function F;^(x) is based on transforming a uniform random variable Y in the interval [0, 1], which can be readily generated. The transformation Y = Fx{X) has an inverse because of the monotone nondecreasing nature of the function F^, so that X = F^^{Y)\ further, for any number x, X ^ xoY ^y = F^ix), so that Pr{X ^ x} = P r { 7 ^ y} = y. Hence Y is uniform. The case of discrete X is left for the student. The generation of random vectors with statistically independent components immediately follows from the preceding discussion. However, the case of correlated components clearly requires further considerations.

4.6. Monte Carlo Simulation Methods

1 77

relatively to the others. At this stage of the simulation, we recognize that the quiescence has been disturbed and the next step is to identify the event responsible for it. The identification of the disturbing event is made rather simply by using the rules of probabihty theory, viz., Pr[Particle exit \A^, '^'\=7, 6

Pr[Particle entry \A,, T] = N.qii:\A^, V

Pr[Particle break-up \A„ T] = ^

iA^^

fo(xJ, V— 1

Pr[Particle aggregation 14, T] = + X

V

Z

a(x^, x^.)<'^l^t>-

(4-6.9)

The probabilities represented above are conditional not only on the state Af, but also on the quiescence interval T. Since the four discrete events that can disturb quiescence have probabilities as listed in (4.6.9), a random number can be generated to identify the event that occurs at this stage in the sample path.^"^ Once the disturbing event is identified, the next step (towards updating the system at t + T) is to find the specific particle concerned with the disturbance. Thus, for particle entry, it remains to identify the state of the particle that entered, for exit the particle that left the system, for aggregation the pair that underwent aggregation, and for breakup the particle that suffered breakage. Again the probabihties of each of these events are easily determined. The cumulative distribution function for the state of the particle that entered is obtained from the stipulated density function /(x). For particle exit each of the v particles has an equal probability of exit so that the probability of any specific particle being the one that exited is 1/v. For particle aggregation, the probability of the pair that has undergone aggregation being the specific pair (say of states x^ and x^- or more simply i —j) is readily seen from the elementary rules of probability to be given by {z(x-

Pr[i —j I Af, T, aggregation event] = •:^z:^ ^—

Z

r=l

Z

X-^

,

(4.6.10)

^(^r. Xj

s=r+1

where the conditional nature of the probability in (4.6.10) includes the fact 34

Shah et al. (1977) neatly handle these issues by defining indicator random variables that can be conveniently used for computer algorithms. The reader is referred to this article on this and other matters relating to computerization of the simulation algorithm.

1 78

4. The Solution of Population Balance Equations

that an aggregation event has already been identified. If, on the other hand, a breakage event has been identified, the probabihty of that having occurred to the ith particle is given by P r [ / | 4 , T, breakage event] = ^ ^^'^ .

(4.6.11)

At this stage, if entry, exit or aggregation has disturbed the quiescence, the state of the system can be exactly specified at time t + T; the consequence of aggregation events is generally known exactly. The only remaining uncertainty concerns the (binary) breakage event since the states of the resulting fragments are known through the density function P(x|Xj) granting that it is the ith particle that underwent breakage. Thus, the cumulative distribution function corresponding to P(x|Xj) can be used to identify the new fragments. ^^ The state of the system at r -h T can now be completely determined. The simulation procedure can be continued in the same way for the next quiescence interval. Equations (4.6.8) through (4.6.11) have been obtained for the case in which the particle states do not change with time during the quiescence interval and the cumulative distribution function for the quiescence interval is given by (4.6.7). The corresponding results for the case in which particle state changes in accord with the differential equation (4.6.1) are readily obtained by using the cumulative distribution for the quiescence interval given by (4.6.6). In either case, a sample path of the simulation produces a sequence of quiescence intervals T^, T2,..., T^^^, from the initial time t = 0 and ending the sample path at, say the instant t such that Z j i j Zj ^ t. Suppose we have a set of say S sample paths, each providing discrete data of the kind Au ^ {N'^\t,)

= V,; X,{t,) = xr i = l,2,...,v,},

fc

= 1,2,...,M,.

We now show how the average number density can be calculated from the sample paths in prediscretized (time-invariant) domains of the particle state space. This calculation is considerably easier for the case where particle state does not vary with time during the quiescence period. Denote the 35 If breakage occurs into a random number of fragments with a stated distribution, a further step is required in the simulation to identify this number before identifying the states of the new fragments. For such a simulation, more detailed statistics of the breakage process would be necessary.

4.6. Monte Carlo Simulation Methods

1 79

prediscretized domains by coj with volume AP^ ^ where j = 1,2, Suppose we desire from a specific sample path the number density averaged over each domain coj at time t defined by nj{t) =

n{x,t)dV^,

^Kj

7=1,2,....

We examine the specific sample path for the latest data on particle states at k = M^ and assign particles to the discrete domains {coj; j = 1,2,...}. Let Vj particles be found at the instant in question in domain coj. Then we may write for this sample path

"^^'^^VXlT'

E 0 = VM.

(4.6.12)

where the division by V^ is because of the fact that the number density is defined per unit spatial volume. An estimate of the expected number density in domain cOj, denoted Enj{t) =f^j{t\ is obtained by averaging over all the S sample paths. To represent this mathematically, we let the data of the 5th sample path be distinguished by superscript {s) so that fuj = ^ i

nf(t)

(4.6.13)

An estimate of the variance of the number density Vnj{t) in the domain comay also be estimated by using the relation

Vnj{t) = - i - i Infit) -f,j{t)y. *^ ~ ^ s = l

(4.6.14)

Thus, the technique of Shah et al (1977) also generates stochastic information on the population, as claimed earlier. Of course, the mean and variance of other quantities associated with the population may also be calculated in a similar manner. If /(x) represents an extensive property of interest associated with particles of state x, then the mean and variance of the property associated with particles in cOj defined by Fj{t)=

f{x)n{x,t)dV Jcoj

may be estimated from the simulations by writing

180

4. The Solution of Population Balance Equations

where the inner sum has Vj elements because there are as many particles in the domain coj. The preceding calculations which were specific to the population in the domain cOj can also be extended to the population in any other state domain (such as that obtained by summing all or some of the cOjS) by replacing the summand in (4.6.13) by its sum over co/s. For the case where particle state changes continuously during the period of quiescence, the calculation of the mean and variance of quantities associated with prediscretized particle state domains at an arbitrary instant t is not as conveniently done. Suppose t occurs somewhere within a quiescence interval. The distribution of particles in the fixed domains {cOp j = 1,2,...} may vary at any instant during the period of quiescence because particles can commute across the boundaries of each cOj. Thus the formulas (4.6.13) and (4.6.14) must be modified by evaluating the particle states exactly at time t and determining the values of Vj apphcable precisely at time t. This information would be already available to a fair degree of approximation, since the changes in particle states would have been solved for in generating the quiescence interval distribution. In describing the simulation technique of Shah et al (1977), we have laid stress on the underlying probabilistic arguments rather than giving precise guidelines for the formulation of the simulation algorithm. However, such guidelines are exactly the focus of the article of Shah et al. (1977), which is strongly recommended for the reader interested in directions for computer software. The technique has been successfully applied to a variety of problems in the literature.^^

EXERCISE 4.6.1 It is convenient to use the Monte Carlo technique based on the quiescence interval technique discussed in this section on steady-state continuous systems. For example, consider the continuous extraction process in Section 3.2.4 retaining all the assumptions therein. 36 See Shah et al. (1976), Shah et al. (1977) of Chapter 3. Simulation of a small population of particles with stochastic effects is presented by Ramkrishna et al. (1976). See also Bapat et al. (1983). For a somewhat complex situation in a fluidized bed reactor which investigates the effect of bubble coalescence on chemical reaction conversion, see Swe et al. (1987), Muralidhar et al. (1987) both referenced in Chapter 3.

4.6. Monte Carlo Simulation Methods

181

The strategy is to consider the fate of each entering drop and its descendents through breakup until all of them have been washed out of the extractor. Repeat this for several entering drops, statistically assembhng the size and concentration of the exiting descendents from which the bivariate distribution in size and solute concentration can be estimated. This distribution of course apphes to that in the vessel and to the exiting stream because of perfect mixing. In order to realize this strategy, proceed as follows: (i) Calculate the first quiescence time interval during which the drop neither breaks nor washes out. Show how you will update the concentration in the drop during this initial quiescence interval. (Note this can be done regardless of the other drops in the vessel since the behavior of each drop is independent of the behaviors of all others). What is the probabihty distribution for the two events given that the first quiescence period has just ended because the drop has either exited or split into two equal sized drops? (ii) Show how the analysis in part (i) can be continued until all the descendents have been purged out of the vessel. Regard the quiescence interval at any stage as applicable to the washout or breakage of all the descendents prevailing in the vessel. (iii) Present formulas for the steady-state bivariate distribution of size and concentration in the vessel. How would you estimate the concentration distributions in each generation that are displayed in Fig. 3.2.2?

4.6.3

The Exact Simulation Technique versus Techniques with Discrete Deterministic Steps

We now briefly pause to examine the simulation strategies in Section 4.6.1 which use a fixed discretization interval for the quiescence period in the light of the statistically exact technique presented in Section 4.6.2. The discussion is considerably simpler for the case in v^hich no particle growth occurs. If the discretization interval, say h, is considerably smaller than the average quiescence time as given by (4.6.8), i.e., ~^/Z « 1, then the probability that the quiescence is disturbed betv^een t and t -\- his given by Fjihlt)

= 1 - exp[--i/!] «

-I/I

+ OiP),

(4.6.15)

which reflects the inherent assumption that at most one event can disturb

182

4. The Solution of Population Balance Equations

the quiescence at a time, the Hkehhood of multiple events being of order h^. We recall that the one event may be entrance (denoted D^), exit (D2) or breakup (D3) of one of the v particles, or aggregation (D4) of one of the v(v — l)/2 pairs present at time t. We let D denote the joint occurrence (during t to t -\- h) of the four disturbing events just cited although no two of them are deemed to occur simultaneously.-^^ Clearly, P r ( D | 4 , h) = iT\A,}-'h

+ 0(h^).

(4.6.16)

The negation of D, represented by D\ implies status quo of the state A^ which occurs with probabihty given by Pr(Z)'|y4„ h) = l - (T\A,}-'h

+ 0{h^).

(4.6.17)

In the absence of growth, no change occurs in any quantity associated with the population when D' is true. Thus, the expected change in this quantity during t to t + h is given by the change due to the disturbance event, D multiplied by the probability (4.6.16). The objective of the simulation is to seek the average quantities of the system at time t + h (and thereafter) given the state A^ at time t. Since in the foregoing process, D does not specify the disturbance event in full, it does not suffice to calculate the change in the population at t + h. Toward a complete identification of each disturbance event, we introduce the following events: D^^: D2j'^3^^: D4 j-^:

A particle of state x (more precisely in a small neighborhood about x) enters. The jth particle exits. The7 th particle breaks to yield a particle of state x and the other of state x^- — X. The particles of states x^- and x^^ aggregate.

The change in the population during t to t -^ h depends upon the foregoing events. For the average change, we need their respective probabilities conditional on A^. These have already been defined and are recalled as P r ( D , , J 4 ) = N^qfixjh; Fr{D,jJA,) 37

FviD,j\A,)

= b{xj)Pix\xj)h;

=^h;

Pr(D3,^.|^,) = b{Xj)h;

Fv(D^j^,\A,) = a{xj, x,)/i,

In the algebra of sets, we may write D = D^ + D2 + D^ -\- D^; DiDj = O for / =^j and D:D: = D: froffi whlch it also follows that DD, = D,.

4.6. Monte Carlo Simulation Methods

183

where we have omitted the terms 0{h^) in each of the above probabihties. The complete specification for the change in the population to occur during the time interval t to t -\- h is given by the mutually events D^^, D2J, D^j^ and D^^jj^. Each of these implies a change in the population and therefore quantities associated with it. Because of the smallness of the time interval, the change occurs by only one particle. In what follows, we focus on the number density in spatial and internal coordinates. Thus, it will be of interest to calculate the expected change in the number density during the time interval t to t + h (conditional on A^), which we shall denote by £[An|^J where An = n(x, t + h) — n(x, t). The particle added to or eliminated from the system would have a specific state for a specific realization of the disturbance event, and the corresponding change in the number density would be a Dirac delta function in internal coordinates. Table 4.6.1 shows both the actual change suggested by the specific event the average change (conditional on Af) under all possible Table 4.6.1 Disturbing Event Particle of state x' enters D^^, Particle of state x^- exits the system D2J Particle of state x^- breaks into a particle of state x' and another of state Xj — x' ^j -

Probability of Disturbing Event Nfqhfix')

>

Actual Change in Number Density

Average Change in Number Density

l^(x-x')

^ V ( x )

-4-5(x-x,)

T^h Y VjO

b{Xj)Pix'\Xj)h

^[-^(x-x,.) r

+ ^(x - x^. + x') + S{x - x')]

S{x-X:)

j ^ ^

7 i MX,) *^r j = l

X [ - ^ ( X -X^.)

+ 2P(x|x,)]

X' ^ 3 J,x'

Particle of state Xj aggregates with particle of state Xfc D^j^j^

a{xj,x^)h

~l-3{x-xj) -(5(x-Xfc) + ^(X-X^. -Xfe)]

Sum

An

h ^'^

77 I

Z «(x„x,)

X ~3(x

— Xj)

-^(x-xj H-<5(x-x^.-xJ] E[An|^J

184

4. The Solution of Population Balance Equations

outcomes of that event. This average change is obtained by multiplying the actual change by the probabihty of the event and summing or integrating (with respect to primed coordinates) over all possible realizations. The total change in the average number density during the interval t to t -\- his given by the sum of the terms appearing in the rightmost column in Table 4.6.1. Note that this measured change in the population density is conditional on the state Af of the population at time t. Thus, the average change in the number density at the instant t + h, conditional on the state of the population at t = 0 (denoted A^), is obtained by ElAn\AJ

= X Pr[AJ^]E[An|^J

(4.6.18)

At

where the "summation" over all possible population states A^ is somewhat vague and requires the development in Chapter 7 for proper elucidation. However, it should become apparent at this stage that £[A«|^o] calculated from (4.6.18) and Table 4.6.1 is in fact a numerical solution of the population balance equation which has resulted as a degenerate case because of the choice of a fine interval of discretization. In this regard, the deliberations in Section 7.2.3 may provide further clarification. The foregoing treatment was based on the assumption that h, the length of the time discretization step, was very small compared with the average quiescence interval so that the need for simulation of the random variables was obviated. However, the efficacy of the Monte Carlo methods of Section 4.1.6 depends not on the choice of such a fine interval, but rather on negotiating a compromise among various random features of the process. If the time scales of different disturbance events are disparate, the slowest of them may be allowed to dictate the discrete time step (of the order of its mean recurrence time). During this step, the faster events may have occurred numerous times to be represented reasonably well by the mean number of recurrences with negligible fluctuations so that their random nature can be ignored. The stochastic features of the slower process may then be retained for random simulation. Here, too, an average quiescence interval, independent of the population state at each instant, simplifies the calculations enormously. The change in the population following the interval may then be simulated by an appropriate random number generation. As an example, consider a pure binary breakage process in which an average breakage time is selected as the discretization step. Of course if breakage slows down, as particles become smaller, the discretization step

4.6. Monte Carlo Simulation Methods

185

can be increased in stages. Although the breakage events are distributed deterministically in time, the identification of the fragmenting particle and the sizes of the fragments may be determined by random number generation. Thus, considerable flexibihty exists in the use of Monte Carlo simulation methods. The approximate simulation methods are very important for carrying out economic computations. However, in the absence of any supporting intuitive understanding of the process, one must be conscious of their disadvantages, which he in their inherent uncertainty, and also their inability to afford stochastic aspects that are often important in the simulation of small populations.

4.6.4

Simulation of Agglomerating Populations

The discussion in Sections 4.6.1 to 4.6.3 purported to include agglomeration processes, as the aggregation frequency did enter the calculation of the quiescence interval. Consider, for example, an agglomerating population in a bounded domain across which no particles leave or enter the domain. Assume further that all the particles can access each other during an infinitesimal time interval for the purposes of aggregation, the aggregation frequency per pair remaining the same at all times regardless of the population size. The foregoing assumption implies that the entire volume is "well-mixed." The simulation of such a system, starting with the actual number of particles in the system can be performed using the technique of Shah et al. (1977) discussed in Section 4.6.2. If aggregation proceeds until the population dwindles to small numbers (in fact, all the way down to a single particle!), the mean distribution of particle size, as well as fluctuations about the mean, can be estimated from such a simulation procedure. However, whether the population balance equation itself produces a satisfactory description of the average system behavior is itself an issue in doubt. As we observed earlier, the treatment of small populations is properly in the domain of stochastic analysis, which is the subject of Chapter 7. In general, however, the population about any point in the domain may be well mixed only in a local neighborhood whose extent should be of interest to us. The size of this neighborhood may be estimated from physical considerations of the mechanism of relative motion between particles over a time scale small compared with the time scale in which noticeable change in the population takes place by aggregation. If the volume of mixing is very small, allowing only a very small population, then one is faced with the

186

4. The Solution of Population Balance Equations

situation considered in Chapter 7, raising the possibihty that the population balance equation may itself not be equipped to predict the dynamics of the system. We return to this issue in Section 7.4 of Chapter 7, but for the present, we assume that such a contingency will not arise. Thus, the issue of present interest is only one of simulating an aggregating system without running the risk of encountering very small particle populations. In such a case, it is possible to envisage either a constant volume simulation or a constant number simulation. The constant volume simulation will progressively drop in particle numbers, eventually causing problems with continuation of the simulation. The constant number simulation, on the other hand, is based on the assumption that as the number density of particles decreases by aggregation, the well-mixed volume expands to accommodate more particles. This problem has been dealt with at length by Sampson (1981), and by Sampson and Ramkrishna (1985, 1986).^^ The constant number simulation is carried out as follows. A large number of particles (more than 1000) are distributed among a certain number (MJ of "cells" each containing, say, ATj^ix particles. The technique of Shah et al. (1977) is then used to simulate in parallel the aggregation events in each of the cells. As the number of particles decreases in the different cells, the particles in all the cells are redistributed in M cells of approximately N^-^^ particles each and the simulations continued in each of the new cells as before. ^^ As observed earlier, we will have opportunity to revisit agglomerating populations in Chapter 7 from a considerably deeper perspective than in this section. We shall therefore attend to other interesting aspects of the Monte Carlo simulation method.

4.6.5

The Single Particle Simulation Method

That mathematical abstraction can often be a great asset to finding solutions to physical problems has not been an issue of significance in engineering science. This section provides an opportunity to demonstrate how the recasting of an equation allows a considerably simpler interpretation of a relatively more complex physical situation, thereby affording a very simple solution of the problem. 38 39

Sampson and Ramkrishna (1986) is referenced in Chapter 7, since it is more relevant to that Chapter. A somewhat similar procedure has been communicated by Smith and Matsoukas (1998).

4.6. Monte Carlo Simulation Methods

187

Consider, for example, a population of particles that randomly undergo breakage at various times with the broken fragments possessing a size distribution related to the sphtting parent. The particles are distributed according to their mass, which is conserved during each breakage. This problem has been considered in detail in Section 3.2.2 and the reader is advised to reread it at this stage. Equation (3.2.7) represents the population balance of interest. Suppose we want to simulate this system using the method of Shah et al. (1977). This involves starting with the initial population, generating a quiescence interval for breakage, identifying the particle that has spht by generating another random number, and then generating a random number to identify the sizes of the broken fragments. If the breakage is not binary, one is now forced to ask for more statistical information on the breakage process, such as the number of particles that are formed. Furthermore, it is essential to collate the fragments after each breakage event and redistribute them in the population in order to keep track of the number density at each instant of time. As shown in Section 3.2.2, the foregoing problem may also be described in terms of the cumulative mass fraction, F^{x, t) of particles as defined by Eq. (3.2.9). The population balance equation (3.2.7) can be expressed in terms oi F^{x, t) as shown in (3.2.13), which is reproduced below. dF,{x, t) dt

b{x')G{x\x')8^,F,{x\

t).

(4.6.19)

where the function G(x|x') is the volume fraction of particles with mass at most equal to x formed from breakage of all particles with mass x' that break at any instant. Suppose we seek the value of F^{x, t), given that at time t = 0, we have F,{x, 0) = F^^\xl

(4.6.20)

which displays on the right-hand side the initial volume fraction of particles with mass at most equal to x. A detailed simulation of this system has been presented by Das (1996). Our objective here is to show a considerably more effective method of simulating this system based on the following alternative view of the system. We begin with examining Equation (4.6.19) and recognize that the cumulative volume fraction F^{x, t) of particles of mass at most x is a cumulative distribution function that can be attributed to a random process X{t), which may be interpreted as the mass of a single particle at time t. We imagine that this particle, whose initial size is a random variable with

o

188

4. The Solution of Population Balance Equations

Particle of muss x^ al lime /

Particle of imiss x^ al tiitic / + T

Particle after transition to nmss X

FIGURE 4.6.2 Schematic of the imaginary Single Particle Size Reduction Process.

cumulative distribution function F^^\x\ is randomly undergoing a discontinuous erosion process to smaller sizes in accord with the probabilistic law FrlX{t -f dt) ^ x\X{t) = x'] = B{x\x')dt,

(4.6.21)

where we have set B{x\x') = b{x')G{x\x'). A schematic of this random erosion process is represented in Fig. 4.6.2, in which T represents the time at which erosion occurs by a random amount. We confirm this view of Eq. (4.6.19) by a fresh derivation of the equation for the process shown in Figure 4.6.2. Let X{t) represent the mass of the particle at time t, and let F^ix, t) = FYlX(t)

^xl

In order to derive an equation for Fj(x, t), we consider the probability that time t + dt the particle has mass at most x. Clearly this probability is given by Fi(x, t + dt). We envisage how this situation could have arisen at time t + dt starting from that at time t; there are only two exclusive ways in which this could occur. First, the particle had at most mass x at time t, the probability of which is obviously F^{x, t). Since erosion will only lower particle size, the particle mass will continue to be at most of mass x whether or not erosion occurs. Second, the particle may be of mass larger than x at time t but undergo erosion to a mass less than x during the time interval t to t + dt with probability dt

B{x\x')d^,F^{x',

t) + 0(dt^).

We thus have on equating the probability at time t -{- dt to the sum of the two probabilities above at time t Fi(x, t-hdt) = Fi(x, t) +

B{x\x')d,F^{x\

t)dt + 0{dt^\

4.6. Monte Carlo Simulation Methods

189

Transposing terms, dividing by dt, and letting dt -> 0, we again arrive at Eq. (4.6.19)! We have thus shown that this simple random single particle discontinuous erosion process is equivalent to the fragmenting population considered earlier. The advantage of this finding lies in using the method of Shah et al (1977) in simulating this process in a very simple and effective way. If we define the quiescence interval T to be the time during which no particle erosion occurs then, using the same arguments as in Section 4.6.2, the cumulative distribution function for the quiescence interval is obtained as P r [ T ^ x\X{t) ^x''] = \ - exp{l -

B{x\x')x],

which can be used to generate the quiescence time. At the end of quiescence, the particle mass can be updated using the distribution function r( I ., _-g(^I^O ^''^''^B{x'\x')~

_B{x\x') b{x') '

These two random variables suffice to obtain a sample path of the process {X(t)} over a prescribed time interval (Ramkrishna et al. (1995). Figure 4.6.3 reproduces the results of the simulation by Ramkrishna et al. (1995) that demonstrates the accuracy of the technique by comparison with the numerical solution of a discretized form of Eq. (4.6.19). The breakage functions used were from Narsimhan et al. (1984) (referenced in Chapter 5). Consider the application of the foregoing idea of a single particle process for a purely agglomerating population. Let the frequency for agglomeration between particles of masses x and x' be given by a{x, x'). Thus, we envisage a particle population distributed according to their mass engaged in pairwise aggregation whose dynamics is described by the population balance equation (3.3.5). We again focus on the equation for the cumulative mass fraction F^{x, t) that is shown in Section 3.3.2 to satisfy Eq. (3.3.8) which is reproduced below. dFJx, t)

d^F,{^,t)

r

^ ^ ^ 5 ^ F i ( w , t),

(4.6.22)

where jti^ is the first moment of the number density function or the mass density of the particle phase. Our next task is to interpret (4.6.22) as an appropriate single particle process. Thus, we envisage a single particle with mass X{t) which randomly increases with time in discrete steps. This random increment is viewed as follows. We create another particle of random mass with the same distribution as the existing particle and coalesce

190

4. The Solution of Population Balance Equations 1-

] g

0.8

Jr

\

f

•

Sim analytic


>

> 0.4

I

] r

\i

^0.2

1

1

1

1

1

1

1

1

1

1

20

1

40

1

1

1

60

Drop Volume

FIGURE 4.6.3 Comparison of simulation of a pure breakage process by the single particle technique of Ramkrishna et al. (1995) with numerical solution of the discretized form of Eq. (4.5.19). (Reproduced with permission of the American Institute of Chemical Engineers. Copyright (Q 1995 AIChE. All rights reserved.)

it with the particle present. We stipulate that this coalescence conserves mass. We define the transition probability for the particle with mass x at time t to increase to x + x' during the time interval t to t -j- dt to be given by d^,F^(x\ t)a(x,

x')dt/x'.

We now derive an equation for F^{x, t) to describe the single particle process just outlined and show that it is the same as Eq. (4.6.22). As before we consider the probability of how at time t + dt the single particle may come to possess a mass of at most x starting from its situation at time t. Note first that since the particle's mass can only increase, the only way for its mass to be at most x at time t -\- dt is for it to be so at time t and not undergo, during the time interval t io t + dt, an increment that increases the particle mass beyond x. Suppose the particle mass at time t is x' (which lies somewhere between 0 and x). Then the probability that its mass increase during t to t + dt exceeds x — x' is given by dt

5^"F^(x", t)a{x\

x")lx".

4.6. Monte Carlo Simulation Methods

191

so that the probabihty that the particle's increase in mass during t to t + dt does not increase beyond x is clearly 1

d^.F^{x'\ t)a{x', x")lx"

-dt

Thus, we must have for the single particle process F^{x, t -{- dt) =

S.'F,{x\t)

d^>rF^{x", t)a{x\ x")lx"

1-dt

Transposing terms suitably, dividing by dt, and letting dt ^ 0 we obtain the same equation as (4.6.22) as required. Although the single particle process has been identified, we are faced with the inconvenient circumstance of having to increment the particle size by an amount whose distribution is itself the quantity to be calculated. Thus, the situation here is not as desirable as that in the breakage process considered earlier. However, let us proceed to consider the simulation of this process using the technique of Shah et al. (1977). The cumulative distribution function for the quiescence interval T during which time no increment occurs in the particle is given by Pr[T^T|X(t) = x ] = 1 - e x p ^ l

-

du

d^rF^{x\ t + u)a{x, x')/x' (4.6.23)

which contains the unknown distribution function F^. Updating the particle mass is accomplished by generating a random number for the mass increment whose cumulative distribution function can be obtained by standard probability arguments to be d^>F^{x\ t + T)a(x, x')/x' d^>F^{x\ t-\-T)a{x,

(4.6.24)

x')/x'

which also clearly displays the unknown distribution function F^. In order for the simulation to be initiated, some initial approximation for F^ is needed in order to calculate the sample paths of the process from which a new value can be calculated by averaging for the function F^. Denoting the nth approximation by F^"^ and substituting into (4.6.23) and (4.6.24), random numbers can be generated for the quiescence interval as well as increments in particle size yielding sample paths for the computation of the

192

4. The Solution of Population Balance Equations

next approximation F i " ^ ^ \ The procedure is clearly iterative until convergence is accomplished. This iterative simulation procedure has been presented by Ramkrishna et al. (1995). These authors compare the simulation with the analytical solution for a constant kernel, which shows excellent agreement. It is the author's belief that the speed of such iterative simulation techniques may be vastly enhanced by improved strategies for approximating the distribution functions.

EXERCISE 4.6.2 Show that the single particle simulation of a breakage process can be extended to the case where particle growth occurs in accord with the function X{x). Elucidate the simulation strategy by calculating the quiescence interval distribution. (See Ramkrishna et al 1995 for application to a mass transfer problem in a stirred liquid-hquid contactor).

EXERCISE 4.6.3 A liquid-liquid dispersion is produced in a well-stirred continuous vessel fed with an entering stream containing drops with a number density of mass distribution /i,oW while the dispersion is withdrawn simultaneously at the same volumetric rate at which drops enter. Assume that the average residence time of the drops is 6. The drops do not coalesce but undergo breakage, the functions characterizing breakage being the same as in Section 4.6.4 or Section 3.2.2. Starting from the population balance equation, derive the equation for the cumulative mass fraction of drops. Conceive of a single particle process and propose a simulation procedure for the same.

REFERENCES Bapat, P. M., L. L. Tavlarides and G. W. Smith, "Monte Carlo Simulation of Mass Transfer in Liquid-Liquid Dispersions," Chem. Eng. Sci., 38, 2003-2013, 1983. Batterham, R. J., J. S. Hall and G. Barton, "Pelletizing Kinetics and Simulation of Full Scale Balling Circuits," in Proceedings of the 3rd International Symposium on Agglomeration, Nurenberg, W. Germany, p. A136, 1981. Blatz, B. J., and A. V. Tobolsky, "Note on the Kinetics of Systems Manifesting Simultaneous Polymerization-Depolymerization Phenomena," J. Phys. Chem., 49, 77-80, 1945.

References

193

Bleck, R., "A Fast Approximate Method for Integrating the Stochastic Coalescence Equation," J. Geophys. Res., 49, 77-80, 1970. Coddington, E. A., and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. Collins, S. B., and J. G. Knudsen, "Drop Size Distributions Produced by Turbulent Flow of Immiscible Liquids, A.I.Ch.EJ., 16, 1972-1080 (1970). Courant, R., and D. Hilbert, Methods of Mathematical Physics, Vol. II, Interscience Publishers, New York, 1956. Das, P. K., "Monte Carlo Simulation of Drop Breakage on the Basis of the Drop Volume," Comp. Chem. Eng., 20, 307-313 (1996). David, R., J. Villermaux, P. Marchal and J. P. Klein, "Crystallization and Precipitation Engineering—IV. Kinetic Model of Adipic Acid Crystallization," Chem. Eng. Sci., 46, 1129-1136 (1991). EnukashviH, I. M., "On the Solution of the Kinetic Coagulation Equation," Izv. Geophys. Ser. English Transl. Bull. Acad. Sci. USSR, No. 10, 944-948 (1964a). EnukashviH, I. M., "On the Problem of a Kinetic Theory of Gravitational Coagulation in Spatially Heterogeneous Clouds," Izv. Geophys. Ser. English Transl. Bull. Acad. Sci. USSR. No. 11. 1043-1045 (1964b). Finlayson, B. A., The Method of Weighted Residuals and Variational Principles, with Application in Fluid Mechanics, Heat and Mass Transfer, New York, Academic Press, 1972. Gelbard, F. M., and J. H. Seinfeld, "Dynamics of Source-Reinforced, Coagulating, and Condensing Aerosols," J. Colloid Interf Sci., 63, 426-445 (1978a). Gelbard, F. M., and J. H. Seinfeld, "Coagulation and Growth of Multicomponent Aerosol," J. Colloid Interf Sci., 63, 472-479 (1978b). Gelbard, F., and J. H. Seinfeld, "Numerical Solution of the Dynamic Equation for Particulate Systems," J. Comp. Phys., 28, 357-376 (1978c). Gelbard, F., Y. Tambour and J. H. Seinfeld, "Sectional Representation of Simulating Aerosol Dynamics," J. Colloid Interf Sci., 76, 541-556 (1980). Gillespie, D. T., "An Exact Method for Numerically Simulating the Stochastic Coalescence Process in a Cloud," J. Atm. Sci., 32, 1977-1989 (1975). Golovin, A. M., "On the Spectrum of Coagulating Cloud Droplets. II," Izv. Geophys. Ser. English Transl. Bull. Acad. Sciences, USSR 9, 880-884 (1963a). Golovin, A. M., "On the Kinetic Equation for Coagulating Cloud Droplets with Allowance for Condensation. Ill," Izv. Geophys. Ser. English Transl. Bull. Acad. Sci. USSR 9, 880-884, 10, 949-953 (1963). Hounslow, M. J., R. L. Ryall and V. R. Marshall, "A Discretized Population Balance for Nucleation, Growth and Aggregation," AJ.Ch.E.J., 34, 1821-1832 (1988). Hulburt, H. M. and T. Akiyama, "Liouville Equations for Agglomeration and Dispersion Processes," Indust. Eng. Chem. Fundls. 8, (319-324) 1969. Kendall, D. G., "An Artificial Reahzation of a Simple Birth-and-Death Process," J. Roy. Stat. Soc. Ser. B, 12, 116-119 (1950).

194

4. The Solution of Population Balance Equations

Kostoglou, M. and A. J. Karabelas, "Evaluation of Zero Order Methods for Simulating Particle Coagulation," J. Colloid Interf. ScL, 163 (420-431) 1994. Kumar, S., and D. Ramkrishna, "A General Discretization Technique for Solving Population Balance Equations Involving Bivariate Distributions," Paper No. 139c, AIChE Annual Meeting, 1995, Miami Beach, FL, November 12-17, 1995. Kumar, S. and D. Ramkrishna, "On the Solution of Population Balance Equations-I. A Fixed Pivot Technique," Chem. Eng. ScL, 8, 1311-1332 (1996a). Kumar, S. and D. Ramkrishna, "On the Solution of Population Balance Equations by Discretization II," Chem. Eng. ScL, 51, 1333-1342 (1996b). Kumar, S. and D. Ramkrishna, "On the Solution of Population Balance Equations by Discretization-III. Nucleation, Growth and Aggregation of Particles," Chem. Eng. ScL, 24, 4659-4679 (1997). Landgrebe, J. D. and S. E. Pratsinis, "A Discrete Sectional Model for Particulate Production by Gas Phase Chemical Reaction and Aerosol Coagulation in Free Molecular Regime," J. Colloid Interf. Sci., 139, 63-86 (1990). Liou, Jia-Jer, F. Srienc, and A. G. Fredrickson, "Solutions of Population Balance Models Based on a Successive Generations Approach," Chem. Eng. Sci. 52,1529-1540(1997). Marchal, P., R. David, J. P. Klein and J. Villermaux, "Crystallization and Precipitation Engineering—I. An Efficient Method for Solving Population Balances in CrystalHzation with Agglomeration," Chem. Eng. Sci., 43, 59-67 (1990). Michaelsen, M. L., and J. V. Villadsen, Solution of Differential Equation Models by Polynomial Approximation,"" Prentice-Hall, Englewood Cliffs, N.J., 1978. Naylor, A. W., and G. R. Sell, Linear Operator Theory in Engineering and Science, Holt Rinehart and Winston, New York, 1971. Petrovsky, I., Lectures on the Theory of Integral Equations, English Translation, Graylock Press, 1957. Ramkrishna, D., "Solution of Population Balance Equations by the Method of Weighted Residuals," Chem. Eng. Sci., 26, 1134-1136 (1971). Ramkrishna, D., "On Problem-Specific Polynomials," Chem. Eng. Sci., 28, 13621365 (1973). Ramkrishna, D., B. H. Shah and J. D. Borwanker, "Analysis of Population Balance — III. Agglomerating Populations," Chem. Eng. Sci., 31, 435-442 (1976). Ramkrishna, D., "Analysis of Population Balance — IV. The Precise Connection Between Monte Carlo Simulations and Population Balances," Chem. Eng. Sci., 36, 1203-1209 (1981). Ramkrishna, D. and N. R. Amundson, "Linear Operator Methods in Chemical Engineering," Prentice Hall, Englewood Cliffs, N.J., 1985. Ramkrishna, D., A. Sathyagal, and G. Narsimhan, "Analysis of Dispersed Phase Systems: A Fresh Perspective," AIChEJl, 41, 35-44 (1995). Rhee, H. K., R. Aris, and N. R. Amundson, First Order Partial Differential Equations. Theory and Applications of Single Equations, Vol. 1, Prentice-Hall, Englewood Cliffs, NJ, 1986.

References

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Sampson, K., "An Investigation of Particle Size Correlations and the Effect of Limited Mixing in Brownian Coagulation," Ph.D. Thesis, Purdue University, West Lafayette, 198L Sampson, K. J. and D. Ramkrishna, "A New Solution to the Brownian Coagulation Equation through the Use of Root-Shifted Problem-Specific Polynomials," J. Colloid & Interf. Sci. 103, 245-254 (1985). Scott, W. T., "Analytic Studies of Cloud Droplet Coalescence I," J. Atmos. Sci. 25, 54-65 (1968). Shah, B. H., "A Simulative and Analytic Study of Particulate Systems," Ph.D. Thesis, Institute of Technology, Kanpur, 1974. Shah, B. H., J. D. Borwanker, and D. Ramkrishna, "Monte Carlo Simulation of Microbial Population Growth," Math. Biosci., 31, 1-23 (1976). Shah, B. H., D. Ramkrishna, and J. D. Borwanker, "Simulation of Particulate Systems Using the Concept of the Interval of Quiescence," AIChEJ, 23, 897-904 (1977a). Singh, P. N. and D. Ramkrishna, "Transient Solution of the Brownian Coagulation Equation by Problem-Specific Polynomials," J. Colloid Interf. Sci., 53, 214-223 (1975). Singh, P. N., and D. Ramkrishna, "Solution of Population Balance Equations," Comp. Chem. Eng. 1, 23-31 (1977). Smith, M., and T. Matsoukas, "Constant-Number Monte Carlo Simulation of Population Balance," Chem. Eng. Sci., 1777-1786 (1998). Spielman, L. and O. Levenspiel, "A Monte Carlo Treatment for Reacting and Coalescing Dispersed Phase Systems," Chem. Eng. Sci., 20, 247-254 (1965). Subramanian, G., D. Ramkrishna, A. G. Fredrickson and H. M. Tsuchiya, "On the Mass Distribution Model for Microbial Cell Populations," Bull. Math. Biophys. 32, 521-537 (1970). Subramanian, G., and D. Ramkrishna, "On the Solution of Statistical Models of Cell Populations," Math. Biosc. 10, 1-23 (1971). Zeithn, M. A., and L. L. Tavlarides, "Fluid-Fluid Interaction and Hydrodynamics in Agitated Dispersions: A Simulation Model," Can. J. Chem. Eng., 50, 207-215 (1972a). ZeitUn, M. A. and L. L. Tavlarides, "Dispersed Phase Reactor Model for Predicting Conversion and Mixing," AIChEJ, 18, 1268-1271 (1972b). Zeitlin, M. A. and L. L. Tavlarides,"On Fluid-Fluid Interactions and Hydrodynamics in Dispersed Phase CSTR's: Prediction of Local Concentrations, Transfer Rates and Reaction Conversion," in Proceedings, 5th European (2nd International) Symposium on Chemical Reaction Engineering, Amsterdam, 1972c.

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CHAPTER 5 Similarity Behavior of Population Balance Equations

Our concern in this chapter is of certain "similarity" properties of the solution of population balance equations. These properties are of considerable value not only to the characterization of experimental data, but also to the identification of key model parameters associated with system behavior, and frequently in the elucidation of behavior at the particle level from population data. The property of similarity manifests in the form of what is often described as a self-similar or self-preserving solution associated with the behavior of many partial differential and integro-partial differential equations. 5.1

THE SELF-SIMILAR SOLUTION

Broadly, the self-similar solution identifies "invariant" domains in the space of the independent variables along which the solution remains the same or contains a part that is the same. Consider, for example, the number density function f^{x, t) that may satisfy a population balance equation such as (3.2.8) or (3.3.5). By a self-similar solution of either of these equations we mean one to be of the form Mx, t) = gimrj),

n ^ -^

(5.1.1) 197

198

5. Similarity Behavior of Population Balance Equations

where the functions g{t\ h{t) and \l/{rj) are as yet unknown but assumed to be nonnegative, smooth and bounded functions for the present. Note that the preceding solution contains the part il/{rj), which remains the same along the invariant or curve on the (x, t) plane defined hy rj = c, a, constant or X = ch{t). By varying the value of the constant c, one obtains a family of invariants. In this case, the "self-similar" form is assumed by the function il/{f]) (although we still refer to /i(x, 0 as determined by (5.1.1) as the self-similar solution) because the form of il/{r]) is time-invariant.^ If we were interested in the cumulative function such as F^{x, t), defined by (3.2.9), then it is readily shown that the self-similar form above for /^(x, t) will lead to Jo r]ilj{rj)dr]

Thus, the cumulative function is itself time-invariant along ^ = c so that it has a self-similar form. There is some question as to the existence of the integrals in (5.1.2), which we shall ignore for the present with a promise return to it presently. Clearly, the functions g(t), h{t\ and \l/{rj) must depend upon the population balance equation. However, it is easy to show that the functions are related to the zeroth moment /^^{t\ and the first moment //^(r), as defined in Section 4.4. Thus Q{t) = —7-,

h{t) = — — ,

(5.1.3)

r^xjj{f])dr] = 1,

(5.1.4)

provided that we set f 00

il/(r]) df] = 0

which can be accomplished without any loss of generality, provided of course that both integrals exist. It may be a source of some surprise that the existence of these integrals cannot be taken for granted even when ^^{t) and lii(t\ which are the corresponding integrals with the original number density function /^(x, t), may be assumed to exist at any given instant t. However, one cannot be presumptuous about the existence of the integrals 1

The concept of a self-similar solution is well known to the student familiar with the development of a boundary layer along a flat plate where the velocity profile remains the same when distance from the wall is scaled with the boundary layer thickness that varies along the direction of flow. Similarly, diffusion profiles in semi-infinite media are known to be self-similar when distance is scaled with respect to the square root of diffusion time.

5.1. The Self-Similar Solution

199

in (5.1.4) since the behavior of il/{rj) at either end of the interval of integration is determined also by h{t)^ As f^{x, t) usually vanishes rapidly for increasingly large x, the nonexistence of the integrals in (5.1.4) is generally associated with the possibility of singular behavior of the similarity solution il/{r]) at the origin. We assume that the order of singularity of \l/{fj) Sit the origin is denoted by s^, i.e., il/(r]) — 0(r\~^°) or alternatively lim rf°\\i(r]) = constant,

(5^ > 0).

If 5^ < 1, both integrals exist in (5.1.4) so that (5.1.3) is appropriate. Suppose now that s^^ 1. Then (5.1.3) is no longer valid, since the integral to the extreme left of (5.1.4) does not exist; in this case we seek a new similarity variable rj associated with a higher integral moment /^^(f) calculated as fiM

rj'ikiri) drj,

= g{t)h{tf

(5.1.5)

0

where the integral on the right-hand side of (5.1.5) is presumed to exist. In other words, if we let s^ denote the order of singularity of \l/(rj), then the integral in (5.1.5) will exist if s^ — /c < 1. The relationship (5.1.5) obviously applies for fik+i{t) also, so that we have h ( t ) = ^

(5.1.6)

where we have further arranged to have the similarity variable r] and the similarity solution il/{rj) defined such that r]^^^il/{ri)dr] = 1,

rj il/{rj) drj = 0

(5.1.7)

0

which is clearly reminiscent of (5.1.4). Indeed (5.1.6) represents the generalization of (5.1.3) for h{t). The function g{t) can be obtained by substituting for h(t) from (5.1.6) into (5.1.5). Thus d(t)=r

rU^l^

(5-1.8)

which generalizes (5.1.3) for g{t). Since, under these circumstances, the In fact it should be evident that dynamic data over a finite time cannot be adequate to span the entire range of scaled particle size (the similarity variable) unless the initial particle size distribution is exactly compatible with the self-similar distribution over the whole positive interval. This is seldom the case.

200

5. Similarity Behavior of Population Balance Equations

integral to the right of (5.1.4) may not exist, the cumulative distribution function cannot be calculated from (5.1.2). In summary of the preceding discussion, we have defined the similarity variable and the self-similar solution by (5.1.6) through (5.1.8) where k is the smallest integer such that Sj^ — k < 1, where Sj^ is the order of singularity of il/(rj). Recall again that k = 0 when s^ < 1. The actual determination of k must proceed by trial and error. It can be accomplished, however, by testing for self-similarity with integral values of k starting from 0. If self-similarity is perceived with /c = 0, further quest is redundant. If no self-similarity is evident with k = 0, the quest is continued with the similarity variable until the procedure yields self-consistent results. In what follows, we assume that both integrals in (5.1.4) exist so that the similarity variable is as defined in (5.1.3). If x represents particle mass and is conserved then the first moment ^^(t) becomes a constant, and the functions g{t) and h{t) are entirely determined by the zeroth moment alone. Furthermore, since the zeroth moment fi^{t) is the total number of particles per unit volume, h{t) turns out to be the average particle mass. Since we are interested in dynamic analysis starting from some initial conditions which can be arbitrary, it is clear that one cannot associate a self-similar solution from the very beginning of the process except for an initial condition that happens to be compatible with the self-similar solution. Thus, the question of a self-similar solution basically arises when the system has evolved away from the initial condition. There is thus a sense of independence of the self-similar solution from the initial condition. This independence may, however, apply only for a class of initial conditions outside of which no self-similar solution may be attained. Questions in regard to the conditions under which a self-similar solution exists for a population balance equation, and the class of initial conditions for which the solution can approach such a self-similar solution, are indeed mathematically very deep and cannot be answered within the scope of this treatment.^ On the other hand, numerical solutions can be examined for their approach to self-similarity. What will be of interest to us in this chapter is whether a similarity solution is feasible for a population balance equation. In other words, can the population balance equation admit a self-similar solution? Feasibility is of course necessary (but far from sufficient!) for the existence of a self-similar 3

The existence of a self-similar solution for the coagulation equation has been addressed by the following: Lushnikov (1973), Ziff et al. (1983), van Dongen and Ernst (1988).

5.2. Similarity Analysis of Population Balance Equations

201

solution. An attempt to answer this question is made through what is known as similarity analysis, which also leads to the calculation of the functions g{t) and h{t) and the derivation of the equation to be satisfied by il/{r]). Friedlander (1960, 1961) is the first to have conceived of self-similarity for population balance equations in connection with his investigation of the coagulation of aerosols. His ideas were inspired by Kolmogorov's discovery of universal scaling behavior in energy spectra of turbulent flows of fluids at high Reynolds numbers.

5.2

SIMILARITY ANALYSIS OF POI^ULATION BALANCE EQUATIONS

It will be of interest to consider both pure breakage processes and pure aggregation processes. A somewhat different form of self-similarity exists when both breakage and aggregation processes are present, which will also be of interest to us. Self-similarity in the presence of particle growth processes also deserves further investigation.

5.2.1

Pure Breakage Processes

We begin our considerations with a pure breakage system by recalling the population balance equation for the same from Section 3.2. In view of the fact that the cumulative fraction is a direct candidate for the self-similar form, we recafl the equation (3.2.13) 5Fi(x, t) dt

b{x')G{x \x')d^,F^{x\t).

(5.2.1)

Similarity analysis of Eq. (5.2.1) consists in assuming first the self-similar form (5.1.2), viz., F,{x, t) =
(5.2.2)

202

5. Similarity Behavior of Population Balance Equations

where K and a are positive parameters indicating higher rates of breakage of larger particles, a frequent characteristic of breakage processes. We further assume that the broken fragments from a parent particle possess a common statistical relationship relative to the size of the parent. Mathematically, this implies that the bivariate function G is such that G{x\x')=g{^.

(5.2.3)

Since for each x\ G(x \ x') is a cumulative distribution function of x, it is monotone increasing with respect to x such that G{x' |x') = 1, from which it follows that g is also a cumulative distribution function with respect to its argument in the interval [0,1] with ^(1) = 1. In the light of (5.1.2), (5.2.2), and (5.2.3), Eq. (5.2.1), on transformation of the integration variable to Yi' = x'/h{t), becomes • ^nri) T7^

= ^ I ^9 ( -,) d
(5.2.4)

where the primes on functions have been used to denote their derivatives with respect to their respective arguments. By virtue of its explicit dependence on time, the left-hand side of Eq. (5.2.4) is notably inconsistent with the right-hand side, which is free from any temporal dependence. Thus, we must have for consistency h\t) = -ch(tf^'

(5.2.5)

where c is a positive constant to be chosen. The negative sign is inspired by the physical consideration that /i, being the average particle mass, must decrease with time. Equation (5.2.5) is a differential equation in h that is readily solved to get -hity = k-\- ct, a where k is another constant to be taken as non-negative. The foregoing solution for h implies that the similarity variable defined in (5.1.1) yields ,y« = oc{k + ct)x'',

(5.2.6)

which suggests that (k + ct)x'' is also a similarity variable. If k is set to unity and c is taken equal to the rate constant K defined in (5.2.2) (the advantage of which lies in canceling K in (5.2.4)), we get the similarity variable z = (1 + Kt)x%

(5.2.7)

5.2. Similarity Analysis of Population Balance Equations

203

which Fihppov (1961) presented in an early mathematical study of the splitting process. Letting 0[z] = O(^) in which the square bracket is designed to account for the difference in the argument, transforms the integral equation (5.2.4) to zO'[^] =

(5.2.8)

d^{_z'\

zg

where again the square brackets for the function g are to distinguish its argument from that associated with g in Eq. (5.2.4). Equation (5.2.8) can be shown to have a unique solution, which shows that a self-similar solution is feasible for the chosen breakage functions. As we had pointed out earlier, this does not necessarily imply that a self-similar solution exists because it requires proving that the transient solution does approach the self-similar solution. Filippov (1961) has shown that such convergence is possible under suitable conditions on the breakage functions."^ We shall now see how the number density function /^(x, t) is related to the self-similar form ^ [ z ] . Noting the relationship between/^(x, t) and from Eq. (3.2,9), we obtain X

OX

rj

ex

fXy

Y]

fi^

which, besides containing the self-similar form, also has the time-dependent component in the ratio involving moments ix^ and ^i^. Note that the actual similarity variable to use is a negotiable quantity. For example, the similarity variable z in (5.2.7) is proportional to the ath power of the similarity variable Y}. Practical consideration may warrant other choices of the similarity variable. Thus, for instance, a fixed value of the cumulative mass fraction F^{x, t) at, say, O^ corresponds at instant t to a (unique) particle m^ass denoted by X^{t) such that

F,(XM

t) = 0„.

During self-similar behavior, we may invoke Eq. (5.1.2) to extract a unique

4

Particularly noteworthy of the conditions presented by Fihppov (1961) are that: (i)

Jo ^

dk converges.

(ii) g be monotone increasing. Analytical asymptotic results are presented for the specific case, g(X) = AX^, where P > 0.

204

5. Similarity Behavior of Population Balance Equations

value r]^ of the similarity variable rj such that so that X^{t) is also a scaling variable yielding x/X^{t) as another similarity variable. We consider below some applications.

5.2.1.1

Applications

Kapur (1972) has presented evidence of self-similar behavior of comminuted solids from experimental data. The model formulation is precisely as elaborated previously. The similarity variable was chosen to be "80% fines" which corresponds to O^ = 0.8. His results show that self-similarity is preserved over a notably long period of time. In an interesting earlier communication, Kapur (1970) also shows that such self-similar behavior leads to the well-known laws of grinding due to Rittinger and Bond (Orr, 1966) by attributing different values to the exponent a. For another application, we turn to the breakup of liquid drops in a well-stirred liquid-liquid dispersion (see Ramkrishna (1974) of Chapter 3). Clearly, the scenario surrounding the breakage of liquid drops is substantially different from the comminution of solids. Consider drops distributed according to their volume denoted x with Eq. (5.2.1) representing the evolution equation for the cumulative volume fraction. This situation could be realized by allowing the liquid-liquid dispersion to be "lean" in the dispersed phase so that the frequency of drop coalescence is neghgible. We further assume that the breakage frequency is of the form (5.2.2) and the size distribution of breakage fragments satisfies (5.2.3). It is of interest to see whether the similarity variable (5.2.7) is applicable to the cumulative volume fraction dynamics of the dispersion in various size ranges. The author performed a test (Ramkrishna, 1974) of such experimental data obtained by Madden and McCoy (1969). At large enough times for drops of sizes such that Kix""» 1, x'^t could play the role of a similarity variable if (5.2.7) is one. It is then possible to identify from a plot of the F^{x, t) versus x at various times t the relationship between x and t for numerous fixed values of F^. For each fixed value, of F^ it follows from (5.1.2) that O must have the same fixed value, implying the same for the similarity variables z and (for large enough times) the variable x'^t. Thus, we obtain ^ 1 = 0 = constant => x^'t = constant.

5.2. Similarity Analysis of Population Balance Equations

205

L :UMULATIVE VOLUME FRACTION 2

8

b90°/.

cc 6

-

^ m ^ "S?

~ ..1

1

1

1

1

TIME (MINUTES)

FIGURE 5.2.1 Evidence of self-similarity in experimental data of Madden and McCoy (1969) on drop size distributions from Ramkrishna (1974). Evolution of drop size is assumed to be by breakage process only. Reprinted with permission from Elsevier Science.

from which it is readily inferred that a plot of x on (the logarithmic scale) versus t (on linear scale) for every fixed value of F^ must produce a straight line of slope a. The plot, reproduced in Figure 5.2.1, shows a remarkably parallel set of straight lines for a wide range of fixed cumulative fractions, thus presenting evidence of self-similar behavior. The slope a is seen to be approximately 2. Figure 5.2.2 shows the result of plotting the cumulative fraction (in terms of percent cumulative volume) against the drop diameter times the one-sixth power (i.e., (3a) ~^) of t, which is also clearly a similarity variable. The collapse of all the data at different times provides further confirmation of the self-similar behavior of the breakup of liquid drops. The foregoing behavior of self-similarity can be generalized somewhat to accommodate a more general breakage frequency (maintaining the monotone increasing dependence on particle size), provided we assume the distribution function G{x \ x') for the fragments of breakage to be of the form G{x \x')

=g

bjx) fo(xO

(5.2.9)

which is equivalent to (5.2.3) if the breakage frequency satisfies (5.2.2). In other words, the case of the power law breakage frequency is subsumed by (5.2.9). The form (5.2.9) postulates that breakage favors more fragments in the more breakable range. It can be shown that the variable z = b{x)t can be a similarity variable. A test of dynamic experimental data for this form of self-similarity is based on recognizing the invariance of the cumulative

206

5. Similarity Behavior of Population Balance Equations

200

300

AOO 500 600 S i m i l a r i t y V a r i a b l e - t 1/6. "d

700

FIGURE 5.2.2 Self-similar distribution using experimental data of Madden and McCoy (1969) on drop size distributions from Ramkrishna (1974). Evolution of drop size is assumed to be by breakage process only (Reprinted with permission from Elsevier Science.)

fraction F^{x, t) along curves defined by h(x)t = z = constant. Thus, ^dx\ dt 'F,

_ /ax\ _

dz/dt

h(x)

V^'^/z

dz/dx

th\x)'

which is readily rearranged and solved as a differential equation in b{x) to give b{x) = b(xj exp

rinx

dint 51nx

Jinx

(5.2.10)

where we have set x^ to be some reference particle size and changed the integration variable by transforming it to the logarithm of particle size. The generality of (5.2.10) can be sensed from how it reduces to the power law case when the integrand in the foregoing is a constant, which follows from the situation in Figure 5.2.2. Thus the negative constant of - a for the integrand in (5.2.10) leads to the proper power law expression for b{x). The usefulness of (5.2.10) lies in its possible applicability for more general forms of b{xl however. The similarity variable, say z\ for a test of the data for self-

5.2. Similarity Analysis of Population Balance Equations

207

similarity can be calculated as z =

b{x)t

bix„)

texp

since the right-hand side of the foregoing equation can be estimated from the data. Data on drop size distributions obtained by Narsimhan et al. (1984) are displayed in Figure 5.2.3, which shows that the similarity

2 10-2

5

12 la'

5

r(v)t/r(vo)(min) FIGURE 5.2.3 Evidence of self-similarity in drop size distributions evolving purely by breakage obtained by Narsimhan et al. (1984). (Reproduced with permission of the American Institute of Chemical Engineers. Copyright © 1984 AIChE. All rights reserved.)

208

5. Similarity Behavior of Population Balance Equations

transformation does indeed work rather well. Similar data have also been obtained by Sathyagal et al. (1996). We shall have more occasion in the next chapter to discuss this issue in connection with inverse problems for breakage systems.

5.2.2

Pure Aggregation Processes

It is again expedient to use the equation in terms of the cumulative volume fraction presented in Chapter 3, viz., Eq. (3.3.9). Thus we have dF^{x, t)

8,F,{i, t)

- ^ 1

It

Jo

a{^, u) d,F,{u, t) u Jx-i

where a{x, y) is the aggregation frequency. Substituting for f i(x, t) the self-similar solution <^{r]) from (5.1.2), we obtain t]^'{r]) =

1

W)

d^(r]')

d^iri")

a{n'h{t), r]"h(t))

(5.2.11)

where h'{t) is the derivative of h(t) and the primes on the similarity variable do not signify any differentiation. From the foregoing equation, the feasibility condition for self-similarity is given by d_

Jt

1

d^r]')

dO(/y")

a{ri'h{t), ri"Kt)) = 0.

It is now possible to show that the differentiation can be carried inside the integration and to obtain the condition d_ -aitl'hjt), r,"h{t)) = 0 h'(t) Jt

(5.2.12)

which is necessary for self-similarity. Hence we may impose the general requirements that a{rj'h(t), r,"h{t)) = aW, n")H(hit)), h'{t) = cH(h{t)),

(5.2.13)

where c is a positive constant and H is any positive-valued function that remains to be specified. The positivity requirement on c and H is imposed so that the scaling size is increasing with time, since it is presumably related to increasing the average particle size in an aggregating population. In (5.2.13) we have let a(>/, t]') be a function different from a{ri, rj'). Indeed, a homogeneous frequency, which is a homogeneous function of the particle sizes, viz., a{2.x, Xy) = A'"a(x, y),

m ^ 0,

(5.2.14)

5.2. Similarity Analysis of Population Balance Equations

209

m being the degree of homogeneity, satisfies the requirement specified in (5.2.12) with H being given by the power law function, H{h) = If. For this case, the differential equation in (5.2.13) is immediately solved to get h{t) oc t^i^^'"^^ as an exphcit similarity variable. Moreover, we have a{r], rj') = When (5.2.13) holds, the self-similar cumulative distribution function from (5.2.11) can be seen to satisfy the integral equation dQ)irj')

d^{rj')

cr]Q>'{r]) = 0

Jrj-rt'

cc{rj\ f]").

(5.2.15)

V

The corresponding equation in the self-similar solution ^{Y]) is readily identified since from (5.1.2) we gave dO(f/) = Yi\j/{r]) drj, given (5.1.4). Thus we obtain

cfyVW

rj'il/{r]') df]'

\jj{f]")oi{f]\

f]") df]".

(5.2.16)

The constant c is evaluated by realizing the constraint to the right of (5.1.4) that yields c =

^"^ dn 0

n

rj'il/{rj') df}'

il/{ri")(x{rj\ n") drj".

(5.2.17)

A quick demonstration may be made of the constant aggregation kernel, which allows exact calculation of the self-similar solution. We let a{x, x') = a^, and recognize it to be homogeneous as described by (5.2.14) with m = 0 so that the scaling size h{t) is proportional to t. Equation (5.2.16) may be solved by Laplace transform, which is left as an exercise to the reader. However, the reader may more readily verify that \l/{fj) =^ e~'' for this case by substituting it into (5.2.16) and (5.2.17). It is of some interest to reflect on this exponential nature of the self-similar solution for the constant kernel. If the initial condition for the aggregation process had the number density vanishing at the origin, then clearly the scaled number density at finite times will not display the required behavior at the origin until the scaling size h{t) is large enough to produce small enough scaled sizes. Wang (1966) has investigated and established the convergence of the transient solution (for the constant kernel case) to the self-similar solution for a class of gamma distributions as the initial particle size distribution.^ The problem of establishing self-similar solutions for more complicated kernels is more difficult, however. 5

See also Drake (1970).

210

5. Similarity Behavior of Population Balance Equations

Wang (1966) has considered the "sum" kernel (a{x, y) = x + y) and the "product" kernel (a(x, y) = xy) for their self-similar forms and found them to be generalized functions, viz., Dirac delta functions thus ruling out the possibihty of observable self-similar behavior. However, this conclusion was clearly in error, as it is now known that both the sum and product kernels have the respective self-similar solutions

(Sum kernel)

(Product kernel)

both of which possess a singularity at the origin. Thus, the similarity variable (5.1.1) cannot be defined by using the scaling variable h{t) in (5.1.3) but instead should be defined by (5.1.6) with /c = 1 for the sum kernel, and k = 2 for the product kernel (Ernst, 1985). Numerical simulations of the population balance equation for the sum frequency performed by Wright and Ramkrishna (1992) are displayed in Fig. 6.2.7 in Chapter 6, which show self-similar behavior. Self-similar behavior from numerical calculations have been shown for the case of Brownian motion by Friedlander and Wang (1966)^ for which the aggregation kernel is given by 2kT a{x, X') = ^ [ x - ^/^ + x'-"'Xx'" 3/i

+ x'^'^l

The self-preserving size distribution is attained following a short lag during which the effect of the initial distribution is eliminated. Such lags have been estimated for log-normal initial conditions by Vemury et al (1994). 5.2.2.1

Experimental Evidence of Self-Similarity

The earliest experimental evidence of self-similarity in an aggregating system was provided by Swift and Friedlander (1964) and was obtained in the coagulation of hydrosols. Similar self-similar distributions have been obtained in the granulation of solids. Droplet size distributions obtained by Wright and Ramkrishna (1994) in a purely coalescing stirred liquid-liquid dispersion produced in a baffled mixing vessel are shown in Fig. 5.2.4. The data for the dispersion of a neutrally buoyant mixture of benzene and carbon tetrachloride in water are 6 See also Wang and Friedlander (1967).

5.2. Similarity Analysis of Population Balance Equations 1

1

•

0.9 0.8

•5 0.5 .|0.4

o^

•»

A t«10min

+

r

+ t«20 min

i

•

1

AI *

o t«30 min

I 0.3

mm—1

t«5 min

0.7 J © 0.6-d

211

'

Q

t«40 min O

<3o.2 0.1

a"

—. 1x10*^

c j q i ^ IM 1x10''

«»|^ > 1 1 1 M l

1x10*

1

1 1 > 1 1n

1x10°

' 1

1 > 111 I I I

1x10^

FIGURE 5.2.4 Self-similarity in coalescing drop size distributions from Wright and Ramkrishna (1994). (Reproduced with permission of the American Institute of Chemical Engineers. Copyright © 1994 AIChE. All rights reserved.)

presented in the form of a plot of the cumulative volume fraction of droplets at any time versus drop volume scaled by the number averaged drop volume at the instant in question. The figure clearly shows a collapse of the plots at various times into a single self-similar curve. In the foregoing experiments, no drop breakup was evident since the initial drops were rendered deliberately small (by prestirring at a much higher speed than that used in the follow-up coalescence experiment). Figure 5.2.5 shows even a more dramatic case of self-similarity since it shows a single self-similar curve encompassing the numerous cumulative volume distribution curves obtained in different experiments by Wright and Ramkrishna (1994) for a range of volume fractions of the dispersed phase and stirring speeds. A possible explanation for such "universal" self-similarity may He in a common degree of homogeneity of the coalescence kernels in all the different experiments. This extended self-similarity could not, however be found in experiments performed with another water-organic system (acetophenone in water). Self-similar behavior has also been observed in computer simulation of aggregation processes. Thus aggregates of colloidal particles in diffusionlimited aggregation processes have been found to display self-similar behavior (Meakin, 1983).

212

5. Similarity Behavior of Population Balance Equations 1000 o 1%. 1100-200 100

">( 6 ^

• 1%, 1200-200

,j>

%

S.

A. 5%. 800-200 * 5%. 1200-400

."1^

o 10%. 800-200 • 15%. 1300-200

\

*1

a 15%. 1400-400 • 25%. 1400-200

+

X 25%.1400-400 1x10^

IxlO"*

1x10"

1x10^

FIGURE 5.2.5 Universal self-similarity in coalescing liquid-liquid dispersions (neutrally buoyant benzene-carbon tetrachloride mixture with water). Data of Wright and Ramkrishna (1994). (Reproduced with permission of the American Institute of Chemical Engineers. Copyright © 1994 AIChE. All rights reserved.)

EXERCISE 5.2.1 Estabhsh directly by solving Eq. (5.2.16) via the method of Laplace transforms for the case of constant aggregation frequency, given by a{x, x') = a^ the self-similar solution {//{r]) = e'"'. (Hint: Recognize the convolution on the right-hand side of (5.2.16). Letting 4> = ^\ where i/^' is the derivative of the Laplace transform ij/ of ij/ respect to the transform variable s, obtain and solve a (separable) differential equation for the derivative of 0 with respect to i/^).

EXERCISE 5.2.2 Starting from Eq. (5.2.16), show that

^ - f + 2iA

a(r], rj')il/{t]') drj' -

-

oi{rj — rj\ rj')\l/{rj — rj') drj'.

drj

(Hint: Use the symmetry property of the aggregation kernel).

5.3. Self-Similarity in Systems with Breakage and Aggregation Processes

5.3

213

SELF-SIMJLARITY IN SYSTEMS WITH BREAKAGE AND AGGREGATION PROCESSES

A somewhat different form of scaling behavior is observable in systems in which both aggregation and breakage processes occur together. We are concerned here with systems in which the two processes occur such that they can be described by a population balance equation featuring an additive combination of the breakage and aggregation rates. Thus we may write the equation dMx, t) dt

1 2

a{x — x\ x')f^{x — x', t)f^{x\ t) dx' a{x, x')f^{x\ i) dx' 0

+

v{x')b{x')P{x I x')/i(x', t) dx - b{x)Mx, t).

(5.3.1)

When both breakage and aggregation processes occur, the question first arises as to whether a steady state distribution can emerge in which the number density remains constant, representing a dynamic balance between the two processes. Whether such a dynamic balance can occur must depend on the relative time scales of the two processes. If breakage occurs much more rapidly than aggregation, then a dynamic balance is not possible, since particles fragment to arbitrarily small sizes without aggregation having the opportunity to compensate for this effect. However, the size-specific nature of the time scales of the two processes calls for a more sophisticated argument on the issue. Broadly, a steady state can arise favoring sizes in the range in which the time scales of the two processes can balance if such a range were to exist. The question has been addressed by Vigil and Ziff (1989) at some length for an ad hoc class of aggregation and breakage kernels. In many physical systems, the smaller particles are much harder to break so that their time scales increase progressively making it possible for a match with aggregation time scales and consequently for a steady state to exist. An example of such a situation is the steady-state size distribution of dispersed phase droplets in a liquid-liquid dispersion, in which both breakage and coalescence processes occur, calculated by Zeitlin and Tavlarides (1972a).^ 7

Referenced in Chapter 4.

214

5. Similarity Behavior of Population Balance Equations

Our concern here in regard to the possibiHty of self-similar behavior is restricted to systems in which a steady-state balance is possible between aggregation and breakage processes. The aggregation kernel is assumed to be homogeneous with degree m, i.e., satisfying (5.2.14). In order to restrict the time scale of the breakage process, a small parameter, say fe, is used to characterize the breakage frequency. We assume further that breakage is binary^: b(x) = fcx^ a > 0;

P{x\y)=-^p(^].

Following Meakin and Ernst (1988), we assume that the transient period well before the attainment of steady state is aggregation dominated.^ Suppose we now allow the breakage parameter k to vary (continuously) from one system to another among a (continuous) collection of systems characterized by k. (The system in which no breakage occurs belongs to this family of systems with k = 0.) Then we shall be concerned with the number density /^(x, t;k) of particles of size x at instant t in system /c, which satisfies the population balance given by dMx,t',k) dt

^1 2

a{x — x\ x')/i(x — x\ t\ k)f^(x\ t; k) dx'

Mx,t;k) -f 2/c

a{x, x')f^{x\

^ " " P ( - ) fM\

t;k)dx'

t\ k) dx' - kx%{x, t\ k). (5.3.2)

A similarity analysis of the foregoing equation with the form Mx,t;k)

=g{t-k)il/{r],TX

rj =

h{t;k)'

T

=

lik)

Vigil and ZifF (1989) dispense with the assumption of binary breakage in their analysis of self-similarity but appear to assume a constant mean number of fragments independently of the size of the fragmenting particle. These authors point out the further restriction of excluding "gelling" aggregation kernels and "shattering" breakage kernels from this analysis. This requires that m ^ 1 and a ^ — 1.

5.3. Self-Similarity in Systems with Breakage and Aggregation Processes

215

yields on substitution into the population balance equation (5.3.2)

Q

1/dg g\dt 1 2

ij/irj, T)

rj fdh\ dil/ 1 dij/ + • h\dtjj^drj l{k) dx

a{Yi — r]\ Yi')\j/{r] rj', T)il/{r]\ T) drj'

kk"

+ •

-txjj

9

^-2

a{Y], r]')\j/(Y], T)^(Y]\ T) dri'

n

n'"~'p[^A^in\^)dri'

(5.3.3)

The invariance we seek is of the scaled number density associated with different systems of the same total mass. Thus the scaled distribution is the same for all systems at the same scaled time T and the similarity analysis must require that the quantities encountered in (5.3.3) that are combinations of terms involving unsealed time t and k must be independent of k (and dependent only on scaled time T). However, before we proceed with that step, we note that (5.1.3) must hold for each k so that h\t;k)g{t;k)

= fi,.

(5.3.4)

the right-hand side of which has the first m o m e n t fi^ which must remain constant, through the conservation of mass for each system. In view of (5.3.4), the different time-dependent coefficients in (5.3.3) may be assembled as follows: Left-hand side of (5.3.3)

Right-hand side of (5.3.3) fc/i^

Ml

fij(k) On setting the middle term in the left-hand side equal to a constant and switching the differentiation to that with respect to scaled time, we obtain the same requirement as that on the bottom term on the left-hand side, viz., U-m+l

fiM

function only of T (but independent of k)

216

5. Similarity Behavior of Population Balance Equations

from which it follows that h = [/(/c)]^/^^~^^^/f(T). Requiring the right-hand side term to be independent of fe, one obtains kll{k)T~"'^'^^^'~'"^ - 1

or

l{k) =

fe-d--)/(«--+!),

so that

Lastly, the requirement that the top term in the left-hand side column be independent of k leads to ^

^ l{k)W^^'g^

^ /,-2/(a-.n+l) 2

oX

Q^

k^l^^-"^^'^H{T)-\

where consistency is enforced with (5.3.4). In the foregoing treatment, we have not been concerned about the specific values of the exponents a and m. In particular, the case of m = 1 requires special treatment but one that is easily done and is left to the reader. It further transpires that the exponents must satisfy a constraint in order that the steady state between aggregation and breakage is attained eventually. We consider this next. Since the process (for each k) must be aggregation dominated at the outset, the function g must decrease with time while h must correspondingly increase in order to maintain (5.3.4). Thus //(i) must consequently increase with time. Further, if we arrange that T ^ 0 as /c -^ 0, then the resulting purely aggregating system admits a self-similar solution that we take to be \I/{Y], 0). We wish further to see that t ^ oo as /c -^ oo in such a way that T is constant in order that the self-similar solution \jj{r}, i), while representing the specific solution for a system of parameter k at time t, also encompasses the steady state solution at t = co. Since gelling kernels are forbidden (see footnote 9) we have m ^ 1. Consequently, for the desired relationship among the variables t, k, and i, we must require that a — m + 1 > 0. This inequality has been regarded as a stability condition for the attainment of equilibrium between aggregation and breakage. Computational demonstrations have been made of the existence of the similarity distribution il/{r], T) by Meakin and Ernst (1988) (see footnote 9). The importance of this form of self-similar behavior does not appear to have been realized in experiments. A particularly fruitful area of application lies in the experiments of Wright and Ramkrishna (1994) with hquid droplets in a stirred liquid-liquid dispersion without the restriction imposed by these authors to purely coalescing dynamics, i.e., with the inclusion of droplet breakup as well.

5.4. Self-Similarity in Systems with Growth

5.4

217

SELF-SIMILARITY IN SYSTEMS WITH GROWTH

Instances of self-similarity in the presence of particle growth have been relatively rare in the literature. However, the author has discussed the possibility of self-similar behavior in the dynamics of microbial populations (Ramkrishna, 1994; Ramkrishna and Schell, 1999). We present a simple example to illustrate the broad ideas from the foregoing development. We consider a population of cells distributed according to cell mass x with growth rate X{x). The cells further undergo binary division at the rate b{x) into daughter cells whose mass distribution is given by P{x \ y) where y represents the mass of the mother cell and x is the mass of either daughter cell. The population balance equation may be written as ^

^

+ ^ [X(x)/i(x, t)] = -b{x)fM,

b(x')P(x

t) + 2

\x')fiix',t)dx'. (5.4.1)

We assume the following homogeneity relations for the X{x) and b(x): X{^x) = rX{xX

b{lx) = l^b{x).

(5.4.2)

We also impose the equivalent of condition (5.2.3) on the function P{x \ y), viz., (5.4.3)

Pix\y)=^^p(^\

In order to seek self-similar behavior of the number density function /^(x, t), we let it be of the form (5.1.1), i.e., /^(x, t) = g{t)il/{f]) so that (5.1.3) must be true. Substituting this expression in Eq. (5.4.1) we obtain, in view of the homogeneity conditions (5.4.2), and the condition (5.4.3),

= -b{n)ijj{n) + 2

rj

\rj

where the prime on any function is used to denote differentiation with respect to the argument of that function. The left-hand side of this equation depends on time and rj while the right-hand side is a function of rj alone; this situation can only be corrected by setting the combination of time-

218

5. Similarity Behavior of Population Balance Equations

dependent quantities to be in fact independent of time. Thus we require that g' = gh^c,

h' = -h^^^c^,

a = jS + 1

(5.4.5)

where c and c^ are constants so that we have differential equations in the functions g and h. The constants c and c^ are related through an overall number balance of cells obtained by integrating Eq. (5.4.1) with respect to X between 0 and oo. The procedure yields on recognition of the self-similar solution c=

b{y)il/{y)dy + c^

The condition on the exponents a and P appears to be overly restrictive but one to which we shall return presently. On solving the differential equation for h, and using the relationship (5.1.3) to set gh = /IQ, we obtain -1

g{t) = fioit) ^ + pc,t

-|i//^

•1//?

,

h{t)

ht

+ Pc^t

(5.4.6)

where h^ is the "initial" value of h at some reference time r = 0 (at the onset of self-similar behavior). If it happens that for some interval of time the time-dependence of h in (5.4.6) is not perceptible then we may interpret this as what is known as "balanced" growth in the microbiological Hterature during which the population density ixj^t) increases exponentially.^^ This situation is, however, not one of true self-similarity, as the scaling cell mass is not time-dependent. It is the circumstance of perceptible dynamic variation in h that constitutes true self-similar behavior. We now return to the conditions in (5.4.5), which were necessary for self-similarity. It was observed that the last condition on the exponents a and P is overly restrictive.^^ In the absence of this equality, however, the time-dependence of the third term in the left-hand side of (5.4.4) is incompatible with its time-independent right-hand side. In this case, consistency with self-similarity can come about only by the diminishing of the term ^a-A-i relative to the other terms in the left-hand side of (5.4.4). Thus, if h decreases with time (which occurs when c^ is positive) then self-similarity can occur only when the exponents a and P satisfy the inequality a — jS — 1 ^ 0. This inequality is reversed if h increases with time. Self10 See for example, pp 142-143 of Davis et al (1968). 11 Equality constraints were used by the author in a multi-dimensional setting (Ramkrishna, 1994).

References

219

similar behavior has been observed computationally by Ramkrishna and Schell (1999) on using growth and cell division models for a cell population distributed with respect to two cellular constituents which satisfy the requirements corresponding to (5.4.4). N o computational demonstrations have been made of the possibility of self-similar behavior when the exponents satisfy inequality constraints.

REFERENCES Davis, B. D., R. Dulbecco, H. N. Eisen, H. S. Ginsberg, and W. B. Wood, Jr., Principles of Microbiology and Immunology, Harper & Row, New York, 1968. Drake, R. L., "A General Mathematical Survey of the Coagulation Equation," in Topics in Current Aerosol Research, (Part 2), (G. M. Hidy and J. R. Brock, Eds.), p. 315, Pergamon Press, New York, 1970. Ernst, M. H., in Fundamental Problems in Statistical Mechanics VI, (E. G. D. Cohen, Ed.) North Holland, Amsterdam, 1985. Filippov, A. S., "On the Distribution of Sizes of Particles which Undergo SpHtting" (translated by N. GreenleaO, Theory of Prob. and its Applns., 6, 275-294, (1961). Friedlander, S. K., "Similarity Considerations for the Particle Size Spectrum of Coagulating, Sedimenting Aerosol," J. Meteor., 17, 479-483, (1960). Friedlander, S. K., "Theoretical Considerations for the Particle Size Spectrum of the Stratospheric Aerosol," J. Meteor., 17, 753-759, (1961). Friedlander, S. K., and C. S. Wang, "The Self-Preserving Particle Size Distribution for Coagulation by Brownian Motion," J. Colloid Interf Sci., 22, 126-132, (1966). Kapur, P. C , "A Similarity Solution to an Integro-Differential Equation Describing Batch Grinding," Chem. Eng. Sci, 25, 899-901, (1970). Kapur, P. C , "Self-Preserving Size Spectra of Comminuted Particles," Chem.Eng.Sci., 27, 425-431, (1972). Lushnikov, A. A., "Evolution of Coagulating Systems," J. Colloid Interf. Sci., 45, 549-556, (1973). Madden, A. J., and B. J. McCoy, "Drop size in stirred liquid-liquid systems via encapsulation," Chem. Eng. Sci., 24, 416-420, (1969). Meakin, P., "Formation of Fractal Clusters and Networks by Irreversible DiffusionLimited Aggregation," Phys. Rev. Lett., 51, 1119-1122 (1983). Meakin, P. and M. H. Ernst, "Scaling in Aggregation with Break-up Simulations and Mean Field Theory," Phys. Rev. Lett., 60(24), 2503-2506 (1988). Narsimhan, G., D. Ramkrishna, and J. P. Gupta, "Analysis of Drop Size Distributions in Liquid-Liquid Dispersions," AIChE J. 26, 991-1000 (1980). Narsimhan, G., Nejfelt, G. and D. Ramkrishna, "Breakage Functions for Droplets in Agitated Liquid-Liquid Dispersions," AIChE J., 30, 457-467 (1984).

220

5. Similarity Behavior of Population Balance Equations

Orr, C. Jr., Particulate Technology, Macmillan, New York, 1966. Ramkrishna, D., "Drop-Breakage in Agitated Liquid-Liquid Dispersions," Chem. Eng. Set, 29, 987-992 (1974). Ramkrishna, D. "Towards a Self-Similar Theory of Microbial Populations," Biotech. & Bioeng., 43, 138-148, 1994. Ramkrishna, D. and J. Schell, "On Self-Similar Growth," J. Biotechnology, 71, 255-258, 1999. Sathyagal, A., G. Narsimhan, and D. Ramkrishna, "Breakage Functions of Droplets in a Stirred Liquid-Liquid Dispersion from Experimental Drop Size Distributions," Chem. Eng. Sci., 51, 1377-1391 (1996). Swift, D. L., S. K. Friedlander, "The Coagulation of Hydrosols by Brownian Motion and Laminar Shear Flow," J. Colloid Sci., 19, 621-647 (1964). van Dongen, P. G. J., and M. H. Ernst, "Scaling Solutions of Smoluchowski's Coagulation Equation," J. Stat. Phys. 50, 295-329 (1988). Vemury, S., K. A. Kusters, and S. E. Pratsinis, "Time-Lag for Attainment of Self-Preserving Size Distribution by Coagulation," J. Coll. & Interf. Sci., 165, 53-59 (1994). Vigil, R., and R. M. Ziff, "On the Stability of Coagulation-Fragmentation Population Balances," J. Colloid. Interf. Sci., 133, 257-264 (1989). Wang, C. S. "A Mathematical Study of Particle Size Distribution of Coagulating Disperse Systems," Ph.D. Thesis, California Institute of Technology, Pasadena, CA, 1966. Wang, C. S., and S. K. Friedlander, "The Self-Preserving Particle Size Distribution for Coagulation by Brownian Motion: II, Small Particle Slip Correction and Simultaneous Shear Flow," J. Colloid Interf. Sci., 24, 170-179 (1967). Wright, H., and D. Ramkrishna, "Solutions of Inverse Problems in Population Balances-I. Aggregation Kinetics," Comp. Chem. Eng., 16, 1019-1038 (1992). Wright, H., and D. Ramkrishna, "Factors Affecting Coalescence Frequency of Droplets in a Stirred Liquid-Liquid Dispersion," AIChE JL, 40, 767-776 (1994). Ziff, R. M., M. H. Ernst, and E. M. Hendriks, "Kinetics of Gelation nd Universality," J. Phys. A: Math. Gen. 16, 2293-2320 (1983).

CHAPTER 6

Inverse Problems in Population Balances

We address, in this chapter, problems of fundamental importance in the application of population balances. The population balance equation is based on a number balance that arises from the consideration of single particle behavior. Since the particle behavior must be considered in the local population setting, it is often not an experimentally accessible quantity, for it calls for observation on specific particles that can be readily obscured by the presence of its numerous neighbors. Consequently, the approach has been one of assuming the validity of single particle behavior obtained in isolation from its neighbors either from experiment or theory. Obviously, in order to obtain better characterizations of single particle behavior in a population, experimental observations must be made on the population and a method must be found to extract the behavior of single particles from such measurements.^ We shall refer to this as the inverse problem approach, its main advantage being that it is not committed to any specific form of the model function under investigation. When an available model form is inappropriate, parameter-fitting procedures will at best lead to compromise choices of the parameters resulting in inadequate particle models. 1

The problem thus stated is akin to the famiUar question in fluid mechanics of how Eulerian observations, that are more conveniently made, can be converted to Lagrangian information that is often more relevant and the desired quantity. This is because Eulerian observations are made at a fixed point in space whereas Lagrangian measurements require tracking a specific particle in motion.

221

222

6. Inverse Problems in Population Balances

The single particle behavior pertains both to continuous changes such as particle growth processes in which a particle retains its identity, and processes such as aggregation, breakage, and nucleation in which termination and/or initiation occurs of the identity of particles. Thus, we shall be interested in particle growth rates as well as the phenomenological quantities associated with the description of breakage and aggregation rates from suitable experimental observations on the population. The experimental observations are generally dynamic measurements on the population that evolves in time.^ Furthermore, it is desirable to consider situations in which only one of the different particle processes is present so that the inversion is accomplished without unduly risking loss of uniqueness. Thus, the measurement of number densities in particle size at various times in a pure breakage or pure aggregation process constitutes an example of population data. The inversion of such data, however, represents a difficult problem, since it is generally ill-posed, by which is meant that small errors in the input data produce large errors in the extracted information. This calls for some presmoothing of the input data, thereby substantially raising the required amount of data. In this connection, our approach relies on the exploitation of self-similar behavior, dealt with in Chapter 5, which when applicable, allows for more effective use of experimental data.^ This is because the input data in such a case is the self-similar distribution that collapses all the dynamic data into a single self-similar curve providing for a large number of points to facilitate the presmoothing process. It will emerge that the solution of the inverse problems is greatly assisted by the use of any analytic information that is available on the nature of the self-similar distribution such as its asymptotic properties. In discussing the different processes individually, this feature will become apparent.

6.1

THE INVERSE BREAKAGE PROBLEM: DETERMINATION OF BREAKAGE FUNCTIONS

The modeling of a breakage process has been discussed in Section 3.2 of Chapter 3. We assume that no particle growth occurs and that aggregation There are examples in the Hterature of fitting parameters to single particle models in both aggregation and breakage processes until an experimentally measured "equiUbrium" particle size distribution is closely matched by the solution to the population balance equation. The rationality of such a procedure is much in question, as it is clearly not sensitive to the time scales of breakage and aggregation. Furthermore, numerical "regularization" procedures, to be referred to subsequently, are required to restore well-posedness to the inversion problem.

6.1. The Inverse Breakage Problem: Determination of Breakage Functions

223

events are absent from the system so that the population balance equation of specific interest here is Eq. (3.2.8). However, we shall prefer the form (3.2.13) in the cumulative volume fraction, which is dF (x t) r°° %' = 1 b{x')Gix\x')d,,F,{x',t).

(6.1.1)

The mathematical statement of the inverse problem is as follows: Given measurements of F^{x, t), the cumulative volume (or mass) fraction of particles of volume ( ^ x) at various times, determines, b{x), the breakage frequency of particles of volume x, and G{x \ x'), the cumulative volume fraction of fragments with volume ( ^ x ) from the breakage of a parent particle of volume x'. Obviously, the experimental data on F^(x, t) would be discrete in nature. We assume that G{x \ x') is of the form (5.2.9) and rely on the development in Section 5.2.1.1 using the similarity variable z = b(x)t. Self-similarity is expressed by the equation F^{x, t) = 0(z), which, when substituted into (6.1.1), yields the equation zO^z) =

g(j^]z'nz')dz^

(6.1.2)

where the prime on the function 0(z) represents its derivative with respect to z. The function g{x) is connected to G{x \ x') through (5.2.9), which is

Since the discussion in Section 5.2.1.1 goes into how the experimental data can be subjected to a similarity test, we avoid its repetition here."^ However, we mention here a feature that has been added by Sathyagal et al (1995) to the foregoing test. Since self-similarity implies that the cumulative fraction F^ be invariant on z = b{x)t, the relationship between t and x at constant z (or equivalently constant F^ can be obtained from the data of F^ versus x at various t. Further, as it is readily seen that dXnt d\nx

^

, (6.1.4) dmx it follows that the left-hand side of the preceding relationship must inherit its independence of F^ from that of the right-hand side, which depends only on the particle volume x. Thus, the left-hand side of (6.1.4), obtained from the cumulative fraction curves at different times, must also show the same 4 This test was originally due to Narisimhan et al, (1980, 1984) referenced in Chapter 5.

224

6. Inverse Problems in Population Balances

dependence on particle size as the right-hand side. The slope of the In t versus \nx curves at various F^ must depend only on x so that by translating the curves along the In t coordinate (i.e., along constant x lines) we expect that the curves must merge into a single curve. This curve must span some reference size x^ up to the maximum possible size x^^^ for which data could be collected. The merging family of curves will of course contain overlapping particle size intervals over which the smoothness of the merger will provide for a test of the similarity. The entire procedure is handled by Sathyagal et al (1995) by fitting the different In t versus In x data for each F^ to a smooth (quadratic) curve from which the slope and curvature of the fitted curve could be calculated at each particle size and examined for consistency. This procedure will lead to calculation of the left-hand side of (6.1.4) for the particle size range {x^,x^^^). Equation (5.2.10), which arises from integrating (6.1.4) and is reproduced below for ready reference. dint

b{x) = b{x^) exp

^Inx

(6.1.5)

yields the breakage frequency up to the unknown multiplicative constant b{xj. Since b{x) is not known exactly, neither is the similarity variable z; it can, however, be modified to z defined by . b{x)t z = -— = exp

In Xo

d\nt 51nx

d\nx

(6.1.6)

which is explicitly known by virtue of the expression on the extreme right. Thus, in case self-similarity is observed experimentally, a further test of it lies in a plot of F^ versus z at different times showing a single collapsed curve. In terms of the modified similarity variable z, Eq. (6.1.2) becomes

mz) = b{xj

g(-)z'nz')dz\

(6.1.7)

where $ ' is the derivative of 3) with respect to z. Transforming variables, may rewrite (6.1.7) as z^'(z) = p

-^(t>'(z)g{u) du,

P = b{xJ.

(6.1.8)

In what follows, we let u represent the ratio of the breakage rate of the fragment to that of the parent particle. The statement of the inverse problem lies in calculating the unknown function g{u) over the unit interval and the constant P given the self-similar curve in the form of O' versus z. Since g{u)

6.1. The Inverse Breakage Problem: Determination of Breakage Functions

225

is a cumulative distribution function over the interval 0 ^u ^ 1, the solution of Eq. (6.1.8) for the function Pg{u) at w = 1 automatically yields the value of jS as ^(1) = 1.

6.1.1

Solution of the Inverse Problem

The solution of Eq. (6.1.8) is accompHshed by Sathyagal (1995) by expanding the function Pg{u) in terms of an appropriate set of basis functions as rib

Pg{u) = X ^j^M)

(6-1.9)

where n^ is the number of basis functions and a^, ^2? • • • ? ^n^' which we shall jointly denote by a vector a, are the coefficients of expansion to be evaluated. The similarity coordinate z is discretized into several (m) points {zj to get a corresponding discrete version of the self-similar distribution {2^0' (z^), i = 1,2,..., m}, which we shall represent, by a vector O. If we denote by a the vector whose components are {aj'J = 1,2,..., n^} then the discrete version of the inverse problem becomes
(6.1.10)

where X is a matrix whose components are given by Xij^

u

' <X)' I - I Gj{u) du, \u

i = 1 , 2 , . . . , m; j = 1, 2 , . . . , n^

The solution of the discrete inverse problem is to seek the vector a by minimizing the magnitude (or "norm") of the residual vector Xa — O,^ i.e., min | | X a - 0 | | .

(6.1.11)

For a terse introduction to the mathematical background and the associated Hterature on the solution of inverse problems, the reader is referred to the publication of Sathyagal et al, (1995) and to Wright and Ramkrishna (1992). In regard to the mathematical literature on inverse problems we refer to Tikhonov and Arsenin. (1977). The solution of inverse problems of the type $ = Xa is best estabhshed for situations in which the input data represented by the vector 0 has noise and the operator X is noise-free. Since, in the context of our discussion, X inherits the noise of the self-similar data, the presmoothing process discussed in the text is an important issue. The quaUty of the inversion can of course be assessed by evaluating the ability of the "forward" problem to predict back the data used for inversion.

226

6. Inverse Problems in Population Balances

Such an inverse problem (as the continuous version 6.1.8) is ill-posed in the sense that errors in the input vector O greatly amplify those in the solution vector a. The regularization strategy of Tikhonov and Arsenin (1977) seeks to cure the ill-posedness by minimizing the norm of the residual suitably weighted with the norm of the solution vector, i.e., min[||Xa-0||+2,,g||a|r], where A^gg is a regularization parameter and the prime on the norm of a is used to indicate the possibility of using one that is different from that for the residual vector. This strategy is motivated by having to penalize unreasonable fluctuations created in a for the residual vector to match the noisy data €>. The value of the regularization parameter A^eg depends on the appUcation. In the situation at hand, we have other sources of regularization such as the monotonicity requirement on the unknown function g{uX since it is in fact a cumulative distribution function on the unit interval. Sathyagal et a/., (1995) used the constraints g{u) > 0,

g'(u) ^ 0,

^'(1) = 0,

the last equality constraint being somewhat specific to the application considered by these authors.

6.1.1.1

Choice of Basis Functions

The choice of the basis functions depends crucially on the behavior of the self-similar distribution (see footnote 5). For example, suppose that the self-similar distribution zO'(z) has the asymptotic behavior z^ (/z < 1) in the region of z close to zero. Then it is possible to show (see Appendix of Sathyagal et a/., 1995) that the function g{u) is approximated by u^ for u close to zero. In other words, g{u) is of order 0{u^).^ Consequently, g{u) is not analytic at w = 0, and a very large number of basis functions in the expansion (6.1.9) are required to describe adequately the behavior near the origin. This problem can be overcome by choosing basis functions that have the same dependence on u near w = 0, as g{u) does. Incorporating as much known analytical information as possible about the nature of the solution is an important aspect of the solution of inverse problems. Let us see how 6

g{u) is said to be 0{u^) implies that lim„^o+^M/"'' < ^ -

6.1. The Inverse Breakage Problem: Determination of Breakage Functions

227

this asymptotic behavior can be incorporated into the trial functions. Since g{u) is defined over the unit interval, we first choose a Hnear space ^ of functions defined on [0,1] with an inner product between any two (realvalued) elements (j){u) and il/{u) by w{u)(j){u)il/{u) du,

<4>,^}^

u,vE^.

(6.1.12)

Further, consider a set of functions, {Jj{u)}, each of which is of order 0(1) for small u and orthonormal with respect to the inner product (6.1.12). In other words. fO, j ^ k w{u)Jj{u)Jj^{u) du = 6jj^ = < ' . _ ;o L^^ J — ^ We seek as our basis functions

a,-,A>-

^

G.{u) = u^Jjiu)

(6.1.13)

(6.1.14)

so that each Gj{u) is of order 0{u^) and choose a new inner product on the linear space if defined by (
r{u)(l){u)il/{u) du

s

(6.1.15)

0

in which the weight function r{u) is to be chosen such that the basis functions {Gj} are orthonormal with respect to the inner product (6.1.15). Thus, we may write {Gi, Gj) =

pi

p r{u)Gi{u)G^{u) du = r{uy^J,{u)J^{u) du - d,^, (6.1.16) Jo Jo

in which we have inserted (6.1.14). Comparison of (6.1.16) with (6.1.13) leads to the obvious choice of r{u) = w{u)u~^^. Sathyagal et al. (1995) made the choice of the Jacobi polynomials^ orthogonal with respect to (6.1.13), for which w{u) = u so that r{u) = w^"^^ Sathyagal et al (1995) give a striking demonstration of the solution of the inverse problem using both computer-simulated data and as experimental data with hquid drops in a stirred hquid-liquid dispersion. The computer-simulated examples provide for a direct comparison of the 7

See page 774 of Abramowitz and A. Stegun (1964). The Jacobi polynomials belong to a family characterized by two integers p and q. The polynomials chosen in this context corresponded to p = g = 2.

228

6. Inverse Problems in Population Balances

inverted information with that used in the simulation, while those using actual data can be checked by solving the forward problem for comparison of the data with those predicted using the breakage functions obtained by inversion. We consider here an example each of both types.

6. / . 2 . /

Computer-Simulated Data

We reproduce here an example considered by Sathyagal et al. (1995). The breakage frequency^ is assumed to be of the form b{x) = 1.2exp[0.12(lnx + 3.5) - 020(\nxf

- 12.25],

(6.1.17)

which was obtained from Narsimhan et al (1984) from experimental data on drop breakage. The cumulative distribution function g{u) for the broken fragments was assumed to be g(u) = | v ^ - | w ^ • ^

(6.1.18)

The breakage process was simulated by Sathyagal et al. (1995) using the single particle technique of Section 4.6.4 to obtain the cumulative volume fraction F^ of drops of various sizes x at different times t. Their test of similarity, made through a plot of In t versus In x for 14 different values of the cumulative fraction F^ is represented in Fig. 6.1.1 below. That conformation to self-similarity is excellent is evidenced by a plot of the arc length versus particle size curve shown in Figure 6.1.2. This plot was obtained by assembling different segments of the curve for different cumulative volume fractions by fitting a smooth quadratic curve to In t versus In x from all of the data at hand. At this stage, it is possible to calculate the breakage frequency up to the multiplicative constant P or the similarity variable z by performing the integration in (6.1.6) using the curve fitted to the In t versus In x data. Figure 6.1.3 provides direct evidence of self-similarity of the data since it shows a plot of zO'(z) versus z. The approach to self-similarity is quick over the time scale of the experiment.

The expression in (6.1.13) above was obtained from the correlation developed by Narsimhan et al, (1984) (referenced in Chapter 5) from drop size distribution data on the breakage of liquid drops in a manner very similar to that of Sathyagal et al (1995). Although their methodology of Narsimhan et al (1984) was similar to that of Sathyagal et al (1995), their solution to the inverse problem was vastly improved in the latter publication.

•

o •

A

•

o

•

•

•

F = 0.02

*

F = 0.6

o

F = 0.05

»

F = 0.7

•

F = 0.1

•

F « 0.8

A

F = 0.2

X

F « 0.9

•

F = 0.3

*

F»0.92

1

F»0.4

•

F-0.95

+

F = 0.6

«

F.0.97

A T 1 • * » • i 4 # • » lt>^^

A ^

c

•

1 • * s • MHk-tH

•

A

3

•

O

•

A

•

O

T 1 4 * m • A

•

UHf*

T 1 * *• 9 % U-A-m

2

•

11

O

•| -9

-8

-7

A

• 1 « 4t 9 •

' ' • ' 1

Inx

-6

1

.

.

.

mirMf

J

-5

FIGURE 6.1.1 Similarity test of simulated data on the t-x plane at fixed cumulative volume fractions (from Sathyagal et al. 1995) (Reprinted with permission from Elsevier Science.)

•

F « 0.02

*

F « 0.6

o

F«0.06

«

F « 0.7

•

F«0.1

•

F « 0.8

A

F«0.2

n

F « 0.9

•

F.0.3

•

F-0.92

1

F«0.4

•

F-0.95

+

F«0.5

*

F«0.97

B

-10

-15

I

-9

I

I

I

I

I

-8

I

I

r

I

-7

-6 In X

FIGURE 6.1.2 Plot of arc length versus In (particle volume) showing a single smooth curve in confirmation of self-similar behavior (from Sathyagal et al, 1995. Reprinted with permission from Elsevier Science.) 229

230

6. Inverse Problems in Population Balances

1x10-*

1x10"^

1x10"®

1x10^ 1x1(r* 1x10-^ 1x10-2 Similarity Variable, C,

1x10-^

1x10°

1x10^

FIGURE 6.1.3 Plot of zQ)'(z) versus z showing self-similarity over a wide range of the simulated dynamic data (from Sathyagal et a/., 1995). (Reprinted with permission from Elsevier Science.)

Finally, we inquire into the inverse problem for the determination of the function g(u). Since the inverse problem is in terms of the function Pg{uX we examine the results with respect to it. In Fig. 6.1.4 is shown the solution to the inverse problem as a function of n^, the number of basis functions. Plotted alongside is the actual function used in the simulation. It is interesting to note that n^ = 3 or 4 provides the best solution to the inverse problem while the solution for 5, 6, and 7 show progressive deterioration. The demand on accuracy goes up with an increasing number of functions, and even minor errors in the self-similar distribution will cause significant deviations of the inverted solution from the actual function as seen in Fig. 6.1.4. The sensitivity at higher number of basis functions may be curable by regularization. Thus, Fig. 6.1.5 shows that a small regularization parameter quells the oscillatory behavior at the right end, enabhng a good estimate of the parameter jS.

1.8

j

Lk'''i

1.6

1

1.2

4(^^"^

[0.9 0.6

•

„ -^' r ^

j/

actual

• • -

nb = 3

— —

nb = 4

— ••

nb = 6

0.3 0

Db»6 rib = 7 1

1 • t

1

1

1

»'

1

0.2

1 ' ' ' ' 1

1

0.4

0.6

1

r

1

>

0.8

FIGURE 6.1.4 Comparison of inverse problem solution with the actual cumulative size distribution for breakage fragments (from Sathyagal et al, 1995). (Reprinted with permission from Elsevier Science.)

1.8 1.6

•' 1.2

.

-

r-

—

actual 0.61 - - • Xreg«0 0.31 - - Xreg = 0.002

)~

1

1

1

1

'

0.2

1

1

1

1

1

1

• 1

0.4

0.6

'

•

'

'

1

•

•

'

•

'

0.8

X

FIGURE 6.1.5 Effect of regularization on the inverse problem (from Sathyagal et al, 1995). (Reprinted with permission from Esevier Science.) 231

232

6. Inverse Problems in Population Balances 1.4-

j

actual

J • • •

pred

J c

I

• J

©0.8 J oc |0.6

]

0.4

"1

0.2 1

1

1

1

1

"^T""^""'*^^^

100

1

1

t

200 300 Drop Diameter, ^m

• t

400

FIGURE 6.1.6 Breakage frequency predicted from solution of the inverse problem compared with the true one used for simulation (from Sathyagal et al, 1995). (Reprinted with permission from Elsevier Science.) With the parameter jS determined as the value of inverse problem solution for Pg{u) at w = 1, the complete inversion of the problem is at hand. Figure 6.1.6 shows the breakage frequency function b{x), which is indistinguishable from the function used to simulate the data. 6.1.2.2

Experimental Data on Drop Size Distributions from a Purely Breaking Dispersion

Sathyagal et al. (1995), have obtained experimental data on lean liquidliquid dispersions in a stirred mixer in which the evolution of drop size distributions occurred virtually without any significant coalescence. The cumulative drop size distributions, obtained by image analysis of dispersion samples carefully withdrawn from the mixer, are shown in Fig. 6.1.7. Treating the experimental data to the same similarity test recounted eariier, the similarity variable z is calculated and a plot of zQ>\z) made against z, which is shown in Fig. 6.1.8. In view of the collapse of the data in Fig. 6.1.8, it is evident that the data are indeed self-similar. Proceeding with the solution of the inverse problem,

J •

t*0.5

J o

t-1

A

t«2

1

1 Q

t-5

E

1 « t«15

0.8 i

OT1

-:L WW ur-cr • •

•

1

«0.6 J • ^"^0

>

5

1 •

3

3

^^f a -Sir 1

E

o

t-30

0.4

0.2

•do f'

1 M i l l '^BBHHMB

1x10'^

1x10-*

1 1 1 I I I Hi"•

1x10"* 1x10"^ 1x10"^ Drop Volume, \i\ii

' "1' » 1 m i j

1x10"'

1x10-^

FIGURE 6.1.7 Cumulative volume fractions of drop sizes in a lean liquid-liquid dispersion evolving by pure breakage (from Sathyagal et a/., 1995). (Reprinted with permission from Elsevier Science.) 0.35

0.3 H

0.25

E 0.15

E 0.05

1x10'^ 1x10"® 1x10-^ 1x10-^ 1x10"^ 1x10-2 1x10-^

1x10°

1x10^

Similarity Variable FIGURE 6.1.8 Evidence of self-similarity in dynamic evolution of drop size distributions during breakage (from Sathyagal et a/., 1995). (Reprinted with permission from Elsevier Science.) 233

234

6. Inverse Problems in Population Balances 4.23.5 2.8 -5 2.1

1 >/c^

1 1

x nb = 3

/

1.4

1

,,^rrT.'.nT\

Jf

...

nb « 4

— —

nb«5

0.7

1 lU « O

nb-7 r

1 1 1 1

1

0.2

1

1

1

1

0.4

I

1

u

l'

1 ' ' ' '

0.6

1 1 I

1 1 1

0.8

FIGURE 6.1.9 Cumulative distribution for the size of breakage fragements from the solution of the inverse problem for various choices of the number of basis functions (From Sathyagal et al, 1995). (Reprinted with permission from Elsevier Science.) Note that x here represents the ratio of the breakage rate of broken fragment to that of the parent particle. the cumulative size distribution of the breakage fragments is displayed in Fig. 6.1.9 for various choices of the number of basis functions. Since the actual breakage functions are unknown in this situation, the test of inversion lies in recovering the dynamic data by forward simulation using the identified breakage functions. By choosing an early measurement of the size distribution as the initial distribution, the prediction of its evolution with time is shown in Fig. 6.1.10. The predictions are reasonably close to the measured distributions even at times far removed from the initial time. When self-similarity is observed, the solution of the inverse problem provides a very satisfactory estimate of the breakage functions. Sathyagal et a/., (1996) (see Chapter 5) show the applicabihty of self-similarity in drop breakage under a variety of experimental conditions. The calculation of breakage functions has also been of interest to metallurgical engineers in the past. Gardner and Sukanjnajtee (1972) designate the inverse problem approach as the back-calculation method. These authors consider time-dependent as well as time-independent breakage functions in grinding dynamics free from self-similarity assumptions. Their

6.2. The Inverse Aggregation Problem

235

0.016

0.012 c

a

c •

o 0.008 0)

E o

>

0.004

100

200 Drop Diameter, jim

300

400

FIGURE 6.1.10 Prediction of the temporal evolution of drop size distributions using the breakage functions from the solution of the inverse problem. The initial condition for the solution is the measured distribution at some initial time (from Sathyagal et al, 1995). (Reprinted with permission from Elsevier Science.)

discussion covers direct methods using tracers on a single specific particle size range and analyzing for the products among different sizes after a prescribed period of comminution. A more recent technique due to Berthiaux and Dodds (1997) employs an identification procedure based on what the authors refer to as a sequential differentiation method, the advantage of w^hich appears to be the recovery of the breakage functions using a minimum of experimental data.

6.2

THE INVERSE AGGREGATION PROBLEM: DETERMINATION OF THE AGGREGATION FREQUENCY

In Chapter 5, we observed that self-similarity is observed in many pure aggregation processes. It will be the objective of this section to show that

236

6. Inverse Problems in Population Balances

the self-similar solution in fact possesses information about the aggregation frequency sufficient to allow its recovery through the solution of the inverse problem. The first attempt at solution of this inverse aggregation problem was made by Muralidhar and Ramkrishna (1986). They assumed that the self-similarity must have arisen from the homogeneity of the aggregation kernel. In a subsequent paper, however, this assumption was replaced by a somewhat more specific point of view, which may be understood as follows (Muralidhar and Ramkrishna, 1989). The population balance equation for an aggregating population with aggregation frequency for a particle pair of sizes X and y, respectively, may be appropriately written as df(x t) 1C"" ^^^ = 2 ^(^ -y^ y)f2i^ - y^ y^ t)dy-

r°°

a{x, y)/2(x, y, t) dy (6.2.1)

where /2(x, y, t) represents the density of pairs of particles with sizes x and y. The reader is referred to the discussion in Section 3.3.1 for an introduction to the density /2(x, y, t) and to Chapter 7 for a full elucidation of its origin. The population balance equation as normally written makes the assumption that the pair density may be written as the product of the population densities for each size. Although this assumption is more appropriate for populations that are sparsely distributed in space, it is always possible to replace (6.2.1) by a population balance equation with an aggregation frequency redefined as follows. We let ~,

,,

a{x,y)f2{x,y,t) /i(^, 0/i(y, t)

Notice in particular the acquired time dependence of the newly defined aggregation frequency. Equation (6.2.2) converts (6.2.1) into the population balance equation 5/i(x, 0 ^^

1 a{x - y, y, t)f^{x - y, t)f^{y, t) dy 2_ - /i(^, t)

a{x, y, t)f^(y, t) dy.

(6.2.3)

Equation (6.2.3) is viewed to free the analysis from the constraint of dilute populations through the time dependence of the equivalent frequency (6.2.2) so that denser populations can be admitted to the scope of this treatment. The feasibihty condition for self-similarity (5.2.12), derived in Section 5.2 of

6.2. The Inverse Aggregation Problem

237

Chapter 5, may now be rewritten for the equivalent frequency a{x, y, t) as 8_ ~a(tj'h{t), n"h{t), t) = 0 hit) dt

(6.2.4)

where h{t) is the scahng particle size for self-similarity introduced in Section 5.1. Following further the analysis in Section 5.2, we recall the general form satisfying (6.2.4) as a(rj'h{t% rj"h{tl t) = a{rj\ rj")H{h{t)l

h\t) = cH{h{t)\

(6.2.5)

which is a restatement of (5.2.13). The assumption of absolute homogeneity of the aggregation kernel is circumvented in favor of the more specific postulate of (6.2.5) from which the time-dependent aggregation kernel is obtained by estimating the time-independent scaled frequency a{Y]\ v]"). Note that a(?/', f]") is not necessarily the same as a{r\\ rj"). The integral equation to be solved for the frequency a(^', rj") is given by cr]^\l/{rj) =

\j/{Y]")(l{f]\ Y]") dv]"

f]'^{f]') dn'

(6.2.6)

where c is a constant^ associated with the evolution of the scaling size, is given by h\t) (6.2.7) c = H(h{t)) ' and must be estimated from experimental data. We must also avail ourselves of the normalization conditions (5.1.7) for the self-similar distribution, il/(rj) with /c = 1, which makes provisions for possible singular behavior of ij/ at the origin in accord with the discussion in Section 5.1. Consequently, the scaling particle size h{t) is given by the ratio of the second to the first moment of the population density, i.e..

hit) 9

MO i"i(f)

Note that this constant is also given by

^ Jo

Jn-t]'

which is Eq. (5.2.17) and follows from the deliberations of Section 5.2. In relating the notation presented here to that in the article of Wright and Ramkrishna (1992), note that the function xf/ here is denoted <S> in the cited article, so that the O used here is not the same as that used in the article.

238

6. Inverse Problems in Population Balances

The first of the normahzation conditions (5.1.7) with k = 1 yields the following expression for the constant c: c =

n

n'W) dn'

il/{rj'')a{ri\ rj") drj".

The second normalization condition in (5.1.7) together with (6.2.6) can be shown to yield c =

drj

drj'r]rj'il/{rj)il/{r]')(x{rj, rj') = ,

so that c is the same as the average value of the scaled aggregation frequency (or the average aggregation rate during self-similar behavior) which we have denoted previously by . The estimation of c by fitting transient data on the scaling particle size to the dynamic behavior represented in (6.2.7) thus directly determines the value of . In the rest of the discussion we shall dispense with the notation c and instead deal only with is more readily facilitated.

6.2.1

Solution of the Inverse Problem

Wright and Ramkrishna (1992) state the inverse aggregation problem by rewriting (6.2.6) in terms of the volume fraction ^{Y]) = (!>'{rj) = rj\l/{rj),^^ the equahty to the right arising from differentiating (5.1.2) with respect to rj, to obtain

^(t>M =

drj'cpW)

df]'

(l){rj") (x{r]\ ri") ' rj'

(6.2.8)

in which the unknown function to be extracted is (x(rj\ ^")/. On calculation of this function, the scaled frequency a(f]\ri") is obtained from an 10 We caution the reader to be alert to the differences in notation between the treatment here and that of Wright and Ramkrishna (1992). For example, the symbol / in the cited article is the same as here, while <^ in the article is the same as xj/ here; the symbol 0 here has no parallel in the article except in terms of other symbols.

6.2. The Inverse Aggregation Problem

239

estimate of from the dynamics of h(t). The original unsealed aggregation frequency then becomes available if we use the scaled frequency in (6.2.5). The solution of (6.2.8) proceeds by the methods of Section 6.1.1. Thus, we write rib

Z ajA^{n\nl


(6.2.9)

where {A^{Y]\ r]")] is a set of n^ basis functions to be chosen suitably, an issue to which we shall return presently. As before we discretize the similarity coordinate rj into the set (f/^; i = 1,2,..., n^ and define the matrix X whose coefficients {X^-y} are given by X,,^\

Jo

d#(^)

Jrii-r,

f = 1,2,..., n„ 7 = 1,2,..., n,.

drj'^Aj{ri,riy, ^

The discretized version of the inverse problem (6.2.8) may now be written as

where a is the unknown finite dimensional vector of the constants {^1, ^ 2 , . . . , a„ J representing the discretized (scaled) aggregation frequency (x(r]\ f/")/, and O is a vector (in n^-dimensional space) whose components are the known discrete version of the self-similar data {^i(/>(^i)} with dimension depending on the number n^ of discrete points chosen on the similarity coordinate. There is also an issue connected with the actual positioning of the discretization points, which will be addressed later. We recall our discussion in Section 6.1.1 of the ill-posedness of the foregoing problem, because of which large errors are encountered in the vector a for relatively small errors in O, and of its possible cure through the process of regularization. Thus, we seek to solve the quadratic minimization problem min{||Xa-0||?+A,,J|a||^}

(6.2.10)

where || ||i and || II2 are norms selected in the respective finite dimensional spaces 9?"'^ and 9l"^ The norms are indicated to be different as an additional source of flexibihty in the minimization process. The norms of the finite dimensional vectors in (6.2.10) are of course related to those of their continuous counterparts as tib

= Z

rib

Z ^j^k

j=lk=l

f* 00

Jo

1*00

drj

Jo

drj'w{r],rj')Aj{rj,r]')A^{rj,rj') (6.2.11)

240

6. Inverse Problems in Population Balances

where the integral under the sum on the right-hand side represent the inner product between Aj{r], r]') and Aj^{rj, rj') with w{r], rj') as the weight function. The flexibihty of the norm of the vector a on the left-hand side is associated with the flexibility in the choice of the weight function w{rj, rj'). The advantage of this flexibility lies in the potential for preferentially weighting particle size ranges for which the variation in the aggregation frequency is believed to be significant. In this connection, we return to the choice of basis functions {Aj{f]', rj")}.

6.2.1. /

Choice of Basis Functions

Our choice of basis functions is designed so that the trial solution (6.2.9) has the desired asymptotic properties of the aggregation frequency. Such properties are often available through knowledge of their relationship to the asymptotic properties of the self-similar distribution. What we mean by "asymptotic properties" and how those of the frequency are related to those of the self-similar distribution are elucidated in the discussion that follows. Most aggregation frequencies in population balance models are such that they have the property^ ^ (x{rj, rj') ^ ocrj^r]'",

r\' » v],

(6.2.12)

where a is a constant. For example, for the constant kernel case, /i and v are both clearly zero. For the sum kernel, we write a(^, rj') = rj -\- rj' ^

Y]\

rj' »

Y],

so that jU = 0 and v = 1. For aggregation due to Brownian motion, we set (x{Y],r]') = (f/^/^ + rj"'^)(rj-'/^ + rj"'^^) ^ f]-'^^rj"^\

rj' » rj,

from which fi = —1/3, and v = 1/3. For the product kernel, it is readily perceived that /^ = I, v = 1.

11

Of course, this relationship actually holds for a{rj, t]') since, invariably, homogeneous frequencies are involved. However, the relationship between its asymptotic properties and those of the self-similarity solution is determined by the integral equation (6.2.8), which is the same whether we are dealing with a or a.

6.2. The Inverse Aggregation Problem

241

Table 6,2,1 Sign of Exponent /i

r = Order of Singularity^ of i/zfTj) near rj = 0 T= l+/iH-v, T = 2 - p^a/,

fi> 0 ju = 0

—00

T=

fi < 0

p^ = oo Pv < ^o

The behavior of the self-similar distribution il/(rj) for small values of rj can be shown to depend on the sign of the exponent ju}^ The results of the analysis for different signs of/i are presented in Table 6.2.1. In Table 6.2.1, f]^il/{r])df],

Pk = 0

a =

lim (ri'/rj)^

arj ^rj'

CO

the second being the dimensional constant excluded in the asymptotic behavior of oc(rj,rj') represented in (6.2.12). In particular note that the self-similar distribution is not singular when the exponent ja is negative. In fact, for this case, the self-similar distribution vanishes at ?/ == 0 faster than any power of rj. Consider as examples the self-similar distributions for the constant, sum, and Brownian aggregation frequencies that were evaluated earlier for the values of the exponents fi and v. For the constant aggregation frequency, since the exponent /^ == 0, we have a singular self-similar distribution il/(rj) with an order of singularity equal to 1. Note that the function (/)(f/), however, is not singular as a result. For the sum frequency, the order of singularity rises to 3/2, which implies an order of singularity of 1/2 for the function 0(f/). For the Brownian aggregation frequency, which has fi = —1/3, neither il/(r]) nor (j){rj) is singular. The function (/)(^) for the three different aggregation frequencies shown earlier is plotted in Figure 6.2.1.^^ 12 The analysis here is mainly inspired by the papers of Leyvraz (1986) and van Dongen and Ernst (1988), referenced in Chapter 5. The adaptation of the arguments of van Dongen and Ernst is readily made to the case in which the homogeneity assumption is replaced by condition (6.2.5) required for self-similarity and has been accomplished by Wright and Ramkrishna (1992). 13 This figure has been replotted from Wright and Ramkrishna (1992) which erroneously depicts the self-similar distribution displayed in their Figure 1 as \l/{r]) instead of 0(?7).

242

6. Inverse Problems in Population Balances

I

0.001

I M nil

0.01

FIGURE 6.2.1 The self-similar distribution function (/>(?/) for (i) the constant aggregation frequency (dotted line), (ii) the sum frequency (continuous line), and (iii) the Brownian aggregation frequency (dot-and-dash line). (From Wright and Ramkrishna, 1992. Reprinted with permission from Elsevier Science.)

The function (pirj) shown in Figure 6.2.1 is vastly different for the different frequencies, particularly with respect to the behavior at the origin. The methodology of the inverse problem is of course based on exploitation of these differences in identifying the aggregation frequency. We now return to the issue of the choice of basis functions for solution of the inverse problem (6.2.8). The behavior of the aggregation frequency that relates to the small-rj behavior of the function (/)(^) is the issue of specific interest. We choose to fit (pirj) with y-distributions that can accommodate either a singular or nonsingular nature of the self-similar distribution near the origin and accordingly set {ri)= I

(6.2.13)

c,rj^^-'e-^'<^

k=i

where a^ and ^^ are constants subject to the normalization constraints (5.1.7) with fc = 1, viz., Pr

rf\j/{r]) df] = \

yf ^^{yj) drj = 1,

r = 1, 2.

We now return to a consideration of the inner product in (6.2.11) and the

6.2. The Inverse Aggregation Problem

243

choice of the basis functions. Toward greater emphasis on the particle size region near the origin, the weight function w(f/, rj') is chosen as w{rj,rj') = e-<^ + '^'\

(6.2.14)

which lays the maximum stress at the origin. Further, the basis functions {Aj{f]', rj")} are chosen to be M^\

j = (p - l)n + g,

Vl = Lp{r})L^{r]%

p, g = 1, 2 , . . . , n

where L^ is the (p — l)st Laguerre polynomial. ^"^ We have set n as the number of Laguerre polynomials to be employed so that the number of basis functions in the expansion (6.2.9) is given by n^ = n^. The preceding relationship among the indices j , p, and q allows for uniquely identifying p and q in terms of 7 as

P = U/nl + 1,

q=j

-nU/n]

where [x] is the function defined in Chapter 2 below Eq. (2.11.9), viz., the largest integer smaller than x. The choice of Laguerre polynomials and the inner product in (6.2.11) using the weight function (6.2.14) makes the set {Aj{r]\ rj")] orthogonal. Thus, some analytical computation is facilitated for the matrix coefficients Xij. It is thus possible to show that

x,=

(t){rj)L [ji„^ + i(f]) Yj _ „[ji„lrj i-ri)dr]

(6.2.15)

0

where the function Yj^{rj) is defined by

YM = The calculation of this function can be reduced further in view of the availabihty of a semi-analytical form using the expansion (6.2.13). It is found to be (fc-l)! (5!)^(/c - 5 -

y X + 5-l,i8,/7) 1)!

14 For an introduction to Laguerre polynomials, see pages 93-97 of Courant and Hilbert (1956), referenced in Chapter 4. The functions {Aj{rj', rj")} may be regarded as from the tensor product space of L2IO, 00; e"'') with itself, having an inner product as defined in (6.2.11). For a definition of the tensor product of function spaces, see Ramkrishna and Amundson (1985), referenced in Chapter 4.

244

6. Inverse Problems in Population Balances

where y^ is the incomplete gamma function. ^^ In carrying out the solution of the inverse problem, by constrained quadratic minimization of (6.2.10), Wright and Ramkrishna (1992) have made explicit use of the positivity and symmetry constraints of the aggregation frequency. The positivity of the frequency is imposed as a linear inequality constraint on the constants {ayj = 1,2, . . . , n ^ } , while the symmetry is ensured by requiring that the constants defined by a^^ = a^^p^^^n^ + q satisfy a^^ = a^p. The numerical integration of (6.2.15) was accomphshed with 64 quadrature points. In each of the cases studies, 80 discretization points and 4 Laguerre functions (n = 4) were found to be satisfactory. With the application of the inequality constraints a total of 10 independent coefficients of expansion were encountered in the solution. Following Wright and Ramkrishna (1992), we consider the application of the inverse problem strategy for data produced by simulation of the aggregation process using known frequencies that were cited earlier. Thus, we shall use the constant and sum kernels in the process and evaluate the inversion process by comparing the aggregation frequencies obtained with those used in generating the data. In considering these applications it is of particular interest to evaluate the role of noise in the input data on the solution to the inverse problem.

6.2.1.2

Constant Frequency

As observed in Section 5.2, the self-similar distribution can be obtained analytically for this case. Thus one finds that (/>(f/) — 4r]e~'^''. The aggregation frequency is assumed to be 1 for all particle sizes so that we also have = 1. The average particle size evolves in accord with the linear law h{t) = 1 + f. Our goal is to extract from the known self-similar distribution, the scaled aggregation frequency (x(rj\ r]")/{(x} and evaluate how close it is to its actual value of unity. We assume at first that the function (j){rj) is known exactly, but subsequently evaluate the effect of 10% randomly added noise. The analytically known function (/)(^) can be expressed exactly in terms of expansion (6.2.12) by having m = 1, c^ = 4, a^ = jS^ = 2.

15 The incomplete gamma function y^{a, x) is defined as follows:

.ia,x) =

e'Y-'dt

6.2. The Inverse Aggregation Problem

245

1.2 1-1 0.8 A

I 0.6H ^ 0.4-

o

Xreg =0

•

A,reg=lG-8

•

Xreg=1e-5

r

>,reg«1e-2

w

^reg =1

•

>.reg =10

0.2-J TTrr|—

0.001

0.01

0.1

10

FIGURE 6.2.2 Comparison of the aggregation frequency from the inverse problems with the actual (constant) frequency for various values of the regularization parameter when the self-similar distribution is known exactly. Note that the most accurate estimate of the aggregation frequency is obtained with no regularization. (From Wright and Ramkrishna, 1992. Reprinted with permission from Elsevier Science.)

Figure 6.2.2 shows the results of the calculated frequency a(0,f/)/^^ versus rj for the situation where the self-similar distribution was known exactly. The matrix X is obtained from (6.2.14) and (6.2.15) and the scaled aggregation frequency is obtained for various values of the regularization parameter A^^g. The plot shows the most accurate calculation for 1^^^ = 0. This computation shows that no regularization is required when the input data are known exactly. However, it is seldom that such data can be known so accurately. Figure 6.2.3 shows the self-similar data on (j)(r]) when 10% noise is added to the exact self-similar distribution. In this situation, the worst results for the inverted aggregation frequency are obtained in the absence of regulariz16 The frequency evaluated at one of the particle sizes equated to zero need not always exist! For example, the Brownian aggregation frequency is unbounded when one of the particle sizes approaches zero. This is a reflection of the fact that aggregation is virtually certain between two particles when one of them is much smaller than the other and consequendy capable of very rapid diffusional motion relative to the other much larger particle. In this case a(0, ^)/ cannot be expected to exist.

246

6. Inverse Problems in Population Balances u.»-

0.8^

n

le

with error

0.7^ fit

0.6-^

fn ^

0.5^ 0.4-: 0.3-^ 0.2-i

n

I

0.1^ u-

0.001

1 1 i T rrrri

0.01

1

1 » 1 1 Mil

0.1

1

1 1 1 1 IMj

1

1 1 1 1 1

IT|

10

FIGURE 6.2.3 Self-similar distribution for the case of constant aggregation frequency in the presence of 10% random noise. (From Wright and Ramkrishna, 1992. Reprinted with permission from Elsevier Science.)

ation as shown in Fig. 6.2.4. Increasing the value of the regularization parameter /l^eg improves the quality of the inversion up to a stage beyond which the solution deteriorates again. The calculations of Wright and Ramkrishna clearly demonstrate that regularization is a significant step in solving for the aggregation frequency from self-similar data with noise.

6.2.1.3

Sum Frequency

As a second example, we consider self-similar data for the sum frequency for which it is known that (t)(f]) = {2nr])~'^'^e~'^'^. The expansion (6.2.13) holds exactly for this 0 with m= I, c^ = l/^/ln, aj = jS^ = 1/2. On the addition of 10% noise, however, the expansion for 0 holds with m = l,c^ = 0.39091, ^1 = Pi = 0.49109. The function displaying a notable singularity at the origin appears in Fig. 6.2.7. The exact value of is calculated from the self-similar distribution to be 0.1788. The average particle size has the dynamics h{t) == 83.625^^-^'^^^^

actual

o

Xreg - 0

•

Xreg =1e-8

•

X,reg=1e-5

T

X.reg=1e-2

w

Xreg =1

•

A.reg»10

FIGURE 6.2.4 Comparison of the aggregation frequency from the inverse problem with the actual (constant) frequency for various values of the regularization parameter when the self-similar distribution is known with 10% error. Regularization improves the quality of the inverted solution up to a certain value of /l.^^. (From Wright and Ramkrishna, 1992.) (Reprinted with permission from Elsevier Science.)

ST

FIGURE 6.2.5 Three-dimensional plot of the aggregation frequency in the absence of regularization for the case of constant aggregation frequency. (From Wright and Ramkrishna, 1992. Reprinted with permission from Elsevier Science.) 247

FIGURE 6.2.6 Three-dimensional plot of the aggregation frequency with X^^^ = 10~^ for the case of constant aggregation frequency. (From Wright and Ramkrishna, 1992. Reprinted with permission from Elsevier Science.)

I -

K K

6^

K

with error

5^ fit

Si

4^

S 3-

ici

-

-e-

2-

\r\

U-

1

I 1 11""T

' ~ ^T-TTTTT]-

''

I 1 11 rTr~^^^""T'TT"T"^nj

10

FIGURE 6.2.7 Similarity data for the sum frequency with 10% random error. Note in particular the singularity at the origin. (From Wright and Ramkrishna, 1992. Reprinted with permission from Elsevier Science.) 248

6.2. The Inverse Aggregation Problem

249

actual o

Xreg =0 A,reg =le-8 Xreg =1e-5 Xreg=1e-2 X.reg =1 Xreg=:10

0.001 Tl

FIGURE 6.2.8 Comparison of the aggregation frequency from the inverse problem with the actual (sum) frequency for various values of the regularization parameter when the self-similar distribution is known exactly. Note that the most accurate estimate of the aggregation frequency is obtained with no regularization. (From Wright and Ramkrishna, 1992. Reprinted with permission from Elsevier Science.)

The results of the inverse problem are recounted later (i) with no added error, and (ii) with 10% random error to the exact average particle size as it evolves through time as well as to the self-similar data. Figure 6.2.8 shows the results obtained by Wright and Ramkrishna (1992) for a(0, ?7)/ from the solution of the inverse problem with no added error. It further shows that the best estimate of the aggregation frequency with error-free input data is obtained with no regularization, i.e., A,^g = 0. On the other hand, the addition of 10% noise leads to the calculation shown in Fig. 6.2.9. As in the case of the constant frequency shown in Fig. 6.2.4, it is found that the regularization parameter must be chosen appropriately to get the best estimate of the aggregation frequency. In the examples just presented, the quality of the inversion could be assessed by directly comparing the aggregation frequencies obtained with the actual frequencies used to generate the self-similar data. However, in deaUng with experimental data from aggregating systems for which the aggregation frequency is unknown, the test of the inversion lies in being able to predict from "forward" simulations the transient size distribution data

250

6. Inverse Problems in Population Balances actual o

Xreg =0

•

Xreg-1e>8

A

Xreg =1 e-5

T

Xreg=1e-2

"^

Xreg =1

•

^reg=10

FIGURE 6.2.9 Comparison of the aggregation frequency from the inverse problem with the actual (sum) frequency for various values of the regularization parameter when the self-similar distribution is known with 10% error. Regularization improves the quahty of the inverted solution up to a certain value of ^^^g. (From Wright and Ramkrishna, 1992. Reprinted with permission from Elsevier Science.)

used in the solution to the inverse problem. A rationale for the choice of the regularization parameter is desirable. We discuss here one that has emerged from Wright and Ramkrishna (1992). Broadly, the value of the regularization parameter is dictated by the inaccuracy in the input data. Exact data require no regularization, but with increasing error, the regularization parameter must be assigned larger values, the effect of which is to reduce the dependence of the inverted aggregation frequency on the uncertain regions of the input data. On the other hand, when the aggregation frequency obtained with a larger regularization parameter is used in the forward simulations, the predicted transients in particle size distribution will deviate from the observed ones more significantly. Thus, the regularization parameter to be fixed should reflect the compromise between the required accuracy in forward predictions and the extent to which errors in input data must be ignored. Because of the sensitivity of the nature of the singularity of the self-similar distribution at the origin to the aggregation frequency, the uncertainty in the similarity data is concentrated in the small rj region of asymptotic

6.2. The Inverse Aggregation Problem

251

behavior of il/{r]). The data near rj = 0 are apt to be noisy so that inversion to recover the aggregation frequency calls for a suitably sensitive strategy. This strategy is based on analysis of the relationship between the aggregation frequency and the singularity properties of the self-similar distribution. Reference to Table 6.2.1 will refresh the reader as to the different alternatives in this regard. We consider them in some detail below. (i) If il/{rj) clearly approaches zero as rj approaches zero (as is the case with the data generated from the process with constant frequency), the exponent ^, defined in (6.2.11), must indeed be negative. It is now possible to find a range of values for the exponent ft using an asymptotic form^^ to fit the self-similar data. We may now solve the inverse problem using various values of the regularization parameter (e.g., /Ireg = 0, 1 0 - ^ 1 0 - ^ 10-^). Since (6.2.11) implies that



the left-hand side of which can be estimated from the solution to the inverse problem it becomes possible to fit the right-hand side of this equation and determine the range of values for the exponent /a. For a preliminary estimate of the regularization parameter, the strategy proposed by Wright and Ramkrishna (1992) seeks its maximum value for which the range of ju estimated by fitting the inverse problem result from (6.2.17) is subsumed by the range of ji obtained from fitting the asymptotic form of the self-similar distribution. If the self-similar data show a distinctive drop of the self-similar distribution to zero, both ranges of fi may be negative. If, however, the data are too noisy for an unambiguous interpretation of their trend, it might be necessary to consider the other asymptotic behaviors, viz. that of singularity, in order to make a preliminary estimate of the regularization parameter. Since this situation requires the same strategy as that required when singular behavior of the self-similar distribution is apparent, we consider this together, (ii) If \l/{rj) is singular at rj = 0, we seek to determine the order of singularity T through a fit of the log-log plot of il/{rj) versus r] near 17 See Eq. (64) of Wright and Ramkrishna (1992) which is given by \l/(ri) - e x p — — < by

252

6. Inverse Problems in Population Balances

the origin. From Table 6.2.1, we note that the order of singularity is related to the parameter v. Under these circumstances, we may choose one or more values of a scaled size rj^ and calculate (x{r]^,rj)/<^(x} for rj »r]^ from the inverse problem solution for specific values of the regularization parameter, which satisfies the relation

^

=^,.V

(6.2.8,

SO that the right-hand side can be fitted to the inverse problem solution for the parameter v. If from the estimated order of singularity (T ^ 1 + /z + v) and the exponent V, one finds that fi> 0 and that p^-^ cc, internal consistency is assumed and the determination of the aggregation frequency is pursued under the situation of /i > 0. Moreover, the preliminary estimate of the regularization parameter is the largest values for which the foregoing internal consistency is preserved. If, on the other hand, these constraints hold only for a limited range of the regularization parameter, then the case of jU = 0 is considered. For this case, (6.2.18) may be used with /x = 0 for estimating the value of the exponent v as well as that of a/ by fitting the inverse problem results. From Table 6.2.1, one has Pv 7 ^ - 2 - 1 .

(6.2.19)

An alternative estimate of a/ is also available from (6.2.17) with fi = 0. The preliminary estimate of the regularization parameter A^eg is determined in this case to be the maximum so that internal consistency as expressed by (6.2.19) is maintained within the uncertainty of the order of singularity T of the self-similar distribution il/(rj). We have dwelt at length on the strategy of estimating the preliminary value of the regularization parameter. The strategy has been built around the issue of internal consistency established by analysis of the relationship between the self-similar distribution and the aggregation frequency and to make certain that qualitative trends are not ignored while reducing rehance on uncertain parts of the input data. For a demonstration of the details of the evolution of this strategy for the constant and sum frequencies, the reader is referred to Wright and Ramkrishna (1992).

6.2. The Inverse Aggregation Problem

253

The final phase of the inversion procedure consists in carrying out forward simulations of the aggregation process by adopting, as the initial condition, one of the measured number densities at an early instant and comparing the predicted number densities at subsequent time with the measured ones. This comparison is made with frequencies obtained for various regularization parameters in the vicinity of its preliminary value arrived at earlier. The regularization parameter is fixed at the value that yields successful comparison of the predicted number densities with the measured ones within the measurement error. The aggregation frequency that corresponds to this optimal regularization parameter is then regarded as the actual frequency. We now provide an example of such an inversion from the work of Wright et al. (1992) in which spatial computer simulations were used to generate data on the aggregation of fractal clusters formed by Brownian motion of colloidal particles. We consider three-dimensional diffusion under two circumstances: (i) that in which the diffusion coefficient of the cluster is independent of its mass and (ii) that in which the diffusion coefficient, decreases with increasing mass.^^ The simulated process automatically produces noisy data and the number density in cluster mass is presented in Figure 6.2.10 at three different times for both cases (i) and (ii). As mentioned earlier in this section, the scaling size h{t) is taken to be the ratio of the second moment to the first moment. That self-similarity is attained in this diffusion-limited aggregation process is clear from Fig. 6.2.11 which shows the collapse of the scaled transient data. The application of the inverse problem procedure outlined in this section leads to the strikingly different aggregation frequencies for the two cases in Fig. 6.2.12. In particular, the inverse problem procedure has been able to show the large aggregation rates in case (ii) for pairs of particles with highly discrepant sizes and consequently disparate diffusion coefficients. Wright and Ramkrishna (1994) have also shown the utility of the inverse problem procedure to estimate the size-sensitive coalescence frequencies of liquid droplets in a stirred liquid-liquid dispersion from experimental data.^^

18

More precisely, the difFusivity, D^ of a cluster of mass x is given by D^ = x~'^''^^ where djis the fractal dimension of the cluster. From the three-dimensional spatial simulations this fractal dimension was found to be 1.78. 19 See references in Chapter 5.

0.09 •

t=250.7

o t=558.4 « 0.06.

•

t=1243.5

•

t=2769.2

- 0.03O

if>«i A .ib^AV Q^n^iAQ , oAo.'q A o . o A o * o | 60 120 180 240 Cluster Mass (m)

(i) 0.075 t=592

A

o t=1757

a> 0.05-J

E

t=5154

0.025-4 CJL. '^n

B8O o ^ 0

60

120 Cluster Mass (m)

180

(ii) FIGURE 6.2.10 Number density of fractal clusters as a function of cluster mass at different times obtained from computer simulations: (i) mass-independent diffusion; (ii) mass-dependent diffusion. (From Wright et ai, 1992.)

254

0.8

0.6 H

•

I

t=592

o t=1757 •

<

0.4-4

t=5154

0.2

'•.

,m, (^ m ,•»

• I •,

(i) • _j 1.2 • 1 0.8 0.6

L

h -J

0

t=250.7

- t=558.4

i

0.4 0.2

•

•

t=l243.5

-*»
%

J

^%

>

'•••'»..

1 ri

(ii) FIGURE 6.2.11 Self-similar behavior of aggregating fractal clusters obtained by scaling. The scaling size is obtained by taking the ratio of the second moment to the first: (i) mass-independent diffusion; (ii) mass-dependent diffusion. Notice different properties of the self-similar distribution at the origin for the two cases. (From Wright et a/., 1992.)

255

(i)

(ii)

FIGURE 6.2.12 Aggregation frequency obtained by soving the inverse problem for the aggregation of fractal clusters; (i) mass-independent diffusion; (ii) mass-dependent diffusion. (From Wright et al, 1992.)

256

6.3. Determination of Nucleation and Growth Kinetics

6.3

257

DETERMINATION OF NUCLEATION AND GROWTH KINETICS

The inverse problems discussed in Sections 6.1 and 6.2 were addressed in the absence of nucleation and growth processes. In this section we investigate inverse problems for the recovery of the kinetics of nucleation and growth from experimental measurements of the number density. It is assumed, however, that particle break-up and aggregation processes do not occur. Determination of nucleation and growth rates is of considerable practical significance since the control of particle size in crystallization and precipitation processes depends critically on such information. We will dispense with the assumption of self-similar behavior, as it is often not observed in such systems. Also, we provide here only a preliminary analysis of this problem, as it is still in the process of active investigation by Mahoney (2000). We consider particles distributed according to a characteristic length / and let the number density be f^{l, t). Nucleation and growth can occur only in the presence of supersaturation of the crystallizing solute in the solution. Supersaturation must generally be produced by cooling of a solution saturated at a higher temperature, by chemical reaction, and so on. The occurrence of crystallization or precipitation must of course result in the reduction of supersaturation of the solution, necessitating the simultaneous analysis of the dynamics of supersaturation and that of particle nucleation and growth. Thus, the kinetics of nucleation must be sought as a function of supersaturation and that of growth as a function of both supersaturation and particle size. Nucleation includes primary as well as secondary nucleation. In primary nucleation, there is spontaneous generation of new particles in the absence of existing particles. Secondary nucleation includes the formation of nuclei in the neighborhood of an existing particle, and those microparticles that are formed by breakage by impact with the impeller or the container walls (in other words, first-order processes), or by particle-particle colhsions (second-order processes). Denoting the supersaturation at time t by (T{t), we may represent the primary nucleation rate by j6^((7) and the particle growth rate by L{1, a). Mahoney (2000) has proposed the following general formulation for the nucleation rate:

i5M = i 5 » +

p,il,<^)fiil,t)dl + 0

dlf.il t) 0

0

dl'M',tW2ihl',
258

6. Inverse Problems in Population Balances

The left-hand side represents the total nucleation rate as an explicit function of time alone in view of the number density and the supersaturation being functions of time. The function p^ characterizes first-order nucleation processes while P2 represents second-order nucleation processes. The population balance equation will of course feature the kinetic expressions for nucleation and particle growth while a mass balance for the solute in the continuous phase will serve to identify the differential equation for o-(t), the supersaturation. Indeed, the equation for supersaturation will be coupled with the population balance equation, since the latter must account for the depletion of supersaturation due to the growth of all the particles in the medium. Further, as the supersaturation might feature other concentration variables, it is conceivable that a complete set of equations will emerge only with equations identified for all the relevant chemical species in the solution. One is thus left with a rather open situation with a consequent potential for very diverse behavior. It will therefore serve the purpose of generahty to circumvent the details of supersaturation dynamics and instead view the particle nucleation and growth rates as explicit functions of time and inquire into whether they can be identified as such. To this end, the left-hand side of (6.3.1) already has the nucleation rate in this form. For the particle growth rate we let L[/, t] = L(/, a(t)) at any instant t following the onset of a crystallization or precipitation process with specified initial supersaturation and number density. We may then write the population balance equation for a batch system together with initial and boundary conditions.^^ ^ ^

+ |WU]/1(U)}=0

fiih 0) = AJll

/i(0, t)U0, r] = Pltl

(6.3.2)

(6.3.3)

The boundary condition in (6.3.3) reflects the assumption that the nuclei are assumed to be of size "zero." The problem is to estimate at first, given measurements of the number density f^(l, t), the functions j8[t] and L[/, t]. The particle growth rate kinetics L(/, a) can be obtained from L[/, r] by estimating the supersaturation from measurements of appropriate concentrations. In order to obtain the primary and secondary nucleation rates, it is necessary to calculate the functions P^i^), Piil, a) and P2il, I', (j)> This is a 20

It is possible that the mechanism for creating or renewing supersaturation may include continuous bubbling of, say a gas. Thus, the "batch" system considered here refers to the system being "closed" with respect to the particle slurry.

259

6.3. Determination of Nucleation and Growth Kinetics

difficult problem, but Mahoney (2000) points out that for most real situations (6.3.1) may be represented by the approximate equation

iS[t] « plia)

P\(l)M,t)dl,

(6.3.4)

which is obtained from (6.3.1) by dropping the first term on the right-hand side since it is likely to be negligible in the presence of a large number of precipitate particles and the second-order term because crystal breakage by colhsion between crystals may often be neglected. Thus, the problem of determining nucleation kinetics reduces to that of obtaining the functions Pl{(7) and jSi(/) from plQ and f^{l t), once jS[t] has been obtained from /i(/, t). We now return to the problem of determining jS[f] and L[/, Q from measurements of f^{l, t). It is assumed that data are available on the number density as a function of particle size at various discrete times. Since particle size measurements are generally over discrete size ranges, Fig. 6.3.1 shows an example of the

0.8-1:;:

Particle Size

Time

FIGURE 6.3.1 Data on particle size distributions for the determination of nucleation and particle growth rates generated by forward simulations of the population balance equation with the addition of 3% normally distributed noise. (From Mahoney 2000.)

260

6. Inverse Problems in Population Balances

same. These data were obtained by Mahoney (2000) through forward simulations of a system with known nucleation and particle growth kinetics to which normally distributed 3% noise was added. We begin our quest for the functions jS[t] and L[/, t] by assuming that the latter may be written as L [ U ] = Li{l)L,{t)

(6.3.5)

which is derived from a similar separability constraint imposed on L(/, a) expressed as L(/, a) = Li{l)L^{a) so that L^(0 = L^{G{t)). In the absence of particle aggregation and breakage, the number of particles in any size range defined by boundaries moving with the local particle growth rate must remain invariant with time. Furthermore, the total number of particles must increase only because of nucleation. Consequently, the cumulative number density of particles above any size /, denoted F[{1, t) and expressed in terms of the number density function as

F\{lt)^

[^

M\t)dl\

cannot change by nucleation at any instant. This population clearly represents that which was present at the initial time t = 0. Suppose F[{1, t) were plotted along the ordinate versus / along the abscissa at various times. Then the intersection of the Fj-curves with any horizontal line F[ = constant would yield the temporal evolution of particle size or, in other words, a specific characteristic curve. Thus, an entire family of characteristic curves can be generated from allowing the Fj-curves to intersect several suitably spaced constant F[ hues. Figure 6.3.2 shows Mahoney's plot of the F[curves, while Fig. 6.3.3 shows the characteristic curves. Following Mahoney (2000), we now show that the quantity j\{l, t)Li{l) is invariant along the characteristic, a property that is at the crux of the inversion technique for the particle growth rate.

- [/,(/, r)L,(/)] ^ - [/,(/, t)Um + ui, t) - iMh t)L,m = UD | / i ( U ) + | ( / i ( U ) L [ U ) ] )

= 0, (6.3.6)

the last of which follows from the population balance equation (6.3.2). The invariant depends on whether the characteristic originates from t = 0 (representing a crystal that was present initially in the experiment with some

10-1 CD

E ZJ

0

.> '•*->

E o

2 3 4 Particle Size

^

""

^

FIGURE 6.3.2 A plot of cumulative number fractions (F\) above each particle size at different times calculated from the data in Fig. 6.3.1. (From Mahoney, 2000.)

• ••••« f

444444AA . . • • • • ; N lA

•*

A f_ tA O^ OAo O o O O O O O f

A A

2

^AAAA^AA

1f

0.5

1

1.5 Time

2.5

FIGURE 6.3.3 Particle growth histories (characteristic curves) obtained from Fig. 6.3.2. (From Mahoney, 2000.) 261

262

6. Inverse Problems in Population Balances

size, say / J or from / = 0 (representing a crystal that appeared in the experiment by nucleation at some point in time, say Q . In the former case, the invariant is simply fx^oiD^iih)^ whereas in the latter case the invariant is /i(0, t)Li{0) which is iS[t]/L^(0 from the boundary condition in (6.3.3). The foregoing invariance property can be exploited to estimate the size-dependent part of the growth rate, L^(/), from measurements of the number density at sizes along the characteristic curves. Once L^(/) is estimated, the time dependent part of the growth rate Lj(t) can be obtained as follows. Since the characteristic curve is defined by ~ = W,tl

(6.3.7)

we have, in view of (6.3.5) dl

mi)

L,{t) dt.

(6.3.8)

Equation (6.3.8) provides the route to determining Lj{t) from L^(/) at least as discrete approximations. Thus, the complete growth rate L[/, t] is obtainable by this approach. The property of invariance enables the "mapping" of the measurement of particle size distribution at each instant into that at the initial time. The consistency with which the different calculations "collapse" into a single distribution provides a test of the efficacy of the process. The result shown in Fig. 6.3.4 does indicate that the method has considerable potential since the estimated initial distributions collapse to that assumed. The route to determination of the nucleation rate is through the boundary condition in (6.3.3), which yields p[Q = LMLi{0)MO,t)l

(6.3.9)

The quantity in square brackets is an invariant along the characteristic emanating from / = 0. Because of the difficulty of directly observing nuclei (i.e., measuring /i(0, t)) and particles of very small sizes, it is essential for number density measurements to be made at sizes somewhat larger. From N points {li,ti) on a characteristic curve originating from (0, Q we can obtain the mean value of the quantity in square brackets in (6.3.9) to obtain the nucleation rate at time t^. Thus, we have the nucleation rate as Plt^=-^lLSt)A{lt,td^

(6.3.10)

6.3. Determination of Nucleation and Growth Kinetics

263

1.4 1.2h

2

3 4 5 6 Particle Size FIGURE 6.3.4 Collapse of the estimated initial size distributions from data on number densities at various times. (From Mahoney, 2000.)

Figure 6.3.5 shows the calculated nucleation rate at each time from each point chosen on the appropriate characteristic. The calculated nucleation rate is reasonably close to the assumed nucleation rate (represented by the dashed lines) in the generated data. Figure 6.3.6a is a plot of the estimated L^(/), the size-dependent part of the particle growth rate, while Fig. 6.3.6b shows the estimated timedependent part, Lf{t). The favorable comparison of the estimated ones with those used in the simulation of the data, shown alongside as dotted lines, is a testimonial to the promise of Mahoney's approach (2000). The time dependency of the particle growth and nucleation rates can be converted to their rightful dependence on supersaturation when estimates of the latter are available from experimental measurements at different times. Among other interesting attributes of this technique, the use of inverse problem strategies to identify the characterization in (6.3.4) from estimated nucleation rates deserves special mention. Thus, based on the assumption that the supersaturation dependence can be described by a power law, identification of the optimum exponent and the function p\{r) have been possible. The method is under active investigation by Mahoney (2000).

264

6. Inverse Problems in Population Balances 5

^^ J ^ 1

4.5 4

*'•

^.,x/

S "(3 3.5 en c

O

*7

0

•• r

1 v^-^* ^,,71

TO 2.5 0)

^ Z

2

TI^M

1.5 1

itfaM^*^

0

' 0.2

; 0.4

1

1

J

0.6

0.8

1

Time FIGURE 6.3.5 Nucleation rates determined as a function of time from data in Fig. 6.3.4 and compared with the actual rates used in simulation (dashed lines). (From Mahoney, 2000.)

6.4

OTHER INVERSE PROBLEMS

There are other examples of inverse problems in the literature that are worthy of mention. Thus, the determination of growth rates of cells from size distributions during balanced growth of a microbial culture is an interesting example which we treat briefly in the next section.

6.4.1

Growth Rates of Cells during Balanced Growth

The method discussed here is due to Collins and Richmond (1964), whose development, however, is vastly different from that presented here. Consider the evolution of the number density of cells distributed according to their lengths. The cells multiply by binary division. We let h{l) be the division frequency of a cell of length /, and p{l \ I') be the length distribution of daughter cells born of a mother cell of length /'. The growth rate of cells of length /, denoted L{1\ is assumed to occur independently of the environmental medium (containing nutrients in excess). The population balance

0.8

0.6

"D ' C

Q

0.2h Reconstruction True Value

2

3 Particle Size

4

5

6

(a) 2.5

— . . . .

0.5

Reconstruction True Value

1.5 Time

2.5

(b)

FIGURE 6.3.6 Comparison of estimated particle growth rate with actual value used in simulations. Size-dependent part (a); time-dependent part (b). (From Mahoney, 2000.) 265

266

6. Inverse Problems in Population Balances

equation is then given by

5/i(/, 0 ^ 1.(1^1) f^^i^ f)] = -b{l)Ml t) + 2 dt

f 00

dV

b{l')p{l\l')f,{l\t)dl'.

(6.4.1)

We are concerned with the determination of the functions L(/), fc(/), and p(/1 r), from suitably designed population data to be determined from experiments. We assume that the cells are growing in a batch culture under balanced exponential growth conditions. Under these circumstances we have /i(/, t) = N^e^^fQX where /(/) is the time-invariant probabihty distribution for cell length and /i is the exponential growth rate constant. On substituting the foregoing into the population balance equation, one obtains

^(/.(/)+^^[L(/)/(0] = -ft(/)/(0+2

b(l')p{l I /')/(/') dl'.

(6.4.2)

If Eq. (6.4.2) is integrated with respect to cell length, we obtain the result that

-i:

b{l)mdl.

(6.4.3)

Since the right-hand side of (6.4.3) is the unconditional cell division rate, the exponential growth constant also inherits this interpretation.^^ Alternatively, lidt represents the probabihty that there is a division between t and t + dt regardless of the size of the dividing cell. In what follows, we first address the determination of the growth rate L(/).

6.4.1.1

Determination of Growth Rate

We begin by examining the right-hand side of Eq. (6.4.2). The first term represents the a priori division rate of a cell of length / at time t. This term can also be written as the product of the unconditional division rate /x and the posteriori length distribution of cells that have just divided, which we denote by, say (/>(/). Similarly, the integral in the last term, represents the division rate of a cell (of length larger than /) times the length distribution of daughter cells length /. This can also be written as the product of the unconditional division rate fx and the posteriori probability distribution for 21

This follows from the total probabihty theorem. See, for example, pp 64 of Gnedenko (1964).

6.4. Other Inverse Problems

267

the length of daughter cells just formed by division, which we denote by i/^(/). Thus, we write

b{l')p(l\l')mdl' = -m

b{i)f{i)=-4>il),

(6.4.4)

At

so that (6.4.2) may be rewritten as d ^, wo/(0] = -i^m - b[i)m + 2/^^(0, which may be integrated readily to obtain the growth rate of cells as

^'^MI' [2.A(/') -

4>{l') - /(/')] dl'.

(6.4.5)

Thus, the size-specific growth rate of individual cells is obtained by the measurement of distributions /(/), (/>(/), and xj/Q).

6.4.1.2

Determination of Cell Division Rate and Daughter Size Distribution

The cell division rate can be obtained directly from the first of the relations (6.4.4) since

urn' The daughter size distribution p{l \ I') can be obtained by assuming that it is of the form

p(IIO-ii>(^). which says that the daughter cell length is related through a monovariate distribution of its length scaled with respect to that of the mother. This assumption makes it possible to calculate readily the moments of the distribution P{x) defined by P„=

x"P(x)rfx.

(6.4.6)

These moments are calculable from the second of the relations (6.4.4). On multiplying that relation by /" and integrating over the semi-infinite interval,

268

6. Inverse Problems in Population Balances

one obtains l"dl

b{nmj,Pi-]dl'-

b{r)mj,p(j,]di'

/>(/) dl.

Since P{l/l') vanishes for / > /', the foregoing becomes

l"P I J, 1 dl

b(/')/(/')d/' = - I " / > ( / ) d/.

Transforming variables in accord with / = xl', we obtain /"/(/) dl,

„ =

/"
Clearly, the accuracy of calculation of the moments of P{x) would depend on the accuracy with which the moments i/^„ and (/>„ can be estimated. The method, originally due to Collins and Richmond (1964), has been discussed by Ramkrishna et al. (1968) from the perspective of population balances. It has been exploited in substantial measure by Srienc and co-workers (Lavin et al, 1990; Srienc and Dien, 1992; Dien and Srienc, 1991) using flow cytometric data.

6.4.2

Hindered Settling of Particles in a Polydisperse Suspension

Our concern in this section has been the determination of the growth rate of individual particles from dynamic measurement of particle size distributions. Insofar as "growth" refers to particle convection along the size coordinate, the problem of determining hindered motion of particles differs only in the coordinate (spatial) along which convection occurs. Although we had excluded the mechanics of dispersions from the scope of this book, the reader wifl note that the analysis of hindered motion, as presented next, eminently serves the purposes of this section. The problem at hand is the velocity of a particle in a polydisperse suspension, which is averaged over both the continuous and dispersed phase variables. The exact nature of this averaging may be understood by deriving the population balance equation for a population of particles distributed according to size and their spatial coordinates from the master density equation in Chapter 7. In gravitational settling, the spatial coordinates contain only the vertical coordinate, given that the suspension is well mixed along transverse coordinates although there are several issues in this

6.4. Other Inverse Problems

269

connection that are outside the scope of this discussion. For a detailed discussion of such issues, the reader is referred to Kumar et al. (2000). For the present, it suffices to recognize that the velocity of any given particle is not only a function of its own size but also of its neighborhood, consisting of both the fluid and fellow particles. Our objective is to determine this velocity through a population-associated measurement. The merit of this approach is that it circumvents the numerous, complex issues associated with multibody particle-particle and fluid-particle interactions that makes the detailed mechanical theory very difficult, particularly for dense dispersions. The experiment of Kumar et al. (2000) consists of continuously feeding the polydisperse suspension through a vertical column in the weU-mixed state and aflowing the relative motion of particles to exit at an outlet located at a suitable distance from the point of entry. The relative motion of particles wiU have established a steady state, spatially uniform distribution of particles with an exit number density that can be measured by a device such as a Coulter counter. The population density, /^(z, i;) in vertical coordinate z and particle size described by volume v, satisfies the population balance equation ^[Z(t;,...)/,(z,t;)]=0,

(6.4.7)

where Z{v,...) is the velocity of the particle of volume v relative to a fixed frame of reference.^^ The dots in the argument refer to the dependence of the hindered velocity on the polydispersity, whose characterization is left open. If we denote the hindered velocity relative to the fluid by VJ{V,...), then we may write Z(i;,...) = i i , + F,(t;,...) where u^ is the area averaged velocity of the continuous phase fluid in the vertical direction, which must remain constant. AUowing for assumptions justified in detail by Kumar et al. (2000), the transversely well-mixed state of the dispersion is maintained throughout the column so that the population flux remains constant in the column. If the inlet volumetric flow rate of the dispersion is Q with a number density oi f^^{v\ then the uniform particle flow through the column is given by Qfi^i{v). Thus at any cross-section in 22 The reader must note the difference in notation for the hindered velocity between that used here and in the work of Kumar et al. (1999).

270

6. Inverse Problems in Population Balances

the zone, where the number density is spatially uniform at the exit density of, say fi^eiv), we have e/l.-M = /l.e(fM[»z + ^(1^, • • •)]

(6.4.8)

where A is the cross-section of the column. If the total volume fraction of the particles in the inlet suspension is 0^ and that in the exit stream is 0^, then the conservation of the continuous-phase liquid yields (2(1 - 4>i) = Au^il - U

(6.4.9)

Substituting (6.4.9) in (6.4.8), we get

QhM = fxA^) AVXv,..) +

2(1 -
which, on rearrangement, yields the explicit formula for the hindered setthng velocity (6.4.10) The results of experiments performed by Pirog (1997) using formula (6.4.10) for creaming polydisperse emulsion particles (lighter than the aqueous continuous phase) are shown in Fig. 6.4.1 (for the low-volume fraction) and in Fig. 6.4.2 (for dense volume fraction). It is interesting to note that the Stokes limit is indicated as required for the lean emulsion in Fig. 6.4.1 (the scatter for larger sizes being due to the limited number of particles in that range). On the other hand, in Fig. 6.4.2 the smallest particles in the dense dispersion are creaming at a rate faster than even the Stokes velocity while the largest particles are moving as predicted by a well-known correlation due to Richardson and Zaki (1954). Kumar et al. (2000) point out that the motion of the small particles exceeds their Stokes velocities because of the dragging effect of the faster-rising larger particles. The foregoing inverse problem for the hindered velocity was analytically solvable by a key measurement from experiment. The important issue of this chapter has been the extraction of single particle behavior from the measurement of population-based quantities. This is an issue of the greatest importance in the practical application of population balances.

10 c

I

1 >

column holdup 0.2% Stokes velocity

1 0.1

C

*E CO

0)

0.01 [

0.001 100

1000

o

10000

drop volume (\i^ ) FIGURE 6.4.1 Hindered creaming rate versus drop volume for a 0.2% paraffin oil emulsion stabilized with 0.5wt.% Sodium lauryl sulfate. (From Kumar et al, 2000. Reprinted with permission from Elsevier Science.)

column holdup 23.8% Richardson & Zaki (23.8%) Stokes velocity

c

I ^

1 > c E CO

0.1 [

0.01

p

0.001 100

1000 drop volume

10000

FIGURE 6.4.2 Hindered creaming rate versus drop volume for a 23.8% paraffin oil emulsion stabihzed with 0.5wt.% Sodium lauryl sulfate. (From Kumar et al, 2000, Reprinted with permission from Elsevier Science.) 271

272

6. Inverse Problems in Population Balances

REFERENCES Abramowitz, M. and A. Stegun, Handbook of Mathematical Functions with Examples, Graphs and Mathematical Tables, National Bureau of Standards Applied Mathematics Series 55, Washington D.C., 1964. Berthiaux, H. and J. Dodds, "A New Estimation Technique for the Determination of Breakage and Selection Parameters in Batch Grinding," Powder Technol 94, 173-179 (1997). Collins, J. F. and M. H. Richmond, "The Distribution and Formation of Penicillinase in a Bacterial Population of Bacillus licheniformis," J. Gen. Microbiol, 34, 363-377 (1964). Dien, B. S. and F. Srienc, "Bromodeoxyuridine labeling and flow cytometric identification of rephcating Saccharomyces cerevisiae cells: lengths of cell cycle phases and population variabiUty at specific cell cycle positions," Biotech. Prog. 1, 292-298 (1991). Gardner, R. P. and K. Sukanjnajtee, "A Combined Tracer and Back-Calculation Method for Determining Particulate Breakage Function in Ball Milling. Part I. Rationale and Description of the Proposed Method," Powder Technol. 6, 65-74 (1972). Gnedenko, B. V., The Theory of Probability (English translation by B. D. Seckler), Chelsea Pubhshing Co., New York, 1962. Kumar, S., T. W. Pirog and D. Ramkrishna, "A New Method for Estimating Hindered Creaming/Settling Velocity of Particles in Polydisperse Systems," Chem. Eng. Sci. 55, 1893-1904 (2000). Lavin, D., P. C. Hatzis, F. Srienc and A. G. Fredrickson, "Size effects on the uptake of particles by populations of Tetrahymena pyriformis cells. J. Protozool. 37,157163 (1990). Leyvraz, F., On Growth and Form: Fractal and Non-Fractal Patterns in Physics (H. E. Stanley and N. Ostrowsky, Eds.) Martinus Nijhoff, Dordrecht, 1986. Mahoney, A. W. "Investigation of Population Balance Models towards Control of Particle Size Distribution," PhD. Thesis, Purdue University, West Lafayette, In, 2000. MuraUdhar, R. and D. Ramkrishna, "An Inverse Problem in Agglomeration Kinetics," J. Colloid Interf Sci., 112, 348-361 (1986). Muralidhar, R. and D. Ramkrishna "Inverse Problems in Agglomeration Kinetics-II. Non-homogeneous Kernels," J. Colloid Interf Sci. 131, 503-513 (1989). Pirog, T., "Dynamics of Destabilization of Food Emulsions: Measurement and Simulation of Gravity Driven Particle Velocities in Polydisperse Dispersions," Ph.D. Thesis, Purdue University, 1998. Ramkrishna, D., A. G. Fredrickson and H. M. Tsuchiya, "On Relationships Between Various Distribution Functions in Balanced Unicellular Growth," Bull Math. Biophys. 30, 319-323 (1968).

6.4. Other Inverse Problems

273

Richardson, J. F. and W. N. Zaki, "Sedimentation and Fluidization: Part 1," Trans. Instil. Chem. Eng. 32, 35-53, 1954. Sathyagal, A. N., D. Ramkrishna and G. Narsimhan, "Solution of Inverse Problems in Population Balances —11. Particle Break-up," Comp. Chem. Eng. 19, 437-451 (1995). Srienc, F. and B. S. Dien, "Kinetics of the cell cycle of Saccharomyces cerevesiae," Ann. N.Y. Acad. Sci. 665, 59-71, 1992. Tikhonov, A. N. and V. Y. Arsenin, Solution of III-Posed Problems. V. H. Winston and Sons, Washington, D.C., 1977. Wright, H., and D. Ramkrishna, "Solution of Inverse Problems in Population Balances — I. Aggregation Processes," Comp. Chem. Eng. 16, 1019-1038 (1992). Wright, H., R. Muralidhar and D. Ramkrishna, "Aggregation Frequencies of Fractal Aggregates," Phys. Rev. A. 46, 5072-5083, 1992.

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CHAPTER 7

The Statistical Foundation of Population Balances

In this chapter, we inquire into the statistical foundation of the population balance equation. As a smooth function capable of differentiation with respect to time as well as particle state coordinates, the number density in the population balance equation is clearly an averaged quantity. Thus, the population balance equation is to be viewed as an "averaged" equation designed to predict average information. Further, it is of interest to inquire into the nature of this averaging, the circumstances under which such averaging produces the usual population balance equation, those under which it does not, and the methodology for dealing with the same.^ However, the issues in this chapter should not be solely viewed as constituting an intellectual quest, as there are applications for which the developments in this chapter are indispensable. Broadly, the theory is of interest to small populations, which behave randomly because randomness in the behavior of single particles does not average out among a small number of particles. Small populations are encountered in a variety of engineering 1

The population balance equation has often been erroneously viewed as a "stochastic" equation in the literature. Although the behavior of any single particle in a population is regarded as random, the behavior of the population is viewed as deterministic in that deviation about average behavior is negligible when the number of individuals in a population is large. An estimate of what constitutes a small population may be found from examining the average number of individuals in a "pure birth" process in which the "fluctuations" in the population are of average value EN with an average fluctuation of yJEN. Thus a 30% fluctuation is implied in a population of approximately 10 individuals. 275

276

7. The Statistical Foundation of Population Balances

applications. For example, the production of fine particles within the confines of a microemulsion droplet such as a reverse micelle presents a situation in which the number of particles will be very small and random because of the inherently random nucleation process. The objective of sterilization processes is to reduce the population of contaminating organisms to as near extinction as possible. In such processes, the objective of analysis is to seek the average behavior of the system as well as the average fluctuations about the average behavior, thus calling for a truly stochastic analysis. In view of the numerous issues involved for discussion, the organization of this chapter deserves some explanation. First, the master density function is introduced in Section 7.1. The scalar particle state is discussed in detail in Section 7.1.1 with directions for generalization to the vector case in Section 7.1.2. Couphng with the continuous phase variables is ignored in the foregoing sections, but the necessary modifications for accommodating the environmental effect on the particles are discussed in Section 7.1.3. Thus, from Section 7.1, the basic implements of the stochastic theory of populations along with their probabilistic interpretations become available. These implements are the master density, moment densities that are called product densities, and the resulting mathematical machinery for the calculation of fluctuations. Second, Section 7.2 provides the derivation of equations in the master density for some of the particulate processes discussed in Chapter 3 and shows the combinatorial complexity of their solution. The objective of this section is to show how Monte Carlo simulations of a process eliminate the quantitatively less significant combinatorial elements of the solution by artificial realization. Thus, the discussion here is of largely conceptual value. Third, Section 7.3 goes into the direct derivation of stochastic equations of population balance. These equations are also obtainable from averaging the master density equations of Section 7.2, but are best obtained by using the methodology of Section 7.3. Some applications of stochastic analysis are shown in this section, which are of focal interest to the subject of this chapter. Fourth, in Section 7.4 we examine the closure problem arising in aggregating systems as well as in random environments and some approximations that have been used to obtain closure. Finally, in Section 7.5 generalizations are introduced for accounting for correlated behavior of particles that have a common origin at birth. This is

7.1. The Master Density Function

277

particularly useful in the analysis of biological populations in which the progeny of a common parent behave in a highly correlated manner.

7.1

THE MASTER DENSITY FUNCTION

We shall deal with a system of particles distributed according to a vector particle state as considered in Section 2.1. However, we shall not seek to distinguish between internal coordinates x and external coordinates r, and instead regard the joint particle state as z.

7.1.1

The Scalar Case

It will ease our discussion considerably to deal with the scalar case first, so that we denote the particle state by z. Further, we neglect any dependence on the continuous phase variables and consider this in Section 7.1.3. The strategy is to address the state of the entire particle population at any instant and consider the probability associated with the state of each of these particles. We deem the particles to be indistinguishable other than by the specification of their chosen states. We regard the state to be continuous so that we are concerned with the probabilities of the particles in the system being in some intervals of particle state. Also, we assume that only one particle can be found in an arbitrarily small interval at any given time. This assumption is quite unnecessary except for eliminating some needless generality at this stage. It will turn out, however, that the final results are independent of this assumption. Thus, we consider the joint probabihty at instant t that there are v particles in the system with one each in the v infinitesimal intervals {Zi^z^-h dz^), / = 1,2,..., v. A very convenient way of mathematically representing this situation is to define the actual number density of particles by n{z, t) and write it as n{z,t)=

i(5(z-z,),

(7.1.1)

The actual total number of particles in the system is obtained by integrating the number density over all possible states of all particles. Thus, iV(t) =

n{z,t)dz

(7.1.2)

278

7. The Statistical Foundation of Population Balances

where the positivity of z is a dispensable constraint. If (7.1.1) is substituted into (7.1.2), one obtains N{t) = v as one should. We now define the master density function^ as follows: Pr{There are at time t, v particles in the system with one each in l,2,...,v}

{Zi,Zi + dZili=

V

= Pr{n(z, t) = X ^(z-Zi)}

= J^{z ^, Z2,..., z^;t) dz^dz2,... ,dz^

1= 1

The probability density function J^ is symmetric with respect to all the particle states in the sense that its value is unaltered by the permutations of the z/s. This follows from the particles being indistinguishable except through their chosen states. The proper normalization condition for the density function J^ may be arrived at by ordering the particle states as Zi > Z2 > • • > z„ in order to avoid redundancies so that 00

r 00

Z

d^r

dz,J,(zi,Z2,...,z,;t) = 1,

dz-j

(7.1.3)

where of course no integration is involved for " = 0. Since the integration range in (7.1.3) is awkward, we may exploit the symmetry of the integrand to rewrite (7.1.3) as 00

2

y -

dz^

dz2 •''

dz„J,(zi,Z2,...,z„;0 = 1.

(7.1.4)

This multiple integration is most conveniently done since they are not of the iterated type. Since the integration range in all the state variables is the same, it is convenient in the sequel to introduce the product symbol in representing the multiple integration in (7.1.4) as 1 i;^n

dz,J,(zi,Z2,...,z,;0 = 1,

which has the advantage of looking somewhat more compact than (7.1.4). The summand in (7.1.4) may be recognized as the probability that there are a total of V particles in the system regardless of their individual states, which we restate as dz^J^{zi,Z2,...,z^;t). r — 1 c/

2

This density is known as the Janossy density (1950).

(7.1.5)

7.1. The Master Density Function

279

Equation (7.1.5) shows how the probabihty distribution for the total number of particles in the system can be obtained from the more detailed consideration of their individual states. Note also that Po(0 = «/o(0» the probability that there are no particles in the system at time t. We next consider the calculation of expected or mean quantities associated with the population.

7.1.1.1

Expected Quantities

The expectation of any quantity associated with the population can now be calculated as follows. Let F{ ) be any quantity depending on the state of the population at time. Then the statistical expectation of F is denoted EF and is defined by 00

1

V

/"OO

EF=Y.~,U

v = 0 ^' r=l

Jo

dz,F{ )J,{z„z^,...,z,;t).

(7.1.6)

We now obtain the expected number density of particles at any instant by seeking the average of the actual number density n{z, t) (the actual number of particles in (z, z + dz) being n(z, t) dz). Thus, using the general formula (7.1.6) we obtain V

00

En{z, t) = EY^d{z -Zi)= i=l

1

V

/ * 00

V

\ dz^Y. ^(^ " ^i)«^v(^i. ^2. • • •. ^.'^ 0-

Y^-Yi

v=l^-r=lJo

i=l

Using the properties of the Dirac delta function, it is not difficult to see that the foregoing can be written as 00

1

V

v=l

^'

i=l

£n(z,0= E - T Z

V

n

r=l,i

0

Since the master density function is insensitive to the permutation of the particle state arguments, the integrals in the inner sum are all the same so that 00

£n(z,f)=/i(z,0=

1

V—

Ez—n?n

1

dz, J,(zi, Z2,..., z,_ 1, z; 0. (7.1.7)

The term to the extreme right above has a probability interpretation, viz., on multiplying by dz, it represents all the mutually exclusive and exhaustive circumstances under which there is a particle between z and z + dz. Thus, we conclude that /^(z, t\ which is the expected population density, is also

280

7. The Statistical Foundation of Population Balances

amenable to a probability interpretation.^ However, it is not a probability density because on integration over any range of particle states it produces the expected number of particles in that range. Thus, EN{t) =

f^{z, t)dz;

Mz,t)dz.

EN{a,b;t) =

(7.1.8)

0

The first of these relations is the expected (or mean) total number of particles in the system, and the second represents the expected (or mean) number of particles in the interval {a, b). The probability interpretation of /i(z, t) is particularly useful in the derivation of equations to be satisfied for any particulate process. If the quantity F considered in (7.1.6) is defined by F{t)

(7.1.9)

f{z)n{z, t)dz,

which arises when F is obtained as the cumulation of a particle-specific property /(z), the expectation of F can then be obtained using (7.1.1) in (7.1.9) and substituting into (7.1.6). The procedure, which is identical to that used in the derivation of (7.1.8), leads to the result EF{t)

(7.1.10)

f{z)Mz,t)dz. 0

We now consider expectations leading to higher order moments of the number density. Thus, consider disjoint intervals (z, z + dz% (z', z' + dz') and ask for the expectation of the quantity n(z, t) dzn(z\ t) dz'. Dropping the differentials for the present, En{z, t)n{z', t) = E t

S{z - z,) ^ S(z' - Zj)

= Et3{zi= 1

z,)S{z' - 2,.) + £ i

t

^(2 - ^M^' - ^h

i = 1 j = 1 ,i

where the comma in the summation range for 7 in the second term is used to exclude its being equal to the index / that follows it. Using (7.1.6) one 3

This dual interpretation of/^(z, t) is a source of much confusion to the uninitiated. It arises as a result of restricting an infinitesimal interval such as (z, z + dz) to contain at most one particle. Thus the expected number of particles in the interval (z, z + dz) is given by unity times the probabihty that there is one particle so that the probability and expectation are the same! In Section 7.5, we deal with the more general case of an infinitesimal interval accommodating any number of particles and the anomaly under discussion no longer exists.

7.1. The Master Density Function

281

obtains from the preceding En(z,t)niz\ t) = d{z-z')

n

1 X "f Z V! .

dZyJ^(z^,

Z2,...,

-^i-i, z, Zi + ^,...,

z^it)

+ x^i: I n v = 0 ^' i=l

j=l,i

dZ^J^{Z^,

r=l

Z2, . . . , ^ i - i ? Z, Z-+ I, . . . , Zj-i,

Z •> Zj+V

• • • ? -^v' U-

Notice in particular the first term on the right-hand side where a Dirac delta function appears because of integration with respect to Zj. The second term on the right-hand side has been identified as such based on the assumption that z # z\ since Zj ^ z^ while integrating the previous equation. We now make use of the symmetry properties of the master density function. In both terms on the right-hand side of the preceding equation, the summands within the inner sums are independent of the index of summation so that we may write En{z, t)n{z\ t) = 5{z - z') ^

1

\iy-nR

dzJ^{z^,Z2,...,z^_^,z;t)

+ 1

1

n

v-2

rfz,J,(zi,Z2,...,z,_2,z,z';0. The first term has already been identified in (7.1.7). The second term on the right-hand side considers the probability under all possible circumstances of a particle being found at instant t between z and z + dz, and another particle between z' and z' + dz\ Thus, we define for z ^ z'

/^(z, z\ 0 - I

1

v-2

—— n dz^J^iz^, ^2^..., ^v-2. z, z'\ t)

(7.1.11)

282

7. The Statistical Foundation of Population Balances

where /2(z, z', t) dz dz' represents the probabihty that at time t there is one particle in the interval (z, z + dz) and one particle in the interval (z', z' -\-dz'). It should be understood that the presence of this particle pair has no implications for either the existence or states of other particles in the system. Thus, the function /2 should not be regarded as a probability density. Also note that the symmetry properties of the master density imply that /2 is symmetric with respect to particle coordinates. We may now write En{z, t)n{z\ t) as En{z, t)n{z\ t) = f,{z, t)d{z - z') + f^{z,z\ t).

(7.1.12)

The second moment of the number of particles in any interval (a, h) is now obtained by integrating (7.1.12) with respect to z as well as z' over the interval (a, b)'."^ EN{a,b;tf

=

dz

Cb

dz'En{z, t)n{z\ t) =

f^{z,t)dz

+

Cb

dz

dz'f^iz, z\ t). (7.1.13a)

This may also be written as dz

dz'f^i^, z\ t) = ElN{a, b; t){N{a, b; t) - 1)].

(7.1.13b)

Thus, the integral of the second-order product density over any particle size interval (a, b) gives the expected number of pairs that can be formed of particles in the interval at time t. Similarly, the second moment of the total population density N{t) is given by ^ =

"oo

»0

dz

dz'En(z, t)n{z', t) J

fiiz, t)dz+

\

dz \

dz'f^iz, z', t)

(7.1.14a)

The expressions on the right-hand sides of (7.1.13) and (7.1.14) are often confusing at first because of their apparent dimensional inconsistency. The first term has the dimension of "number" while the second (as also the left-hand side) has that of "number square." This inconsistency can be reconciled, as it is the consequence of constraining the number of particles in any inifinitesimal interval to be at most 1. The expressions generalized to include more particles, appearing subsequently, are free from such apparent inconsistency.

7.1. The Master Density Function

283

with the following form corresponding to (7.1.13b): dz

-1)].

dz'Uz,z',t)=Emt){N{t)

(7.1.14b)

This is the expected total number of pairs in the population at any instant t. The second moment of any quantity F as defined in (7.1.10) associated with the population is obtained by using the same procedure to derive (7.1.13) and (7.1.14). Thus, we have EF{tY =

dz'f{z)f(z')En{z,

dz

fizrf,iz,t}dz

+

t)n{z', t) f{z)dz\

f{z'}dz'Uz,z',t).

(7.1.15)

Formula (7.1.13a) also allows us to calculate the cross correlation between particle numbers in different state intervals such as {a, b) and {a\ b'). It is left for the reader to show that the result is given by EN{a,b\t)N{a\b'\t)

f,{z, t) dz +

=

dz

rb'

dzUz,z\t)

(7.1.16)

where the first term on the right-hand side is an integral over the interval (c, d) representing the intersection of the intervals (a, b) and (a', b'). Similarly, the cross correlation between any population-associated quantity such as (7.1.9) associated with intervals (a, b) and (a', b') can be written as f{z)n{z, t)dz

f{z')n{z\

-r

t)dz

+

f{zfU{z,t)dz

/(z) dz

f{z')dzUz,z\t).

(7.1.17)

In particular, note that the first term on the right-hand side will not appear if the intervals {a, b) and {a\ b') do not have an interval of intersection such as (c, d). Since the first and second moments of quantities have been identified, their variances are readily calculated. An example is the variance of the population in the interval (a, b) at time t, is given by VN(a, b; t) = EN{a, b; tf - E^N{a, b; t) =

f,{z,t)dz

+

dz

dz'lUz,

z', f ) - / i ( z , t)Mz', t)l (7.1.18)

284

7. The Statistical Foundation of Population Balances

Similarly, the variance in the quantity F{t), as defined in (7.1.9), is given by

f{zrf,{z,t)dz

VF{t) =

+ 7.1.1.2

dzfiz)

dz'f{z')lf,{z,

z\ t) - Mz, t)Mz\ t)l

(7.1.19)

Product Densities

The development in the previous section produced the functions /^(z, t) and /2(z, z\ t\ which were very important in the calculation of the first and second moments of the population densities and other properties derived from them. Their probability interpretations were also recognized, although they were clearly not probability densities. Following developments in the early physics literature, we will refer to them as product densities.^ Thus, /i(z, t) is called the first order product density. The term "product" refers to the product taken of the number density, and the "order" refers to the number of times the product is taken of the number density. Thus, /2(z, z', t) is called product density of order 2. More generally, it is possible to define the product density of order r as (7.1.20)

f,{z„z,,...,z,,t)^EY\n{z,,tl k=l

which when multiplied by dz^ dz2 ••- dz^ is the probabihty of finding at time t one particle in each of the r intervals (z^,z^ + Jz^^), /c = 1,2,.. .,r. As before, it must be recognized that this situation is unmindful of all the other particles that may be present in the system. Hence, /^ should not be regarded as a probability density. In fact using combinatorial arguments, it is possible to estabhsh that (see A. Ramakrishnan, 1959)

EN{a,b;tY= Y. <^'U

dzj,{z,,z,,.,.,z,,t),

(7.1.21)

which provides the /cth moment of the population in any interval. In (7.1.21) {C^/,r = 1, 2 , . . . , /c} are known as Stirling numbers of the second kind and 5

The concept of product densities was developed originally by A. Ramakrishnan (1959). See also Bhabha (1950).

7.1. The Master Density Function

285

arise commonly in combinatorial treatments.^ Similarly the feth moment of quantity F, as defined in (7.1.9), is given by EF{tf = i

C:; n

r=l

s=1

[ ' dzJ{z,)Uz„

z,,...,

z„ t).

(7.1.22)

Ja

Thus higher order fluctuations of the population in any interval or property associated with it can be calculated from the product densities.

EXERCISE 7.1.1 Show that the rth order product density is given by

/,(z„z3,...,z„t) = i : — - n 00

1

^

V

"^r + k'^v(^l''^25 • • • ' ^ f ^ r + 1 ' ^ r + 2 ' • • • ' ^V ^)0

It is of particular interest to note that the result of Exercise 7.1.1 has an "inverse" counterpart relating the master density to the product densities: 00

/

1V~^

^

j,(zi,z2,...,z„o = E - ; — ; r n

^zJ,(z„Z2,...,z„t).

(7.1.23)

The foregoing result is due to Kuznestov and Stratanovich (1956). One may integrate the above equation (in view of (7.1.5)) to also get the result 00

/

iy~^

**

nt) = E V-4i n ,

dzJXz„z^,...,z,,t).

(7.1.24)

0

We shall have subsequent occasions to recall Eq. (7.1.24) in applications where it will be seen to be particularly significant.

7.1.2

The Vectorial Case

The extension of the development of Section 7.1.1 to the vectorial case is indeed straightforward. All the results derived carry forward with only minor changes. The scalar state z is replaced by the vector state 6 See Abramowitz and Stegun (1964) referenced in Chapter 6 for values of the StirUng numbers.

7. The Statistical Foundation of Population Balances

286

z = (zi, Z2,..., z„) so that the master density J^ is defined for v vector states Zi, Z2,..., z^, where Zj = {zj^^, z^ 2? • • • ? ^j,n)- The integral with respect to any scalar particle state must be replaced by a volume integral in n-dimensional space over the domain in which the specific particle is considered. Thus, dz:

j

dZi,

JzeQ

where Q is the domain of integration in n-dimensional space for the ith particle and dz^ is a differential volume element about Zj. All the formulas presented in the sequel are exactly applicable to the n-dimensional vectorial case with the changes just mentioned.

7.1.3

Environment-Dependent Case

We consider here the required changes in the treatment for the case in which the environment, characterized by continuous variables, affects the behavior of the particles. As in Section 7.1.1, we will only consider a scalar particle state and restrict consideration to a single continuous phase variable, say, Y{t\ which is further assumed to be uniform in space. We are mainly concerned with Y{t) becoming a stochastic process as a result of interaction with random changes in the number of particles in the system. Although other sources of randomness in Y{t) could be envisaged and incorporated into this analysis, we prefer not to be concerned with such issues here. The master density should be concerned not only with the population of particles and their states, but also with the state of the environment. Thus, we define the function J^yi^i^ ^2» • •»^v y''> 0 such that Pr[There are, at time t, v particles in the system with one each between z^ and Zi + dZj-, where i = 1,2,..., v and y < Y{t)
I

'zi

"00

•*Zv-

1

dyJ^Yi^^,Z2,...,z^,y;t) = 1. (7.1.25) ^ The expectations of quantities associated with the population are carried out with the integration range in (7.1.25). Thus, for the property F we have v= 0 » 0

dz^

dz^

dz2 • • •

0

EF-l-M vTov!,=

0

ij

dz.

dyF{ )J,Yizi, ^2. • • •. ^v. y; 0-

(7.1.26)

7.1. The Master Density Function

287

We now investigate the product densities arising out of this master density function.

7.1,3.1

Product Densities. The Environment-Dependent Case

As in Section 7.1.1.2, the product densities arise from taking expectations of products of the number density (7.1.1) as defined in (7.1.20). Let us first observe that the product densities now depend on the environmental variable y. Furthermore, we have the zeroth order density 00

1

V

f* CO

foAy, t)^Y~,Yl\

v = 0 ^- r=l Jo

dz.J^z„z„...,z„y;t).

(7.1.27)

The integration in (7.1.27) is over all possible particle states so that foriy^ 0 dy is the probability at time t that the environmental variable Y{t) is between y and y + dy. Unlike the higher order densities, /^^(y, 0 is a true probability density satisfying the normalization condition (7.1.28)

' f„y{y,t)dy^L

Equation (7.1.28) is merely a restatement of (7.1.25). It is useful to define the first-order product density, which yields the expected population density and is defined by CX)

-j

V— 1

fM^,y,t)= Z - j n

v = l ^- r = l

;o

dz^J^yi^^, Z2,..., z,_ 1, z, y; t),

(7.1.29)

which, on multiphcation with dzdy, represents the joint probability that Y{t) is between y and y + dy, and that there is a particle between z and z -h dz. It is not a true probability density since integration with respect to z eliminates the probability interpretation. It is also convenient to define a conditional product density /i]y(z, t\y) by Eln{z, t) I Y{t) = y] = f,^riz, t\y) = 'T/. J foriy, t)

(7-1-30)

so that we obtain ElN{a, b; t) \ Y(t) = y] =

f,,Jz,t\y)dz,

(7.1.31)

The definitions (7.1.29) and (7.1.30) can obviously be extended to the higher

288

7. The Statistical Foundation of Population Balances

order product densities. Thus, 00

X

0

dz, +

1

V —r

fcJ,y(zi,Z2,...,z„z,^i,z,

Yln{z„t)\Yit)=y

+ 2.---.^v'}^;0

= fr\y{z 1,^2,--^^Z^^t

foviy^ 0

\Y{t)

(7.1.32)

=y)

(7.1.33)

The probabihty interpretations of f^y ^^^ fr\Y ^^^ the same as those of f^y and /i|y, respectively, except for the difference of the former being both concerned with the states of r particles in conjunction with the state of Y{t) We now define the unconditional product densities as follows: En{z, t) = f,{z, t) E n n{Zf,, t) = f,{z^, Z2,..., z,, t) =

f,y{z,y.t)dy

(7.1.34)

/,y(zi,Z2,...,z,,};,0rfy. (7.1.35)

The unconditional product densities have the same probability interpretations as the product densities encountered in Section 7.1.1.2 without environmental effects. Furthermore, the expected moments of the population density are obtained as in (7.1.21). Similarly, the formulas (7.1.21) and (7.1.22) also carry over to this case with the product densities defined by (7.1.34) and (7.1.35). The equation for the master density for the class of processes considered in Chapter 3 is readily obtained using standard probability arguments. The probability interpretation of the product densities is also the basis for the derivation of equations representing the behavior of model systems. In the following sections, we shall demonstrate the application of this methodology.

7.2

THE MASTER DENSITY EQUATION FOR PARTICULATE PROCESSES

We now demonstrate how the equation for the master density is derived for the processes considered in Chapter 3. We shall select only one of them,

7.2 The Master Density Equation for Particulate Processes

289

however, since the methodology will hold for other processes as well. The objectives of this section are as follows. Standard probabihstic arguments are used to first derive the master density equation. We then recover, by the process of averaging the master density equation, the population balance equation presented in Chapter 3. Next, we provide a formal solution of the master density equation and expose its combinatorial complexity. Finally, we conclude the section by presenting the role of Monte Carlo simulations from the perspective of the solution to the master density equation.

7.2.1

The Master Density Equation for the Breakage Process

We consider the breakage process of Section 3.2.2 which considers the population as distributed according to say, particle mass x with a growth rate X{x).^ From Section 3.2.2, the breakage frequency is denoted b{x\ and the size distribution of fragments from breakage of a particle of mass x' is denoted P{x \ x'), where x denotes the mass of the fragment. We neglect dependence on the environment, although its inclusion is rather straightforward. Since we are concerned with a stochastic population in a breakage process, we are forced to consider the detailed statistics of fragmentation. However, our task is considerably simplified by assuming binary breakage (as in a microbial population in which cells multiply by binary division), thus facilitating our demonstration without excessive complication. This demonstration is concerned with first deriving the master density equation and then showing that, by averaging it, the population balance equation in Section 3.2.2 can be obtained. In order to derive the growth term in the master density equation we adopt an approach that is only apparently different from that used in Chapter 2 for the derivation of the population balance equation. We begin the derivation by addressing the state of the population at time t. We wish to address the probability that at this instant the system has v particles with masses between x- and x^ + dxi, i = 1,2,..., v. Our strategy is to identify the scenario at instant t — dt that would lead to the situation envisaged at t. Since the particles must grow during the interval dt, the particle of mass x^ at time t must have had mass x[ at time t — dt and survived breakage during the interval dt with probability 1 — b{x[) dt. 7

See Ramkrishna (1981). See Fox and Fan (1988) for a stochastic treatment of breakage and coalescence using the master equation approach.

290

7. The Statistical Foundation of Population Balances

Further, as at most one breakup event may occur during the time interval dt with probabihty of order 0{dt), at most two of the v particles at time t may be freshly formed fragments. This pair of fragments, which could be any two of the V particles present at time t, must necessarily arise from the breakage of a particle whose mass at time t — dt must equal the sum of the masses of the pair selected. Thus,

[Probabihty that at time t there are v particles with one in each of the intervals (x^, x^ + dxi), i = 1, 2 , . . . , v] = J^{x\, ^ 2 , . . . , x'^; f — dt) dx\ dx2 • • • dx'^ [Probabihty that at time t — dt there are v particles with one in each of the intervals (x|, xj + t/x-), / = 1,2,..., v] 1 - X Hx'i) dt

+

[Probabihty that during t — dt to t none of the v particles undergo breakage] V — 1

+

2J J=l

V

2J " ^ V - l ( ^ l 5 • • • ? ^ J - 1 ? ^ j "'" -^k? ^ j + 1 ? • • • ) ^ k - 1 ? ^ k + 1 ? • • • ? ^v? ^ k=j+l V

— dt){dXj -h rfXfc) Yi

^^i

i=l,j,k

[Probability that at time t — dt there are v — 1 particles with one in each of the intervals (x,, x^- + dx^) where i ^j, i # k and one in the interval (x^- + x^^, x^- + x^ H- dXj + dx^)] X b(xj + Xj^) dtP{Xj I x^- + XJ dXj

[Probability that during t — dt to t particle of mass (x^ + x^) breaks into one particle of mass between x^ and x^ + dXj another of mass between x^ and x^^ 4- ^ x j (7.2.1) There are several clarifications to be made in regard to the last expression on the right-hand side of the foregoing equation. First, when examining the terms in the sum the particle size with subscripts 0 and v + 1 must be ignored, since they do not exist. Second, the summation on j and k is such as to automatically avoid double counting of particle pairs. Third, the notation i = lj,k in the product range is used to exclude / =j as well as

7.2 The Master Density Equation for Particulate Processes

291

i = k. Fourth, note that in the particle sizes in the master density are not dated back to those at time t — dt because in the hmit a,s dt ^0 the sizes would be those envisaged at t. The symmetry property of the function P, viz., P(Xj I Xj + Xj^) = P(x^ I Xj + Xfc), and the symmetry of J^_ ^ with respect to permutation of its arguments help us to write the last term on the right-hand side of (7.2.1) as

J=lk=j+l

: b(xj + Xk) dt Yl dXiP{Xj

\ Xj + x^).

Transposing the first term on the right-hand side of (7.2.1) to the left, dividing by dt, and letting dt -^ 0, we obtain

l\dx, = -

Dt

Z

*^v('^l? -^2' • • • ' -^v5 ^) 1 1 dX^

Kx^)J^{x^,X2,...,X^\t)

1=1

v-1

+ 2 2^ j=l xb{xj

2 J -'v-lV^l? • • • > ^ j - 1 ? ^7 + ^k? ^ 7 + 1 ' • • • ? % - 1 ' ^ k + 1 ' • • • > ^ V ' 0 k=j+l +

(7.2.2)

Xj;)P{xj\xj-{-Xk)

where we have set Dt

dt

dX:

(7.2.3)

a derivative evaluated along the joint trajectory of all the growing particles. This derivative arises naturally on dividing by dt and letting dt tend to 0, the difference between a function of time and particle states (whose dynamics are determined by the growth rate X{x)) at t and at t — dt. From the calculus of transformations, it is well known that^

Y\dx, .i=l

8

See, for example, Aris (1962).

'R. Udx, = I X'{x,) Dt

(7.2.4)

292

7. The Statistical Foundation of Population Balances

where X' is the derivative of X with respect to its argument. The combination of Eqs. (7.2.2) to (7.2.4) leads to the following differential equation in the master density function:

V— 1

j=l

V

k=j+l

X b{xj + x,)P{xj I Xj + X,),

V ^ 1.

(7.2.5)

Equation (7.2.5) provides the most detailed statistical description of the breakage process in question. The boundary and initial conditions for the master density are easily identified. For example, if particles of "zero" size are produced by nucleation at a rate, say, h^, then we may write the boundary condition as J,(Xi,...,x,_i,0;r)X(0)-n,J,_,(xi,...,x,_i;0,

(7.2.6)

which reflects the addition of the vth particle of size 0 by nucleation. The initial condition requires detailed specification of all the particles at the inception of the process. The reader will note that Eq. (7.2.5) holds for v ^ 1. For V = 0, the process gets nowhere, since there are no particles to break, a stage that cannot be reached unless it starts (and hence ends) that way! A similar derivation is possible for the master density function of an aggregation process but is left to the reader. Instead, we will consider the derivation of equations for an aggregation process in Section 7.3 directly using product densities.

7.2.2

The Population Balance Equation via Averaging of the Master Density Equation

We wish to derive the equation for the expected population density f^{x, t) for the breakage process considered in Section 7.2.1 by averaging the master density as in (7.1.7). However, since the second term in the left-hand side of (7.2.5) involves differentiation with respect to particle coordinates, it is desirable to perform the averaging of this term by using the definition that /i(x, t) is the expected value of (7.1.1) with z replaced by x. The averaging

7.2 The Master Density Equation for Particulate Processes

293

requires evaluating the integral ^(^ - ^i) ^

—

^ 2 , . . . , Xi_ 1, Xi, x , . + 1 , . . . , X,; 0 ] dXi

lM^i)Jv(^i^

\^X(x)J y(Xj^, ^ 2 , . . •, x^_ j^, X, Xj+ j ^ , . . . , x^; t)].

The average of the growth terms in (7.2.5) is then obtained by summing the foregoing over i, integrating over all the particle coordinates (excepting x^), dividing by v!, and summing over all v. The procedure leads to the average growth term 1 _5_ ^ W ZJ ~7 11 v = ov!, = i dx

dx^ ZJ

Jyi^i^i^'•'^^i-u^^^i+u-"^^vl^)

The symmetry of J^ leads to elimination of the summation with respect to i in the foregoing term and replacement of v! by (v — 1)!. The sum over v represents the definition of/^(x, 0 so that the average growth term becomes

The averaging of the remaining terms in (7.2.5) may be performed in accord with the definition (7.1.7). Thus, we set x^ = x, and integrate Eq. (7.2.5) with respect to Xj^, X2,..., ^v-i? ^^^h over the semi-infinite interval, to obtain

= -b{x)Mx,t)-

+2 1

v=i u

1

X

n

dx, X b{x,)J,

^^,(v-iy.x\ 1 -L;. r=i

JO

7=1

k = j+l 00

•.b{xj + x,)P{x,\xj X I

+ x^) + 2 X

zn

v - l v - l

v=i(v-l)!j=i/=i

CIX^J ^_ ^(X i^,X2', ' ' ' , Xj_ ^,Xj-T

X b{xj + x)P(x IX • + x).

1

X,X, j ^ ^ , . . . ,

X^_^lt)

(7.2.7)

294

7. The Statistical Foundation of Population Balances

In (12.1), the third and the fourth terms on the right-hand side were obtained from the double sum in the second term on the right-hand side of (7.2.5) as follows. We isolate from the inner sum the term that corresponds to k = V and write it separately as in the last term shown. We now show that the second term on the right-hand side of (7.2.7) exactly cancels the third as follows. The multiple integral in the third term includes the integration with respect to x^ and x^ each over the semi-infinite interval, viz., dxj

0

dx^ '•• =

Jo

0

dx:J I

'dx^...,

Jo

where Xj = Xj + x^. Examining the integrand in the term we recognize that the only part that qualifies for integration with respect to Xj^ is the function P{x^\Xj), which integrates to unity since it is in fact the normalization condition for the function. Thus, the third term on the right-hand side of (7.2.1) becomes 00

1

V— 1

^00

V - 1

V - 1

uXjD(Xj)J y_ i(Xj,..., Xj_ J, Xj, Xj^ 1,..., x^_ J, Xjij^. 1,..., x^_^, x; t). (7.2.8) If we rename Xj as Xj and rewrite the sum with respect to j and k as v-1

v-1

v - l v - 1

21 I -=Z j=l

k=j+l

j=l

1-,

fc=l

the expression (7.2.8) becomes

' n

v- 1

v= i(v-l)!,ii,fc 0 X

v-1

v-1

^

2^

dx^

0(Xj)J^_l(X^,.,.,Xj_^,Xj,Xj^l,...,Xj^_^,Xj^^l,...,X^_^,Xlt).

j=lk=l

Again, in view of the symmetry of J ^ . j , the integrals under the sum with respect to k are the same. Thus, we may write the preceding expression after renaming particle size coordinates in serial order: 00

Z 7

1

V—

^

n

I

f* CO

v = 2 l^ ~ ^i- r=l Jo

V— 1

dx^T,

j=l

b{xj}J,_^{x^,X2,...,x,_^,x;t).

7.2 The Master Density Equation for Particulate Processes

295

If in the foregoing v is replaced by v + 1, one finds that it is the same as the second term on the right-hand side of (7.2.7) with the sign reversed so that the desired cancellation is accomplished. What remains to establish the population balance equation is to treat further the last term on the right-hand side of (7.2.7). Toward this end we set x' = Xj -\~ x and, using the symmetry property of J^_i, we may rewrite the term in question as )

dx'b{x')Pix\x') £ -—- n 00

uX^J

1

^_ ^yX ^, X2? • • • ? •^j-

V— 1

1 ' -^ ? ^ j + 1? • • • ? - ^ v - 1 ' V-

0

In Hght of (7.1.7), the term within the sum may be recognized to be fi{x\ t), thus yielding from (7.2.8) the population balance equation df (x t) d ' + -^ St dx [X(x)/i(x, 0] = -h{x)f,{x,

0+2

r°°

b{x')P{x I x')f,{x\ t) dx\ (7.2.9)

The boundary condition (7.2.6) at x = 0 can be similarly averaged to yield Z(O)/i(O,0 = n,. We have thus estabhshed the population balance equation and its boundary condition rigorously from the master density equation. It is possible in an entirely analogous manner to also derive equations in the higher order product densities by appropriately averaging Eq. (7.2.7) and thus facilitate the calculation of fluctuations. We do not take this route here because we shall derive the product density equations directly from their probabihty interpretations in Section 7.3. In the preceding derivation, we had assumed either the absence of environmental variables or their effect on particle behavior. The inclusion of environmental effects leads to difficulties connected with closure of the set of equations as we shall discover in Section 7.3. 7.2.3

Solution of the Master Density Equation

It is interesting to observe that Eq. (7.2.5) in the master density equation can be solved somewhat readily in a conceptual sense. The issue has been dealt with by the author at length in a publication referenced in footnote 7. We shall discuss only the broad features of this development here.

7. The Statistical Foundation of Population Balances

296

Consider the evolution of the breakage process as viewed by (7.2.5) from the instant t' to t. We regard this time interval to be suitably small in order that the population increases by at most one particle. Thus, if we envisage V particles at time t with masses x^, ^ 2 , . . . , x^, the population at time f must consist of no less than v — 1 particles with masses that must be determined by solving backwards the differential equation dx = X{xX du

x{ti) = Xi, i = 1,2,..., V.

u < t;

We let the solutions be represented by X{u \ Xj, t) with the property that X{t I Xj, t) = Xj. In order to compute J^ at time t, we rewrite the differential equation (7.2.5) along the foregoing characteristics as

-J. I X'ix,)

Dt

i= 1 V -

1

j=lk=j+l

x^(x^. + xJP(x^.|x^. + Xfc). The solution of this differential equation is rather straightforward and may be written as

=^J,{X{t' Ixi,t),X{t'IX2,t),...,X{t'Ix„0;0 X

^XiX{t'\x,,t)) -.

+ t j=i

+

t k=\,j

exp

-I

\ dt"J.- mt"\x

h{X(u |x,-, t)) du

„ t), X(t"\x„ f),..., X(t"\xj_ „ t), X{t"\xj, t)

Ji'

X{t"\x„t),X{t"\xj^i,t),...,X{t"\x,.i,t),X{t"\x,^i,t),...,X(t"\x,_„t);t')

n

^~'X(X(t"\x„t)) X{x,

X b(X{t"\Xj, t) + X(t"|xt, 0) V

X P{X{t"\x^, t)\X(t"\x^, t) + X{f\x^, t)) X exp L

b(X{u\x^,t)) du t'

i= = ll JJt'

J

(7.2.10) The two terms on the right-hand side represent two different ways of

7.2 The Master Density Equation for Particulate Processes

297

evolving to the state at time t starting from that at time f. The first is the probabihty that there are v suitably sized particles at time f which successfully grow without breakage of any of them to the sizes envisaged at time t. The second is the probability there are only v — 1 at time f and that one pair of the particles at time t has come about by the breakage of one of the v — 1 particles at time t'. It is interesting at this stage to recall the development of Section 4.6.2, in which the concept of the "quiescence interval" was introduced in connection with an exact method for Monte Carlo simulation of particulate systems. Accordingly, let us observe that the quiescence time, say, T in the breakage process under discussion is the time elapsed since the time f of present reckoning. Clearly, T is a random variable whose distribution is a function of the state of the population at time t'. In fact, it is readily inferred from Section 4.1.6 that Pr{T>T|yl,,} = exp

b{X{t' + u IX-, f)) du 0

where A^. represents the detailed statement of the state of the population at time f. The right-hand side of this equation differs from the exponential terms in Eq. (7.2.10) only with respect to the instant at which the state of the population is specified. Suppose we consider the state of the population as exactly specified at time t\ i.e., the number and sizes of particles are known exactly to be as specified in A^,. Then at the instant t\ J^ is the product of Dirac Delta functions at the different particle sizes specified and J^_ 1 is equal to zero. Thus, the solution (7.2.10) will consist only of the first term on the right-hand side. The characteristic curve will be a smooth curve in v-dimensional space with the parametric representation {X{u I x;, t% / = 1, 2 , . . . , v},

u> t'.

(7.2.11)

During the quiescence interval following instant t\ the foregoing curve will extend smoothly in time until a random breakage event occurs, at which point the characteristic curve must "jump" to a point in the higher (v + l)-dimensional space. The location of the point to which the characteristic curve jumps is of course random, with a probability distribution that is contained in the second term on the right-hand side of (7.2.10). More precisely, the probabihty distribution for identifying the particle that disturbs the quiescence by breakup, and the distribution for identifying the sizes of breakage fragments can both be obtained from (7.2.10). For details, the reader is referred to Ramkrishna (1981).

298

7. The Statistical Foundation of Population Balances

FIGURE 7.2.1

Evolution of the characteristic curve for a breakage system.

The foregoing situation of the characteristic evolves is conveniently represented in Figure 7.2.1 for the case of a population starting with one particle initially and randomly increasing in number with respect to time. Thus, the initial state of the population is the point O from which the particle grows to the location A without breakage. At A the particle suffers binary breakage to put the system at B, from which the two particles grow along the curve BC until the point C is reached, when breakage occurs again. The new state of the system is shown to be at D, following which the three-particle system evolves along the characteristic curve DE. Although this geometric demonstration cannot survive another breakage event, the evolutionary trend of the system is clearly established. The role of the Monte Carlo simulation procedure is to locate through random number generation sample positions of the points B, C, D, E, etc. The curves OA, BC, DE etc., are of course the particle path curves (7.2.11). Thus, a sample path of the simulation is the characteristic curve that jumps into spaces of increasing dimension at times and locations as determined by random number generation. The solution (7.2.10) of the master density equation, on the other hand, can be traced back to the actual initial time by successively substituting for the master density on the right-hand side and marching regressively in time. We thus have an analytical solution to the master density of the breakage process but because of its combinatorial complexity it cannot be evaluated without the discriminating aid of Monte Carlo simulation techniques.

7.3 Stochastic Equations of Population Balance 73

299

STOCHASTIC EQUATIONS OF POPULATION BALANCE

As pointed out earlier, the stochastic equation (7.2.5) is the source of all stochastic equations of the breakage process considered. Thus, the product density equations of concern to us in this section may all be obtained by appropriate averaging of Eq. (7.2.5) in a manner akin to the derivation of the population balance equation (7.2.9). However, because the product densities have probability interpretations, we are able to derive them directly for any process. The product density equations have considerably less combinatorial complexity than the master density equations because they have already been subjected to an averaging process. Although the master density equations were derived only for the breakage process, we deal with both breakage and aggregation processes in this section. We are first concerned with the case of particles without environmental effects, and we then consider that in which particle behavior is affected by a single environmental variable. The extension to the vector case is usually self-evident.

7.3.1

Product Density Equations for the Breakage Process

Consider again the breakage process without environmental effects which was dealt with in Section 7.2.1. We are concerned with finding at time t a particle between x and x -\- dx without regard to all the other particles in the system.^ As was done with the derivation of the master density equation, our strategy is to construct the scenario at time t — dt oi various situations that can lead to that envisaged at instant t. For the process in question, a particle of size between x and x -h dx a,t time can arise in two ways. First, the particle exists at time t — dt with mass between x' and x' + dx' during the time interval t — dt to t and it grows to mass between x and x + dx without suffering breakage. Second, a larger particle exists at time t — dt that breaks during the time interval (t — dt, t) to yield one particle between X and X + dx. Thus we may write /i(x, t) dx Pr[There exists at time t particle between x and x + dx]

= fi{x\ t — dt)dx' Pr[There exists at time t — dt a. particle between x' and x' + dx'] 9

See Ramkrishna and Borwanker (1973).

300

7. The Statistical Foundation of Population Balances

X [1 - b{x') dt] Pr[Particle does not suffer breakage during interval (t — dt, t) but adds mass X{x') dt]

+ 2

d^M^,t)biOdtP{x\i)dx

Pr[Particle of mass larger than mass x breaks during interval (f — dt, t) into two fragments one of which has mass between x and x + dx~\

Transposing the second term on the right to the left, dividing by dt, and letting dt tend to zero, we obtain I Mx, t) + Mx, t)^j^

(dx') = - bix) Mx, t) + 2

b(i)P{x\OA{Lt)di, (7.3.1)

where we have recognized that as dt -> 0, dx' -^ dx, and set

1 = 1^ Xix) — If we recognize further that

dt

dt

dx

^^(dx')=X'{xl dx dt Equation (7.3.1) yields the usual population balance equation dMx, t) ^8 ^^^^y^^^^^ j)-| ^ _fc(^)/^(x, t) + 2 dt dx

h{i)Pix\i)Mi,t)di (7.3.2)

The boundary condition at x = 0 is obtained by argument that the probability there is a particle (nucleus) between 0 and X(0) dt on the size coordinate during time t and t + dt can be obtained in two different ways. First, it is given by the left-hand side of the equation appearing below by definition of the first-order product density. Second, it is also given by the right-hand side by definition of the nucleation rate, which is the transition probability for the appearance of a nucleus in the time interval (t, t + dt). Thus, f,{0,t)XiO)dt

= hJt,

(7.3.3)

so that the boundary condition is also established. The initial condition simply relates the product density to the initial probability of finding a particle in any infinitesimal particle state interval and is therefore unchanged

7.3 Stochastic Equations of Population Balance

301

from that used for the population balance equation. We now have the result that the population balance and its boundary and initial conditions are the same as those obtained from both the master density equation and the product density formulation. The reader should bear in mind that this equivalence was established for the case where environmental variables did not play a role in determining particle behavior. Equation (7.3.2), however, provides only the average behavior of the system. It is of interest to observe here that the average behavior of the system could be obtained by dealing only with the first-order product density, viz., the expected population density. A truly stochastic formulation must consider, however, the higher order densities in order to calculate the average fluctuations about the mean behavior. The calculation of fluctuations was the subject of Sections 7.1.1.1 and 7.1.1.2. Since the higher order densities were the basic implements of this calculation, it will be our objective to first formulate the second-order product density equation for the breakage process under discussion. Recall that the product density, /2(x, (^, 0 when multiplied by dx d^, represents the probability that there is, at time t, a particle between x and X + dx, and another between S, and £, + d^. Note again that this probability is disregarding the sizes of all other particles that may be present in the population. This probability interpretation will now be used to derive an equation for the breakage process in question. The strategy of derivation is to investigate the circumstances under which the situation envisaged at time t will emerge from that at t — dt. We shall enumerate these. First, we consider at t — dt two particles, one of size between x' and x' + dx\ and another of size between ^' and S,' + d^', both of which survive breakage during the time interval t — dt to t and grow to the required sizes at time t. Second, we envisage one of the particles to grow from its neighboring size to its present size without breakage while the other results from breakage of a larger particle. There are two such terms, clearly, since this divided history may be shared by either particle in the pair. Third, we must consider the situation of a particle that is exactly the size of the sum {x + ^), which may suffer breakage during the interval {t — dt, t) to form the required pair of particles envisaged at time t. We itemize these below with their corresponding mathematical representations: /2(x, (^, t) dx d^ Pr[There are at time t two particles one between x and x + dx, and another between ^ and ^ + d^']

302

7. The Statistical Foundation of Population Balances = f2{x\i\t)dx'di'

X

Pr[There are at time t — dt two particles one between x' and x' + dx', and another between ^' and ^' + d^'^

11 - b{x') dt']ll

-b{^')dt']

Pr[Neither particle present at time t — dt breaks during the time interval {t — dt, t) but instead grow to the sizes in the respective intervals (x, x + dx) and {^, ^ -\- d^)']

driUr], ^', t)b(rO dtP{x\r]) dx

+2

Pr[There are two particles at time t — dt, one larger than x and the other between ^' and t,' + d^', the former suffering breakage during the interval {t — dt, t) to produce a particle in (x, x + dx) and the latter surviving breakage to grow into a particle in

{L ^ + ds,)-] + 2

dnU(x', n, t)h(n) dtP(^\n) d^

Pr[There are two particles at time t — dt, one larger than ^ and the other between x' and x' + dx', the former suffering breakage during the interval (f — dt, t) to produce a particle in {<^, ^ + dt) and the latter surviving breakage to grow into a particle in

(x, X + dxy] + 2j\{x + (^, r)fo(x + (^)rfrP((^|x+ ^) dxd^ Pr[There is one particle at time t — dt, of size between {x + 0 and ^(x + ^) which during the interval {t — Jr, f) to produce a particle in (x, x + dx) and another in

Rearranging the terms above, dividing by dt and letting dt -^ 0, we obtain the following equation:

dt

dx^

' '•"'

' " '-•

= -lbix)+b(mf2(x,^,t) +2

dy'

+2

b(n)P(x\n)UnA,t)dn

b{n)Pmt]) f2(x, n, t) dn + 2b(x + ^)P{x\x + a/i(x + ^, t). (7.3.4)

7.3 Stochastic Equations of Population Balance

303

We have thus derived the second-order product density equation for the pure breakage process based purely on probabihstic considerations. It could also have been derived by averaging the master density equation (7.2.5), using the definition (7.1.11). The boundary conditions for the second-order product density can be obtained in much the same manner as the first-order product density by accounting for the formation of nuclei. Thus, recognizing the symmetry of the product density, we have the boundary condition /^(x, 0, t)X{0) = /,(0, X, t)X{0) = /,(x, t)n,.

(7.3.5)

The higher order product density equations can be derived in an entirely analogous manner. It is noteworthy that Eq. (7.3.4) in the second-order product density contains only product densities of order 1 and 2. Thus, the average behavior of the population can be obtained by solving only the first-order product density equation, i.e., without requiring knowledge about fluctuations. Similarly, the second-order density equation can be solved without knowledge of the higher order densities. In the next section, we consider the derivation of product density equations for an aggregation process.

7.3.2

Product Density Equations for an Aggregation Process

We are concerned with a particle population distributed according to their mass X with growth rate X{x) independent of any environmental variables. ^° Aggregation takes place only between particle pairs with a frequency denoted by a{x,x')}^ The population is assumed to be uniformly mixed in space throughout the system so that external coordinates do not enter the analysis. This assumption of spatial uniformity is questioned towards the end of this section as well as in Section 7.4, and the reader is advised to take subsequent note of the discussion therein. We shall first address the first-order product density or the expected population density, f^{x, t). As before, the strategy is one of considering various circumstances at the instant t — dt that will lead to the presence of a particle at time t with mass between x and x + dx. First, there may be a particle at t — dt with mass between x' and x' + dx' that may grow to the 10 The considerations in this section are due to Ramkrishna and Borwanker (1973). 11 This assumption views aggregation among more than two particles as a rapid sequence of binary aggregation steps.

304

7. The Statistical Foundation of Population Balances

size envisaged alt time t without aggregating with any other particle. Second, there may be an aggregation between two particles present at time t — dt with appropriate masses to yield a single particle of mass between x and X + dx. Thus /^(x, t) dx Pr[There exists at time t particle between x and x + dx] = /i(x', t - dt) dx' Pr[There exists at time t — dt a. particle between x' and x' + dx'l

-dt

a{i,x')f2{i,x\t)d^ 0

Pr[Particle between x' and x' + dx' at time t — dt aggregates with another of any size]

1 ^2

a{x-^,i)f2{x-i,i,t)d^dt

Pr[There exist at time t — dt two particles of appropriate sizes that aggregate to form a particle at time t of size between x and x + dx]

Note in particular that the second term on the right-hand side is subtracted from the first in order to calculate the probability that the particle of mass between x' and x' + dx' is not lost by aggregation with particles of any size during the time interval (t — dt, t). Transposing terms, dividing by dt and letting dt -> 0, we obtain the following aggregation equation for the expected population density

^

4 :*)/,.. o]^i

0

aix-^^Ofii^

-L^.t)d^

X)

a(x,^)Ux,^,t)dt

(7.3.6)

If there is nucleation, Eq. (7.3.6) must be subject to the boundary condition (7.3.3). The foregoing equation is significantly different from the population balance equation because of the presence of the second-order product density on the right-hand side. The population balance equation can be obtained from Eq. (7.3.6) only under the assumption of statistical independence of particle states in each pair, viz., f2{^A.t) = h{x,t)m,t).

(13.1)

Since Eq. (7.3.6) must be subject to an initial condition, the question arises as to whether or not there is initial correlation between particle sizes. If there

7.3 Stochastic Equations of Population Balance

305

exists no initial correlation, the issue of interest is whether particle size correlation does indeed develop in the course of the aggregation process. For example, suppose we are concerned with Brownian aggregation in which particles of unlike sizes aggregate very rapidly. In due course of time, one may expect a correlation to develop between particles of like sizes. Thus, size-sensitive aggregation rates may give rise to the growth of correlation between specific size pairs. When such correlation develops, the assumption of independence for all particle pairs may well be subject to question. We shall return to this issue at a subsequent stage. The problem of correlation has been discussed at length by Ramkrishna et al. (1976). The presence of the second-order product density in the right-hand side of (7.3.6) makes it impossible to solve for the population density. An equation is needed for the second-order product density to "close" (7.3.5), although the reader will have guessed that such a quest for closure is doomed to failure since the risk of featuring a third order density is imminent in the process. We now directly present below the equation for the second-order product density assuming that the detailed methodology of prior derivations will have developed a facihty for this beeline: df,{x, ^'i) ^l_

iX{x)f,{x,

1

=+2

a(x', X — x')f^(x', X — x', ^, t) dx' 0

1 f^

+ -

^ In

0

1,0] + ii [^(^)/2(x, 4 0]

a(x', ^ — x')/3(x', X, (^ — x\ t) dx'

[a(x, x') + <2(^, x')~\f^{x, (^, x\ t) dx' - a(x, ^)f2{x, ^ t).

(7.3.8)

The terms on the left-hand side are readily interpreted. The first two terms on the right-hand side represent the formation of the particle pair in question with one of them arising from aggregation while the other does by pure growth. The third term represents the loss by aggregation with particles of all sizes, and the last term denotes the destruction of the pair by mutual aggregation. As expected, the lack of closure is perpetuated by appearance of the third order product density. The rth-order density

306

7. The Statistical Foundation of Population Balances

equation is readily written down by inspection. d

*" d 1 ''

'

ci(x , Xi X jjj,^. ^(x j^, X2,..., x^_ ^, Xj- X , Xj.|. ^,..., x^, X rj ax X a{Xi, x')/,+ i(x 1, X2,..., X,, x', 0 dx'

-fr{x^,X2,...,x„t)

Z

Y, j=i

^(^r^fc)-

(7.3.9)

k=j+i

Thus, the product density equations are clearly unclosed, making it impossible to solve them without a suitable closure hypothesis. If it is known that there are no more than N^ particles in the system initially, the closure is automatic at r = N^, since in the equation for fj^^ ^/N^+I = 0. One may also identify equations in the master density, J^ using probabilistic arguments as in the breakage process. The resulting hierarchy of equations is slightly different from (7.3.9): — J,(xi,X2,...,x„r)+ X ^[^(•x,)J,(Xi,X2,...,x,,0] , = 1 dxi

ot

^ + —2^

I

^ i =1 Jo

ClyX , Xi

X jc/y^ j^(X J, X2, • • . , X j _ J, XjV -

-J,(xi,x2,...,x,,r) X

1

X , Xj.|. J, . . . , Xy, X , t j a X

V

X

^(^j^^kl

(7.3.10)

J=lk=j+l

The difference between (7.3.9) and (7.3.10) is in the absence, in the latter, of a term similar to the second one on the right-hand side of (7.3.9). The reason for this difference lies in the master density having to take responsibility for the entire population at any given instant. Thus, it cannot ignore, as (7.3.9) does, the status of the particle formed by aggregation between the particle of mass x' and any of the v other preexisting particles of sizes x^, X2,..., x^. Obviously, the preceding hierarchy of master density equations can also be closed at v = N^. However, the product density equations may allow closure at a considerably lower value of r, which makes them much more attractive to solve than the master density equations. As pointed out earlier, even an analytical solution to the master density equation is not particularly valuable because of its combinatorial complexity.

73 Stochastic Equations of Population Balance

307

A discrete version of the master density equations (7.3.10), without particle growth, has been solved by Bayewitz et al (1974), and later by Wilhams (1979), to examine the dynamic average particle size distribution in an aggregating system with a constant kernel. When the population is small (EN < 50) their predictions reveal significant variations from those predicted by the population balance equation. However, the solution of such master density equations is extremely difficult even for the small populations of interest for nonconstant kernels. It is from this point of view that a suitably closed set of product density equations presents a much better alternative for analysis of such aggregating systems. We take up this issue of closure again in Section 7.4. The reader should note that the population balance equation for an aggregating system predicts that the number of particles should eventually dwindle to zero (rather than a single lump of the entire mass). Of course, the model may cease to be vaHd for other physical reasons before the particle population gets to very low values. For example, the aggregation frequency itself may cease to be vahd when the particles become large enough. Not withstanding the foregoing issue, it is worth noting that the product density hierarchy will account for the system converging towards a single particle in due course of time. Before closing this section, however, it is worth noting that the equations of this section were concerned with aggregation occurring in a closed system, in which the particles were mixed uniformly in space. It will turn out from the deliberations in Section 7.4 that for a more reahstic analysis of aggregation problems in a large system, such as a cloud of particles, the assumption of spatial uniformity may not hold for the entire system. Instead, the particles may be uniformly mixed only in a local volume of mixing, V^^^, which could be changing with time in some prescribed manner. Thus, the population balance equation as well as the product density equations (7.3.6), (7.3.8), and (7.3.9) must be modified to account for the exchange of particles between the well mixed volume and its surroundings. The issue is taken up further in Section 7.4.

7.3.3

Product Density Equations for the Environment-Dependent Case

In Sections 7.3.1 and 7.3.2 our concern was restricted to situations in which the particle behavior did not depend on its environment. Often one encounters situations in which this assumption does not apply. In particular, the nucleation and particle growth processes are indeed functions of

308

7. The Statistical Foundation of Population Balances

supersaturation in the continuous phase necessitating the incorporation of several environmental variables. Since the extension to the vector case is self-evident, we will entertain only one environmental variable in the sequel, however. Thus, we shall assume as in Section 7.1.3 the presence of a single environmental (stochastic) entity Y{t) whose values are denoted by the variable y. The required product density hierarchy is presented in Section 7.1.3, and the purpose of this section is to present the methodology for the derivation of equations satisfied by the product densities. As before, the basis of their derivation is their probabilistic interpretation. We shall first introduce the process of interest. Since the methodology for derivation of breakage and aggregation terms has been discussed at length in prior sections, we shall neglect these processes to avoid repetition. We assume a system consisting of particles that are formed by nucleation and are subjected to growth. In a subsequent apphcation, we shall be more specific about the nature of the physical process, but for the present we leave the setting somewhat at an abstract level since it is the mathematical formulation that is the focus of our discussion. The dispersion is uniformly distributed in space so that the number density of particles as well as the concentration of the environmental variable is independent of position. The number of particles in any size range varies randomly, causing random changes in the residual environmental variable. The nucleation rate is described by h^{y) and the growth rate of particles of size x by X(x, y). We assume further that Y is the rate at which the environmental variable is changing in the system. In applications, Y must depend on the environmental variable as well as the population. Generally, the environmental variable is some conserved quantity, which is exchanged between the continuous phase and the particles in the system. It can be expressed in the form Y = tiy)

+ Uy) i

(t>{Xi)X{x,,y)

(7.3.11)

which is inspired by the following considerations. In the absence of all particles, transport and reaction processes may lead to a rate of change of the environmental variable in the continuous phase given by Y^{y). Further, a change occurs because of the presence of the ith particle of size x^- which is proportional to the rate of growth of the particle with proportionality "constant" that contains a size-dependent part (/>(Xi) as well as an environment-dependent part Y^iy). Thus, (j){Xi) may refer to the surface area of the particle and Y^{y) may represent the rate of transport of the environmental

7.3 Stochastic Equations of Population Balance

309

entity to the particle from the continuous phase. An actual application will undoubtedly throw more light on the form (7.3.11). In deriving the product density equations, we shall take the route of first identifying the master density equation and obtain the former by averaging. We prefer this route to that of direct derivation of the product density equations in this case because the rate of change of the environmental variable is given by (7.3.11) which involves all the particles in the system. We address the evolution of the population from the state at time t — dt to that at t given by v particles with sizes between Xi and x^ + dxi, i = 1,2,..., V, and y < Y{t) < y -\- dy. If, as in prior sections, the neighboring intervals are assumed to be (x •, xj + dXi) for the particle states and (y\ y' + dy') for y(t — dt\ then probabilistic arguments directly lead to the result vyV 1' ^2? • * • 9 ^v'3^' / ^ ^ 1 ^ ^ 2 *** (XXyay

= J^y(xi,X2,.. .,Xv'y;t —Jt)dxi(ix2---(ix^rf/ X II — h^iy) dt'], which reflects evolution purely by "growth" with allowance for avoiding a nucleation event since it would change the total number of particles. Transposing and taking limits as dt -> 0, one arrives at the master density equation ^^ ^

+ Z ^ . iMxi, y)J.y-] + I ; [ t i v y ] = -KiyV.Y.

V ^ 1. (7.3.12a)

For V = 0, this equation becomes

^

+ 1 ; "^^-^"^^ = -KiyVoY,

(V.3.i2b)

which only needs an initial condition. Equation (7.3.12), on the other hand, must be supplemented with a boundary condition for the appearance of nuclei. In the absence of secondary nucleation by existing particles, the boundary condition is readily identified as j^yix 1, X2,..., X,, 0, y\ 0^(0, y) = J(v-I)F(^i, -X:2, • • •, ^v. y; ^)^oiy\

^^ 1 (7.3.13)

In the presence of secondary nucleation by existing particles, the nucleation rate could depend on the sizes of the existing particles, and a boundary 12 The author's derivation of the master density equation for a microbial population with environmental effects is available in Ramkrishna (1979).

7. The Statistical Foundation of Population Balances

310

condition more complicated than (7.3.13) will result. The formulation of initial conditions is generally straightforward, for suppose that there are no particles initially and the initial value of Y is Y^; then we may write 5{y-Y,l 0,

J,(xi,X2,...,x„};,0) =

v=0 V ^ 1'

Equations (7.3.12) and (7.3.13) may now be averaged to obtain the product density equations. The first density function of interest is /^yCy, t), which is the only true probability density in the set of product densities. Using the definition for this density, Eq. (7.3.13) may be directly integrated with respect to all the particle coordinates, divided by v! and summer over all V to yield 1 dt ^

.^,

n

dx, X lM^i^y)'^vY

+ dy tW/or+nW

(/)(x)X(x, y ) / i y ( x , y , f )

(7.3.14) The first term and the first within the third term in (7.3.14) arise by integration over all the particle coordinates and using definition (7.1.27). The second within the third term arises after substitution for Y from (7.3.11) and taking expectation of the sum. We presently show that the second term and the fourth cancel each other. In the second term, quantities evaluated at the upper limit of infinity must all vanish because no particles exist to arrive at infinite size, making J^y vanish there. The terms at the lower limit of zero may be written as 00

1

V

/ * 00

v=l^'r=lJo

V

1=1

The symmetry of J^y with respect to all the particle coordinates makes the V integral terms in the summation over / all equal so that the boundary condition (7.3.13) converts the foregoing expression to

-Kiy) 1

1

i(v-l)!

*^(v- n y i - ^ 1? ^ 2 ' •' • •> ^v") y-' ^) ^^r-'

which, from the definition of /oy(y, 0 ^^ accord with (1A.27), yields — h^{y)f^y{y, t) so that the desired cancellation in (7.3.14) is accomplished

7.3 Stochastic Equations of Population Balance

311

= 0.

(7.3.15)

to give St '^ dy yo{y)f„Y+Uy)

(j){x)X(x,y)f^y{x,y,t)

Interestingly, this brings the first-order product density into the equation, thus initiating a closure problem. Averaging the boundary condition (7.3.13), one obtains (7.3.16)

f,My,t)X(Q,y)=fXy,t)nM)-

In accord with the definition (7.1.29) for the first-order product density, we integrate Eq. (7.3.12a) over all but one particle coordinates to obtain

t

+ |[i(x,,)/.,] + dy t W / i r + ^ i W

(l>m{^,y)f,{x,U)d^

= 0.

(7.3.17)

Note that the nucleation term disappears in the same way as it did in Eq. (7.3.15). The preceding equation contains the second-order product density, which confirms the closure problem encountered in this situation. ^^ This closure problem arises because of the mutual coupling between the particulate and the continuous phase variables. The boundary condition for (7.3.17) is obtained by averaging (7.3.13) to get /2y(0, X, y, 0 ^ ( 0 , y) = Ari^, y, t)n,iy).

(7.3.18)

From (7.3.16) and (7.3.18), it is clear that the lack of closure also manifests in the boundary conditions. Unless a suitable closure hypothesis is advanced at this stage, there is no way to solve Eqs. (7.3.15) to (7.3.18). Although we address this issue in a subsequent section, our next endeavor will be to determine the conditions under which conventional population balance description will be apphcable for computing the mean behavior of this stochastic system. In order to average over the environmental variable, we integrate the differential equation (7.3.15) with respect to y. If we define the event that at time t there is a particle in the population of size x by A^{t), then the result 13 Through an error in the derivation of the first order product density, the author did not recognize the appearance of the second-order product density in Ramkrishna (1979).

312

7. The Statistical Foundation of Population Balances

of integrating (7.3.15) may be written as

^ ^ i ^ + ^ {ElMx, VIA Am Aix, t)} = 0

(7.3.19)

where the conditional expectation £[X(x, y^)|^^(0] is defined by £[X(x, YMM

- ioXix,y)AA^,y,t)dy

^^ ^ ^^^

Note that Y^ is the random environmental variable at time t. Similarly, integrating (7.3.16) with respect to y, we obtain MO, t)ElX{0, YMo(t) = ElMY,)l

(7.3.21)

in which £[n<,(y,)] is defined by E\.K{Y,):\ = 0

no(y)fo(y,t)dy.

(7.3.22)

In the usual population balance approach, one replaces Eq. (7.3.19) with ^ ^ ^ + j ^ lX(x, EVyUix, t)-] = 0

(7.3.23)

and Eq. (7.3.21) with the boundary condition /,(0,fWO,£i;) = n,(£y,).

(7.3.24)

In (7.3.23) and (7.3.24), the particle growth and the nucleation rates are evaluated at the expected value of the environmental variable and are thus clearly not the same as (7.3.19) and (7.3.21), respectively. If the growth rate is not dependent on y, then from (7.3.20) it is readily apparent that (7.3.19) is equivalent to (7.3.23). Further, the boundary condition (7.3.21) will be equivalent to (7.3.24) if the nucleation rate is a linear function oi y}^ Since the dependence of h^{y) on y is generally nonlinear, the validity of the population balance approach for calculating mean behavior cannot be taken for granted except when the average fluctuations can be considered negligible, and consequently the environmental variable changes in a virtually deterministic manner. 14 It is tempting to assume that if the growth rate is a hnear function of the environmental variable y, then (7.3.19) is equivalent to (7.3.23). However, this is not true because EY^ and £[yj^^(t)] are not necessarily the same.

7.3 Stochastic Equations of Population Balance

313

Before closing this section, we briefly consider the case in which there are several environmental variables forming a vector Y^ with realizations described by the vector y. The relevant equations for the scalar case are readily modified by replacing y with y, and Y with Y, and replacing the partial derivatives with respect to y in Eqs. (7,3.12), (7.3.14), (7.3.15), and (7.3.17) with the partial divergence

V/CY()]^X^[^()1 '

i Syt

The unfilled parentheses are either for the master density as in (7.3.12) or for the product density functions as in the other equations. The vector case is now fully identified. In Section 7.4.2, we discuss an application of the foregoing development. We now consider some apphcations of the stochastic theory just presented. We are at first concerned with applications, which do not feature the problem of closure encountered in Sections 7.3.2 and 7.3.3. Subsequently, in Section 7.4 we address the closure problem and consider some applications therein.

7.3.4

Applications of Stochastic Population Balance

We address applications here in which closure problems are not encountered. Thus, the average behavior of the population can be obtained from solving the first-order product density equation, and average fluctuations (of any order) about the mean can be calculated progressively by solving higher-order product density equations. In order to elucidate the nature of what can be obtained from such a theory, we shall consider a simple enough example for which analytical answers can be found. It is followed by a second example which has potential application to the study of cell death kinetics and hence to sterilization processes.

7.3.4.1

Stochastic Age Distribution of a Population

Consider a population of individuals distributed according to their age. Each individual of age T may give birth to a new individual of age zero with a birth frequency ^(T). The age of the parent may be assumed to be unaltered by the birth process. Further, assume an expiry rate e{T) for an individual

314

7. The Statistical Foundation of Population Balances

of age T. No spatial coordinates are involved since the population is spatially homogeneous. Note that such a population has the possibiHty of total extinction and it would be of interest to determine its probability among certain other calculations. The average number of individuals in any age bracket is obtained by the first order product density. The coefficient of variation about the mean requires the calculation of the second-order product density /2(T, T\ t). We assume that the initial average number of individuals to be N^ with an age distribution given by g^ir) and that initially their ages are not correlated. Neither assumption is a necessary attribute of the analysis. The first-order product density must satisfy the following partial differential equation and initial condition ^

^

+ ^

^

+ e(T)Mt, t) = 0,

Uz, 0) = N„g,iT). (7.3.25)

Furthermore, /^ must also satisfy the boundary condition /i(0, t) =

(7.3.26)

b{T)MT,t)dT.

Integration of Eq. (7.3.25) along the characteristics (f = T + constant), leads to the result /I(T - r,0)exp

e{T')dT'>,

T> t (7.3.27)

/l(^, t) = /i(0, r - T ) e x p < | -

e{T') dx' \ ,

T
Equation (7.3.26) may be rewritten in view of (7.3.27) as /i(0, t) =

b,{T)f,(0,t-T)dT

+ h(t\

h{t)^N,

biit -\-T)g^{T)dT (7.3.28)

where we have set ^^(T) = b{T) exp< — ^(T') dz' >. in defining the function h{t), we have used the initial condition in (7.3.25).

7.3 Stochastic Equations of Population Balance

315

The Volterra integral equation (7.3.28) can be solved by successive substitution (or otherwise) for the unknown function /^(O, 0 to obtain /i(0, t) = h{t) +

BiT)h{t - T) dT

(7.3.29)

where the function B(T) is given by B(T)

^ X bM

b,{T) =

fofc_

I(T')/?I(T

—

T')

dz',

/c = 2, 3, —

The recursive relations in the b^'s are even more conveniently represented in terms of their Laplace transforms, since we have b,^{s) = ^^-1(5) -6(5) = b{sf. Thus, the solution is completely determined for this case for arbitrary initial conditions through Eq. (7.3.29). The methods of this section are readily employed to obtain the differential equation for the second-order product density, /2(T.T', t). It is written as

8f^

dt

+

'J^ ex

The initial condition for

+

'-h^

OT

/2(T.T',

/^(T.T',

+ [e(.) + e(.')] /,(..', 0 = 0. (7.3.30)

i) is given by

0) = NSNo -

1)^I(T)^ I(TO.

(7.3.31)

Note that this initial condition was arrived at by using the uncorrelated state of the population. Also integrating over both T and T' must produce the number of particle pairs that can be formed. The boundary condition is obtained as /^(T, 0, 0 =

/2(T, T\ t)b{T') dT' + /,(T)fo(T).

(7.3.32)

The first term on the right-hand side shows that the birth event associated with the particle of age T' paired with another of age T must contribute immediately to the pair with age T and age zero. The second term on the right-hand side of (7.3.32) is that caused by the birth of an offspring from the particle of age T itself, which deserves separate consideration, although it might appear to have been included in the first. This is typical of product density analysis.

7. The Statistical Foundation of Population Balances

316

The method of characteristics can be used to obtain from Eq. (7.3.30) the result.

fii^. ^\ t) /2(T-f, T ' - t , 0 ) e x p

e{T2)dT2>, -t

= {

r

f2{0,T'-Z,t-T)QXp<-

e{T^)dT^-

e(T2)dT2>,

(7.3.33)

where we have tacitly assumed that i' > i since the symmetry of /2(T.T', t) with respect to i and T' makes it unessential to identify the relationship in (7.3.33) that corresponds to T' < T. Note also that the second of the relationships above is insensitive to whether i' > r or i' < t. The secondorder product density equation can be solved for in a manner analogous to that used for the first-order density for fairly general models of birth and death functions. However, we shall be content with the simplification that the functions ^(T) and e{T) be constants p and s, respectively. Such a simplification has the merit of a rapid solution but the demerit of missing out on interesting size-sensitive correlation effects. The product density equations can be solved by Lapalace transform with respect to age. Thus, defining

Ms, t) =

MT,t)e

''dx,

/2(Si,52, 0 =

dxe'

dT'e-'^''f2{T,T\t\

where the double overbar has been used for the Laplace transform with respect to the two age variables in the second-order product density. In obtaining the actual solutions, however, we will only focus on the total population, which can be obtained by setting the Laplace transform variables to zero. Thus, at any time t, /^(O, t) represents the expected total population and /2(0,0, t) represents the expected total number of particle pairs. In order to calculate the foregoing quantities, we take Laplace transform of Eqs. (7.3.25) and (7.3.30), to get dhjs, t) + (5 + £)/,(s, t) = MO, tl dt 3/2(51,52,0

dt

+ (5i +52 + 2e)/2(5i, 52, t) = f2{s,, 0, t) + /2(0, 52, t). (7.3.34)

7.3 Stochastic Equations of Population Balance

31 7

Next we obtain Laplace transform versions of (7.3.26) and (7.3.32) as

MO, t) = pm t) ] Ms,, 0, t) = PlMs,, 0, t) + Ms,, 0] . MO, s„ t) = piMO, S2, t) + /i(s „ m J

(7.3.35)

Although the differential equations (7.3.34) can be easily solved and the Laplace transforms inverted following the use of boundary conditions (7.3.35), we take a slightly simpler route, as our present interest is only in the total population. By setting s = Sj = Sj = 0 in (7.3.34) we obtain using (7.3.35) MO, t) = /i(0, 0)e - <= - - ^>'

or

EN{t) = N„ e " <-=" ^>'

MO, 0, t) = MO, 0,0)e-2<=-''>' + 2p I MO, t>-2(e-^,(,-o^('. (7336) Jo

From Eq. (7.1.14), the second moment of the population, EN^ = /2(0, 0, t) + f^{0, t), so that on substitution from (7.3.36) we get

from which the variance VN in the total population is calculated as VN = N, (^-^)

[e-^'-f"^' - e-^^'-^^'l

(7.3.37)

We thus have the first two moments of the population by solving the first-order and second product density equations for the given process. The coefficient of variation of the total population, denoted COVN, which is obtained by calculating uVN/EN, is seen to be

The foregoing expression implies that a small population would give rise to a large relative fluctuation which would continually increase with time. When the birth and death rates balance exactly i.e., e = jS, the mean population remains the same as that initially, and the variance (by taking the limit of VN appropriately) is given by 2jSt, which progressively increases with time. In solving the foregoing specific model, we sought the easy route of dealing only with the total population because the birth and death rates

318

7. The Statistical Foundation of Population Balances

were independent of age. Indeed, such a treatment does fall short of realizing our desire to demonstrate how the mean and fluctuations can vary with particle state (age). The redeeming feature lies in the fact that both the formulation and the method of solution presented here can be applied to the more general problem of age-sensitive behavior. It is important to note that in this example the boundary conditions (7.3.28) and (7.3.32) provide a source of generating correlations between cells because of the relationship that newborns of age zero are linked to parents of all ages. This is not so in the two examples that follow.

7.3.4.2

Modeling of the Dynamics of Cell Death

Instances abound in human endeavor where the death of cells in a population by some lethal agent or the other is a process objective. Sterilization of fermentation equipment is a well-known example. In clinical practice, the extinction of a cancerous tumor by radiation or chemotherapy is a significant example in which the death of cells down to the smallest possible population is a desired goal.^^ Often the process time is desired to be minimized for a particular extinction level so that the intensity of the lethal environment needs to be maintained suitably. The objective of this section is to show that the methodology of this chapter has all the ingredients of the required mathematical framework for analyzing such problems. We assume that the probability of cell death per unit time is determined by the amount (say, x) of some component within the cell as well as on the local intensity of the lethal environment, which we denote by /. This intensity might be a function of position as determined by the actual spatial configuration of the population, which will not be of concern to us here. If the cells are assumed to be static, the effect of spatial variation in / will merely parameterize the population density in terms of spatial coordinates. Let us consider a population of cells distributed uniformly in space in the sense that no explicit spatial coordinates appear in the particle state. We further assume that the cellular component governing cell death varies at a rate denoted by X that could depend on x, / and other cellular components. Although these complexities can be accounted for, in principle, we assume 15 It is known, for example, that susceptibility of cancerous cells to radiation is affected by "oxygen tension" within the cells.

7.3 Stochastic Equations of Population Balance

319

that the dependence of X is only on x and /. We denote the cell death probability per unit time by e{x, I). It is also assumed that no cell proliferation occurs in the lethal environment. The first-order product density /^(x, t) must satisfy the following differential equation and initial condition ^ ^ i | ^ + | - [X(x, /)/,(x, 0] + e{x, I)Mx, t)=0

/,(x, 0) = N^g.ix). (7.3.39)

Here N^ is the initial population density and g^{x) is the initial distribution of the cellular component in the cells. One could also insert a spatial dependence without introducing any extraordinary compUcations, but we shall consider this at a subsequent stage. Since it is reasonable to assume that X must vanish at x = 0, there is no need for a boundary condition for Eq. (7.3.39).^^ The second-order product density /2(x, ^, t) satisfies the differential equation.

^M^M + i_ i;i:(^, /)/^(^, ^, t)] + ^ [x(4 i)Mx, i, m + [e(x, /) + e{^, /)] Ux, ^, t) = 0.

(7.3.40)

The initial condition for this equation may be written as Ux, 4 0 ) = iV„(Ar„ - l)g,{x)g M

(7.3.41)

For the same reason that Eq. (7.3.39) did not necessitate a boundary condition, Eq. (7.3.40) also requires no boundary condition. Thus, the initial condition (7.3.41) is sufficient to determine the solution to Eq. (7.3.40). The solution to Eq. (7.3.39) can be conveniently represented by the method of characteristics. The characteristic curves are defined by the solution of the differential equation ^ = X(x, /), at

x{0) = X.

(7.3.42)

16 This may be understood from the point of view of solving Eq. (7.3.39) by the method of characteristics. The characteristic curves on the {t, x) plane originate only from the x-axis, since the one starting at x = 0 will He along the t-axis. Thus, the initial condition is sufficient to determine the solution on the entire (f, x) plane.

7. The Statistical Foundation of Population Balances

320

The solution curve to (7.3.42) represents the characteristic originating from the point t = 0, x = x^ on the (t, x) plane. If we let X{t; xj be the solution and let F^{t;xJ = f^{X{t; x j , t\ then (7.3.39) implies that 'dX{X{t;x,lI) dx

dt

+

e(X{t;xJ,I) Fu

F,iO;x,) = N,g,{xJ, (7.3.43)

which has the solution F^{t;xJ

=N^g^{xjQxp

'dXjXjt'; x,l I) + e{X{f; x,X I) dfy dx (7.3.44)

so that the solution at the point [X(r; xJ, r] has been obtained for the population density. By selecting several initial states x^, the solution for /i(x, t) will have been determined over the entire (t, x) plane. On the other hand, if a solution is directly desired for a specific choice of (r, x), then we may solve the differential equation (7.3.42) backwards and represent the solution at, say, T (which runs between 0 and t) as X(T ; x, 0- This solution has the property that X{t;x,t) = x. Thus, from (7.3.44) we may write the solution as Mx,t)

=

N,g,{X{0;x,t))Qxp^-

'dX{X{T;xj)J) dx

+

e{X(T;x,tlI) dx}. (7.3.45)

The solution of Eq. (7.3.40) can be seen to be the product /^(x, 0/i(^? 0 since it can be readily shown to satisfy the partial differential equation (7.3.40). If we let both /^(x, t) and /^((J, t) be described by (7.3.45) then in one of the solutions one must have N^— \ in place of N^. The dynamics of the process produces no correlation between any two cell states. However, if there were an initial correlation, it would persist thereafter, but the solution can be obtained the same way as before by incorporating the correlation in the initial condition. In demonstration of this, assume that the initial correlation is represented by M , O ) = (N„-l)0i(^W

7.3 Stochastic Equations of Population Balance

321

where g^i^lx) is the conditional distribution of the initial population. Then M^, t) = {N,-l)g,{m

xexp

X, t)\X(0; ^, 0) 'dx{X{TU,tii) dx

e{XiT;i,t),I) dx)

+

(7.3.46)

so that the second-order product density in this case is obtained by taking the product of (7.3.45) and (7.3.46). We may write the rth order product density for this case as

X exp <] - X

dx

-+e{X{T;xj^,tlI)

dTh (7.3.47)

where we have disregarded any initial correlation between cell states. Thus, the product density of any order can be calculated from the first-order product density. In other words the system is stochastically complete with the specification of the first-order product density. It is now of interest to see how this model can be of value to the application for which it was considered. The cancerous cells are spatially distributed among normal cells so that the foregoing density functions are in fact a function of external coordinates. The external coordinates occur only as parameters, since the cells are considered to be fixed at their physical locations in the tumor. Furthermore, such a spatial dependence can also account for the possibility that the cellular constituent under consideration may vary with where the cells are in the tumor. Thus, the rth-order product density must feature a spatial vector r because of the dependence of the initial distribution g^ on r. Alternatively, we may regard the functions / . and g^ in (7.3.47) as already integrated over the tumor domain so that N^ represents the mean number of cells in the entire tumor at time t = 0. We may now apply formula (7.1.24) for the probabihty that the total number of live cancerous cells in the tumor at the end of irradiation time,

7. The Statistical Foundation of Population Balances

322

say, tjr is zero, since that is in fact our objective of irradiation, and obtain f* 00

Jo

X exp ^ - X

dx

+

e{X(T;Xk,t,XI) dx>. (7.3.48)

Expression (7.3.48), which must of course be evaluated numerically, is capable of accommodating any objective of the irradiation process. Suppose, for example, we want to be 1 % certain that all the cancerous cells have been killed. This implies that the left-hand side of (7.3.48) is 0.01. If the irradiation intensity / is known, then (7.3.48) can be used to calculate the irradiation time ^ir- Another alternative is to calculate the irradiation intensity for a fixed time of irradiation. A third even more interesting alternative is the manipulation of the initial distribution g^, which will imply some form of pretreatment of the tumor, so that one can maximize the probability of extinction at the end of irradiation. Clearly, this is an open problem that has several possibilities for future research.

7.3.4.3

Applications to Industrial Sterilization Processes

One may also envisage applications of the stochastic theory in this chapter to sterilization processes in the fermentation, food, and other industries in which contamination by microorganisms cannot be permitted. ^^ A problem of engineering interest may be in calculation of the size of, say, a tubular sterilizer from knowledge of the death rate of the contaminating organisms as a function of temperature. We consider a relatively simple scenario that will demonstrate how the analysis is to be made, rather than labor to expose the generality of the methodology. Suppose the contaminating organisms are contained in a process fluid (such as a food product), which is in fully developed flow along the z-direction through a peripherally heated circular tube with velocity v^(r), where r is radial distance perpendicular to the tube axis. Assume that 17

Fredrickson (1966) was the first to point out the importance of stochastic analysis in dealing with sterihzation processes. Unfortunately, however, in spite of his pubHcation several years ago, biotechnologists have paid little attention to this issue.

73 Stochastic Equations of Population Balance

323

the inner surface of the tube is maintained at a constant temperature, say T^. The number density of organisms in the entering fluid is estimated to be N^. The cells, which are assumed to be small with gravity playing a negligible role, display Brownian motion (with diffusion coefficient D) through the predetermined fluid velocity and temperature fields. The death rate of a ceU is described by the function k{T), where T is the local temperature of the fluid. Note that in this treatment no internal coordinates have been used for the ceU to which the death rate can be related. We shall assume that the sterilization process is operating under steady-state conditions so that time derivatives need not be considered. The particle state includes only spatial coordinates along the radial and axial directions. We let the first-order product density be /^(r, z) and let the second-order product density be /2(r, z; r', z'). The differential equations are readily identified for the foregoing functions. Thus,

0
(7.3.49)

0
The relevant boundary conditions are readily found in the chemical engineering literature.^* These are

miljmA^,^ or

0
(7.3.50)

or

we let Q be the volumetric flow rate of the contaminated fluid given by Q ^ 271 j^v,(r)r dr. Similarly, one may identify the differential equation and boundary conditions for the second (and higher) order product densities. However, it is not essential to go through this process because the model does not recognize any production of correlation between the locations of ceUs.^^ Thus, the product density of any order can be calculated from the solution of (7.3.49) because of the stochastic completeness of the system with the specification 18 See, for example, Aris (1965). 19 This is of course strictly not true because of the possibility of correlated movement of particles which are close to each other.

324

7. The Statistical Foundation of Population Balances

of the first-order density. Hence, we may write r

fri^U ^i; ^2. ^2; • • • ; ^r. Z,) = f ] M^k^ h) k=l

(7.3.51)

where /^{rj^^z^) satisfies the partial differential equation (7.3.49) with r^ replacing r and z^ replacing z. The boundary condition at z = 0 must feature N^ — k -\- 1 instead of AT^; otherwise, the boundary condition (7.3.50) are the same for each k. The extinction probability for the total number of viable contaminants at the cross-section z = / may be obtained from formula (7.1.24) where the rth-order product density is given by (7.3.51) with z^ = / for each k and v = 0; the integration with respect to each of the r^'s is performed with the weight factor 2nrj^: GO

Po(0 = l + Z(-27rr r=0

\

/N

"

\ ^

''

n

r,drj\{r„l).

(7.3.52)

/k=l

A design criterion for the sterilizer may be to calculate the length I of the heating section so that the actual number of contaminants in the exit stream is zero with, say, probability equal to 0.99. This implies that the right-hand side of (7.3.52) must be 0.99. Indeed, the solution of (7.3.49) must be substituted into (7.3.52) for suitably large values of / until a value of 0.99 is reached. A more interesting engineering criterion might be an optimal consideration for the length of the sterilizer in which the cost of sterilization may be accounted for in order to maximize the profit from the process which is increasingly threatened with increase in contamination. We shall not dilate on this issue any further.

7.4

ON THE CLOSURE PROBLEM

In dealing with stochastic problems, it became clear from Section 7.3 that one is frequently faced with lack of closure, especially in situations where interaction occurs between particles or between particles and their environment. Such lack of closure arises because of the development of correlations between particle states promoted by preferential behavior between particle pairs of specific states or between the particle and its environment The population balance equation, which generally comes about by making the crudest closure approximation, does not make accurate predictions in such cases of the average behavior of the system. The question naturally arises as to whether one can find other mean field descriptions by making more refined closure approximations on the unclosed product density equations.

7.4. On the Closure Problem 7.4.1

325

Aggregating Systems

We shall first address aggregating systems in which the aggregation frequency is given by a{x, x') in terms of particle mass or volume. Let us recall Eq. (13 J) as the closure approximation that had to be made to recover the population balance equation from the product density equation (7.3.6). We examine the implication of this approximation to the mean number of pairs of particles in any interval (a, b) given by the right-hand side of Eq. (7.1.13a). Approximation {13.1) determines the left-hand side of (7.1.13a) as [EN{a, b; t)Y, which is different from the mean number of pairs by the mean number of particles in the interval. This discrepancy becomes less serious at large populations since the population density then would be substantially smaller than its square. Conversely, we surmise that the seriousness of the discrepancy manifests at low population densities. Consider, for example, the aggregation kernel for Brownian motion according to which particles of most unlike sizes aggregate at the highest rate. Examine the population in a suitably large interval (a, b) after the aggregation process has occurred for a certain period of time. In this interval particles that are sufficiently far apart are unlikely to exist as a pair, since the likelihood of aggregation would have been extremely high. Thus, one expects high statistical correlation between particles of comparable sizes in the size interval (a, b). More precisely, given that there exists a particle at one end of the interval, the probability of finding another far enough away in the same interval becomes highly unlikely. If the second-order product density were exactly known somehow, we would predict the number of pairs for aggregation in the interval through formula (7.1.13a). Making the closure assumption (7.3.7), on the other hand, would yield an answer differing from that arising from (7.1.13a) by the mean number of particles in (a, b). If the mean number of particles were small, the relative error between the two estimates may be deemed to be large. Thus, we conclude that the closure approximation (7.3.7) is likely to fail for small populations. Sampson and Ramkrishna (1985) investigated aggregation by Brownian motion in order to examine the effect of correlation between particle sizes on the particle size distribution. In order to report their findings here, we consider again the population balance equation 3/i(x, t) St

1 2_

a{x — x\ x')f^{x — x\ t)f^{x\ t) dx'

- fii^, t)

a{x,x')f^{x\t)dx' 0

(7.4.1)

326

7. The Statistical Foundation of Population Balances

where we have dropped the growth term, since it makes Uttle difference to the nature of arguments to be made. If we wish to apply this equation for a large cloud of particles, then for small enough time scales, it is not reasonable to expect the system to be uniformly mixed over the large scale of the cloud. In other words, particles far apart in the cloud at any instant could not qualify for aggregation in the infinitesimal time interval following that instant. However, over a suitably sized local volume of mixing, denoted Fjnix, at each point in space, one may expect the population to be uniformly mixed so that Eq. (7.4.1), after accounting for any temporal changes in V^^^, may be applicable there. We reproduce here the estimate of V^^^ by Sampson and Ramkrishna (1985) based on a scaling argument that the time scale of mixing (as determined by Brownian diffusion) within l^^ix must exceed the time scale of aggregation (as determined by the rate of Brownian aggregation in V^^J. In other words, the time over which the population undergoes significant aggregation is much larger than the time required for a particle to move about in the volume of mixing. The time scale of mixing, denoted t^^^, is estimated for Brownian diffusion in the volume V^^^ while the time scale of aggregation for a single particle may be estimated by the reciprocal of the average aggregation rate, say a, with its companions in V^.^. Thus, f.ix « V^i^/D,

t,^^ a Lav^JV^^J -'

(7.4.2)

where a may be assumed to be a{x,x) in which x is the average particle volume. For Brownian aggregation (see Section 3.3.5.1) a turns out to be 8/cr/37r/i and is a constant at a fixed temperature. The Brownian diffusion coefficient in (7.4.2) is given hy D = kT/6njuR, where R is the radius of the diffusing particle. If we further assume that t^^^ oc r^gg, we obtain the following proportionality:

r^ _ ' mix_

-1/3

'A'

-1/3

(7.4.3)

_ V m•i x _

The expression to the extreme right of (7.4.3) arises from the volume fraction 4* = ^mix^/Knix' which remains constant with time. Thus, it follows from (7.4.3) that v^j^ oc 0"^^^ and hence that the number of particles in the volume of mixing must remain substantially constant with respect to time. This implies that K^j^ oc x0^^^ and since the average particle volume must increase with time because of aggregation, the volume of mixing must correspondingly increase with time. Sampson and Ramkrishna (1985) have performed spatial Monte Carlo simulations indicating that for relatively

7.4. On the Closure Problem

327

smaller volume fractions (which correspond well to experimental values for both aerosols and hydrosols) the inverse square root dependence of v^i^ on (/) is sustained well. Consequently, it appears reasonable to assume that V^^^ must expand to maintain the volume fraction and the number of particles in the volume constant with the progress of aggregation in time. The expanding volume of mixing calls for a population balance equation different from (7.4.1). We may arrive at the required change as follows. We integrate the population balance equation (7.4.1) over the volume V^^^ and write /i(x, t) = V^^J^{x, t) to obtain V^^^a{x - x', x')fi{x - x\ t)f^{x\ t) dx'

^mix|[Kn;x7l(^,0]=^

M^, t)

V^,:a(x,x')Mx\t)dx'.

Dividing this equation by V^^^ and setting a{x, x') = V^^^ a{x, x'), we obtain the following equation in /^: a/i(x,t)

^^^^^

^

1

-/i(x, 0

a{x — x\ x')/i(x — x\ t)f^{x\ t) dx'

a{x, x')Mx\

t) dx'.

(7.4.5)

Here we have set for convenience the "expansion coefficient" of V^^^. 1 dV . dt Equation (7.4.5) is notably different from (7.4.1) because of the first term on the right-hand side. Since the number of particles in V^-^^ is to remain constant, we must supplement the preceding equation with the condition dt

/i(x, t) dx = 0.

(7.4.6)

The foregoing condition may be used to determine ^(t) in (7.4.5). Equations (7.4.5) and (7.4.6) then consist of the population balance model for aggregating systems with a local volume of mixing as defined in this section. It is our objective next to compare the prediction of this population balance model with Monte Carlo simulations using the interval of quiescence in Section 4.6 as well as with predictions made from the product density equations using various closure approximations. The Monte Carlo approach provides one

328

7. The Statistical Foundation of Population Balances

with the proper numerical evaluation of the process model just described and hence can be used as a standard against which can be compared predictions based on (i) the population balance model, and (ii) the product density analysis using suitable closure procedures. What is left in the sequel is the identification of the product density equations for the process above. Following Sampson and Ramkrishna (1986), we directly write the equations in the product densities integrated over the volume V^-^ so that the densities are only in particle size coordinate but not in spatial volume and recognize them with a hat on top. This implies that we must use the redefined aggregation frequency a{x, x') in describing the aggregation process. The expected population density or the first-order product density must satisfy

'^

= mMx,t) + \ —

a{x — x\ x')f2{x — x\ x\ t) dx'

a{x, x')/2(x', X, t) dx\

(7.4.7)

which holds within the mixing volume anywhere in the system. Also, /^ must satisfy Eq. (7.4.6). Note that we have made no distinction in notation between /^ in the population balance equation and in the product density equations, since both refer to the expected population density, although different estimates will emerge from them. As Eq. (7.4.7) is not closed because it involves the second-order product density /2, we now identify an equation in /2. In writing this we shall assume that the pairs which enter K^ix are uncorrelated. This is in fact a closure assumption, but one that is not sufficient to close the equations.

^-h^^^

= mUx,t)Ux\t) + \ + •

1

a{x ~ x'\ x")f^(x — x'\ x'\ x\ t) dx" a{x — x", x")f^{x — x'\ x\ X, t) dx" [a(x, x") + a{x\ x")]/3(x, x', x'\ t) dx"

- a(x, x')/2(x, x', 0-

(7.4.8)

In the foregoing equation, the right-hand side shows along the top row a term that represents inclusion of pairs from the surroundings into V^^^,

7.4. On the Closure Problem

329

along the second row terms that represent the creation of the specific pair in question (i.e., of particles with sizes x and x'), and along the bottom row terms that represent the disappearance rate of pairs by aggregation with other particles as well as between themselves. Indeed, the appearance of the third-order product density shows the need for closure, which we shall abstractly represent by Mx,x\x\t)=H,

(7.4.9)

Equation (7.4.9) denotes the closure hypothesis to be identified later. Sampson and Ramkrishna (1986) have evaluated several closure hypotheses. The simplest one that meets the symmetry property of the third-order product density is given by Hi = Mx, t)Mx', t)Mx", t). It is unlikely that the preceding closure hypothesis has much substance over that contained in the population balance equation itself, however. Other closure hypotheses considered by Sampson and Ramkrishna (1986) are recounted below. ^ 2 = iC/l(^. t)f2{x\ X'\ t) -h /i(x', t)/2(x, X'\ t) + f,{x\ t)f2(x, x', t)] ^ 3 = ik^. ^\ t)Mx\ X\ t)Ux\

X, t)\"^

^ 4 = [/l{^. 0/2(:^^ ^\ t) + /i(x', t)Ux, X\ t) + /i(x", i)Ux, X\ i) - 2/,(x, t)Ux\ Hs

t)Mx\

0]

/2 (X, X\ t)f2{x\ X'\ t)f2{x'\ X, t)

/i(x, t)f^{x\ t)f^{x'\ t)

Although the closure hypotheses just considered are not exhaustive, they represent a reasonably diverse set of formulations for investigation. For a detailed discussion on each of them, the reader is referred to Sampson (1981), and Sampson and Ramkrishna (1986). Detailed calculations were made by these authors show that the hypothesis H^ is superior to all the others, especially when the correlation effects are strong. The closed set of product density equations is given by Eqs. (7.4.7) through (7.4.9) in which H is any one of the set {H^,H2,H^,H^,H^]. The solution of the set of product density equations is, however, most efficiently done by recognizing a set of integral constraints that arise naturally. Taking

330

7. The Statistical Foundation of Population Balances

the first moment of (7.4.7), we obtain through conservation of mass xf^{x, t) dx = ^{t)

dt

x/i(x, t) dx,

xMx,t)dx=V^,,c^.(lA.10)

Multiplying the second-order product density equation (7.4.8) by xx' and integrating over the semi-infinite interval with respect to both x and x\ we get on recognition of the conservation of mass (which eliminates the aggregation terms) the following integral constraint: poo

d_ dt

xYiix, t) dx + 'o

=m

^

poo

dxx

Jo

dx'x'f2{^^ ^\ 0

^^^(x, t)dx + 2

x/i(x, t) dx

.

(7.4.11)

In a similar vein, we multiply the third-order product density equation by xx'x"and integrate over the semi-infinite interval with respect to x, x' and x" to obtain the third integral constraint: d_ dt

dxx-^

x^f^{x, t)dx + 1> -h I

)o

= ^(0

dxx

0

+3

Jo

dx'x'f2{^^ ^\ t)

dx''xy^{x, x\ x", t)

dx'x' 0

xyi(x, t)dx + 6 dxxf^{x, t)

Jo

dxxYiix, t) dx'x'

Jo

dx'x'f^{x\ t)

dx"x"f2{^\ ^'\ t)

(7.4.12)

Equation (7.4.12) involves the third-order product density, which can be eliminated by using the closure hypothesis H^ given earlier, but in order to meet (7.4.12) exactly it is desirable to modify H^ by introducing a timevarying normalization factor C{t) by

f^ix, x', x", t) = C(0

/^(X, X', t)f2{x\ X\ t)f2{x'\ X, t) /i(x, t)f,{x\ t)f,{x'\ t)

(7.4.13)

For a complete discussion of the computational details, the reader is referred to Sampson (1981). We review here the salient results of those calculations. Calculations were made for Brownian aggregation for both a constant volume of mixing and a constant number of particles v^j^ in the volume of

7.4. On the Closure Problem H 3 .

/ // /

— PBE A

1 55.

O

"-——

T

Nmix = 10 J PDts

/ / /

E

///

Confidence Limits (95%)

1

30.

/// ///

(U

/

/// / /// // ///

25.

n

/// /

Mf /

Ol

0

a

>

/

/ /

//' / ///' '' /// -''

3

O

u

//% •' / / / / / / / /// $• /// /

•Nmix = 2 0 ^

c o

^

f-^/

Nmix = 5 ^ 1 Monte Niiiix = 10 S carlo 1 Simulation Nmix -' 2 0 ^

D

-

40.

331

20.

o

A /' /// >• /////// /


o To

5 15.1 ra

i

10.[

A'' ////

r 5.

0.1

1

\

\

10. 15. Dimensionless time

1

20.

1

FIGURE 7.4.1 Temporal evolution of the average particle volume in Brfownian aggregation for a constant number of particles in the volume of mixing from Monte Carlo simulation compared with (i) population balance equation, (ii) product density analysis using closure hypothesis (7.4.13). (From Sampson and Ramakrishna, 1986.) mixing. We do not report the calculations for the constant V^^^ case here, since the conclusions were in favor of the latter case, i.e., the case of constant numbers. Figure 7.4.1 shows the temporal evolution of the dimensionless average particle volume versus dimensionless time for v^^^ = 5, 10 and 20 obtained in three different ways, viz., (i) by the population balance equation

332

7. The Statistical Foundation of Population Balances

(7.4.5), (ii) by Monte Carlo simulation, and (iii) by the product density analysis using Eqs. (7.4.7), (7.4.8), the normalized closure hypothesis (7.4.13), and the integral constraints (7.4.10) through (7.4.12). Note that the size of the population does not affect the prediction of the average particle volume by the population balance equation. At small times, the average particle size is predicted reasonably well by the population balance equation, regardless of the number of particles. However, as time progresses, the predictions of the population balance equation deviate significantly from the simulations. This deviation is more pronounced as the number of particles in the mixing volume is lowered. On the other hand, the prediction of the product density equations agrees very well with the Monte Carlo simulations all along, even for the smallest populations, showing the efficacy of the closure approximation. In further corroboration of this conclusion. Fig. 7.4.2 shows the comparison of the cumulative number fraction for v^j^ = 5 at a chosen instant removed from the start. While the population balance equation is unable to predict the cumulative fraction for the larger particles, predictions by the product density analysis are well within the 95% confidence limits of the simulated values.

7.4.2

Precipitation in Small Droplets

We consider here an application of stochastic population balance in which correlation between particle states is caused by interaction between the particles and their environment. Such correlation effects could be considerable if the random behavior of particles is sufficient to induce random fluctuations in their environment. Consequently, the closure problem discussed in Section 7.3.3 is encountered here. This application is abstracted from the work of Manjunath et al. (1994, 1996). The problem of interest is the precipitation (in a small drop of volume V) of a metal hydroxide by reacting a soluble salt of the metal with ammonium hydroxide. The drop is supplied with a flux of ammonia from the continuous phase surrounding it. The precipitation reaction is given by M^^ + ocOH- =>M{OH)^.

(7.4.14)

The metal ions M""^ arise from the ionization of the metal salt whose (monovalent) anion is represented by A~. The hydroxyl ions 0H~ originate

7.4. On the Closure Problem

333

yyyy

— PBE D

-

Monte Carlo Simulation

I

— PDEs Confidence Limits (95%)

T '''

—

.999

/

m/

T'

'

c

0) Q 0)

y

E ,99

/

^ /

-

ZJ

E O

'7 .9 —

-

.0

0.

L_

1.

1

1

2.

3.

J

^.

L

Particle size scaled by average

5.

L.___ 17.

6.

FIGURE 7.4.2 Predictions of the cumulative number fraction in Brownian aggregation for five particles in the volume of mixing from Monte Carlo simulation compared with (i) population balance equation, (ii) product density analysis using closure hypothesis (7.4.13).

from the ionization of ammonium hydroxide represented by

Nm

•NH4, +

0H-,

[iVH;][OH-] K, = -

INHf^l

(7.4.15)

334

7. The Statistical Foundation of Population Balances

where K^ is the equihbrium constant of the reaction and square brackets are used to denote concentrations of species. The drop initially contains the metal salt. The anion is not consumed by any reaction or replenished from the surrounding, so its concentration remains at its initial value. The precipitation occurs by transport of the ammonia (at, say, molar flow rate Fj^^j^) from the surrounding phase into the drop. All species in the drop are assumed to be spatially uniform (although transport effects can be accounted for in a more elaborate treatment). For a detailed discussion of the underlying chemistry, which includes consideration of the solubility product of the metal hydroxide, ionic equilibria and the ionic product, K^ of water, the reader is referred to the work of Manjunath et al. (1994). It will suffice our objective here to say that two environmental variables in the drop are required for determining the nucleation rate and the growth rate of precipitated particles. We let the two environmental variables be y = {yuy^ where ^i = [ O i / " ] and y^ = \^M']y\/K^, where K^ is the solubility product of the metal hydroxide, so that y2 is in fact the supersaturation. The rate vector Y = (7^, Y^, following the form specified in (7.3.11), is given by %{y) = YjAy) + Yj^iiy) Z 0(x,)^(x,, y),

j = l, 2.

(7.4.16)

i=l

In this equation, we have used x as the particle size variable which represents the characteristic length of the particle. Manjunath et al. (1994) obtain the following expressions for the functions in the right-hand side of (7.4.16).

YUy) = FNHM^K;')(lM:[,-y,y;^)-^K^y;' Ki(y)=-P(^+^t'yi)[.{^K^'){lM)^-y,y;^) Y2,o{y) = ^yiyi'K

+ 2K;'y, + i r \

^{x) = x'

+ K^y;' + 2K;'y, + ir'

Y,^,{y)= -yyVK,.

The particle growth rate X{x, y) was obtained by assuming that it was limited by mass transfer to the particle, yielding an implicit relationship of the kind X{x. y) = Ky;ry,

- (1 - aX/X'yJ"^],

where K and K' are constants associated with the mass transfer of metal (or hydroxyl ions as determined by the principle of electroneutrality) to the particle surface. The imphcit nature of the particle growth rate, except for

7.4. On the Closure Problem

335

its contribution to the computational woes, is not of any conceptual significance. The stochastic model of the precipitation process is identified by the vector generalization of Eqs. (7.3.15) and (7.3.17) so that we may write dt

+ V.

Y./.y + Y,


where Y^ = [Y^ ^, 72,0] ^^^ Y^ = [7^ 1, 72,i]- The first-order product density satisfies the partial differential equation

^ + | ^ [ X ( x , y ) A . ] + v , - Y„/ir +Yi

y)/2,(x, i, y, t) d^ = 0.

mXii,

(7.4.18)

The boundary conditions are the same as (7.3.16) and (7.3.18), except for writing the vector y place of the scalar y. The nucleation rate used by Manjunath et al. (1996) is given by fc(a + 1)^

«„(y) = A exp

^2 > 1,

(ln}'2)'

Akconstants.

The unclosed nature of the foregoing equations can be seen from the last term on the right-hand side of (7.4.18). The second-order product density appearing in (7.4.18) may be written as /2Y(X,

(^, y, 0 =

/2/Y(X,

(^, t\y)My,

t).

The closure approximation employed by Manjunath et al. (1996) is given by /2/v(x, L t|y) = /i(x, t\y)m,

t\y).

(7.4.19)

In terms of the original product densities, this closure approximation becomes /2Y(^. ^. y. 0 =

/i(^, y, 0/i(^. y, t) foiy^ 0

(7.4.20)

Manjunath et al. (1996) substituted the closure approximation (7.4.20) into Eq. (7.4.18) and solved the resulting system of partial differential equations. It is of interest now to examine their results. Unfortunately, their calculations are not exact, in that an assumption was made to reduce the dimensionahty of the problem by having only one environmental variable.

336

7. The Statistical Foundation of Population Balances

However, the same assumption was made also in the stochastic simulation technique which was used as a standard to compare the calculations using the closure approximation (7.4.20). The numerical values of the parameters used in the calculations can be found in Manjunath et al. (1996). We use only a sample of their results below. Figure 7.4.3 shows the results when the mean number of particles could rise to the order of 30 before the supersaturation disappears. The mean number of particles is predicted well by the population balance equation since it agrees with both the results of the simulation and the closure approximation. Note that the prediction of the mean number in Fig. 7.4.3a by population balance is very good, even at very short times when the numbers are expected to be very small. The reason for this is that the change caused in the supersaturation by the growth of a very small number of particles is negligible so that the environment is virtually constant (as corroborated by Fig. 7.4.3b). In this situation the population balance equation for the process accurately predicts the mean behavior, even though the fluctuation about the average is very high as seen from the standard deviation about the mean plotted in Fig. 7.4.3c. The standard deviation in the supersaturation at larger times is predicted reasonably well by the closure approximation, as seen from Fig. 7.4.3d. However, the situation is quite different when the nucleation rate is such that it allows only a very small number of particles. Figure 7.4.4 shows the calculations when the mean number of particles is less than 1. From Fig. 7.4.4a, it follows that the closure approximation does considerably better than population balance at later times but tends to deviate at still larger times. The population balance prediction agrees with the stochastic simulation for a short time during which the supersaturation remains substantially the same (see Fig. 7.4.4b). Soon after, the supersaturation drops in value, showing that the stochastic effects of the randomly changing population cannot be handled by the population balance even in predicting mean behavior. The closure approximation does well during this period until, at still later times, it, too, starts to deviate from the simulated value. The standard deviation in particle numbers is described reasonably well by the closure approximation, although it worsens at later times. Figure 7.4.4d shows the standard deviation in the supersaturation, which is not predicted well by the closure approximation. For the same situation as in Fig. 7.4.4, Fig. 7.4.5 shows the cumulative size distribution of the precipitate particles from the closure approximation compared against the simulated results. At very short times the predictions from population balance as well as from the

uu X

v'1

26 -

• SIM — PBE MFE

20 -

A

,y> .>^

^'

?^15 J

V

<^

10

j

/

^

5 0

C

,

,

,

1—

1

1

0.05

FIGURE 7.4.3 Predictions by Manjunath et al. (1996) based on closure approximation (7.4.20) compared against stochastic simulation results, alongside population balance predictions. Results show good agreement of population balance predictions and the closure approximation at higher population densities. (Reproduced with permission from Elsevier Science.)

337

338

7. The Statistical Foundation of Population Balances 1.5

1 i

0.5 H

0.05

0

1 0.2

-| 0.15

0.1

time

r 0.25

0.3

(c) u.o •

0.75 •».

0.7 ^

•

0.65

- SIM

\

'S

MFE

0.6 • 0.55 0.5 • 0

1

1

1

1

1

0.05

0.1

0.15

0.2

0.25

time

0.3

(d) FIGURE 7.4.3

Continued.

closure approximation agree with that from the simulation (Fig. 7.4.5a), but at much later times the closure approximation is considerably better than that predicted by population balance (Fig. 7.4.5b). We have in this chapter dealt with stochastic formulation of population balances at some length. It should have become clear to the reader that there are numerous problems of interest to engineers that remain to be explored in this area.

7.5. Some Further Considerations of Correlated Behavior

339

0.3

(c)

(d)

FIGURE 7.4.4 Predictions by Manjunath et al (1996) based on closure approximation (7.4.20) compared against stochastic simulation results, alongside population balance predictions. Results from using the closure approximation are considerably better than those of population balance when the population size is small. (Reproduced with permission from Elsevier Science.)

7.5

SOME FURTHER CONSIDERATIONS OF CORRELATED BEHAVIOR

This section will investigate the formulation of problems in which one can account for correlated behavior of particles that have descended from a common parent. The problem arises in the treatment of biological popula-

7. The Statistical Foundation of Population Balances

340

X

0.8

\

•'V

if

U-i 0.4 J

/ /-" /

t=2 //

0.8

0.6 i

0.2

-

/ — PBE • SIM MFE (128x128)

|i|0.4 0.2

y/

0

0.5 X

1

1

1

1.5

0 I

//

/

1 1

1 1.5

// / /> / ,' * / // / • SIM if / — PBE n/ -— MFE (128x128) MFE (64x64) ly

Q0.6

f

/

r

()

1 0.5

1 2

1 2.5

:

X

FIGURE 7.4.5 Predictions of the sumulative pareticle size distribution by Manujunath et al. (1996) based on closure approximation (7.4.20) compared against stochastic simulation results, alongside population balance predictions. The predictions from the closure approximation for small populations are considerably better than population balance predictions, which deteriorate progressively with time. (Reproduced with permission from Elsevier Science.)

tions in which the behaviors of siblings are highly correlated. Thus, it is known that the life spans of siblings in bacterial populations are correlated. Powell (1956, 1958) has observed that the life spans of sister cells are positively correlated and that the life spans of mother and daughter are negatively correlated. Similar observations have also been made by Schaechter et al (1962). This has been the basis of criticism of population balance models based on cell age, since there is no mechanism to include the effect of correlation in the models. Subsequently, however. Crump and Mode (1969), using the theory of branching processes, have shown how to include the effects of correlated behavior among sister cells. The objective of this section is the exposition of a model framework that can account for the aforementioned correlation effects from a perspective more fitting with the methodology presented in this book. It has also been presented by the author (Ramkrishna, 1979). We shall begin the treatment with a statistical treatment since it also gives us the opportunity for a generalization that was promised in Section 7.1 where the assumption was made that only one particle could be seen of a particular state in the population at any instant of time. This assumption, which was then introduced for simplicity, will now be eliminated. Thus, the

7.5. Some Further Considerations of Correlated Behavior

341

treatment in this section also serves to provide a more comprehensive view of the statistical underpinnings of the population balance equation. The development in this section closely follows that of Ramkrishna and Borwanker (1974).

7.5.1

The Master Density Function

We now define a more general master density function than that defined in Section 7.1.1. Again, we confine the treatment to the scalar particle state, denoted x, since the extension to the vector case, as seen in Section 7.1.2, is straightforward. If there are / individuals of a particular state, we refer to the collection as an "/-tuple". There can be at most one /-tuple of a given state, so that two i-tuples should rather be viewed as a 2i-tuple. Thus, a particular state may be occupied by one /-tuple where / may take on any integral value in the range, say / = 1 to m. Suppose at time t there are r^ /-tuples, one in each of the intervals (x), x) + dx]) where j = 1,2,..., r^ and / can vary from 1 to m. The actual number density is then given by

A-

(7.5.1)

In this summation the inner sum over the index j does not exist for those / for which r^ = 0. The total number of particles in the system is then obtained as (7.5.2)

N{t) = Z ir,, i= i

which, while directly evident from the population containing r^ /-tuples, / = 1,2,..., m, also follows by integration of the number density in (7.5.1) with respect to x. The master density function is associated with the probability of realization of (7.5.1). Thus, we define a function Xi, X2, • • •, Xf.^1, x^, X2,..., x^^i..., x^, X2,..., ^rw' / sucn tnat J. ' ri,r2,...,rm Pr<|n(x, t) = I

I

iS{x - 4 \ = Jrur,,...,r„ f l UK'

(7-5-3)

The normalization of this density is represented by

E

E•

'•1 = 0 r2 = 0

E 0 ' \.hJ ''2'

1

••• 'm-

dx^ 1 p=l

q=\

= 1

(7.5.4)

342

7. The Statistical Foundation of Population Balances

where the product symbols have again been used to represent multiple integration. In (7.5.4) it is understood that no integration is involved with respect to x^ when r^ = 0. The symmetry of the master density function renders it insensitive to the permutation of any of the m sets of its arguments. Thus, the r^!'s appear in the denominator of (7.5.4) to compensate for redundant configurations. If we now define the probability 71^1,^2,...,r^W for the distribution of m different r/s, it may be recognized as the summand within the summation in (7.5.4). In other words, we may write 1

'^l-'2-

m

rp

/• 00

n q=ln p=l

'm'

Jo

(7.5.5)

dxp,,,,,,

From (7.5.5), the probability distribution for the total number of entities in the population can be obtained as Pv{N(t) = v} ^ P,{t) =

X

7i,.,,„...,,Jf).

(7.5.6)

Note in particular that the summation in (7.5.6) involves multiple summation over the r-'s in such a way that the total number of particles in the population is given by v. The expected number of particles in the interval (x, x -h dx) is obtained by taking the number density in (7.5.1) and integrating with respect to the probabihty density (7.5.3). Thus we write

En{x,t)dx= x E - X 00

00

ri=0r2 = 0 ^00

00

0

m

rm = 0 ' l ' ' 2 ' '" m

nn

1

'm'

rp

p=l

q=l

ri

i=lj=l

With due regard to the Dirac delta functions in the preceding integrand, we can evaluate the integrals to rewrite the result as

En{x,t}dx=Yi{i m

1=1

r

z - z . • • z z n n ^<

ao

CO

00

Ui=0r2 =0 n

X

1

rm = 0'l-'2''"'m-

n

m

rp

s=l

j=l,s

p=l,i

q=l

f* cc

JO

f*oo

11

^^j'^ri,r2,.-,rm^^l^-^2^-'">^ri^^l^^2'>'-">^r2'>''''>

j=l,s Jo = l,s JO

x' x'

ri

x'

X x'

x' '

•

Y"" Y'"

Y'"

• t) dx

(7.5.7)

7,5, Some Further Considerations of Correlated Behavior

343

where the notation p = 1. i (or j == 1, s) impUes exclusion of p = i (or j = s) The expression in the curly brackets on the right-hand side of (7.5.7) represents the probability that there is at time t an i-tuple between x and X-\-dx. We denote this probability by fl{x,t)dx and rewrite (7.5.7) (by dropping dx) as m

En(x, t)) = Mx, t) = X ifi(x, t).

(7.5.8)

i=l

The function //(x, t) is not a probabihty density. In fact, its integral over an particle state interval represents the expected number of i-tuples in that interval. Thus, we may write

£r,(rt =

fl{x, t) dx;

Er^ia, b;t) =

fi{x,t)dx,

(7.5.9)

In view of (7.5.2) and (7.5.9), we have for the mean total number of particles the following relationships: m

m

EN[t) = X iEr,{t\

EN{a, b;t) = ^ ^'^^/(^^ ^' 0-

i= 1

i= 1

The product density of order 1, /^(x, t) defined in (7.5.8), unlike that defined in Section 7.1.1, has no probability interpretation. However, the case in (7.1.1) may be readily understood from the present perspective by allowing m to be only 1 in Eq. (7.5.8); thus the product density of order 1 (when multiplied by dx) is found to be equal to the probabihty that there is one particle between x and x + dx. For properties associated with the population, as defined in (7.1.9), the expected value given by (7.1.10) also holds in the present context. For the fluctuations about average values, one must seek as before the product densities of higher order. We shall spare ourselves the tortuous algebra and state the results directly since they can be understood by analogy with the development in Section 7.1. Also, the details can be found in Ramkrishna and Borwanker (1974). We define the function /2^(x, ^, t) such that Pr[There is at time t an /-tuple in (x, x + dx) and a j-tuple in

(i,^ + dm =

f2\^,i,t)dxdi.

Then we may write for En{x, t)n{^, t) mm

En(x, t)n{^, ?) = I

m

Z ijf^ix, 4 0 + 1 ''//(^' t)d{x - ^), (7.5.10)

i= 1 J= 1

i= 1

344

7. The Statistical Foundation of Population Balances

which bears comparison with (7.1.12). However, Eq. (7.5.10) is considerably more reveahng than (7.1.12) of the second moment appearing in the second term on the right-hand side. The second-order product density may be defined as m

m

Mx, ^, t) ^ [En(x, t)n{^, t ) ] . ^ , = 1 1

ijf^i^^ ^^ 0-

(7.5.11)

Again there is no probabihty interpretation to the second-order density defined in (7.5.11), but for the case m = 1, the probabihstic interpretation obtained in Section 7.1.1 is resurrected. The second moment of the population in any interval may be obtained from (7.5.10) as EN(a,b;tf=

^

^ ij

dx

d^mx.u)+ Y^i'

fi{x,t)dx.

(7.5.12)

Formulas similar to (7.1.16) for the correlation between the number of particles in any two intervals are self-evident. Similarly, for any property, say, F, associated with the population, defined by (7.1.9), formula (7.1.10) for the expectation, EF and formula (7.1.15) for EF^ also hold in the present case. Ramkrishna and Borwanker (1974) have derived equations in the foregoing product densities for breakage and aggregation processes. The resulting equations are the same as those given here in Section 7.3, which were derived for the case m = 1. We do not repeat those derivations here, but proceed instead to the application that motivated the discussion in this section. As observed earlier, in a bacterial population the life spans of the mother and daughter cells are negatively correlated, while those of sister cells are positively correlated. We show next that the negative correlation between mother and daughter is easily accounted for with the usual framework of population balance. The positive correlation between siblings, however, requires the framework just described in Section 7.5.1. We consider first the formulation to account for the negative correlation between mother and daughter.

7.5.2

Modeling of Biological Populations with Anticorrelation between Life Spans of Mother and Daughter Cells

Insofar as this treatment does not require the methodology of Section 7.5.1, it is somewhat disconnected from the rest of the discussion in this section.

7.5. Some Further Considerations of Correlated Behavior

345

Nevertheless, its inclusion is germane to the topic of Section 7.5, which addresses various aspects of correlated behavior. Since it is of interest to account for the mother's life span in assessing the division rate, we define a division rate conditional on the life span of the mother cell from which the cell in question has descended. Thus, we let B{T\TJ be the division rate of a cell of age T given that its mother's life span is T^. In order to account for the negative correlation between the life spans of mother and daughter, B{T\T^) must be chosen to fit the experimentally observed correlation. We shall return to this issue subsequently. We further define the number density function /^(T, T^; t) such that /I(T, T^; t) dxdi^ = Pr[There is at time t a particle between T and x + dx and its mother's life span is between T^ and T^ + dx^. The number density of all cells (without regard to the life span of the mother cell) is then given by /I(T,

0 =

/,(T,T,;0^T,.

(7.5.13)

The unconditional division rate of a cell of age x is then given by

,(,,,)Jo"/A,yO^(^k.)^^.^

(7.5.14)

which must be specially noted for its dependence on time. The function /I(T, T^; 0 in^st satisfy a population balance equation given by ^T

+

^

= -B{x\xJJ^{x, T^, t).

(7.5.15)

The foregoing differential equation must be subject to the boundary condition /,(0,T,;0 = 2fo(T„0/i(T.,4

(V.5.16)

which follows from the binary division of the mother cell of age x^. To solve Eq. (7.5.15), one also needs the initial condition on /^(T, T^; t). The need for this initial condition is understandable, as the model's objective to incorporate the negative correlation between the life spans of mother and daughter can be realized only with the record of when the mothers divided. It also follows that the distribution of life spans of mothers changes with time as

346

7. The Statistical Foundation of Population Balances

the population evolves so that the observed correlation between mother and daughter will also vary with time. Thus, in finding a suitable model for the function 5(T|T J , observations of correlation between mother and daughter must be based on some known distribution for the life spans of mothers belonging to a specific initial generation. The distribution of life spans of daughters, conditional on T^ as the mother's age, can be readily calculated (see Exercise 7.5.1) to be B{T\TJ

B{T'\T,)dT'

exp 0

The correlation between life spans of this generation of daughters and those of the mothers from the previous generation then clearly depends on the distribution of the life span T^. We do not wish to pursue this characterization of correlation between mother and daughter any further. The population balance equation, regardless of the mother's age, can then be obtained by integrating Eq. (7.5.15) with respect to T^ over its semiinfinite range. The result, in view of (7.5.13) and (7.5.14), is given by —T-— + —

= -b(T, t}J^(T, t).

(7.5.17)

The boundary condition for (7.5.17) is obtained by integrating (7.5.14) with respect to T^ over the semi-infinite range to yield M0,t) = 2\

fc(T„0/i(T.,r)rfT,.

(7.5.18)

The foregoing boundary condition was also encountered in Chapter 2 (see Eq. (2.11.8)).

EXERCISE 7.5.1 Consider the population discussed in Section 7.5.2. The probability density for the distribution of life spans of a cell just born from a mother cell of age T^ is desired. To obtain this proceed as follows. Let ^^^(T | TJ be the probability that the life span of the cell in question is larger than T. Derive a differential equation for PJ(T \ TJ and show that the required probabihty density is given by — 5 P J ( T | T J / 5 T .

7.5. Some Further Considerations of Correlated Behavior 7.5.3

347

Modeling for Biological Populations with Correlation between Life Spans of Siblings

The treatment of this problem is that of Ramkrishna (1979). As in Section 7.5.2, the population multiplies by binary division. Since age is reckoned from the instant of birth, we deem sibhngs to be of the same age. Thus, the sister cells, both of the same age, carry on their aging process with each possessing a certain likelihood of its own division. We adopt the framework discussed in Section 7.5.1 with m = 2 for the analysis of this problem. We define two division rates for a cell of age T depending on whether its sibling has yet to divide or has already divided. Thus, we let ^^(T) be the division rate for the cell whose sibling has divided, and b2{T) be that for the cell whose sibling has not divided. One senses that ^^(T) must be larger than b2{T), the difference being governed by the correlation between life spans of sister cells. We shall subsequently address how the measured correlation between life spans of sister cells can be used to determine the relative values of ^^(T) and b2WSuppose there are at time t = 0 n singlets of ages ^]j= 1,2,..., n, and m doublets of ages T^, j = 1,2,..., m. We let the product density of singlets of age T at any time t be denoted by / / ( T , t) and the product density of doublets of age T be denoted by /^{T, t). Based on the probability interpretation of / / ( T , t), we recognize that a singlet of age between T and T -\- dz at time t + dt could have arisen in either of two exclusive ways. First, it may have arisen from a singlet of age between T — dt and T — Jt + dr at the instant t which failed to divide between t and t + dt. Second, it may have arisen from a doublet of age between z — dt and z — dt -\- dz 2ii time t and one of them dividing during the period between t and t + dt. Thus, we may write / / ( T , t + dt) dz = fy{z - dt, t) dz{l -b^(z-

dt) dt'] + 2f^{z - dt, t) dzb2(z - dt) dt.

On rearranging the foregoing, dividing by dt, and letting dt -^ 0, we obtain the partial differential equation

^ % ^ + ^ % ^ = -hir)flir,

t) + 2b,{t)f,\t, t).

(7.5.19)

The equation for doublets is similarly obtained by recognizing that doublets other than of age zero can only arise from younger doublets that do not suffer division. Thus, we obtain f^{z, t + dt) dz = f^{z - dt, t) dzll - 2^2(1 - dt) dt].

348

7. The Statistical Foundation of Population Balances

from which it follows that

a/,^(M)^f^_2M.)/.^(M). 8t dx

(7.5.20)

The boundary conditions for (7.5.19) and (7.5.20) may be identified by recognizing that births produce cells of age zero that necessarily form doublets (no stillborns!), yielding //(O, t) = 0,

f,\0, t) =

[/,^(T, t)b,(T) + 2f,\T, t)b,{T)-] dx. (7.5.21)

Equations (7.5.19) and (7.5.20) must be solved subject to the boundary condition (7.5.21) and initial conditions on both the product densities. The solution has been presented by the author (Ramkrishna, 1979) for the case of constant division rates b^{x) = P^ and ^ 2 ^ = Pi- The more general situation does not present a particularly difficult problem for solution. We shall not pursue the solution here, but instead show how data on correlation between the life spans of siblings can be used to assess the relative magnitudes of the division rates ^^(T) and fo2WSince the behavior of individual cells other than siblings is assumed to be completely uncorrelated, we may investigate the correlation between siblings by considering a single doublet that has just come about through the division of a mother cell at time t = 0. Thus, the initial age of the doublet is zero, and we seek the distribution of times at which the two cells are going to divide. The probabihty density for the distribution of division time, T^ for either sibling (with its sister yet to divide) is given by 2fe2('J^i)exp

-2

b^[T') dx'

where T^ is the specific variable value of the random variable T^. The conditional probability density for the distribution of division time, say T2 (with variable value T2), of the second sibling is obtained easily as foi(T2)exp

b^{z') dx'

>

Ti

References

349

The correlation between life spans of sisters is now given by E{T,T,)=2

dT^T^b2{T^)

xexp

^1212^1(12)

b2{T') dx' +

foi(T') dx'

which must be satisfied by the division rates ^^(T) and b2{x) in order to account for the observed correlation. Since the main objective of this discussion was to display the framework which will account for correlated life spans of sister cells, we do not pursue the analysis of this model any further.

REFERENCES Aris, R., Vectors, Tensors and the Equations of Fluid Mechanics. Prentice-Hall, Englewood Cliffs, NJ, 1962. Aris, R., Introduction to the Analysis of Chemical Reactors. Prentice-Hall, Englewood Cliffs, NJ, 1965. Bayewitz, M. H., J. Yerushalmi, S. Katz and R. Shinnar, "The Extent of Correlations in a Stochastic Coalescence Process (in Clouds)," J. Atmos. Sci. 31, 1604-1614 (1974). Bhabha, H. J., "On the Stochastic Theory of Continuous Parametric Systems and Its Applications to Electron Cascades," Proc. Roy. Soc. (London), ser.A, 202, 301-322, (1950). Fox, R. O. and L. T. Fan, "Application of the Master Equation to Coalescence and Redispersion Phenomena," Chem. Eng. Sci., 43, 655 (1988). Fredrickson, A. G., "Stochastic Models for Sterilization," Biotech. Bioeng. 8, 167-182 (1966). Janossy, L., "On the Absorption of a Nucleon Cascade," Proc. R. Irish Acad. Ser. A, 53, 181-188 (1950). Kuznestov, P. I. and R. L. Stratanovich, "A Note on the Mathematical Theory of Correlated Random Points," Izv. Akad. Nauk. SSSR, Ser Math, 20, 167-178 (1956). Manjunath, S., K. S. Gandhi, R. Kumar, and D. Ramkrishna, "Precipitation in Small Systems — I . Stochastic Analysis," Chem. Eng. Sci. 49, 1451-1463 (1994). Manjunath, S., K. S. Gandhi, R. Kumar, and D. Ramkrishna, "Precipitation in Small Systems — II. Mean Field Equations More Effective than Population Balance," Chem. Eng. Sci. 51, 4423-4436 (1996).

350

7. The Statistical Foundation of Population Balances

Powell, E. O., "Growth Rate and Generation Time in Bacteria, with Special Reference to Continuous Culture," J. Gen. Microbiol. 15, 492-511 (1956). Powell, E. O., "An Outline of the Pattern of Bacterial Generation Times," J. Gen Microbiol. 18, 382-417 (1958). Ramakrishnan, A., "Probabihty and Stochastic Processes," in Handbuch der Physik, Vo.3/2, (S. Flugge, Ed.), pp. 524-651. Springer, Berlin 1959. Ramkrishna, D. and J. D. Borwanker, "A Puristic Analysis of Population Balance," Chem Eng. Sci. 28, 1423-1435 (1973). Ramkrishna, D. and J. D. Borwanker, "A Puristic Analysis of Population Balance — II," Chem. Eng. Sci. 29, 1711-1721 (1974). Ramkrishna, D., B. H. Shah and J. D. Borwanker, "Analysis of Population Balance—III," Chem. Eng. Sci. 31, 435-442, 1976. Ramkrishna, D., "Statistical Models of Cell Populations," in Adv. Biochem. Eng. (T. K. Ghose, A. Fiechter and N. Blakebrough, Eds.) Springer Verlag, Berlin, 11, 1-47, 1979. Ramkrishna, D., Analysis of Population Balance — IV," Chem. Eng Sci. 36, 12031209 (1981). Sampson, K. J. and D. Ramkrishna, "Particle Size Correlations and the Effects of Limited Mixing on Agglomerating Particle Systems," J. Colloid Inter/. Science, 104, 269-276 (1985). Sampson, K. J. and D. Ramkrishna, "Particle Size Correlations in Brownian Agglomeration. Closure Hypotheses for Product Density Equations," J. Colloid Interf. Sci. 110, 410-423 (1986). Schaechter, M., J. P. Williamson, J. R. Hood (Jr.), and A. L. Koch, "Growth, Cell and Nuclear Divisions in Some Bacteria," J. Gen. Microbiol. 29, 421-434, 1962. WilUams, M. M. R., "The Statistical Distribution of Coagulating Droplets," J. Phys. A. 12, 983-989, 1979.

Index

Aerosol dynamics, 78-84 Agglomerating populations, simulation of, 185-186 Aggregation, 3, 4, 18 aerosol dynamics, 78-84 bubbling fluidized bed reactor, 84-92 closure problem, 325-332 from correlated random movement, 100-101 efficiency, modeling of, 102-108 examples of, 70-71 frequencies, modeling of, 92-108 frequency, 71-74, 129-131 frequency determination by inverse problems, 235-256 mass distribution of particles, 74-78 by multiple mechanisms of relative motion, 101-102 product density equations for, 303-307 by random relative motion, 96-99 by relative deterministic motion, 93-96 self-similarity and, 208-212, 213-216 simultaneous, 71 simultaneous, and breakage, 108-114 Anticorrelation between life spans, modeling of biological populations, 344-346 Artificial realization, 167 Asympototic properties, 240-244 Average particle mass, 200

Band-limited noise, 104 Basis functions, choice of aggregation frequency and, 240-244 breakage processes and, 226-228 Birth and death processes, 4, 19 aggregation processes, 70-114 breakage processes, 49-70 importance of, 29-30 net birth rate, 29 rates at boundary condition, 48-49 simultaneous aggregation and breakage, 108-114 Boundary conditions, 20 birth and death rates at, 48-49 for general case, 21-22 integral, 22 Breakage, 3, 4, 18 average number of particles formed by, 51 drop, 67-70 drop size distributions in stirred lean liquid-Hquid dispersions, 56-59 frequency, 50 functions, 50-52 functions, modeling of, 66-70 independent behavior, 49 as an instantaneous process, 51 inverse problems and determination of, 222-235 mass distribution of particles, 52-56

351

352

Index

Breakage (Continued) mass transfer in lean liquid-liquid dispersion, 59-64 master density equation for, 289-292 microbial populations, modeling of, 65 product density equations for, 299-303 self-similarity and, 201-208, 213-216 simultaneous aggregation and, 108-114 transition probability function, 51 use of term, 49 Brownian aggregation, 143 Brownian coalescence frequency, 99 Brownian motion, 96 Bubbling fluidized bed reactor, 84-92 Characteristic curves, 21-22 Characteristics, method of, 119-122 Closure problem, 324-339 Coagulation, 71, 143-144 Coalescence, 71 Brownian coalescence frequency, 99 bubbling fluidized bed reactor, 84-92 -redispersion models, 109-113 uniform redispersion and, 113-114 Coarseness geometric grid, 151-152 moment-specific internal consistency for coarse grids, 152-159 Constant aggregation frequency, 129-131 inverse problems and, 244-246 Continuous coordinates, 3 Continuous phase vector defined, 10 population balance equation for, 24-26 Continuous variables, 7 examples, 8 Convective diffusion equation, 102 Convective processes, 4, 12 Correlation between life spans, modeling of biological populations, 346-348 Cumulative mass fraction, 55 Death processes. See Birth and death processes Deterministic processes, 13 aggregation by relative deterministic motion, 93-96 Differential equation, 164-166 Dirac delta distribution, 135-136 Dirac delta function, 20, 61, 64, 94, 154

Discontinuous erosion, 188 Discrete coordinates, 3 Discrete contributions, 81 Discrete deterministic steps, 169-172 exact simulation versus, 181-185 Discrete formulations, 144-150 geometric grid, 151-152 moment-specific internal consistency for coarse grids, 152-159 moving pivots, 159-162 nucleation and growth and, 162-167 Discrete particle state, 36 Discrete variables, examples, 7 Dissolution kinetics, 30-33 Divergence theorem, 21, 23-24 Domain of external or internal coordinates, 10 inlet versus outlet, 22 Drop breakage, 67-70 Drop size distributions in stirred lean hquidliquid dispersions, breakage and, 56-59 Dynamic morphology, modeling of cells with, 40-44 Environment-dependent case, 286-288 product density equations for, 307-313 Exact simulation of particulate systems, 172-181 versus discrete deterministic steps simulation, 181-185 External coordinates continuous, 3 discrete, 3 domain of, 10 particle state vector and, 10 Finite dimensional state vector, 9 Fokker-Planck equation, 97, 104, 107 Gas holdup in stirred tank, 37-40 Geometric grid, 151-152 Growth rates of cells during balanced growth, 264-265 cell division rate and daughter size distribution, determination of, 267-268 growth rate, determination of, 266-267 Heaviside step function, 34, 82 Hindered settling of particles in polydisperse suspension, 268-271

Index Hydrodynamic theory, 96 Initial conditions, 20 Integral boundary condition, 22 Internal coordinates continuous, 3 discrete, 3 domain of, 10 particle state vector and, 10 Inverse problems advantage of, 221 aggregation frequency, determination of, 235-256 basis functions, choice of, 226-228, 240-244 breakage functions, determination of, 222-235 computer-simulated data for drop breakage, 228-232 constant frequency, 244-246 experimental data on drop size distributions, 232, 234-235 growth rates of cells during balanced growth and, 264-268 hindered settling of particles in polydisperse suspension and, 268-271 nucleation and growth kinetics, determination of, 257-263 sum frequency, 246-256 Ito stochastic differential equations. See Stochastic differential equations Jacobian, 73, 89, 110, 112 Janossy density, 278 Laguerre polynomials, 138-139, 243 Laplace transforms, 35 Laplace transforms method, 128-136 constant aggregation frequency, 129-131 sum frequency, 131-136 Linear differential equation, 130 Linear grid, 150 Liquid-liquid dispersions drop size distributions in, 56-59 mass transfer in, 59-64 Macroscopic balance of particles, 145-146 Markoffian nature, 66 Mass density of particles, 54

353

Mass distribution of particles aggregation process and, 74-78 breakage process and, 52-56 Mass transfer in liquid-liquid dispersion, 59-64 Master density equation, 288 for breakage process, 289-292 population balance equation via averaging of, 292-295 solution of, 295-298 Master density function, 341-344 environment-dependent case, 286-288 product densities, 284-285, 287-288 scalar case, 277-284 vectorial case, 285-286 Mean value theorem, 146-147 Method of characteristics, 119-122 Microbial populations, modeling of, 65 Moments and weighted residuals method, 136-144 Monotonicity, 54 Monte Carlo simulation methods, 92, 167-192 of agglomerating populations, 185-186 based on discrete deterministic steps, 169-172 exact versus discrete deterministic steps simulation, 181-185 single particle simulation, 186-192 statistically exact simulation of particulate systems, 172-181 Navier-Stokes equations, 2 Nucleation, contact, 48 Nucleation and growth discretization and, 162-167 inverse problems and determination of, 257-263 primary, 257 secondary, 257 Number density function, 3-4, 11-12, 76 average, 11 functional of, 118, 122 linear functional of, 119 nonlinear functionals of, 48, 119 Number balance, 4 Number of particles, 1 Orthogonal collocation, method of, 143-144 Orthogonality conditions, 139

354

Index

Particle flux convective, 25 diffusive, 25 through internal coordinate space, 13 through physical space, 13 total mass, 25 Particle growth, self similarity and, 217-219 Particle space continuum, 13-14 Particle state space defined, 3 number density, 3-4, 11-12 Particle state vector, 8-9 continuous, 10 external versus internal coordinates, 10 random rate of change, 26-29 rate of change, 12-13 Picard's iteration, 123-128 Pivotal points, 147 Pivots, moving, 159-162 Polymerization, 158-159 Population balance, applications, 1-2 Population balance equations, 4, 15 boundary conditions for general case, 21-22 for continuous phase vector, 24-26 general case, 19-20 master density equation for averaging, 292-295 one-dimensional case, 16-19 for open systems, 22-24 steady-state, 38-39 Population balance equations, solution of discrete formulations, 144-167 existence of solution, 118-123 Laplace transforms method, 128-136 moments and weighted residuals method, 136-144 Monte Carlo simulation methods, 167-192 numerical, 184 successive approximations method, 123-128 successive generations method, 126-128 Population balance models budding of yeast population, 35-37 dissolution kinetics, 30-33 gas holdup in stirred tank, 37-40 modeling of cells with dynamic morphology, 40-44 synchronous growth of cell population, 33-35

Population of particles, 1 breakage and aggregation, 3, 4, 18 distribution of, 2 - 3 internal versus external coordinates, 3 Precipitation in small droplets, closure problem, 332-339 Probability per unit time, 66 Product densities, 284-285, 287-288 Product density equations for aggregation, 303-307 for breakage, 299-303 for environment-dependent case, 307-313 Prokaryotic populations, 65 Quiescence, 169-170, 173-178 interval of, 174, 297 Random movement, aggregation from correlated, 100-101 Random rate of change, 26-29 Random relative motion aggregation by, 96-99 aggregation by multiple mechanisms, 101-102 Regularity condition, 17, 18 Residence time, 62 Reynolds transport theorem, 14-15 Scalar case, 277-284 Sectional moment, 148 Self similarity behavior analysis of population balance equations, 201-212 applications, 204-208 experimental evidence of, 210-212 pure aggregation processes, 208-212, 213-216 pure breakage processes, 201-208, 213-216 solution, 197-201 with growth, 217-219 Sequential differentiation method, 235 Solute concentration, average, 60 Statistical foundations applications, 275-276 closure problem, 324-339 correlated behavior, further considerations, 339-348 master density equation, for particulate processes, 288-298

Index master density function, description of, 277-288 stochastic equations, 299-324 Steady-state population balance equation, 38-39 Stieltjes integral, 56, 78 Stochastic differential equations, 26-29, 41, 97, 102 closure problem, 324-339 product density equations for aggregation, 303-307 product density equations for breakage, 299-303 product density equations for environmentdependent case, 307-313 Stochastic differential equations, applications of age distribution of a population, 313-318 cell death, modeling of, 318-322 industrial sterilization processes, 322-324 Stratonovich integral, 28, 38 Successive approximations method, 123-128

Successive generations, 126-128 Successive substitution, 126 Sum frequency, 131-136 inverse problems and, 246-256 Symmetry property, 72 Synchronous growth of cell population, 33-35 Taylor series, 83, 103, 134-135 Total volume fraction, 12 Transition probabiUty function, 51 Vectorial case, 285-286 Volterra integral equation, 125, 315 Volume fraction density, 11-12 of broken fragments, 56 Weighted residuals method, 136-144 White noise process, 107 Wiener process, 27, 28, 29, 96-97 Yeast population, budding of, 35-37

355

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