process modeling in the pharmaceutical industry using the discrete element method

8/18/2019 Process Modeling in the Pharmaceutical Industry using the Discrete Element Method

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Process Modeling in the Pharmaceutical Industry using theDiscrete Element Method

WILLIAM R. KETTERHAGEN, MARY T. AM ENDE, BRUNO C. HANCOCK

Pharmaceutical Research and Development, Pzer Inc., Groton, Connecticut 06340

Received 26 February 2008; revised 7 May 2008; accepted 7 May 2008

Published online 18 June 2008 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jps.21466

ABSTRACT: The discrete element method (DEM) is widely used to model a range of processes across many industries. This paper reviews current DEM models forseveral common pharmaceutical processes including material transport and storage,blending, granulation, milling, compression, and lm coating. The studies describedin this review yielded interesting results that provided insight into the effects of various material properties and operating conditions on pharmaceutical processes. Additionally, some basic elements common to most DEM models are overviewed. A discussion of some common model extensions such as nonspherical particle shapes,noncontact forces, and interstitial uids is also presented. While these more complexsystems have been the focus of many recent studies, considerable work must still becompleted to gain a better understanding of how they can affect the processing behaviorof bulk solids. 2008 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci98:442–470, 2009Keywords: pharmaceutical engineering; powder technology; discrete element meth-od; dynamic simulation; molecular dynamics; blending; granulation; milling; tableting;coating

INTRODUCTION

Processes involving particulate matter are pre-valent throughout the pharmaceutical industry.In many cases, however, there are opportunitiesto improve the fundamental understanding of these processes to directly benet equipmentdesign, process efciency, and scale-up. The useof various process modeling tools is becoming increasingly common and is playing a critical rolein efforts to gain insight into these processes.Several computational process modeling tools are

used in the pharmaceutical industry and havebeen reviewed recently. 1,2 These models includecomputational uid dynamics (CFD), the niteelement method (FEM), and the discrete elementmethod (DEM).

Computational uid dynamics is an Eulerianmethod that treats the material as a continuumand numerically solves mass, momentum, andenergy balances. This approach can be extremelyaccurate for systems consisting of uids, and canbe extended to systems consisting of dilute solidsin either of two general ways. In the Eulerian– Eulerian approach, the solids are assumed to be asecond continuous phase for which the aforemen-tioned balance equations are also solved. In theEulerian–Lagrangian approach, the solids aremodeled discretely using DEM, and these resultsare coupled with the CFD uid model. For systems

Correspondence to : William R. Ketterhagen (Telephone:860-686-2868; Fax: 860-686-7521;E-mail: [email protected]) Journal of Pharmaceutical Sciences, Vol. 98, 442–470 (2009)

2008 Wiley-Liss, Inc. and the American Pharmacists Association

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of dense solids however, the material is usuallyassumed to be elastic or elasto-plastic, and othermodeling approaches, such as FEM, are thenrequired.

In contrast to CFD and FEM models, thediscrete element method is a Lagrangian modeland, as such, tracks the positions, velocities, andaccelerations of each particle individually. Thisapproach, reviewed by several authors, 3,4 isadvantageous due to the high level of detail inthe output describing the dynamic behavior of theparticles. The particle positions and velocities canbe used to calculate many particle-scale quantitiesof interest such as local concentrations andparticle phase stresses, as well as examineparticle-level phenomena such as segregation oragglomeration. While this approach does demandsigni cant computational power, with the steadily

increasing speed of computer hardware, the sizeof systems that can be modeled with DEM iscontinually increasing. Some recent DEM modelssimulate systems on the order of a couple hundredthousand particles.

The DEM approach is particularly bene ciallysince predictions can readily be made from simu-lation data that would be dif cult, and possiblyexpensive, to obtain experimentally. Anotherstrong point of DEM modeling is the ease atwhich parametric studies can be executed to studythe effects of particle properties, process condi-tions, or equipment design. Further, since it is adiscrete approach, particles can have a distribu-tion of properties. Thus, the particles with inthe system can have a distribution of sizes, forinstance, or can even have varying materialproperties to model a mixture of various compo-nents. These details can generally not be addres-sed in continuum approaches.

This paper seeks to review the current status of DEM modeling of pharmaceutical processes. Tothis end, some of the fundamental aspects of DEM models are rst reviewed, including thehard- and soft-particle approaches. Next, somecommon extensions of basic DEM models, such asthe incorporation of cohesive forces, nonsphericalparticle shapes, and interstitial uid effects arehighlighted. This is followed by a review of recentDEM modeling work that focuses on somecommon pharmaceutical processes.

The Discrete Element Method (DEM)

There are four main classes of discrete elementmodels, each with their own relative advantages

and state of development. These consist of cellularautomata, Monte Carlo methods, hard-particlemethods, and soft-particle methods. Cellularautomata are simple, deterministic models thathave been used to study ow in silos, 5 downinclined chutes, 6 and in rotating drums. 7 Theyhave also shed light on phenomena such assegregation via percolation 8 and vibration, 9 aswell as the effects of nonspherical particles. 10

In this approach, particles are constrained to alattice, and their movement is governed by simplerules established through experimental observa-tion. While these models are extremely fast, theygenerally only provide qualitative predictions.

The Monte Carlo method is another approachthat has been applied to model granular materi-als. This approach calculates the probability of a random arrangement of particles based on the

energy associated with that arrangement. Likethe cellular automata, Monte Carlo simulationsare rather simple to implement and fairlycomputationally ef cient. Monte Carlo techniqueshave been used to study several aspects of granular behavior, including vibration-inducedsegregation, 11,12 the effect on nonspherical parti-cle shapes on hopper ow, 13 and thedistribution of coating in lm coaters. 14,15 However, the hard-and soft-particle approaches, as discussed in thenext sections, are much more commonly used.These approaches are generally more exible andtypically permit a direct relation between mate-rial properties and model parameters, therebyincreasing the likelihood of obtaining results thatare in quantitative agreement with experiments.

Hard-Particle Approach

The hard-particle approach 16 – 20 assumes thatparticles are rigid so that collisions are instan-taneous and binary. As a result, hard-particlemodels are generally best suited for dilute,collisional ows where these assumptions aregood approximations. Hard-particle models oftenare embedded in event-driven collision detectionschemes that increment the simulation time fromone collision to the next. Consequently, hard-particle, event-driven simulations can be compu-tationally ef cient when the time between particlecollisions is large.

A hybrid, hard-particle-with-overlap model hasalso been developed by Hopkins. 21 This hybridmodel uses the hard-particle collision model des-cribed above, but integrates the particle equationsof motion between collisions at a speci ed time

DOI 10.1002/jps JOURNAL OF PHARMACEUTICAL SCIENCES, VOL. 98, NO. 2, FEBRUARY 2009

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step. Thus, this is referred to as a time-drivenapproach. This time-driven approach can becomputationally more ef cient than the event-driven approach for systems containing manyparticles.

Soft-Particle Approach

In dense systems, the particle contacts aremultiple and enduring. Therefore, the hard-particle interaction is not appropriate and thesoft-particle model must be used. The soft-particleapproach, originally developed by Cundall andStrack, 22 is not limited by the instantaneouscontact time assumption of the hard-particlemodel and can therefore be used to investigatelong-lasting and multiple particle contacts. Thismodel allows for particle deformation which is

modeled as an overlap of the particles. To preventexcessive errors from being introduced, the meanoverlap value must be maintained to levels on theorder of 0.1 – 1.0% of the particle size. 23 This istypically accomplished through appropriate selec-tion of the spring stiffness. Using stiffness valuesthat are too small will result in large overlapvalues, and will potentially lead to the intro-duction of signi cant error through inaccuratedetermination of contact forces and thus post-contact quantities such as accelerations andvelocities. 24 The soft-particle model relies on aforce – displacement (and/or force – displacementrate) relation to determine the interaction forcesfor each particle – particle and particle – wall con-tact. This approach proceeds via small time stepsand is thus referred to as being time-driven. Accurate integration of the resulting particleequations of motion dictates a small simulationtime step and, hence, long computation times.Soft-particle methods are generally utilized fordense, enduring-contact ows, such as thoseoccurring in a hopper, blender, or pan coater.

The DEM soft-particle algorithm is relativelystraightforward and is shown in a owchart inFigure 1. At the onset of a simulation, the particleproperties such as particle size distribution (PSD),density, mass, and moment of inertia are allde ned. If process equipment is to be modeled, thegeometry and dimensions are also de ned. Theparticles are then inserted into the computationaldomain by de ning a position and velocity foreach, often by placing the particles on a latticeor by using a random number generator. Thesimulation then proceeds to a contact detectionstage where all particle – particle and particle –

wall contacts are identi ed. For each contact, thesoft-particle deformation, which is modeled as anoverlap, is calculated. Using the overlap values,force – displacement relations are used to calculatethe forces acting on each particle. The total forcesand moments acting on each particle are thensummed and Newton ’s second law is used tocalculate the translational and rotational accel-erations. The time step is incremented andaccelerations are integrated over time to deter-mine updated particle velocities and positions. After measurement of any quantities of interest,such as velocity pro les, solid phase stresses, orlocal concentrations, the simulation repeats thecontact detection for the updated particle posi-tions and the loop is repeated.

Contact models. At the heart of a DEM simulationis a particle interaction model. This particleinteraction model is used to calculate the forcesacting in either particle – particle or particle – wallcontacts. Each of these two contact types can beresolved using the same contact model, and thematerial properties (e.g. coef cient of restitution,friction coef cient, etc .) for each contact type candiffer so that dissimilar materials can be modeled.These forces are used to calculate accelerations,

Figure 1. A owchart of a general soft-particle DEMalgorithm.

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which are then integrated in time for updatedparticle velocities and positions. The modeledforces in a soft-particle system are typicallysubdivided into normal and tangential compo-nents. A number of different normal force – displacement models have been proposed and thesehave been reviewed by several authors. 3,25 – 27

One of the simplest and most commonly usednormal force model is the linear spring anddashpot (LSD) model. 22,23,28 – 33 Here, the repul-sive portion of the contact is modeled with a linearspring and the dissipative portion of the contactis modeled in parallel with a viscous dashpot.Together, these elements represent the normalportion of a contact fairly well. This commonlyused model is advantageous in that the coef cientof restitution, contact time, and maximum overlapcan be analytically determined. However, the

resulting coef cient of restitution is not depen-dent on the impact velocity.

Others have modi ed the repulsive portion of the LSD model with a nonlinear spring following the Hertz theory of elastic contacts. 34,35 ThisHertzian model is usually coupled with a dashpotto provide dissipation. 36 – 38 However, using thismodel yields a coef cient of restitution thatincreases with impact velocity, which is contraryto experimental data. 39 Thus, later work byKuwabara and Kono 40 and Brilliantov et al. 41

assumed the material to be viscoelastic instead of elastic. With this modi ed dissipation term, thecoef cient of restitution decreases with increasing impact velocity, in accordance with experimentalresults. 40 However, this Hertzian implementation

has no intrinsic time scale for non-zero damping coef cients. 25,26 Therefore, the simulation timestep must be selected by using an estimate of themaximum expected impact velocity.

Another approach was proposed by Walton andBraun 42 who assumed that materials plasticallydeform during contact. They developed a partially-latching, hysteretic spring model shown schema-tically in Figure 2 that uses one linear spring tomodel the repulsive force during loading, andanother, stiffer linear spring to model the forceduring the unloading portion of the contact. Thismodel has been used by many researchers 24,43 – 45

due to several advantages. First, no viscousdamping coef cient is required. This is bene cialas the damping coef cients can be dif cult torelate to particle properties. Second, either aconstant or impact velocity dependent coef cient

of restitution can be speci ed. A number of tangential force – displacement

models of varying complexity have also beenproposed and reviewed by several authors. 3,25,26

One of the simplest models was rst introduced byCundall and Strack 22 and subsequently used byseveral others. 46 – 48 Here, the tangential force ismodeled with a linear spring where the associateddisplacement is integrated from the relativevelocity at the contact point. This tangentialforce determined by the spring is limited by theCoulomb criterion, which speci es that the max-imum frictional force is proportional to the normalforce, where the coef cient of friction is the pro-portionality constant. This tangential force modelis an attractive choice because it is relatively

Figure 2. A schematic of the Walton and Braun 42 hysteretic spring model.




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easy to implement and reproduces experimentalobservations fairly well. 25 It does however requirea tangential displacement value to be stored inmemory for each contact.

Other tangential force models have been basedon the detailed analysis of frictional, elasticspheres by Mindlin and Deresiewicz 49 whichspeci es that the tangential force is dependenton the loading and unloading history of thecontact. The full model considers eleven differentloading/unloading possibilities, but generally,only two are considered in DEM implementations.The implementation of this model is signi cantlymore complex than the Cundall and Strack modeldiscussed previously, but compares only slightlybetter with experimental results, as reported byScha¨ fer et al. 25 Despite the additional complex-ity of this tangential model, it has been used

frequently by several authors.42 – 45,50

Model Extension

Historically, discrete element models have con-sidered only cohesionless particles of circular(in 2D)or spherical (in3D) shape, while neglecting the effects of interstitial uids. These simpli ca-tions have been made primarily due to thecomplexities of contact detection and the exten-sive computational power required to model morecomplex systems. However, as the speed of

computers has increased, the scope of systemsthan can be modeled in a reasonable amount of time has similarly increased. Today, it is notuncommon for researchers to use DEM to modelsystems with secondary forces such as cohesion,nonspherical particle shapes, or interstitial uideffects. Each of these extensions will be discussedin the following paragraphs.

Noncontact Forces

Noncontact forces due to capillary cohesion,electrostatics, or van der Waals interactionsmay signi cantly affect the behavior of bulksolids, especially for systems with small particlesizes or where moisture is present. Figure 3 showsthe magnitudes of these forces and how theybecome quite signi cant relative to particleweight as the particle diameter decreases. Theimportance of these forces has been reviewed bySeville et al. 51 and the application of such forces inDEM models has been reviewed by Zhu et al. 4

Cohesive forces strongly affect the owabilityof powders 52 and can lead to ratholing and

arching during hopper discharge, blockage of pneumatic conveying lines, as well as agglo-meration and caking of powders. 53 Additionally,cohesive forces can signi cantly affect uidizationprocesses by causing agglomeration and subse-quent de uidization. 54 While cohesive forces aresigni cant in most powders, many DEM modelshave not included cohesive forces. 55 Thus, furtherstudy of cohesive systems is clearly needed to gaina better understanding about these effects.

Some of the simplest methods to model cohesionuse a proportionality constant to predict theattractive force between particles. These methodspredict the cohesive force by a spring con-stant, 56,57 a constant proportional to particleweight, 58 – 60 a square-well potential, 53,61 – 63 or byassuming the cohesive force is proportional tocontact area. 64 Each approach is relatively simpleto implement, but uses constants that may notbe directly related to experimentally measurablevalues.

A more rigorous treatment models cohesionwith pendular liquid bridges. The capillary forceof a liquid bridge is speci ed by the Young – Laplace equation; however, this equation gener-ally cannot be solved analytically. Thus, Fisher 65

and, more recently Lian et al., 66 assumed atorodial liquid bridge shape, which allows for anestimation of the capillary force. Lian et al. 66

showed that the error associated with this torodialassumption is less than 10%. While the expres-sions based on the torodial assumption permit anestimation of the capillary force, they are not

Figure 3. A comparison of the magnitude of inter-particle forces for varying particle diameters. This

gure is reprinted from Zhu et al. 4 with permissionfrom Elsevier.




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explicit and so an iterative scheme must be used.This approach has been used in several DEMmodels to study the fundamental behaviors of cohesive materials in shear ow, 52 during pack-ing, 67 and while forming repose angles, 55 as wellas in more complex systems such as a vibratedbed, 68 rotary drum, 69 and high-shear granu-lator. 70 Mikami et al. 54 developed regressionexpressions that provided an explicit calculat-ion of the force as a function of the separationdistance, and therefore provided a more amenablemethod for incorporating cohesive forces intoDEM models. These expressions were laterextended to the polydisperse case. 71 Due to theease of implementation, these regression equa-tions have been used to incorporate cohesiveforces in studies of uniaxial compression, 71 shearow, 72 uidization, 54,73 and hopper discharge. 74

Other attractive forces such as van der Waalsforces and surface adhesion have also beenincorporated in to DEM models. van der Waalsattractive forces have been added to study packing and rearrangement of ne particles 75 as well asuidization. 76 – 78 Thornton and Yin developed atheory for contacts in the presence of adhesion, 79

and later applied this model in the study of agglomerate fracture. 80 A few researchers haverecently applied discrete element models tocolloidal systems as well. For these systems, aDLVO potential, based on the theory of Derjaguin,Landau, Verwey, and Overbeek 81,82 is oftenemployed. This DLVO potential is based on thecumulative effects of electrostatic repulsion andvan der Waals attractive forces. Additionally, anattractive force, based on the theory of Johnson – Kendall – Roberts (JKR), 83 is also included in somemodels for colloidal systems. 84 – 86

Particle Shape

Many studies of granular materials using DEMhave modeled particles with spheres in 3D or disks(circular shapes) in 2D. The use of these simpleshapes allows for straightforward contact detec-tion and contact resolution. However, thesesphere/disk systems are often too idealized toaccurately model some phenomena exhibited byreal granular materials. For instance, it has beenshown that spherical particles have a smallerangle of repose 87,88 and a reduced strength 89,90 ascompared to nonspherical particles. The under-lying cause is that the rotation of spheres is onlyresisted by frictional contacts with neighboring particles whereas for nonspherical particles,

rotation tends to be inhibited by mechanicalinterlocking. 91 – 93

Several different methods have been used inan attempt to reduce the rotational freedom of spheres/disks within DEM models. Ting andCorkum 94 arti cially increased the particle momentof inertia thereby reducing particle rotation.Others have damped 95 or completely xed 96 – 98

particle rotation. Another approach incorporatesa rolling resistance. 38,99 – 101 This approach showspromise however, it also introduces additionalparameters to the model via rolling resistancecoef cients.

Due to the aforementioned differences betweenexperimental particles and DEM models of spheres, a common extension of the basic, non-cohesive, sphere-based DEM models is to incor-porate nonspherical particle shapes. For systems

of spheres/disks, contact detection and contactresolution are relatively straightforward. How-ever, for nonspherical particles, these calculationscan become quite complex. A wide range of non-spherical particles have been implemented inDEM models, and many of these have beenreviewed by Dz ˇiugys and Peters 102 and Hogue. 103

One of the simplest approaches to modelnonspherical particles is by ‘‘gluing ’’ two or morespheres or disks together in either a rigid orexible manner. This clustering approach hasbeen commonly used in 2D with disks 87,88,104,105

and in 3D with spheres. 50,106 – 113 This approachretains the relative simplicity of contact detectionand contact calculation associated with spheres ordisks. Additionally, arbitrary or nonsymmetricparticle shapes can be modeled. However, adrawback is the dif culty in modeling shapesthat have sharp edges. 114,115 Also, a potentialdisadvantage is the roughness of the resulting particles if they consist of only a small number of spheres/disks. It hasbeen shown that the dynamicbehavior of a sphere-cluster particle may besigni cantly different than that of the smoothparticle it is intended to represent. 116

A similar method uses the idea of mathematicaldilation 117 to create spherocylinders and spher-odisks. 13,118 – 123 These so-called dilated shapes arecreated by placing a sphere with a xed diameterat every point on some basic geometric shape. A spherocylinder is created by dilating a linesegment with a sphere while a spherodisk (a 3Ddisk) is created by dilating a 2D disk with asphere. Thus, each particle is modeled by anin nite collection of spheres, resulting in a smoothparticle surface. However, the contact detection




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for dilated shapes becomes somewhat morecomplicated than that for spherical particles.

Another class of shapes is those that can bemathematically de ned. Most commonly, theseinclude ellipses in 2D 89,90,102,124 – 127 and ellipsoidsin 3D 92,128 – 130 as de ned by

xa 2 þ yb 2 þ z c 2 1 ¼ 0 (1)

where a , b, and c de ne the lengths of the threeprinciple axes which are aligned with the local x, y,and z coordinates, respectively.Superquadrics, 131,132

the general class of shapes to which ellipsesbelong, have also been used in 2D 23,130,133,134

and 3D 135 to represent nonspherical particles.The shapes of several superquadrics, shown inFigure 4, are mathematically given by

xa

aþ y

b

aþ z

c

a1 ¼ 0 (2)

where the exponent a > 0 characterizes theblockiness of the particle. For a ¼ 2, the equationof an ellipsoid is recovered. For increasing valuesof a , the corners begin to sharpen and the particlebecomes increasingly blocky. Superquadrics areadvantageous since they can algebraically modela wide variety of particle shapes with variousvalues of the exponent a and the aspect ratios asdetermined by a , b, and c. Additionally, thenormal vector is algebraically de ned. However,for these shapes, contact detection is usuallyprohibitively slow as the intersection of twononlinear functions must be calculated.

Polygonal particles are another class of particleshapes popular in the eld of geomechanics. Arbitrary or symmetric polygons have beenmodeled in 2D 48,115,134,136 – 140 and in 3D. 141 – 145

Additionally, polygons constructed of beams

and triangular elements have also modeledin 2D. 114,146 Polygonal particles pose certain

Figure 4. Images showing some possible particle shapes using 3D superquadrics. Thisgure reprinted from Hogue 103 with permission from Emerald Publishing Group.




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dif culties surrounding contact resolution invol-ving corners and edges. Additionally, modeling round or smooth particles with this method is in-ef cient due to the large number of facets involved.

Potapov and Campbell 147 developed anothershape representation for 2D systems. They con-nected four circular arcs in a continuous mannerto create an oval shape. This representationallows for nonround particles to be modeled withonly a small increase in the computational effort. Additionally, this approach of connected arcs canbe extended to model any other 2D polygon (e.g.triangle, square, pentagon, etc .). A drawback of this method is that there is no simple implemen-tation of this general scheme in 3D. One exceptionis the case of ovals, which have been extended to3D ovoids 148 – 150 by rotating the oval around anaxis of symmetry.

Another approach is the discrete functionrepresentation for 2D shapes, as de ned by Williams and O ’Connor. 137 This method discre-tizes the surface of an arbitrarily shaped particlebased on a single parameter resulting in a list of vertices that can easily be searched for contacts.One example is given by Hogue and Newland 151

who discretized particles via a polar represen-tation. Here, the surface of a given particle isspeci ed by a polar angle and radius for severalvertices on the surface. This approach was alsoused by Wu and Cocks in die lling investiga-tions. 140,152 An extension to 3D has also beenproposed 103 where two parameters are used to

discretize the particle surface. This discreterepresentation approach can represent a widevariety of arbitrary shapes that can be eitherconvex or concave, smooth or angular, and elon-gated or compact.

A nal particle shape worth mentioning is thatof a round, bi-convex tablet. 116,153 This shape isde ned mathematically by the intersection of onesmall sphere that de nes the tablet band and twolarger spheres that de ne each of the convex caps.This shape is advantageous in that it preciselymodels the shape of interest, and it does so in acomputationally ef cient manner.

While each of these nonspherical shape repre-sentations has its relative merits, it is importantto consider the increase in computational timeassociated with each method. Some authors havereported the relative differences in run time for

various shape representations116,147

and severalobserved computational times are summarized inTable 1. While these results are not comprehen-sive, they show that sphere clusters, continuously joined arcs, and bi-convex tablet shapes all requireonly modest additional computational effortbeyond that of spherical particles. In contrast,the computational cost for polygons and super-quadrics may be on the order of two to ten timesthat for a system of spheres. The computationaltimes presented in Table 1 are stated relative tothe case of disks/spheres in an effort to eliminateany differences between number of particles, thelength of time simulated, and the computational

Table 1. Computational Times for Various Particle Shape Representations as a Multiple of the Time Required forDisk-Shaped or Spherical Particles

Dimensionality ShapeIncrease in ComputationalTime Over that for Disks References

2D Quasi-triangle 1.7 1472D Quasi-square 1.8 1472D Superquadric 3 1342D Square 3.9 147

2D Superquadric 12 147

Dimensionality ShapeIncrease in ComputationalTime Over that for Spheres References

3D 10 sphere cluster 1.1 1163D 26 sphere cluster 1.2 1163D Biconvex tablet 1.5 1163D Ovoid 2 1493D Superquadric 2 – 3 333D 66 sphere cluster 2.2 1163D 178 sphere cluster 6.3 116




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resources utilized. However, differences in thecontact algorithm, the degree of optimization, orother factors may still result in widely varying computational times (e.g., the differences for 2Dsuperquadrics). 134,147

Fluid Effects

The effects of interstitial uids are neglected inmany DEM models. For many systems, such asthose with relatively large particles, typicallygreater than 500 m m, this assumption is usuallysatisfactory. 154 However, for systems with neparticles, it may be critical to include interstitialuid effects to obtain accurate model predictions. Additionally, for processes such as uid beddrying and pneumatic transport, the uid phasemust be modeled.

The inclusion of such effects is currently anemerging area of study where an Eulerian CFDmodel is coupled with a Lagrangian DEM model.Thus, the uid phase is treated as a continuum,while the particulates are modeled as discreteelements. Typically, these studies are carried outwith a CFD model coupled with a DEM modelusing either the hard- 155 or soft-particle 156

approach. Further details of these coupled modelsare described in several excellent reviews. 4,157

Several pharmaceutical processes have beenstudied using these coupled models. For example,air effects have been included in the study of

uidized beds, 155,156,158 –

162 spouted beds, 163,164and the effects of cohesion therein. 54,58,165 Thepresence of air has also been shown to affect thedie lling process. 140,166 Finally, the incorporationof air effects is required in pneumatic conveying models. 167 – 169

Validation of DEM Models

The veri cation and validation of computationalmodels are of utmost importance. The termveri cation is used to describe the process of ensuring that the numerical solutions are in factan accurate solution to the model. Thus, this issueaddresses the computational implementation of the mathematical equations. The term validationdescribes the process by which the accuracy of amodel ’s predictions are compared with experi-mental data. This area has been widely discussedin the CFD literature via, for example, text-books, 170 policy statements, 171,172 and reviewarticles. 173 However, the veri cation and valida-tion of DEM models does not appear to be as

rigorous and the procedures certainly are not aswell de ned. This is attributable primarily to thedif culty in obtaining experimental data withwhich to compare. The extent and methods of veri cation and validation used by researchersvary widely, and often these are not fully des-cribed in the literature. A notable exception is thework of Asmar et al. 56 that outlined severalfundamental, two-body contacts to verify theirDEM model. It was also suggested that theseresults be used as a benchmark for veri cation byother researchers.

Discussions of validation studies have appearedconsiderably more frequently in the literaturethan those regarding veri cation. Most validationstudies for DEM models have been qualitative andrely on visualization of experimental ows on thesurface or at a transparent wall. For example,

experimental observations have been used toqualitatively validate ow in a rotating cylin-der, 174 – 176 uidized bed as shown in Figure 5, 162

tumbling mill, 177 vibrated bed, 178 and during heap formation. 179 However, the opacity of bulksolids limits the applicability of these visualiza-tion techniques to quasi-2D systems (e.g. a thinslice of a cylindrical rotating drum). Thus, vali-dation studies of 3D systems that rely onvisualization of the material on a free surface orat a transparent wall in order to assess particlephenomena, such as the ow pattern, occurring within the interior may be tenuous.

One approach to validation for 3D systemsrelies on making measurements of macroscopicquantities. For example, the measurement of hopper discharge rates 180,181 can be compared tothe well established Beverloo correlation. 182

Figure 6 shows one such comparison withexcellent agreement between the DEM andBeverloo predictions. This hopper discharge rateis an example of a simple benchmark test formodel validation. Researchers have also mea-sured other macroscopic quantities for validationincluding the extent of segregation of bidispersematerial during hopper discharge 183 and thepower draw in a tumbling mill. 177

Tomographic techniques have also been devel-oped towards validation of 3D granular systems.These techniques are nonintrusive and are nothindered by the opacity of solids. Therefore,they are ideal methods to probe the internalmicrostructure and particle velocities within 3Dsystems. Nuclear magnetic resonance (NMR) 184

has been used for validation of a long rotating cylinder. 109,185 Also, the blend uniformity in a




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hopper has been quanti ed using gamma-raytomography 186 and the packing of particulateshas been examined using X-ray microtomogra-phy. 187,188 The positron emission particle tracking

(PEPT) technique 189 has also been used to probevarious equipment including high-shear mix-ers, 190,191 a V-blender, 192 a ploughshare mixer, 193

and uidized beds. 194,195 This method tracks asingle radioactively labeled particle for times onthe order of several minutes to an hour to compilethe internal ow velocities, which can then becompared with DEM predictions.

Limitations of DEM Models

The discrete element method is an emerging areaof computational modeling. The primary limita-tion of the technique is its inherent computationalintensity. Since the DEM approach tracks eachindividual particle, any increase in the number of modeled particles, N , increases the computational

time, which typically scales as N Log( N ). Thiseffectively limits the number of particles that canbe modeled. The number of particles in a system of volume V and relative density f is given by

N ¼6V fp d 3 e

V d 3

(3)

where d is the particle diameter. Thus, as thevolume of a system increases or the particlediameter decreases, the number of particlesincreases signi cantly. This is illustrated in

Figure 6. DEM predictions of the mass dischargerate per unit length of a monodisperse material in aat-bottomed, wedge-shaped hopper as a function of the ratio of outlet width to particle size W 0 / d .181 Theline indicates the discharge rate as predicted by theBeverloo correlation 182 modi ed for wedge-shapedhoppers. 298

Figure 5. Comparison of uidized bed simulations (upper) and experiments (lower) atthe given times in seconds. This gure is reprinted from van Wachem et al. 162 withpermission from Elsevier.




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Table 2 where the approximate number of particles in a 1 L volume is tabulated for thegiven particle diameter assuming a relative den-sity of 0.625. Typical DEM models at present arecapable of modeling on the order of 10 5 particles.Thus, to model a 1 L system on a one-to-one basis,the particle diameter could be no smaller thanapproximately 1 – 2 mm.

While the above example demonstrates thatonly select systems with a small number of particles can be modeled on a one-to-one basis,there is utility in modeling larger systems. Forthese systems with large numbers of particlerequired, DEM approaches can still be quiteuseful in elucidating qualitative trends andlearning more about particulate behavior andhow it is affected by certain particle properties,process parameters, and equipment design.

For those systems involving a large number of particles, several different approaches have beentaken to ease the required computational effort.One frequent approach takes advantage of symmetry within a system of interest therebyallowing for the use of periodic boundary condi-tions. 24,70,101,196 The use of such boundariesreduces the size of the computational domainand thus, the number of modeled particles,without affecting the measured results. A secondapproach arti cially increases the size of themodeled particles. For example, 10 m m particlescould be approximated with 10 mm particles in aDEM model which would reduce the total numberof particles within the system for a given solidsvolume. However, caution must be exercisedusing this approach as particle-level phenomenamay differ between the two scales. Further, as in

the example cited above, effects such as cohesionare usually quite signi cant for 10 m m particlesbut typically insigni cant for 10 mm particles.Finally, a third approach partitions the compu-tational domain into smaller regions, but stillmodels the ow in each region. The ow in eachregion can then be modeled with DEM sequen-tially on a single computer 197 or distributed toseveral computers (i.e. parallelized). 23 In contrast,others have only used DEM to model certainregions of interest and have used other appro-aches such as FEM 198 or stochastic methods 199 tomodel the ow in the bulk of the domain.

In addition to the signi cant effect that thenumber of particles has on the computationaltime, certain complexities added to the model canalso increase the computational time consider-ably. Such complexities include nonspherical

particle shapes, as discussed earlier and summar-ized in Table 1. Additionally, the inclusion of secondary forces, such as cohesion, or the coupling of the DEM model with CFD to capture interstitialuid effects will also necessitate a longer compu-tational time.

There are also some other technical concernsregarding DEM models. One common concern isthe dif culty associated with suf ciently validat-ing the models. An advantage of DEM is that itcan probe complex systems where experimentalmeasurements are dif cult or expensive to obtain.This becomes a detriment when it comes tovalidating the model experimentally, as thenecessary measurements may not be available.However, as discussed in the preceding section,researchers are continually devising new techni-ques and using new experimental tools to obtainmeasurements for validation. Another commonconcern is the use of spherical shapes to representnonspherical granular materials. While this is atypical assumption used to reduce the computa-tional time requirements for a given simulation,results from experiments and DEM simulationshave shown that nonspherical particles cansigni cantly affect particulate ow behavior. Thisis also an ongoing area of research with a focus ondeveloping fast contact detection algorithms fornonspherical particle shapes.

APPLICATION TO PHARMACEUTICALPROCESSES

A wide variety of particle handling and processing equipment is typically used in the pharmaceutical

Table 2. The Approximate Number of Particles in a1 L Volume for the Given Particle Diameter and Assuming a Relative Density of 0.625

d (m m) N

10 10 12

20 10 1150 10 10

100 10 9

200 10 8

500 10 7

1000 10 6

2000 10 5

5000 10 4

The dark shading indicates system sizes that are typicallyinaccessible with current DEM models and computationalpower. The unshaded areas are accessible, and the light shad-ing indicates an intermediate regime.




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industry. The discrete element method has beenused with good success to model many of theseprocesses resulting in a better understanding of these systems. In the following sections, high-lights of that body of work are presented in theareas of transport and storage, blending, granula-tion, milling, compression, and lm coating.

Transport and Storage

The transport and storage of bulk solids has beencomprehensively studied using DEM. Much of theresearch has focused on relatively simple geome-tries to permit examination of the underlying granular physics. For instance, numerous studieshave examined the solid phase stresses insimple shear ow with periodic boundaries. 24,32,42

Additionally, others have studied the stressesand solids fractions within ows down anincline, 38,50,200 and during the formation of aheap. 48,201,202 In addition to these relativelysimple geometries, more complex systems suchas hoppers, pneumatic conveying lines, and screwconveyors have also been studied using DEM.

The ow from a hopper is frequently studied dueto the prevalence of hoppers in all solids handling industries. Bulk quantities such as the massdischarge rate 47,203 – 206 are often measured andcompare well with empirical correlations such asthat proposed by Beverloo. 182 Cleary and Sawleyextended this work to show how nonsphericalparticles affect thehopper ow rate. 23 Other work,shown in Figure 7, reported how bidispersematerials segregate during discharge due todifferences in particle size. 101,183 Due to thediscrete nature of DEM models, the positionsand velocities of each particle are known at everytime step. This information has been used toexamine features within the granular bed thatmay be dif cult to obtain experimentally. Forexample, the granular microstructure 186,206 andthe internal stresses and/or ow elds 31,46,207 – 209

have been obtained for various hopper ow sys-tems. Finally, other work has examined unusualhopper geometries, 44 the stresses acting on thehopper walls, 47,203,204,208 and the effects of hoppervibration, 210,211 ow promoting inserts, 197 andinterstitial air on hopper discharge. 154

Discrete element models have also been coupledwith computational uid dynamics (CFD) to studythe effects of interstitial uids. One systemfor which these coupled models are often appliedis pneumatic conveying. In some of the earliest

work, Tsuji et al. 167 developed a model for thepneumatic transport of cohesionless material ina horizontal pipe. Later, other work includedextensions for heat transfer, 212 attrition, 213 andelectrostatics. 214,215 Other researchers have stu-died the saltation of solids that occurs in low-velocity ows when the particles fall out of suspension, 216 and developed phase diagrams todescribe the various ow regimes observed invertical and horizontal pipes. 168 Further workhas examined ow regimes in inclined pipes,and found good agreement with results from anexperimental system. 169

Another solids transport system modeled viaDEM is the screw conveyor. Such models havestudied the conveyance of spherical 33,217 andnonspherical particles. 135 Results have showngood agreement with established empirical corre-lations for transfer velocity and power. 217 Figure 8shows an image from a screw conveyor simulationwhere segregation of polydisperse material isobserved to occur after only two revolutions of the screw. Other relatedwork hasexamined solidsfeeding and conveying within a single-screwextruder 218 and mixing within a twin-screwextruder. 219

Figure 7. Snapshot of a hopper discharge simulationexamining the extent of segregation of bidisperse mate-rial during discharge. 183




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Blending

Blending operations are one of the most commonapplications of DEM models and much of therecent work is highlighted in a review by Bertrandet al. 180 A number of different mixing systemshave been modeled including V-blenders, toteblenders, and bladed mixers. However, one of themost frequently studied mixing systems is that of a rotating drum. For example, studies of rotating cylindrical drums, reviewed by McCarthy et al., 220

typically focus on mixing and segregation in theradial 43,69,196,221 – 224 and axial 175,196 directions.Other studies have reported measurements of the velocity eld, 109,175,224,225 angle of repose, 109

and surface shape. 226 This simple geometryproduces a steady ow where the mixing rateand mechanisms can be readily studied.

The mixing in other rotating devices hasalso been studied. The study of twin shell or V-blenders has focused on the ow and mixing mechanisms, 45 the sensitivity to simulation para-meters, 192 different ll patterns, 227 and the extentof mixing over long times. 228 The effectiveness of a V-blender has also been compared to a double coneblender. 45 Other work has focused on varioustote blenders. Computational mixing results in apyramidal tote blender have compared favorablyto 1:1 scale experiments with glass beads. 229 Themixing mechanisms in a conical tote blender wereexamined as a function of percent loading. 230

Other work compared mixing in a conical toteblender with that in a V-blender. 227 Further, astudy of baf es in a cylindrical tumbler rotating on an axis perpendicular to that of the cylinderhas shown the importance of baf e design. 231

Other studies have examined various mixing equipment with an impeller, such as the vertical

axis cylindrical mixer with a rotating blade shownin Figure 9. Stewart et al. 190 simulated the mixing in such a device and compared the results withexperimental measurements obtained via posi-tron emission particle tracking. The overall owwas successfully predicted by DEM, but the modeldid not closely predict the velocities in all regionsof the bed. Later work by the same authorsexamined the granular microstructure during mixing. 232 Others have determined an optimalll level 233 and examined the effect of differentimpeller shapes. 191

Terashita et al. 234 studied the particle segrega-tion due to density differences while Zhou et al. 235

extended this work and examined segregation dueto both density and size differences. Finally, ahybrid model was proposed and implemented in ahorizontal convective mixer with an impeller. 199

This hybrid model employs DEM to model regionsof high shear and a stochastic compartment modelfor the remainder of the domain.

Two other mixing devices have also beeninvestigated with DEM. A helical ribbon blenderhas been analyzed, 180,236 and results show howthe ow patterns and extent of mixing are affectedfor various rotational speeds and ll levels. 236

Further, the suitability of static mixers for bulksolids has been reviewed 237 and studied using DEM. 238 The observed ow regimes from theDEM study compared well with previous experi-mental results.

Granulation

The granulation process and the manyapproaches to modeling it have been summarizedin a review by Cameron et al. 239 While apopulation balance models have played a key rolein granulation process modeling, discrete elementmodels have also played a small, yet signi cantpart of this modeling effort thus far. Much of thisDEM work has occurred only in the past few years.

One approach used in DEM modeling of granulation was taken by Gantt and Gatzke. 240

In this work, a high shear granulator was modeledusing a DEM model that incorporated three keymechanisms of granulation: coalescence, consoli-dation, and breakage. The rates of each mechan-ism are directly simulated and incorporated tomodel a dynamic particle size distribution (PSD).The results from this DEM model were in goodagreement with previous approaches using

Figure 8. A snapshot from a simulation of a screwconveyor where the particles are shaded according totheir diameter. Size segregation is apparent here afteronly two screw revolutions. This gure is reprintedfrom Cleary 33 with permission from Emerald GroupPublishing.




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Figure 9. Images from a DEM simulation of a bladed mixer after (a) 0.1 s, (b) 0.4 s,(c) 0.7 s, (d) 1 s, (e) 2 s, (f) 3 s, (g) 4 s, and (h) 5 s. This gure is reprinted fromBertrand et al. 180 with permission from Elsevier.



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population balances; 241 however, the DEMapproach also has the ability to model dynamicoperating conditions.

Gantt et al. 242 used another approach where aDEM model with periodic boundary conditionswas used to represent ow in a high-sheargranulator. The particle collision statistics com-piled by the DEM simulation were used to developa coalescence kernel. This kernel was then usedwith a Monte Carlo method to solve a multi-dimensional population balance. 243 Good agree-ment with experiments was observed, but could beimproved with better incorporation of materialproperties such as wet granule yield strength, Young ’s modulus, and asperity size.

The agglomeration of particles is typicallymodeled using one of two methods. In the rstmethod, the discrete elements always represent

the primary particles.244

Thus, the agglomeratesmany consist of many discrete elements. Thismethod retains information on agglomerate por-osity and morphology, and subsequent breakage isreadily modeled. 245 This approach has also beenused to study the impact of two agglomeratesat different impact velocities and binder uidproperties. 246 In the second method, agglomeratesare modeled as a single discrete element, the massand volume of which are determined from those of the constituent primary particles. 247 This methodallows for the simulation of larger systems sincefewer discrete elements are required. The majordisadvantage or limitation is that agglomerate

breakage cannot be readily modeled using thismethod.

A wide variety of different granulation equip-ment has been modeled with the DEM approach.Talu et al. 248 modeled agglomeration and break-age in 2D shear ow of a mixture of ‘‘wet ’’ and‘‘dry ’’ particles showing the effect of the amountof liquid binder, the Stokes number, and theCapillary number on the PSD. Muguruma et al. 70

modeled the centrifugal tumbling granulatorshown in Figure 10 where the liquid was uni-formly distributed. The resulting velocity pro leswere in agreement with experimental data using glass beads of same size. Mishra et al. 249 exam-ined the agglomeration of particulates in a rotarydrum with a model that included a spray zone andalso considered the drying of particles. Granula-tion in uidized beds has also been modeled.

Goldschmidt et al.247

modeled granulation in atop-spray uidized bed. In that work, both theparticles and droplets were modeled using thehard-particle approach. Results showed the devel-opment of the PSD over time as a function of theuidization velocity and the spray pattern. In asimilar study, Link et al. 250 examined granulationin a spout uidized bed.

Dry granulation techniques have also beenexplored by a few authors with mixed successthus far. Odagi et al. 251 modeled a roller compac-tion process using DEM, but the pressures in thenip region were several orders of magnitudesmaller than experimental values. In contrast,

Figure 10. A DEM model of a centrifugal tumbling granulator with (a) particlepositions and (b) particle velocity pro les. This gure is reprinted from Mugurumaet al. 70 with permission from Elsevier.




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Dec et al. 252 modeled roller compaction using thenite element method with good success. Thus, incases where large pressures are present, such asthe nip region of a roller compacter, a continuummodel may be superior to DEM approaches. Additionally, Horio 253 examined dry granulationin a uidized bed. This so-called pressure swing granulation relies on the intrinsic cohesivity of particles to form relatively weak agglomerates.

Milling

The discrete element method approach to model-ing comminution devices has provided insightinto particle fracture patterns and the changein PSD. This work is usually focused on one of two processes. The rst is the study of particle

fracture and/or attrition while the second involvesthe dynamics of materials within the milling device. Several researchers examining particle

fracture have developed models exploring theeffect of impact velocity and bond strength onfracture. The approaches taken to model breakageare varied. One approach, originally developed byPotapov and Campbell, 254 creates a nonporous,2D agglomerate of Delaunay triangles 254 – 256 orarbitrary polygons via a Voroni tessellation. 255 – 257 Agglomerate fracture, as shown in Figure 11,occurs when the tensile strength of the glued joints is exceeded. In another approach, agglom-erates are created with randomly arrangedspherical particles and held together with surfaceadhesive forces. 80,258 – 261 Breakage of the agglom-erate results when tensile forces between spheresexceed the surface adhesive forces. The mode bywhich agglomerates break is dependent on boththe agglomerate strength and the impact velocitywhere large velocities may cause shattering,

but smaller velocities may only lead to a plasticdeformation of the agglomerate. A DEM andexperimental study of weakly agglomerated

Figure 11. Images from a DEM model showing the impact and fracture of a singleparticle consisting of 2043 triangular elements. This gure is reprinted from Potapovand Campbell 254 with permission from Elsevier.




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lactose particles has shown that they tend todeform plastically rather than fracture. 258

A different approach estimates the particlebreakage using an analytic model. For example,Neil and Bridgwater, 262 following the work of Gwyn, 263 attempted to characterize attrition witha single parameter irrespective of the equipmentcausing the attrition. Another model proposed byGhadiri and Zhang 264 models particle damage viaa chipping mechanism rather than fracture as inthe previous approaches. The chipping results insurface damage that is assumed to be a function of impact velocity and the particle size, hardness,and critical stress intensity. Ghadiri et al. 265,266

also model both the fracture and chipping of particulates. It is found that chipping occurs atlow impact velocities while fracture occurs athigher values.

Discrete element methods have also been usedto model the dynamics of granular material and/orgrinding media in comminution devices. Severalstudies have focused only on such dynamics andhave not considered breakage of the particulatematerial. 267 – 270 Other work has incorporated afragmentation or chipping model discussed aboveinto a DEM simulation of particulate motion ina comminution device. These simulations havemodeled breakage of nonporous agglomeratesin various devices including a shear cell, 271 asemiautogenous (SAG) mill, 272 and a Hicommill. 273 Similar work has incorporated Ghadiri ’smodels for particle breakage and chipping instudies of shear deformation 274 and jet milling. 275

Compression

Various aspects relating to tablet compressionhave been thoroughly studied using DEM tech-

niques. The process of die lling 140,152,166,276 – 278 isone aspect that has been studied in both thepharmaceutical and powder metallurgy indus-tries. These studies typically describe the ll rateand particle ow eld into a die and makecomparisons with experimental results. Otherresults show the effects of powder characteristicssuch as friction, 140,152,278 particle-to-die sizeratio, 152 effect of shoe speed, 140 and the effect of air pressure. 140,166 The compression of materialswithin a die has been studied 139,279 – 285 whileothershave investigated compression in a periodicsystem to eliminate die wall effects. 286 – 288 Rear-rangement of particles during the compactionprocess has also been shown to be impor-tant. 75,283,289,290 In much of the compression work,the nite element method (FEM) has been coupledwith DEM models so that systems with large

particle deformations and large relative densitiescan be modeled. 139,280,281,283,285,291 This approachpermits the modeling of materials with a widevariety of mechanical properties, such as thehardness and ductility, as shown in Figure 12.However, this method is more computationallyintensive as individual particles must be meshedand analyzed via FEM.

A few other aspects related to compressionhave also been approached with DEM modeling.Couroyer et al. 292 used a previously mentionedparticle fracture model 265 to examine crushing during compression for various material proper-ties. Martin 293 investigated the springback during the unloading process. Springback was shown tobe affected by the material properties, compactionhistory, and the die wall. In addition to elasticity,the decohesion of particles previously in contactwas also shown to be an important factor during unloading. Sweeney and Martin 294 used DEM todetermine the microstructure of a high-density

Figure 12. Modeling compression of a mixture of hard and soft ductilematerials using a coupled DEM-FEM model. This gure is reprinted from Ransing et al. 285 withpermission from The Royal Society.




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Figure 13. Visualizationof the coating mass at four different times in a DEMmodel of a uidized bed coater. This gure is reprinted from Nakamura et al. 297 with permissionfrom the Pharmaceutical Society of Japan.




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however, this limitation will become less restric-tive. Other concerns, including those that centeron the typical assumptions of spherical particlesand noncohesive interactions, are the focus of many recent and ongoing studies aimed atdeveloping and re ning models that are bothaccurate and computationally ef cient. Incorpora-tion of the results from these efforts will greatlyexpand the scope of systems that can be quantita-tively modeled using DEM, thereby increasing itsutility as a predictive design tool.

ACKNOWLEDGMENTS

The authors wish to acknowledge various P zercolleagues for their helpful comments and sugges-

tions which have been a great aid during thepreparation of this report. Valuable discussionswere held with C. Bentham, D. Kremer, M. Mul-larney, V. Swaminathan, and N. Turnbull.

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