Journal Identification = CHERD Article Identification = 579 Date: May 16, 2011 Time: 8:52 pm

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chemical engineering research and design 8 9 ( 2 0 1 1 ) 519–525

Contents lists available at ScienceDirect

Chemical Engineering Research and Design

journa l homepage: www.e lsev ier .com/ locate /cherd

echanistic analysis and computer simulation of theerodynamic dispersion of loose aggregates

raham Calvert, Ali Hassanpour, Mojtaba Ghadiri ∗

nstitute of Particle Science and Engineering, University of Leeds, Leeds, LS2 9JT, UK

a b s t r a c t

The dispersion of bulk powders is important for a number of applications including particle characterisation, and

the delivery of therapeutic drugs via the lung using dry powder inhalers (DPIs). In recent years the distinct element

method (DEM) coupled with continuum models for a fluid phase to simulate fluid–solids interactions has received

much attention. In this paper these computational techniques are used to investigate the aerodynamic dispersion

in a uniform fluid flow. As intuitively expected, it is seen that with increasing surface energy it is progressively

more difficult to disperse a loose aggregate. However, once the relative particle–fluid velocity goes beyond a certain

threshold, dispersion occurs quickly and approaches a completely dispersed state asymptotically. In addition, as

loose aggregate diameter is increased the necessary threshold velocity reduces. A relationship between the fluid drag

forces acting upon a spherical aggregate and the adhesive force, given by the JKR model, leads to a dimensionless

group termed the dispersion index, which includes the Weber number. It is shown that the effect of surface energy

and loose aggregate diameter on dispersion behaviour can be described by this dimensionless group.

© 2010 Published by Elsevier B.V. on behalf of The Institution of Chemical Engineers.

Keywords: Dispersion; Simulation; Distinct element method; Aggregates; Drag force

acting at the contact point between two particles when accel-

. Introduction

he aerodynamic dispersion of bulk powders is important inany industries such as pharmaceutical, bulk chemical and

ood. Currently, dispersion in the gas phase is receiving muchttention with respect to particle characterisation and drugelivery via the lungs from dry powder inhalers (DPIs). Thebility to control dispersion of a wide range of powders is anrea of great interest and importance. However, it is recognisedhat the complete dispersion of fine cohesive powders is dif-cult due to the relatively large interparticle attraction forces,amely van der Waals, electrostatics and liquid bridge (Visser,989), compared to separating forces. Consequently, a greatmount of energy is necessary to deform and disintegrate par-icle clusters completely to their primary constituents.

Recently, Calvert et al. (2009) reviewed aerodynamic disper-ion of cohesive powders and highlighted the current state ofheoretical and experimental understanding. Loosely aggre-ated particles suspended in fluid flows experience several

inds of forces caused by rapid acceleration, deceleration,

∗ Corresponding author. Tel.: +44 113 343 2406; fax: +44 113 343 2405.E-mail address: m.ghadiri@leeds.ac.uk (M. Ghadiri).Received 22 September 2009; Received in revised form 12 May 2010; A

263-8762/$ – see front matter © 2010 Published by Elsevier B.V. on behoi:10.1016/j.cherd.2010.08.013

turbulent eddies, impact on surfaces etc. When consideringparticle–fluid interactions, the following mechanisms are usedto describe the dominating dispersion process (Gotoh et al.,2006):

(1) dispersion by rapid acceleration or deceleration and/orshear flow;

(2) dispersion of particle clusters by impact onto a stationaryor moving target;

(3) dispersion by other mechanical forces (e.g. fluidisation,mixing, vibration and scraping).

Kousaka et al. (1979) investigated bulk powder disper-sion using various devices and showed that dispersion byimpaction is most effective and that dispersion by acceler-ation can be effective if the velocity difference between thefluid and particles is sufficiently large. A number of theoreticalapproaches are available which describe the separating force

ccepted 26 August 2010

erated in a uniform flow field (Kousaka et al., 1979, 1992; Yuu

alf of The Institution of Chemical Engineers.

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520 chemical engineering research and design 8 9 ( 2 0 1 1 ) 519–525

Nomenclature

C′D drag coefficient (–)

DAgg loose aggregate diameter (m)d particle diameter (m)DI dispersion index (–)DR dispersion ratio (–)F force acting on a spherical particle (N)Fi contact force vector (N)Fn

inormal contact force vector (N)

Fsi

shear contact force vector (N)fc,ij contact force (N)fd,ij viscous contact damping force (N)fint interaction force between particles and fluid (N)g gravitational acceleration (m/s2)Ii moment of inertia (kg/m2)Kn normal stiffness (N/m)ks shear stiffness (N/m)m particle mass (kg)N number of particles in contact with particle i (–)NP number of particles in the loose aggregate (–)N0 initial number of bonds (–)Nt number of bonds after time t (–)np number of particles in a given fluid cell (–)POFF force to break a contact (N)P pressure (Pa)R particle radius (m)Tij particle torque (Nm)U(n/s) contact overlaps for normal and tangential

directions (m)u fluid velocity (m/s)ur velocity of fluid relative to particle (m/s)V volume (m3)v particle velocity (m/s)We Weber number (–)Zt coordination number at time t (–)Z0 initial coordination number (–)

Greek symbols� interface energy (J/m2)ε fluid cell porosity (–)� viscous stress tensor (kg/m s2)� surface energy (J/m2)�f fluid density (kg/m3)ωi angular velocity (1/s)

and Oda, 1983). However, these basic models are not directlyapplicable to dispersion of cohesive particle aggregates by fluidenergy, which forms the theme of this paper.

Recently, computational simulations such as the distinctelement method (DEM) coupled with a continuum modelfor simulating fluid flows have received increasing attention(Tsuji et al., 1993; Xu and Yu, 1997; Moreno-Atanasio et al.,2007). The main advantage of this method is its versatil-ity, allowing easy variation of any material property withoutaffecting others. Additionally, with DEM it is possible to quan-tify the number of interparticle contacts which break withinthe particle assembly (Moreno-Atanasio and Ghadiri, 2006).With respect to deformation and breakage of loose aggregateassemblies in fluid flow fields, a number of computational

studies are available in the literature (Higashitani et al., 2001;Fanelli et al., 2006; Zeidan et al., 2007). In these studies the

simulated fluid is a liquid. Little attention has so far beengiven to the behaviour of loose aggregate clusters dispersedin air. Iimura et al. (2009) investigated the dispersion of par-ticle aggregates in air and water. The investigation showsdispersion behaviour qualitatively; however, a greater under-standing is necessary with respect to particulate dispersion.

In this paper the behaviour of a single loose aggregate clus-ter consisting of a number of particles that are accelerated ina uniform flow field is investigated. Furthermore, a dispersionmodel based on the balance between adhesive and disruptiveforces, with the latter caused by the fluid drag force acting onthe loose aggregate, is introduced. The basis of the model isdescribed below.

2. Simulation method

The distinct element method (DEM) was originally introducedby Cundall (1971) and further developed by Cundall and Strack(1979). The commercial DEM code produced by Itasca calledPFC3D has been used to perform the simulations. DEM is basedon the application of Newton’s second law of motion whichdetermines the translational and rotational motions of eachparticle arising from the contact and body forces acting uponit:

midvi

dt= mig +

N∑j=1

(fc,ij + fd,ij) (1)

Iidωi

dt=

N∑j=1

Tij (2)

where mi and vi are the mass and velocity of particle i, gis gravitational acceleration, N is the number of particles incontact with particle i, fc,ij and fd,ij are the contact and vis-cous contact damping forces, respectively, Ii is the momentof inertia of particle i, ωi is the angular velocity and Tij is thetorque arising from the contacts which causes a particle torotate.

In addition, a force–displacement law is implemented toupdate the contact forces arising from the relative motion ateach contact. The contact force vector, Fi, for particle–particleand particle–wall contacts can be resolved into normal andshear components with respect to the contact plane,

Fi = Fni + Fs

i (3)

where Fni

and Fsi

are the normal and shear component vec-tors, respectively. The normal contact force vector and shearcontact force vector are calculated using the linear contactmodel:

Fni = KnUnni (4)

�Fsi = −ks�Us

i (5)

where Kn is the normal stiffness, ks is the shear stiffness, Un

and �Us are the contact overlaps for normal and tangentialdirections, respectively, and ni is the normal vector.

In these simulations the effect of cohesion has been inves-tigated by including a contact-bond model. In this case the

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chemical engineering research and design 8 9 ( 2 0 1 1 ) 519–525 521

ce

P

wr(

�

wo�

pi

wabc

ε

wtvSa

3a

AoCa

F

wlcpflnc

g(bsf

nation number was achieved (Fig. 1). The interface energy(Eq. (6)) and particle friction are then progressively introduced

Fig. 1 – Schematic illustration of aggregate formation. (a)

ontact-bond force, POFF, is defined by the JKR model (Johnsont al., 1971),

OFF = 32

�R∗ (6)

here � and R* are the interface energy and the reducedadius of the two particles in contact as defined by Eqs. (7)Israelachvili, 1985) and (8), respectively:

= �A + �B − �AB (7)

1R∗ = 1

R1+ 1

R2(8)

here �A and �B are the surface energies of two particles madef different materials, A and B, in contact with each other and

AB is the interaction energy between them.The continuity and Navier–Stokes equations for the fluid

hase in the fluid–solids two-phase model, for an incompress-ble fluid with constant density, are given by Eqs. (9) and (10):

∂ε

∂t+ ∇ · (εu) = 0 (9)

∂(εu)∂t

= −∇ · (εuu) − ε

�f∇P − ε

�f∇ · � + εg + fint

�f(10)

here u is the fluid velocity vector, � is the viscous stress tensornd fint is the force per unit volume, related to the interactionetween the particles and the fluid. The porosity, ε, of a fixedontrol volume is used in Eqs. (14) and (15) and defined as:

= 1 − 1�V

6

np∑i=1

d3i (11)

here �V is the fixed control volume of a fluid cell and np ishe number of particles in the cell. The pressure and velocityectors of the fluid in each cell are calculated by applying theemi-Implicit Method for Pressure Linked Equations (SIMPLE)lgorithm (Patanker, 1980).

. Mechanistic model of dispersion bycceleration

ggregate dispersion occurs when the separating force actingn it exceeds the force which is binding the primary particles.onsidering the motion of a sphere in a fluid, the drag forcecting on an aggregate is given by:

= C′D

D2Agg

4�f u2

r (12)

here F is the drag force acting on the particle, DAgg is theoose aggregate diameter, �f is the fluid density, C′

D is the dragoefficient, and ur is the velocity of the fluid relative to thearticle (Clift et al., 1978). In the range of aggregate size anduid velocity of interest for dispersion, the particle Reynoldsumber is within the Newtonian regime; therefore, the dragoefficient can be regarded as a constant (C′

D = 0.44).To investigate the effect of surface energy on loose aggre-

ate dispersion behaviour, the model of Johnson et al. (1971)JKR model) is used to define the pull-off force necessary toreak an interparticle contact (Eq. (6)). For surfaces of the

ame material �AB is zero and therefore, the resultant inter-ace energy is � = 2�. Furthermore, if the loose aggregate is

composed of monodispersed particles R* = R/2 = d/4, where d isthe primary particle diameter. Therefore, the force necessaryto break a contact can be defined as:

POFF = 38

�d (13)

By defining a dispersion ratio, DR, as the ratio of the fluidresistance force acting on the spherical loose aggregate to theforce necessary to break an interparticle contact, we get:

DR ∝ 23

C′D

�f D2Aggu2

r

�d(14)

Now considering the terms in Eq. (14), it is possible to intro-duce a dispersion index, DI, by dropping the constant termsand grouping the terms which define the Weber number:

DI = �f DAggu2r

�

(DAgg

d

)= We

(DAgg

d

)(15)

Therefore, the dispersion ratio is given by:

DR = f

[We

(DAgg

d

)](16)

The Weber number represents the ratio of the fluid kineticenergy trying to cause dispersion and the work required tobreak the cohesive bonds. It also prevails in other particu-late systems, including aggregate breakage due to impaction(Kafui and Thornton, 1993; Moreno-Atanasio and Ghadiri,2006). Hence, it is of interest to see whether the DI, which con-tains the Weber number, is related to particle dispersion in anaccelerating fluid flow field.

4. Simulations of loose aggregatedispersion

4.1. Simulation details

To investigate the relationship described previously (Eq. (16)),six particle assemblies have been formed in exactly the sameway, but with different particle numbers and hence differ-ent aggregate diameter. For each loose aggregate, the primaryparticles, with properties shown in Table 1, were randomlygenerated and brought together by applying a centripetalacceleration towards a point in space until a stable coordi-

Aggregate formation – velocity vectors indicating direction;(b) stable aggregate.

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522 chemical engineering research and design 8 9 ( 2 0 1 1 ) 519–525

Fig. 2 – Dispersion behaviour of an aggregate which consists of 500 particles in a uniform flow field (aggregate B) as afunction of relative velocity and elapsed time. When the relative velocity, ur, is greater than a critical value dispersionoccurs. The coordination number is given in the brackets.

Table 1 – Particle and fluid properties.

Diameter (�m) 100Density (kg/m3) 2500Normal stiffness (N/m) 7.9 × 104

Shear stiffness (N/m) 7.9 × 104

Contact damping (–) 0.16Friction coefficient (–) 0.3Fluid density (kg/m3) 1.225Fluid viscosity (Pa s) 1.8 × 10−5

Table 2 – Loose aggregate properties. The coordination number

Loose aggregate A B

Number of particles 250 500 7Aggregate diameter (mm) 0.764 0.966 1Contact number 744 (5.952) 1494 (5.976) 22

between particles to the required level. Subsequently, the cen-tripetal acceleration was gradually removed from the particleassembly; this procedure is described by Moreno et al. (2003).Details of the six loose aggregate assemblies which have beeninvestigated are shown in Table 2. All of the loose aggregateshave been bonded with interface energy 0.1 J/m2. However,loose aggregate B has been investigated for a range of interfaceenergies from 0.1 to 1.0 J/m2. For each aggregate the coordi-

nation number is approximately the same. Therefore, it is

is given in the brackets.

C D E F

50 1000 2000 8000.10 1.22 1.52 2.3744 (5.984) 2994 (5.988) 5894 (5.894) 23,608 (5.902)

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chemical engineering research and design 8 9 ( 2 0 1 1 ) 519–525 523

Fig. 3 – Dispersion behaviour of an aggregate which consists of 8000 particles in a uniform flow field (aggregate F) as afunction of relative velocity and elapsed time. For the large aggregate, above a threshold velocity dispersion begins andp e bra

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dftt

4

IpUlap5(ftasnscoib

I(st

eeling is observed. The coordination number is given in th

xpected minute variations in the aggregate structure andacking are not going to influence the dispersion behaviour.hese particle assemblies were then subjected to a uniformow field with various fluid velocities. When dealing withacked assemblies, it is desirable for the fluid cell size to bene enough to resolve the flow of continuum fluid throughhe pores of closely spaced particles (Zhu et al., 2007). In allhe simulations a fluid cell size 2.5 times larger than the parti-le radius has been used. The fluid properties used for air arehown in Table 1.

The dispersion of cohesive loose aggregates is time depen-ent and the rate of dispersion is dependent on the dispersionorce (Masuda, 2009). Consequently, for comparative reasons,he dispersion behaviour is quantified after a given amount ofime, in this case 3 × 10−4 s; unless otherwise stated.

.2. Dispersion behaviour

t is difficult to experimentally investigate the dispersionhenomenon of a single aggregate at the particulate level.sing the computational technique described previously, the

oose aggregate break-up can be qualitatively investigated asfunction of the upstream fluid velocity. In Fig. 2 the dis-

ersion behaviour of a single loose aggregate consisting of00 particles, which yields an aggregate diameter of 0.966 mmaggregate B), is shown at various points in time and for dif-erent fluid velocities. Interestingly, it is observed that unlesshe relative velocity, ur, is above a threshold limit the looseggregate would not deform and disperse but accelerate as aingle entity. The coordination number, defined as the averageumber of contacts per particle, is shown within the parenthe-es. It appears that, if the relative velocity is great enough theoordination number rapidly decreases from an initial valuef 5.976 to nearly zero; suggesting that dispersion is almost

nstantaneous and disintegration dominates the dispersionehaviour.

The dispersion behaviour of large clusters is shown in Fig. 3.n this case the cluster (aggregate F) consists of 8000 particlesDAgg = 2.37 mm, DAgg/d = 23.7). For these large aggregates (the

ize ratio between the aggregates and particles being greaterhan 20), the dispersion behaviour is slightly different. Once

ckets.

again a threshold relative velocity is necessary; but, as indi-cated by the time shown, the dispersion rate is much slowerwith respect to drop in particle coordination number. Thismakes it appear as though the particles are peeling away fromthe aggregate surface. This is attributed to a difference in thetemporal force propagation permeating through the aggre-gate, when compared to smaller aggregates.

4.3. Analysis of dispersion ratio and dispersion index

4.3.1. Effect of interface energyUsing DEM, it is possible to quantitatively investigate the effectof surface energy and aggregate diameter through analysis ofthe dispersion ratio, DR. The dispersion ratio is related to theforce trying to move the aggregate, the drag force, and the forcetrying to resist dispersion, the pull-off force. A value of the dis-persion ratio can be determined in DEM by relating the numberof broken bonds in a specific aggregate after a given amountof time compared with the initial number of bonds, as shownin Eq. (17):

DR = N0 − Nt

N0(17)

where N0 is the initial number of bonds and Nt is the numberof bonds after a given amount of time. The dispersion ratio as afunction of the relative velocity between the particles and thedispersing fluid is shown in Fig. 4 for aggregate B for a rangeof interface energies. At low relative velocities the dispersionratio is not sensitive to the fluid force, but once a thresholdrelative velocity is exceeded, the dispersion ratio quickly risesand approaches unity in an asymptotic fashion. An increase inthe relative velocity results in an increase in the fluid force act-ing on the particles and consequently, more force is availablein the system for breaking contacts. As intuitively expected,with an increase in interface energy it becomes increasinglydifficult to break bonds and disperse the particle assembly.

In Fig. 5, the dispersion ratio, DR, has been plotted as afunction of the dimensionless dispersion index, DI, as given byEq. (16), for loose aggregate B with different interface energies

after 3 × 10−4 s. It is seen that there is a remarkable unifica-tion of the data obtained for the different interface energies.

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524 chemical engineering research and design 8 9 ( 2 0 1 1 ) 519–525

Fig. 4 – Relationship between dispersion ratio and relativevelocity between the fluid and particles for different valuesof interface energy after 3 × 10−4 s.

Fig. 5 – Relationship between the dispersion ratio anddispersion index, for different values of interface energy forloose aggregate B (DAgg = 0.966 mm) after 3 × 10−4 s.

Fig. 6 – Relationship between dispersion ratio and relativevelocity between the fluid and particles for looseaggregates with different diameters shown in Table 2 after3 × 10−4 s. The interface energy is constant at 0.1 J/m2.

Fig. 7 – Relationship between the dispersion ratio anddispersion index for the different loose aggregates shownin Table 2 and interface energy 0.1 J/m2 after 3 × 10−4 s.

Fig. 8 – Relationship between the dispersion ratio anddispersion index, for loose aggregate F (DAgg = 2.37 mm) at

erosion. In Fig. 8 the dispersion ratio as a function of disper-sion index for aggregate F is shown at two different elapsed

This indicates that the DI can be used to define the onset ofloose aggregate dispersion due to acceleration by a fluid flow.Therefore DI may be used as a predictive tool, and that withan increase or decrease in a particular parameter, the rela-tive velocity would need to change accordingly for particledispersion to initiate.

interface energy 0.1 J/m2 at two different periods in time.

4.3.2. Effect of aggregate sizeIn Fig. 6 the dispersion ratio as a function of relative velocityis shown for loose aggregates of different diameters, detailedin Table 2, with constant interface energy of 0.1 J/m2. It isincreasingly difficult to disperse a particle cluster as the aggre-gate diameter is reduced, due to a larger relative velocitybeing needed. As the loose aggregate diameter is increasedthe required relative velocity reduces due to an increase inaggregate cross-sectional area. However, it appears as thoughthe largest aggregate disperses less well at the higher rela-tive velocities. This is because the dispersion process is timedependent, and this plot is at time 3 × 10−4 s after the intro-duction of the aggregate into the fluid flow. Therefore, Fig. 6 issimply demonstrating the change in the onset of dispersiondue to aggregate scale.

In addition to the Weber number, the dispersion indexincorporates a size ratio term which should lead to a unifi-cation of data for different sized aggregates similar to thatshown in Fig. 5. In Fig. 7 the dispersion ratio as a functionof the dispersion index for different sizes of the loose aggre-gates (Table 2) is shown for the interface energy 0.1 J/m2 after3 × 10−4 s of elapsed time after dispersion. A good unificationis observed except for the largest aggregate.

As the aggregate size is dramatically increased the disper-sion behaviour slightly changes from an almost instantaneousdisintegration due to extensive deformation to gradual surface

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chemical engineering research and design 8 9 ( 2 0 1 1 ) 519–525 525

ttmsd

5

Dbaoainbipl

gettlfbtdfpstta

A

Teg

R

C

C

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imes. At the later time, the curve resembles those shown forhe smaller aggregates. Therefore, larger aggregates require

ore time to fully disperse, and if the surface erosion is noto fast they may even manage to accelerate and escape bodyeformation and disintegration.

. Conclusions

ispersion by acceleration in a uniform flow field is seen toe effective when the relative particle–fluid velocity is beyondthreshold value. In this case, dispersion of the aggregate

ccurs quickly and approaches a completely dispersed statesymptotically. The effect of particle surface energy has beennvestigated and as expected a greater relative velocity isecessary to disperse the aggregate with increasing particleond strength. In addition, as the loose aggregate diameter is

ncreased the necessary relative velocity for the onset of dis-ersion reduces, although full dispersion is not achieved for

arge aggregates.To analyse the effect of surface energy and loose aggre-

ate diameter on particle dispersion, a model is developed thatxpresses the extent of dispersion to the ratio of the fluid resis-ance force acting on a sphere to the bond strength bindinghe particles together. This approach leads to a dimension-ess dispersion index which incorporates the Weber numberor the aggregate and a second parameter for the size ratioetween the aggregate diameter and primary particle diame-er. The DEM coupled with a fluid scheme has been used toetermine whether the dispersion index can be used as a toolor describing particle dispersion due to acceleration by a dis-ersing fluid. The results of the numerical simulations havehown a good correlation between the number of broken con-acts and the particle surface energy, loose aggregate diameter,he relative initial velocity between the aggregate and the fluid,s defined by the dispersion index, DI = We(DAgg/d).

cknowledgments

he financial support from the Engineering and Physical Sci-nce Research Council (EPSRC) and Malvern Instruments isreatly appreciated.

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