discrete particle simulation of particulate systems: theoretical developments

Chemical Engineering Science 62 (2007) 3378–3396www.elsevier.com/locate/ces

Discrete particle simulation of particulate systems: Theoretical developments

H.P. Zhu, Z.Y. Zhou, R.Y. Yang, A.B. Yu∗

Centre for Simulation and Modelling of Particulate Systems, School of Materials Science and Engineering, The University of New South Wales,Sydney, NSW 2052, Australia

Received 8 October 2006; received in revised form 10 December 2006; accepted 25 December 2006Available online 23 March 2007

Abstract

Particle science and technology is a rapidly developing interdisciplinary research area with its core being the understanding of the relationshipsbetween micro- and macroscopic properties of particulate/granular matter—a state of matter that is widely encountered but poorly understood.The macroscopic behaviour of particulate matter is controlled by the interactions between individual particles as well as interactions withsurrounding fluids. Understanding the microscopic mechanisms in terms of these interaction forces is therefore key to leading to trulyinterdisciplinary research into particulate matter and producing results that can be generally used. This aim can be effectively achieved viaparticle scale research based on detailed microdynamic information such as the forces acting on and trajectories of individual particles ina considered system. In recent years, such research has been rapidly developed worldwide, mainly as a result of the rapid development ofdiscrete particle simulation technique and computer technology. This paper reviews the work in this area with special reference to the discreteelement method and associated theoretical developments. It covers three important aspects: models for the calculation of the particle–particleand particle–fluid interaction forces, coupling of discrete element method with computational fluid dynamics to describe particle–fluid flow,and the theories for linking discrete to continuum modelling. Needs for future development are also discussed.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Powder technology; Particulate processes; Multiphase flow; Simulation; Mathematical modelling

1. Introduction

Particle science and technology is a rapidly developinginterdisciplinary research area with its core being the under-standing of the relationships between micro- and macroscopicproperties of particulate/granular matter—a state of matter thatis widely encountered but poorly understood. Previous studiesare largely at a macroscopic or global scale, the resulting infor-mation being helpful in developing a broad understanding ofa particulate process of particular interest. However, the lackof quantitative fundamental understanding makes it difficultto generate a general method for reliable scale-up, design andcontrol/optimization of processes of different types.

The macroscopic behaviour of particulate matter is controlledby the interactions between individual particles as well as in-teractions with surrounding gas or liquid and wall. Understand-ing the microscopic mechanism in terms of these interactions

∗ Corresponding author. Tel.: +61 2 9385 4429; fax: + 61 2 9385 5956.E-mail address: [email protected] (A.B. Yu).

0009-2509/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2006.12.089

is therefore the key leading to truly interdisciplinary researchinto particulate matter and producing results that can be gen-erally used. This aim can be effectively achieved via particlescale research based on detailed microdynamic information. Inrecent years, such research has been rapidly developed world-wide, mainly as a result of the rapid development of discreteparticle simulation technique and computer technology. Severaldiscrete modelling techniques have been developed, includingMonte Carlo method, cellular automata and discrete elementmethod (DEM). DEM simulations can provide dynamic infor-mation, such as the trajectories of and transient forces actingon individual particles, which is extremely difficult, if not im-possible, to obtain by physical experimentation at this stage ofdevelopment. Consequently, it has been increasingly used inthe past two decades or so.

Two types of DEMs are most common: soft-particle andhard-particle approaches. The soft-sphere method originallydeveloped by Cundall and Strack (1979) was the first granulardynamics simulation technique published in the open literature.In such an approach, particles are permitted to suffer minute

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H.P. Zhu et al. / Chemical Engineering Science 62 (2007) 3378–3396 3379

180

160

140

120

100

80

60

40

20

0

Num

ber

of public

ations

1990 1995 2000 2005

Year

Fig. 1. Number of publications related to discrete particle simulation inthe recent 20 years, obtained from the Web of Science with the followingkeywords: discrete element method/model, distinct element method/model,discrete particle simulation/method/model, and granular dynamic simulation.

deformations, and these deformations are used to calculate elas-tic, plastic and frictional forces between particles. The motionof particles is described by the well-established Newton’s lawsof motion. A characteristic feature of the soft-sphere modelsis that they are capable of handling multiple particle contactswhich are of importance when modelling quasi-static systems.By contrast, in a hard-particle simulation, a sequence of col-lisions is processed, one collision at a time and being instan-taneous; often the forces between particles are not explicitlyconsidered. Therefore, typically, hard-particle method is mostuseful in rapid granular flows. The two DEMs, particularlythe soft-sphere method, have been extensively used to studyvarious phenomena, such as particle packing, transport prop-erties, heaping/piling process, hopper flow, mixing and gran-ulation. DEM has been coupled with computational fluiddynamics (CFD) to describe particle–fluid flows such as flu-idization and pneumatic conveying.

A survey of the literature indicates that many papers relatingto DEM have been published in the past two decades or so,as shown in Fig. 1. The rapid development and application ofDEM can also be highlighted by the large number of papersbased on DEM at major international conferences, for example,from 29 in 2001 to 80 in 2005 for the International Conferenceon Powders and Grains, and from 34 in 2002 to 92 in 2006 forthe World Congress on Particle Technology. In spite of the largebulk volume, little effort has been made to comprehensivelyreview and summarize the progress made in the past, except fora few relatively focused reviews including, for example, Tsuji(1996) on the work in Japan, Mishra (2003a,b) on tumblingmilling processes, Yu (2004) on the work done in his laboratory,Richards et al. (2004) on environmental science, and Bertrandet al. (2005) on mixing of granular materials.

To overcome this gap, we have recently reviewed the ma-jor work in this area with special reference to the soft-particlemodel. It is based on the publications from the Web of Science

available until the mid of 2006, and covers the development ofsimulation techniques and their application to the study of par-ticle packing, particle flow and particle–fluid flow. This article,as part of the review effort, presents a summary of the majortheoretical developments in DEM, which are for conveniencegrouped into three important aspects: models for the calcula-tion of the particle–particle and particle–fluid interaction forces,coupling of DEM with CFD to describe particle–fluid flow, andtechnique for linking discrete to continuum modelling. Needsfor future development are also discussed.

2. Governing equations and force models

A particle in a granular flow can have two types of motion:translational and rotational. During its movement, the particlemay interact with its neighbouring particles or walls and inter-act with its surrounding fluid, through which the momentumand energy are exchanged. Strictly speaking, this movement isaffected not only by the forces and torques originated from itsimmediate neighbouring particles and vicinal fluid but also theparticles and fluids far away through the propagation of dis-turbance waves. The complexity of such a process has defiedany attempt to model this problem analytically. In DEM ap-proach, it is generally assumed that this problem can be solvedby choosing a numerical time step less than a critical value sothat during a single time step the disturbance cannot propagatefrom the particle and fluid farther than its immediate neigh-bouring particles and vicinal fluid (Cundall and Strack, 1979).Thus, at all times the resultant forces on a particle can be deter-mined exclusively from its interaction with the contacting par-ticles and vicinal fluid for a coarse particle system. For a fineparticle system, non-contact forces such as the van der Waalsand electrostatic forces should also be included. Based on theseconsiderations, Newton’s second law of motion can be usedto describe the motion of individual particles. The governingequations for the translational and rotational motion of particlei with mass mi and moment of inertia Ii can be written as

mi

dvi

dt=∑j

Fcij +

∑k

Fncik + Ff

i + Fgi , (1)

Ii

d�i

dt=∑j

Mij , (2)

where vi and �i are the translational and angular velocities ofparticle i, respectively, Fc

ij and Mij are the contact force andtorque acting on particle i by particle j or walls, Fnc

ik is the non-contact force acting on particle i by particle k or other sources,Ff

i is the particle–fluid interaction force on particle i, and Fgi is

the gravitational force. Fig. 2 schematically shows the typicalforces and torques involved in a DEM simulation.

Various models have been proposed to calculate these forcesand torques, which will be discussed below. Once the forcesand torques are known, Eqs. (1) and (2) can be readily solvednumerically. Thus, the trajectories, velocities and the transientforces of all particles in a system considered can be determined.

3380 H.P. Zhu et al. / Chemical Engineering Science 62 (2007) 3378–3396

vi

ncik

F

tijf

nijf m

ig

rij

m

cijF

βω

i

φ

i

k

j

vj

h

ωj

Fig. 2. Schematic illustration of the forces acting on particle i from contactingparticle j and non-contacting particle k (capillary force here).

2.1. Contact forces between particles

In general, the contact between two particles is not at a sin-gle point but on a finite area due to the deformation of theparticles, which is equivalent to the contact of two rigid bodiesallowed to overlap slightly in the DEM. The contact tractiondistribution over this area can be decomposed into a compo-nent in the contact plane (or tangential plane) and one normalto the plane, thus a contact force has two components: normaland tangential. It is very difficult to accurately and generallydescribe the contact traction distribution over this area and thenthe total force and torque acting on a particle, as it is related tomany geometrical and physical factors such as the shape, ma-terial properties and movement state of particles. Alternatively,to be computationally efficient and hence applicable to multi-particle systems, the DEM generally adopts simplified modelsor equations to determine the forces and torques resulting fromthe contact between particles.

Various approaches have been proposed for this purpose.Generally, linear models are the most intuitive and simple mod-els. The most common linear model is the so-called linearspring–dashpot model proposed by Cundall and Strack (1979),where the spring is used for the elastic deformation while thedashpot accounts for the viscous dissipation. The linear springmodel without inclusion of dashpot has also been used by, forexample, Di Renzo and Di Maio (2004). More complex and the-oretically sound model, Hertz–Mindlin and Deresiewicz model,has also been developed. Hertz (1882) proposed a theory todescribe the elastic contact between two spheres in the normaldirection. He considered that the relationship between the nor-mal force and normal displacement was nonlinear. Mindlin andDeresiewicz (1953) proposed a general tangential force model.They demonstrated that the force–displacement relationship de-pends on the whole loading history and instantaneous rate ofchange of the normal and tangential force or displacement. Acomplete description of the theory of Mindlin and Deresiewicz

can be seen in the recent work of Vu-Quoc and Zhang (1999a,b)and Di Renzo and Di Maio (2004). Due to its complication,however, the complete Hertz–Mindlin and Deresiewicz modelis time-consuming for DEM simulations of granular flows of-ten involving a large number of particles, and is therefore notso popular in the application of DEM.

Various simplified models based on the Hertz, and Mindlinand Deresiewicz theories have been developed for DEM mod-elling. For example, Walton and Braun (1986a) and Walton(1993) used a semi-latched spring force–displacement modelin the normal direction, and an approximation of the Mindlinand Deresiewicz contact theory for the cases of constantnormal force in the tangential direction. Thornton and Yin(1991) proposed a more complex model to simulate the tan-gential force. While adopting the Hertz theory for the nor-mal force, different from Walton and Braun’s model, theirmodel assumes that the incremental tangential force due tothe incremental tangential displacement depends on the vari-ation of the normal force. Both Walton and Braun’s modeland Thornton and Yin’s model for tangential force are thedirect simplifications of the Mindlin and Deresiewicz the-ory. A more intuitive model was adopted by Langston et al.(1994). They used a direct force–displacement relationfor the tangential force and the Hertz theory for the nor-mal force. Due to its simplicity and intuitiveness, themodel has been extensively used to study the dynamicbehaviour of granular matter (Langston et al., 1995a,b;Zhou et al., 1999; Zhu and Yu, 2002). More recent ad-vances on contact force incorporating the plastic deforma-tion have also been made by Thornton (1997) and Vu-Quocand Zhang (1999a,b). However, they need more experimen-tal validation. The above models are often used miscella-neously (Schäfer et al., 1996; Lätzel et al., 2000). Table 1shows the equations for some commonly used force modelsfor spherical particles, including the linear spring–dashpotmodel, the simplified Hertz–Mindlin and Deresiewiczmodel by Langston et al. (1994), and Walton and Braun’smodel.

The interparticle forces act at the contact point between par-ticles rather than the mass centre of a particle and they willgenerate a torque causing the particle to rotate. Generally, thetorque is contributed by two components of the tangential andasymmetrical normal traction distributions as shown in Fig. 3.Compared with the contribution of the tangential component,the determination of the contribution of the normal component,usually called as rolling friction torque, is very difficult and isstill an active research area (Greenwood et al., 1961; Johnson,1985; Brillianton and Pöschel, 1998; Kondic, 1999). The rollingfriction torque is considered to be negligible in many DEMmodels. However, it has been shown that the torque plays asignificant role in some cases involving the transition betweenstatic and dynamic states, such as the formations of shear band(Iwashita and Oda, 1998, 2000) and heaping (Zhou et al., 1999),and movement of a single particle on a plane (Zhou et al., 1999;Zhu and Yu, 2006).

The treatment of the contact force between non-sphericalparticles is far more complicated. Two techniques have been


Table 1Contact force and torque models

Force models Normal force Tangential force References

Linear spring–dashpot fn = −Kn�nnc − Cn(vc · nc)nc ft = −Kt vtc + Ct (vc × nc) × nc Cundall and Strack (1979)

modelSimplified Hertz–Mindlin fn = − 4

3 E∗√R∗(�n)3/2nc ft = −�|fn,e|(1 − (1 − |vt

c|/�max)3/2)vt

c Langston et al. (1994, 1995a,b),and Deresiewicz model −Cn(8m∗E∗√R∗�n)

1/2 +2Ct (1.5 �m ∗ |fn,e| Zhou et al. (1999) and·(vc · nc)nc ·√1 − |vt

c|/�max/�max)1/2 Zhu and Yu (2002)

·(vc × nc) × nc

Walton and Braun’s model Walton and Braun (1986a),and Walton (1993)

fn =

⎧⎪⎨⎪⎩

−k1�nnc, �n �0(loading)

−k2(�n − �n0)nc, �n < 0(unloading)

ft =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

f ′t + k0

t

(1 − ft −f ∗

t

�fn−f ∗t

)1/3�vt

c

if vtc in initial direction

f ′t + k0

t

(1 − f ∗

t −ft

�fn+f ∗t

)1/3�vt

c

if vtc in opposite direction

where ft = |ft |, fn = |fn|.Torque models Rolling friction torque Torque from tangential forces ReferencesMethod 1 mr = −kr�r − Cr d�r /dt mt = R × ft Iwashita and Oda (1998, 2000)Method 2 mr = − min{�r |fn|,�′

r |�n|}�n Zhou et al. (1999),and Zhu and Yu (2002)

Fig. 3. Normal traction distribution exerted on particle i due to the collisionwith particle j (Zhu and Yu, 2003).

proposed to cope with particles of irregular shapes. One is tomodel a non-spherical particle as a collection of spherical par-ticles (Gallas and Sokolowski, 1993; Pelessone, 2003; Bertrandet al., 2005). The advantage of the strategy is that it can beused to handle particles with very complex shapes, and onlya contact model for spherical particle is required. The othertechnique considers that particles are of a given shape, such asellipsoid, polygon and cylinder, and determines whether or notthere is a contact between two such neighbouring particles bysolving the underlying mathematical equations (Kohring et al.,1995; Favier et al., 2001; Wait, 2001; Dziugys and Peters, 2001;Cleary and Sawley, 2002; Munjiza et al., 2003; Langston et al.,2004). The contact force model used in this approach is one

for spherical particles (Favier et al., 2001; Cleary and Sawley,2002) or its modification (Kohring et al., 1995). This approachis more accurate than the first one, but is computationally moredemanding.

As discussed above, linear and nonlinear contact force mod-els have been developed for DEM simulations. Theoretically,the more complex nonlinear models simplified based on theHertz and Mindlin–Deresiewicz theories should be more accu-rate than the linear model. However, in contrast, the numeri-cal investigations conducted by Di Renzo and Di Maio (2004)showed that the simple linear model sometimes gives better re-sults. This may be because theoretical models are often basedon geometrically ideal particles, whereas there are no such per-fect particles in practical applications. Selection of proper pa-rameters also plays an important role in generating accurateresults. Complicated models may also consume computationaltime with insignificant gain in DEM simulations. Moreover,as recently demonstrated by Zhu and Yu (2006), most of theforce models were developed focused on one or two aspects orbased on simplified conditions, their combination in DEM sim-ulation may lead to theoretical or conceptual problems. Theseissues should be considered in the further development of forcemodels.

2.2. Non-contact forces between particles

When fine particles are involved and/or moisture exists, non-contact interparticle forces may affect the packing and flow be-haviour of particles significantly. In the past, the interparticleforces were often evaluated by some empirical indexes suchas Hausner ratio, angle of repose and shear stress (Hausner,1972; Svarovsky, 1987). These indexes may partially interpretthe behaviour of particles (see, for example, Geldart and Wong,1984, 1985; Svarovsky, 1987; Yu and Hall, 1994), but general


Table 2List of a few typical non-contact forces

Cohesive force Origin Formula References

van der Waals force Molecular dipole interaction Fv = − Adp

24h2nij Hamaker (1937),

and Israelachvili (1991)

Electrostatic force Coulomb force Fe = − Q2

16�q0h2

(1 − h√

(R2+h2)

)nij Krupp (1967)

Liquid bridge force (static) Surface tension (capillary pressure Fl = −[2��R sin � sin(� + �) + �R2�psin2�]nij Fisher (1926),and contact line force) Princen (1968), and Lian et al. (1993)

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1 10 100 1000

Particle Diameter (μm)

Inte

rpart

icle

Fo

rce (

N)

Weight

Electrostatic

Capillary Force

Van der Waals Force

Fig. 4. Comparison of the magnitude of interparticle forces calculated basedon: (i) van der Waals forces: A = 6.5 × 10−20 J, h = 1.65 A, (ii) capillaryforce: � = 72.8 × 10−2 Nm−1 (water); (iii) electrostatic forces: �r = 1; and(iv) weight: g = 3 × 103 kg m−3 (Rumpf, 1962).

quantitative application is still difficult. This difficulty can beovercome by DEM because such forces can be directly con-sidered. Often the non-contact forces involve a combination ofthree fundamental forces, i.e. the van der Waals force, capil-lary force and electrostatic force, which can act concurrently orsuccessively to different extents. Table 2 lists the forces, theirorigin and equations for their estimation. Their relative impor-tance varies, depending on a range of variables as illustrated inFig. 4. In this section the mechanisms of these forces and theapplication in DEM modelling will be briefly discussed.

The van der Waals force is the force between moleculeshaving closed shell and consists of several types of interactions(Israelachvili, 1991). The van der Waals force betweenmolecules is proportional to h−6, where h is the separationdistance (Fig. 2). For two particles, the van der Waals force,obtained by integration of all forces for all pair of molecules,decays much more slowly with distance than for two molecules,and is proportional to h−2.

The Hamaker theory (Hamaker, 1937) is usually used inDEM simulation to calculate the van der Waals force. Thistheory is based on the assumption of “pairwise additivity”

and starts from the interactions between individual atoms (ormolecules) and postulates their additive so that the van derWaals attraction between macroscopic bodies can be calculatedby integration over all pairs of atoms (de Boer, 1936; Hamaker,1937). The equation from the Hamaker theory shows the vander Waals force becomes infinite as two particles get into con-tact (h = 0), which induces a singularity problem in DEMsimulation. To solve this problem, a “cut-off” distance is as-sumed in the calculation of this force, and this distance rangesfrom 0.165 to 1 nm (Krupp, 1967; Israelachvili, 1991; Yen andChaki, 1992; Yang et al., 2000).

The contribution due to the dispersion interaction to the totalvan der Waals force decays rapidly as the separation distancebetween the two interaction bodies becomes larger than a fewnanometers, since the time taken for the electromagnetic fieldto propagate is not the same, with the period of the fluctuatingdipole itself at the separation distance larger than a few nanome-ters (Hough and White, 1980; Israelachvili, 1991). This “re-tardation effect” should be taken into account for interactionsbetween macroscopic bodies, especially in a liquid medium. Asimple and accurate expression of the correction factor for theretardation effect in practical applications is given by Zhanget al. (1999a). Real particles are not rigid and will deform elas-tically and/or plastically at the contact point, even under zeroexternal load. For plastic deformation, the van der Waals forcemust include a term for the extended contact area (Visser, 1989;Forsyth et al., 2001).

The classical Hertz contact theory is for the elastic deforma-tion of bodies in contact, but neglects the adhesion force dueto the van der Waals attraction. The JKR (Johnson et al., 1971)and DMT (Derjaguin et al., 1975) models are two commonlyused models developed to improve the Hertz model. The JKRmodel is derived based on contact mechanics and recognizesthat both tensile and compressive interactions contribute to thetotal contact radius. On the other hand, the DMT model han-dles the Hertz deformation and adhesion effect separately. Itis more straightforward to be implemented in DEM simula-tion. In particular, the van der Waals force is calculated accord-ing to the equation for smooth spheres, as listed in Table 2.The so-called Hamaker constant is the only parameter to bedetermined for given geometry and materials. Theoreticallyspeaking, this constant depends on many variables related tophysical and chemical properties such as particle surface rough-ness or asperity, medium chemistry and so on (Israelachvili,1991). It is difficult to evaluate the effects of these variables


comprehensively. In numerical modelling, this constant is of-ten treated as a lumped parameter and determined empirically(see, for example, Yang et al., 2000; Dong et al., 2006).

The electrostatic force exists between charged particles. Theelectrostatic forces can be catagorized into three types: Coulom-bic forces, image-charge forces and space charge forces. Krupp(1967) has given a detailed discussion of various situations.The electrostatic forces between particles are usually approxi-mated by the classical Coulomb equation. Rumpf (1962) com-pared the van der Waals attractive force with the electrostaticforces produced by contact potential or by excess charges. Theresults showed that the van der Waals force is greater by oneorder of magnitude than the electrostatic force for micro-sizedparticle. In some DEM simulations, instead of determining ex-plicitly the non-contact forces such as the van der Waals andelectrostatic forces, the concept of surface energy is incorpo-rated in the simulation (Subero et al., 1999; Antony, 2000;Moreno et al., 2003).

The capillary force is mainly due to the surface tension atsolid/liquid/gas interfaces. For the cases shown in Fig. 2, theforce due to the reduced hydrostatic pressure in the bridge itselfcan be given by (Fisher, 1926; Princen, 1968)

F l = 2��R sin � sin(� + �) + �R2�psin2�, (3)

where � is the liquid surface tension, � is the half-filling an-gle, and � is the contact angle. The reduced hydrostatic pres-sure within the bridge, �p, can be given by the Laplace–Youngequation, which states that the mean curvature of the meniscusprofile is constant and proportional to �p. A number of ana-lytical (Orr et al., 1975; Gao, 1997) and numerical (Erle et al.,1971; De Bisschop and Rigole, 1982; Lian et al., 1993) solu-tions have been proposed. Comparisons between these solutionsand the comprehensive set of experimental data (Mason andClark, 1965) have shown there is a good agreement (Mazzoneet al., 1987).

Fisher (1926) used a toroidal approximation for the shape ofthe liquid bridge. Two different methods have been adopted:the neck or gorge method which estimates the force at the neckof the bridge (Hotta et al., 1974), and the contact method whichconsiders the force at the liquid bridge solid contact region(Adams and Perchard, 1985). Both methods show reasonableaccuracy theoretically (Orr et al., 1975), experimentally (Crossand Picknett, 1963; Mason and Clark, 1965) and numerically(Lian et al., 1993). However, they may underestimate the forceas the separation distance increases (Mazzone et al., 1987).Lian et al. (1993) presented an approach to improve the “gorge”method by introducing simple scaling coefficients.

To provide the capillary force as an explicit function of liq-uid bridge volume and the separation distance which is easyto implement into DEM simulations, some approximate solu-tion procedures or approximate descriptions of the exact so-lutions have been proposed (Simons et al., 1994; Weigert andRipperger, 1999; Willett et al., 2000). The error of these approx-imations is around 4% when the liquid-to-solid volume ratio is0.1% but increases with increasing the volume ratio. The accu-racy can be greatly improved with a considerably more com-plex expression which is valid for volume ratio less than 10%

and gives an error in the force estimate of less than 3% (Willettet al., 2000). To calculate the capillary force between spheresof an unequal size, the Derjaguin approximation (Israelachvili,1991) is a relatively accurate method for small bridge volumesand for separation distances excluding those at close-contactand near-rupture (Willett et al., 2000).

To implement the capillary force in DEM simulation, liquiddistribution among particles has to be determined. Mugurumaet al. (2000) assumed that liquid can transport among particlesand is distributed evenly among all gaps smaller than the rupturedistance. On the other hand, Mikami et al. (1998) assumed thatliquid is distributed evenly among particles and liquid transportbetween particles can be neglected if liquid viscosity is suffi-ciently small. By combining them together, Yang et al. (2003)assumed liquid being distributed evenly and not transferableamong particles. Once the particle gap is smaller than the rup-ture distance, a liquid bridge is formed and the liquid assignedto a particle will be evenly distributed to its liquid bridges.

2.3. Particle–fluid interaction forces

The surrounding fluid will interact with particles, generat-ing various particle–fluid interaction forces, in addition to thebuoyancy force. For example, the movement of particles is al-ways resisted by stagnant fluid. The particle–fluid interactionforce, mainly the drag force, is the driving force for fluidiza-tion. Therefore, particle–fluid interaction forces must be prop-erly considered. To date, a number of such forces have beenimplemented in DEM simulation, including particle–fluid dragforce, pressure gradient force, and other unsteady forces suchas virtual mass force, Basset force, and lift forces (for example,Li et al., 1999; Xiong et al., 2005; Potic et al., 2005).

For an isolated particle in a fluid, the equation to determinethe drag resistance force is well established, described by New-ton’s equation (Table 3). The particle–fluid drag coefficient, Cd ,is dependent upon Reynold’s number, Re, in addition to liquidproperties. There are three regions: the Stoke’s Law region, thetransition region, and Newton’s law region. For each region,the drag coefficient is determined by well-established correla-tions. But for a particulate system, the problem becomes muchmore complicated. The presence of other particles reduces thespace for fluid, generates a sharp fluid velocity gradient and,as a result, yields an increased shear stress on particle surface.The enhancement of the drag is closely associated with particleconfiguration, particle–fluid slip velocity and the properties ofboth particle and fluid.

In general, two methods have been used to determineparticle–fluid drag force. The first one is based on empiri-cal correlations for either bed pressure drop (for example,Ergun, 1952; Wen and Yu, 1966) or bed expansion experiment(Richardson, 1971). The effect of the presence of other par-ticles is considered in terms of local porosity, involving theexponent (see Table 3) and related to the flow regimes or parti-cle Reynolds number (see, for example, Di Felice, 1994). Thevalue of the exponent varies in a rather large range (from −3to 10) (Morgan et al., 1970) and its accurate quantification isimportant. The other method is based on numerical simulations


Table 3Particle–fluid interaction forces

Forces Correlations References

Drag force For an isolated particle moving through a gas,Fd = Cd�f d2

p|u − v|(u − v)/8Effect of surrounding particles is described by avoidage function, f (�f ):Fd = f (�f )Cd�f d2

p|u − v|(u − v)/8Cd = 24(1 + 0.15Re0.687

p )/Rep(Rep < 1000)

Cd = 0.44(Rep > 1000) Rep = f dp�f |u − v|/�f

Fd = pf (u − v)/f Ergun (1952), and

pf = 150(1−�f )2

�f

�f

(�pdp)2 + 1.75(1 − �f )f

�pdp|u − v|(�f �0.8) Wen and Yu (1966)

pf = 34 Cd

|u−v|f (1−�f )

dp�−2.7f (�f > 0.8)

f (�f ) = �−(�+1)f Di Felice (1994)

� = 3.7 − 0.65 exp[−(1.5 − log Rep)2/2]F = F0(�) + F1(�)Re2

p (Rep < 20) Koch and Sangani (1999), andF = F0(�) + F3(�)Re2

p (Rep > 20) Koch and Hill (2001)

F0(�) = 1+3(�/2)1/2+(135/64)� ln �+16.14�1+0.681�−8.48�2+8.16�3 (�< 0.4)

F0(�) = 10�/(1 − �)3 (�> 0.4)

F1(�) = 0.110 + 5.10 × 10−4e11.6�

F3(�) = 0.0673 + 0.212� + 0.0232(1 − �)5

Pressure Fp = −Vp dp/dx = −Vp(f g + f u du/dx) Anderson and Jackson (1967)gradient It is of general validity and all relevant contributions are included whenforce dp/dx is evaluated from the fluid equation of motion.

Virtual FV m = Cvmf Vp(u − v)/2 Odar and Hamilton (1964), and Odar (1966)mass CV m = 2.1 − 0.132/(0.12 + A2

c )

force Ac = (u − v)2/(dpd(u − v)/dt)

Basset force FBasset = 32 d2

p

√�f �f

[∫ t

0(u−v)√

t−t ′ dt ′ + (u−v)0)√t

]Reeks and Mckee (1984), and

where (u − v)0 is the initial velocity difference Mei et al. (1991)Saffman force FSaff = 1.61d2

p(�f f )1/2|�c|−1/2[(u − v) × �c] Saffman (1965, 1968)�c = ∇ × u

Magnus force FMag = �8 d2

pf

[(12 ∇ × u − �d

)× (u − v)

]Rubinow and Keller (1961)

where 12 ∇×u is the local fluid rotation and �d is the particle rotation.

One notes that the lift would be zero if the particle rotation is equal tothe location rotation of the fluid

at a microscale, where the techniques used include the directnumerical simulation (DNS) (Choi and Joseph, 2001) andLattice–Boltzmann (LB) computation (Zhang et al., 1999b).Although rational, but limited by the current computationalcapability, to date the numerical studies have been appliedonly to relatively simple systems. The commonly used correla-tions for determining the fluid drag force are listed in Table 3.Li and Kuipers (2003) did a systematic study to quantify thedifference among these correlations. Their results indicatethat these correlations possess similar predictive capability,although their accuracy may differ.

Other particle–fluid interaction forces have also been consid-ered, particularly when the fluid involved is liquid rather thangas. These include the pressure gradient force, unsteady forceand lift forces (Crowe et al., 1998). In general, the pressure gra-dient force includes not only the buoyancy force due to gravitybut also the acceleration pressure gradient in fluid. There are afew forces in this connection, including the virtual mass force

and the Basset force. The virtual mass force relates to the forcerequired to accelerate the surrounding fluid, and is also calledthe apparent mass force because it is equivalent to adding amass to a particle. The Basset force describes the force due tothe lagging boundary layer development with changing relativevelocity. It accounts for the viscous effects. This term addressesthe temporal delay in boundary layer development as the rel-ative velocity changes with time, and is sometimes called the“history” term. According to Hjelmfel and Mockros (1966), theBasset term and virtual mass term become insignificant undersome conditions, e.g., for small density ratio (f /s ∼ 10−3).The lift forces, including Saffman lift force and Magnus liftforce, on a particle are due to the rotation of the particle. TheSaffman lift force is caused by the pressure distribution inducedby the resultant velocity gradient. On the other hand, the Mag-nus force is developed by a pressure differential on the surfaceof the particle resulting from the velocity differential due torotation. Equations to calculate these forces are listed in Table 3.


Their implementation in DEM simulation will depend on theparticulate system considered but is rather straightforward.

3. Particle–fluid flow

Particle flow is often coupled with fluid (gas and/or liquid)flow. In fact, coupled particle–fluid flow can be observed inalmost all types of particulate processes. Understanding the fun-damentals governing the flow and formulating suitable govern-ing equations and constitutive relationships are of paramountimportance to the formulation of strategies for process devel-opment and control. This necessitates a multiscale approach tounderstand the phenomena at different length and time scales(see, for example, Villermaux, 1996; Li, 2000; Kuipers, 2000;Li and Kwauk, 2003; Bi and Li, 2004). In the past, many stud-ies have been done at either atomic/molecular scales relatedto the thermodynamics and kinetics or large scales related tothe macroscopic performance of an operational unit or plant.However, what is missing is the quantitative understanding ofmicroscale phenomena related to the behaviour of particles,droplets and bubbles. Without this information, it is difficultto generate a general method for reliable scale-up, design andcontrol of particulate processes of different types. Therefore,particles scale modelling of particle–fluid flow has been a re-search focus in the past decade. DEM plays an important rolein this development (Yu and Xu, 2003; Yu, 2004).

3.1. Numerical methods

In principle, for any particle–fluid flow system, the solu-tion of Newton’s equations of motion for discrete particles andthe Navier–Stokes equations for continuum fluid together withboundary and initial conditions will finally determine the solidsand fluid mechanics. In practice, however, there are usually alarge number of particles. Consequently, this requires a verylarge number of governing equations to be solved for the mo-tion of each of the particles and the resolution of the fluid fieldhas to be fine enough to resolve the flow of continuum fluidthrough the pores among closely spaced particles. As a result,depending on the time and length scales of interest, simplifi-cations have to be made when this theoretical approach is fol-lowed.

The existing approaches to modelling particle flow can beclassified into two categories: the continuum approach at amacroscopic level and the discrete approach at a microscopiclevel. In the continuum approach, the macroscopic behaviouris described by balance equations, e.g., mass and momentumas used in the two fluid model (TFM), closed with constitutiverelations together with initial and boundary conditions (see,for example, Gidaspow, 1994). This approach is preferred inprocess modelling and applied research because of its com-putational convenience. However, its effective use heavily de-pends on constitutive or closure relations and the momentumexchange between particles of different type. In the past, dif-ferent theories have been devised for different materials and fordifferent flow regimes. However, to date, there is no acceptedcontinuum theory applicable to all flow conditions. As a re-

sult, phenomenological assumptions have to be made to obtainthe constitutive relations and boundary conditions, which havevery limited application (see, for example, Zhang et al., 1998).The discrete approach is based on the analysis of the motionof individual particles and has the advantage that there is noneed for global assumptions on the solids such as steady-statebehaviour, uniform constituency, and/or constitutive relations.Various methods have been developed in the past. A major typeof discrete approach is based on DEM as discussed above. Themethod considers a finite number of discrete particles inter-acting by means of contact and non-contact forces, and everyparticle in a considered system is described by Newton’s equa-tions of motion. In principle, it is similar to molecular dynamicsimulation (MDS) but the forces involved differ because of thedifference in time and length scales.

The time and length scales for fluid flow can also rangefrom discrete (e.g., MDS, LB, pseudo-particle method (PPM))to continuum DNS, large eddy simulation (LES), and otherconventional CFD techniques including TFM description). Inprinciple, they all can be combined with DEM to describe thecoupled particle–fluid flow. Indeed, many of them have beentried, including, for example, LB-DEM (Cook et al., 2004),PPM-DEM ( Ge and Li, 2001, 2003a,b), DNS-DEM (Hu, 1996;Panet al., 2002), LES-DEM (Zhou et al., 2004a–c), in addition tothe CFD-DEM model which, sometimes referred to as the com-bined discrete and continuum model (CCDM), is most popu-larly used as discussed below. The so-called smoothed particlehydrodynamics (SPH) (Potapov et al., 2001) and front track-ing (FT) (Annaland et al., 2005, 2006) were also attempted.Instead of DEM, other discrete particle models were also at-tempted, such as direct simulation Monte Carlo (DSMC) (Yuuet al., 1997; Tsuji et al., 1998). Smoothed particle (SP) methodwas also tried to replace the TFM to eliminate spatial mesh incomputation (Yuu et al., 2000b; Sugino and Yuu, 2002).

Table 4 lists a few representative combinations of differentlength scales for fluid and particle phases and their relativemerits in different aspects, where for convenience, relative toparticle phase, they can also be categorized into three groups:sub-particle, pseudo-particle and computational cell (Yu, 2005).The advantage/disadvantages of the numerical methods pro-posed can be further understood with reference to the three pop-ular models: TFM, DNS-DEM, and CFD-DEM, as discussed byYu and Xu (2003).

In TFM, both gas and solid phases are treated as interpene-trating continuum media in a computational cell which is muchlarger than individual particles but still small compared with thesize of process equipment so that the number of governing equa-tions is greatly reduced (Anderson and Jackson, 1967). Sincethe first numerical simulation of realistic bubbles by Pritchettet al. (1978), TFM has dominated fluidization modelling fordecades, as summarized by Gidaspow (1994) and others, for ex-ample, Kuipers and van Swaaij (1997) and Arastoopour (2001).However, its effective use heavily depends on constitutive orclosure relations for the solid phase and the momentum ex-change between phases which are not possible to obtain withinits framework; this is particularly true when dealing with differ-


Table 4Typical models for particle–fluid flow and their relative merits (Yu, 2005)

Model type Length scale for fluidphase

Sub-particle (discreteor continuum)

Pseudo-particle (discrete) Computational cell (contin-uum)

Computational cell (contin-uum)

Length scale for par-ticle phase

Particle (discrete) Particle (discrete) Particle (discrete) Computational cell (contin-uum)

Nature of coupling Discrete + discrete orcontinuum + discrete

Discrete + discrete Continuum + discrete Continuum + continuum

Example LB-DEM or DNS-DEM

PPM-DEM CFD-DEM TFM

Closure of equations Yes (but may ex-perience numericaldifficulty for sys-tems with strongparticle–particle inter-actions)

No (difficulty to determinephysical properties of apseudo-particle)

Yes No (constitutive relation forsolid phase and phase interac-tions not generally available)

Incorporation of distributioneffects of dispersed, solidphase

Yes Yes Yes No

Computational effort Extremely demanding Very demanding Demanding Acceptable

Suitability for engineering ap-plication in relation to processmodelling and control

Extremely difficult Very difficult Difficult Easy

Suitability for fundamentalresearch in relation to particlephysics

Most acceptable(particle–fluid inter-action forces can bedetermined and usedfor CCDM)

Acceptable (but only valid forwell-defined PPM system)

Acceptable No

ent types of particles that should be treated as different phases.In fact, development of a general theory to describe granularflow is a research area challenging the whole scientific commu-nity (see, for example, Jaeger et al., 1996; de Gennes, 1999).

In DNS-DEM, the fluid field is resolved at a scale compa-rable with the particle spacing while particles are treated asdiscrete moving boundaries (Hu, 1996). One of the key fea-tures of the method is that the particle–fluid system is treatedimplicitly by using a combined weak formulation. DNS hasgreat potential to produce detailed results of hydrodynamic in-teractions between fluid and particles in a system (Pan et al.,2002). However, one major weakness of this model is its capac-ity in handling particle collisions. In the earlier development,the particle collisions were not modelled at all; if the gap be-tween two approaching particles was less than a preset smallvalue, the simulation had to stop (Pan et al., 2002). In a re-cent development, a repulsive body force was introduced intothe momentum equation to prevent possible collisions betweenparticles (Glowinski et al., 2000; Singh et al., 2000). There-fore, so far, DNS or DNS-based models have mainly been ap-plied to particle–liquid systems where the hydrodynamic in-teraction is dominant and particle–particle interaction is non-violent. This limits its applicability to gas fluidization whereparticle collisions and interparticle forces are significant. Fromthis point of view, LB-DEM is probably a more promisingapproach.

In CFD-DEM coupling approach, the motion of individualparticles is obtained by solving Newton’s equations of motionwhile the flow of continuum gas is determined by the CFD on

a computational cell scale ( Tsuji et al., 1993; Hoomans et al.,1996; Xu and Yu, 1997, 1998). While the governing equationsfor fluid phase are the same as those in TFM—a treatise that hasbeen widely accepted in engineering application, its governingequations for solid flow are mainly based on DEM. Therefore,if focused on the solid phase, CFD-DEM has a feature similarto the so-called molecular dynamic simulation (MDS) (Allenand Tildesley, 1987). The main difference between DEM andMDS is that the forces involved differ because of the differentlength scales concerned. It is also due to the implementation ofinterparticle forces that distinguishes the CFD-DEM from theprevious CFD modelling of gas–solid flow where the interac-tion among particles is often ignored (see, for example, Croweet al., 1977; Tanaka et al., 1996).

As pointed out by Yu and Xu (2003), at this stage of develop-ment, the difficulty in particle–fluid flow modelling is mainlyrelated to solid phase rather than fluid phase. Therefore, theCFD-DEM is attractive because of its superior computationalconvenience as compared to DNS- or LB-DEM and capabilityto capture the particle physics as compared to TFM. Therefore,the present review of particle–fluid flow is mainly along withthe CFD-DEM work.

3.2. Governing equations and coupling schemes

The CFD-DEM approach was firstly proposed by Tsuji et al.(1992, 1993), and then followed by many others. The approachwas rationalized by Xu and Yu (1997, 1998). As mentionedabove, by this approach, the motion of discrete particles is


described by DEM on the basis of Newton’s laws of motionapplied to individual particles and the flow of continuumfluid by the traditional CFD based on the local averagedNavier–Stokes equations. Therefore, for particle phase, thegoverning equations are the same as Eqs. (1) and (2). Forfluid phase, the governing equations comply with the law ofconservation of mass and momentum in terms of local-averagevariables (Anderson and Jackson, 1967) and are the same asthose used in TFM. Two formulations, referred to as Model Aand Model B, have been proposed (Gidaspow, 1994). ModelA assumes that pressure drop shares between the gas and solidphases, and Model B in the gas phase only. Both formulationshave been used in CFD-DEM, given by

for Model A:

��f�t

+ ∇ · (�f uf ) = 0, (4)

�(f �f uf )

�t+ ∇ · (f �f uf uf ) = − �f ∇p − FA + ∇ · (�f �)

+ f �f g (5)

and for Model B:

��f�t

+ ∇ · (�f uf ) = 0, (6)

�(f �f uf )

�t+ ∇ · (f �f uf uf ) = − ∇p − FB + ∇ · (�f �)

+ f �f g, (7)

where u and p are, respectively, the fluid velocity and pressure;�, �f and �V are the fluid viscous stress tensor, porosity andvolume of a computational cell, respectively. FA and FB are thevolumetric particle–fluid interaction forces for the two models.Note that the definition of the total particle–fluid interactionforce can vary depending on how one interprets Eqs. (5) and(7). As the case in TFM, there is a link between FA and FB ,given by FB = FA/�f − f �sg (�s = 1 − �f ) (Feng and Yu,2004a).

For monosized particles, there is little difference betweenModels A and B in the CFD-DEM simulations (Feng and Yu,2004b; Kafui et al., 2004), which is consistent with the TFMsimulation (Bouillard et al., 1989). However, Feng and Yu(2004a) recently demonstrated that for the fluidization of bi-nary mixtures of particles, there is a significant difference be-tween the Model A and Model B simulations. Comparison withthe physical experiments conducted under comparable condi-tions, and the analysis of the numerical scheme to implementa CCDM simulation suggest the Model B formulation is morefavoured. Further studies may be necessary in order to clarifyfully this important issue.

The modelling of the solid flow by DEM is at the individualparticle level, whilst the fluid flow by CFD is at the compu-tational cell level. As shown in Fig. 5, their coupling is nu-merically achieved as follows (Xu and Yu, 1997; Xu et al.,2001). At each time step, DEM will give information, suchas the positions and velocities of individual particles, for the

Discrete Model vi, (x, y, z)i, �i

ui,(u − v)i, Rep,i Continuum Model

fpf, ifpf, i

Fpf

kc

i=1∑

=�V

Fig. 5. Coupling and information exchange between continuum (CFD) anddiscrete (DEM) models (Xu et al., 2001).

evaluation of porosity and volumetric fluid drag force in a com-putational cell. CFD will then use these data to determine thefluid flow field which then yields the fluid drag forces acting onindividual particles. Incorporation of the resulting forces intoDEM will produce information about the motion of individualparticles for the next time step.

In theory, this coupling method is used by most of the in-vestigators. However, examination of the previous implemen-tations by different investigators showed that different schemeshave been employed to couple the two phases modelled at dif-ferent length scales. According to Feng and Yu (2004a), thereare three schemes in the previous CFD-DEM simulation ofgas–solid flow in fluidization:

Scheme 1: The force from the particles to the gas phaseis calculated by a local-average method as used in the TFM,whereas the force from the gas phase to each particle is calcu-lated separately according to individual-particle velocity (Tsujiet al., 1993; Hoomans et al., 1996; Kawaguchi et al., 1998,2000a,b; Ouyang and Li, 1999a,b; Yuu et al., 2000a,b; Binet al., 2003; Limtrakul et al., 2003).

Scheme 2: The force from the particles to the gas phase iscalculated first at a local-average scale as used in scheme 1.This value is then distributed to individual particles accordingto a certain average rule (Mikami et al., 1998; Rong et al.,1999; Kaneko et al., 1999; Kuwagi et al., 2000; Rhodes et al.,2001a–c; Han et al., 2003).

Scheme 3: At each time step, the particle fluid interactionforces on individual particles in a computational cell are cal-culated first, and the values are then summed to produce theparticle–fluid interaction force at the cell scale (Xu and Yu,1997, 1998; Hoomans et al., 2000; Helland et al., 2000; Xuet al., 2000, 2001; van Wachem et al., 2001; Kafui et al., 2002;Li and Kuipers, 2002, 2003; Zhou et al., 2002a,b; van Wachemand Almstedt, 2003; Feng et al., 2003, 2004; Bokkers et al.,2004; Li et al., 2004).

According to the Newton’s third law of motion, the forceof the solid phase acting on the gas phase should be equal tothe force of the gas phase acting on the solid phase but in theopposite direction. Scheme 1 does not guarantee that this con-dition can always be satisfied. Consequently, it is not reason-able. Indeed, this scheme was used only in the early stage of


CFD-DEM development, although it can still be found occa-sionally. Scheme 2 can satisfy Newton’s third law. However,it uniformly distributes the interaction force among the par-ticles in a computational cell irrespective of the differentbehaviours of these particles in the cell. This scheme can-not fully represent reality, as the particle–fluid interactionforces for the particles in the cell should differ for non-uniform particle–fluid flow. In addition, in the calculation ofthe particle–fluid interaction force, a mean particle velocityhas to be used. The appropriate method for calculating thismean particle velocity is still an open question, particularlyfor multisized particle systems. Scheme 3 can overcome theabove problems associated with schemes 1 and 2. Indeed, thisscheme has been widely accepted since its first introduction byXu and Yu (1997, 1998).

4. Transition from discrete to continuum

By use of a proper averaging procedure, a discrete particlesystem can be transferred into a corresponding continuum sys-tem. Extensive research has been carried out to develop suchaveraging methods. Various methods have been proposed toderive the balance equations of the continuum system. Earlierwork often ignored the effect of the rotational motion of parti-cles. Thus, the resultant balance equations are only for mass andlinear momentum. These equations are the same as those in the

Table 5Equations for calculation of stress and couple stress

Methods Equations Availability References

Volume average

Method 1 T = 1

V

∑i

∑j>i

(ui − uj )xij ⊗ fij Valid for quasi-static systems, Drescher and de Josselin de Jong (1972)not sure if satisfying governingequations and valid for other systems

Method 2 T = 1

V

∑i

∑j>i

(uirij − uj rji ) ⊗ fij Rothenburg and Selvadurai (1981),Christoffersen et al. (1981),and Kanatani (1981)

Method 3 T = 1

V

∑p∈V

pV

Cp∑c=1

fc ⊗ lpc Lätzel et al. (2001),and Luding et al. (2001)

M = 1

V

∑p∈V

pV

Cp∑c=1

(lpc × fc) ⊗ lpc

Time–volume average

Method 4 T = 1

V T

∫ T

0

(−∑

i

miv′i ⊗ v′

i +∑i

∑j>i

Iij ⊗ fij

)ds Valid for rapid flows, Walton and Braun (1986a,b)

not sure if satisfying governingequations and valid for other systems

Method 5 T = 1

V T

∫ T

0

(−∑

i

uimiv′i ⊗ v′

i +∑i

∑j>i

uij Iij ⊗ fij

)ds Zhang and Campbell (1992)

M = 1

V T

∫ T

0

(−∑

i

uiIiv′i ⊗ �′

i +∑i

∑j>i

uij Iij ⊗ mij

)ds

classical continuum mechanics, no matter which method is usedto derive them. However, recent studies have illustrated that thegradient of particle rotation in some cases such as shear bandis very high (Oda and Iwashita, 1999). Therefore, additionalquantities should be included in the continuum formulation todescribe this gradient. Based on such consideration, an extraequation has been derived to describe the rotation of particles.Therefore, complete balance equations for a continuum systeminclude those of mass, linear momentum and angular momen-tum. According to Zhu and Yu (2002), these equations can be,respectively, given by

D() + ∇ · u = 0, (8)

D(u) + u∇ · u = ∇ · T + g, (9)

D(��) + ��∇ · u = ∇ · M + M′. (10)

The main macroscopic quantities involved in these balanceequations include mass density , velocity u, angular velocity�, stress tensor T and couple stress tensor M.

These macroscopic variables can be linked to the microscopicvariables in the discrete approaches by means of local averag-ing. In the past, various averaging methods have been proposedto link the macroscopic variables to the microscopic variablesbased on different theoretical considerations. They can be clas-sified into: volume, time–volume and weighted time–volumeaveraging methods. For the volume method, there are two ways


Table 6Equations linking microscopic to macroscopic variables (Zhu and Yu, 2002)

Macroscopic variablesaEquations linking microscopic to macroscopic variables

Mass density =

∫Tt

∑i

himi ds

Velocityu = 1

∫Tt

∑i

himivi ds

Angular velocity� = 1

�

∫Tt

∑i

hiIi�i ds

StressT = −

∫Tt

∑i

himiv′i ⊗ v′

i ds +∫

Tt

∑i

∑j>i

gij dij ⊗ fij ds +∫

Tt

∑i

gbi db

i ⊗ fbi ds

Couple stressM = −

∫Tt

∑i

hiIiv′i ⊗ �i ds + 1

2

∫Tt

∑i

∑j>i

gij dij ⊗ (mij − mji ) ds +∫

Tt

∑i

gbi db

i ⊗ mbi ds

Rate of supply of M′ = 1

2

∫Tt

∑i

∑j>i

(mij + mji )(hi + hj ) ds

internal spin

where hi = h(ri − r, s − t), � = ∫Tt

∑ihiIi ds, v′

i = vi − u, gij = ∫ 10 h(ri + rdij − r, s − t) dr , gb

i = ∫ 10 g(ri + rdb

i − r, s − t) dr .

aThese macroscopic quantities depend on the weighting function h(r, s). A weighting function has been recommended (Zhu and Yu, 2002, 2005b)

h(r, t) ={

12�

√2�

c0Lt

(L2t −t2)Lp(L2

p−r2)exp

[− 1

2

(ln2 Lt +t

Lt −t+ ln2 Lp+r

Lp−r

)], (r, t) ∈ �

0, others

where c0 is the normalized constant of weighting function, Lt , Lp are parameters determining domain �.

to define the stress tensor of an assembly of particles. First,the average stress is expressed in terms of the external forcesacting at the boundary points of the assembly (Drescher andde Josselin de Jong, 1972). Secondly, the stress tensor isexpressed in terms of individual contact forces within theassembly of particles (Rothenburg and Selvadurai, 1981;Christoffersen et al., 1981; Kanatani, 1981). More com-plex expressions of stress and the micro-mechanical def-inition of couple stress have also been considered insome recent studies (Oda, 1999; Lätzel et al., 2000, 2001;Tordesillas and Walsh, 2002). For example, the models ofLätzel et al. (2000, 2001) and Luding et al. (2001) includedmore macroscopic characteristics such as volume fraction andfabric tensor. The volume averaging approaches is mainlydeveloped to describe quasi-static systems. Therefore, theinertial effect is ignored in these approaches. Walton andBraun (1986a,b) proposed an alternative method based ontime–volume averaging in their study of a rapid granularflow, which was later improved by Zhang and Campbell(1992) and Campbell (1993a,b) to include a couple stressterm. The time–volume averaging has been demonstrated to beapplicable to simple shear rapid flows, but its applicability toother granular flows has been questionable (Babic, 1997). It isnot clear if the macroscopic properties obtained by use of thevolume and time–volume averaging conform with those in the

balance equations (8)–(10) in the continuum approach (Ludinget al., 2001). Some commonly used equations for the twoapproaches are shown in Table 5 (for brevity, only stress andcouple stress are included).

Babic (1997) first proposed a weighted time–volume aver-aging method, which is able to overcome the above problems.This method has been further developed by Zhu and Yu (2001,2002) to be suitable for the entire considered domain as shownin Table 6. The concept is shown in Fig. 6. The averagingmethod has been extended to study the constitutive behaviour ofgranular materials (Glasser and Goldhirsch, 2001; Goldhirschand Goldenberg, 2002) and the macro-dynamical behaviour ofgranular flows (Zhu and Yu, 2003, 2005a,b). The developedapproach is suitable for all flow regimes. The selection of thesample size, where the averaging is conducted, is the mainproblem in the application of this averaging method. To date,this problem has not been properly solved.

Nevertheless, the combined approach of discrete approachand averaging method takes into account the discrete nature ofgranular materials and does not require any global assumptionand thus allows a better understanding of the fundamentalmechanisms of granular flow. It has been extensively usedto investigate the dynamics of granular flows. These stud-ies show that it can be used to analyze the macro-dynamicalbehaviour of granular flows under different operational


conditions and micro-properties of granular material (Langstonet al., 1995a; Potapov and Campbell, 1996, Zhu and Yu, 2003,2005b; Heyes et al., 2004), to depict the intrinsic character-istics of granular materials such as the constitutive relation-ship under various flow conditions (Oda and Iwashita, 2000;Alonso-Marroquin and Herrmann, 2005), and to test the contin-uum theories (Makse and Kurchan, 2002). The approach offersa convenient way to link fundamental understanding generatedfrom DEM-based simulations to engineering application oftenachieved by continuum modelling.

0Lp

r

g(r)

Probe point

4πLp3c(σp)

Fig. 6. Concept of weighting for space averaging (the same idea applied totime averaging) (Zhu and Yu, 2002).

-12

-9

-6

-3

0

3

-15 -12 -9 -6 -3 0 3 6 9 12

log

(Len

gth

Scale

) (m

)

log(Time Scale) (s)

Fig. 7. Schematic illustration of the approaches at different time and length scales, their research needs (aims) and links (represented by arrows).

5. Concluding remarks

The multiscale phenomena associated with particulate mat-ter poses a need for multiscale modelling and analysis. Fig.7 schematically shows the approaches at different time andlength scales. Since the bulk behaviour of particulate mat-ter depends on the collective interactions among individualparticles, it is the particle scale modelling and analysis thatplays a crucial role in elucidating the underlying fundamen-tals and linking fundamental to applied research. Indeed, thishas been the major research effort in particulate research inthe past years, as seen from Fig. 1. The extensive investiga-tions under different packing/flow conditions at either macro-or microscopic level by various investigators worldwide, asreviewed by Zhu et al. (2006), clearly indicate that DEM-based discrete particle simulation is an effective way to achievethis goal.

We have seen the rapid development in establishing the the-ories and models for this approach since the pioneer work ofCundall and Strack (1979). In this review, we show this devel-opment by highlighting three important aspects mainly relatedto the modelling of particle or particle–fluid flow. First, the the-ories underlying the DEM and models for calculating the con-tact forces between particles are more soundable, although notperfect yet. Secondly, more forces have been implemented inDEM simulation, which makes the DEM model more applica-ble to particulate research. The most important development inthis direction is the so-called CFD-DEM or CCDM approachwhich can handle the particle–fluid flow widely encounteredin process engineering. Thirdly, the theory to link the discretesimulation to continuum modelling is gradually established,generating a new but natural analytical method to depict thefundamentals governing the behaviour of particulate/granularmatter. This is an important step for continuum-based processmodelling.

While DEM or CFD-DEM, as a state-of-the-art simulationtechnique, is increasingly used to study the physics of particu-late and/or multiphase flow related to various industrial prob-lems, as pointed out by Yu (2004), future effort will be madeto meet the following research needs (also refer to Fig. 1).


• Microscale: To develop a more comprehensive theory and ex-perimental techniques to study and quantify the interactionforces between particles, and between particle and fluid un-der various conditions, generating a more concrete basis forparticle scale simulation.

• Macroscale: To develop a general theory to link the discreteand continuum approaches, so that particle scale informa-tion, generated from DEM or DEM-based simulation, canbe quantified in terms of (macroscopic) governing equations,constitutive relations and boundary conditions that can beimplemented in continuum-based process modelling.

• Application: To develop more robust models and efficientcomputer codes so that the capability of particle scale sim-ulation can be extended, say, from two-phase to multiphaseand/or from simple spherical to complicated non-sphericalparticle system, which is important to transfer the present phe-nomenon simulation to process simulation and hence meetreal engineering needs.

Notations

A Hamaker constant, Jc0 normalized constant of weighting function, dimen-

sionlessCd fluid drag coefficient on an isolated particle, dimen-

sionlessCn damping coefficient, dimensionlessCr viscosity coefficient, dimensionlessCt tangential damping coefficient, dimensionlessCV m virtual mass force coefficient, dimensionlessdp particle diameter, mdij branch vector connecting the mass centres of parti-

cles i and j, mdb

i ray from the mass centre of particle i to ��, mE∗ reduced Young’s modulus, dimensionlessfc contact force, Nfij interaction force between particle i and j, Nfn normal component of interparticle contact force, Nfn,e elastic component of normal interaction between par-

ticles, Nfpf ,i interaction force between fluid and particle i, Nft tangential component of interparticle contact force,

Nf ∗

t initially equal to 0 and set to the value of ft whenevervtc reverses its direction, N

Fcij contact force acting on particle i by particle j, N

Fncik non-contact force acting on particle i by particle k, N

Ffi particle–fluid interaction force on particle i, N

Fgi gravitational force on particle i, N

FA volumetric particle–fluid interaction force in ModelA, N m−3

FB volumetric particle–fluid interaction force in ModelB, N m−3

FBasset Basset force, NFd interaction force between isolated particle and fluid,

N

Fe electrostatic force, NFl liquid bridge force, NFMag Magnus force, NFpf volumetric particle–fluid interaction force, N m−3

Fp pressure gradient force, NFSaff Saffman force, NFv van der Waals force, NFV m virtual mass force, Ng gravitational acceleration, m s−2

h surface gap between two spheres, mI moment of inertia of particle, kg m2

lpc branch vector from the centre of particle p to itscontact c, m

lij branch vector connecting the centroids of particle iand j, m

Kn normal spring coefficient, dimensionlessk1 spring constants for loading, dimensionlessk2 spring constants for unloading, dimensionlesskc number of particles in a computational cell, dimen-

sionlessk0t initial tangential stiffness, N m−1

kr rolling stiffness, N m−1

Kt tangential spring coefficient, dimensionlessLt parameter to determine domain � in weighting func-

tion, sLp parameter to determine domain � in weighting func-

tion, mm mass of particle, kgmr torque from rolling friction, N mmt torque from tangential forces, N mM couple stress tensor, PaMij torque acting on particle i by particle j, N mnc unit normal vector at contactp pressure, Pa�p reduced hydrostatic pressure within the bridge, Paq0 permittivity of vacuum, C2J−1m−1

Q particle charge, Crij position vector of the contact point between particles

i and j, mR vector of the mass centre of the sphere to contact

plane, mR particle radius, mR∗ reduced particle radius, dimensionlessRe reynolds number, dimensionlessRep relative Reynolds number around particle, dimen-

sionlesst time, sT stress tensor, PaT duration of the time periodic cell, su velocity, m s−1

ui equals to 1 if the centroid of particle i is located withV and zero otherwise, dimensionless

V the averaging volume, m3

v translational velocity, m s−1

v′i fluctuation velocity of particle i with respect to the

averaged velocity, m s−1

vtc relative tangential displacement at contact, m


vtc unit vector of vt

c, mVp volume of particle, m3

�V volume of a computational cell, m3

x x direction in a coordinate system, mxij position vector of the contact point between particles

i and j, my y direction in a coordinate system, mz z direction in a coordinate system, m

Greek letters

� coefficient defined by Di Felice (1994), dimension-less

pf momentum exchange coefficient, kg m−3s−1

� liquid surface tension, Pa�n relative normal displacement at contact, m�max the maximum �t when the particles start to slide, m�f volumetric fraction by fluid, dimensionless�s volumetric fraction by solid, dimensionless� contact angle between two particles, dimensionless�r relative particle rotation, dimensionless� sliding friction coefficient, dimensionless�f fluid viscosity, Pa s�r rolling friction coefficient, dimensionless�′

r rotational stiffness, N m mass density for solid phase, kg m−3

f fluid density, kg m−3

s solid density, kg m−3

� stress tensor, Pa� half-filling angle as indicated inFig. 2, dimensionless� solid fraction, dimensionless�p sphericity, dimensionless� angular velocity, s−1

�n the component of the relative angular velocity in con-tact plane, s−1

�n unit vector of �n, s−1

pV weight of the particle’s contribution to the average,

dimensionless�′

i fluctuation angular velocity of particle i, s−1

� average domain, m3s

Subscripts

c contactd dampingf fluidi particle i

ij between particles i and j

j particle j

k particle k

n normal componentt tangential component

Acknowledgement

The authors are grateful to the Australian Research Councilfor the financial support of their work.

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discrete particle simulation of particulate systems: theoretical developments

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