Discrete Particle Simulation

Chemical Engineering Science 62 (2007) 3378 – 3396 www.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 2006 Available online 23 March 2007

Abstract Particle science and technology is a rapidly developing interdisciplinary research area with its core being the understanding of the relationships between 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 with surrounding fluids. Understanding the microscopic mechanisms in terms of these interaction forces is therefore key to leading to truly interdisciplinary research into particulate matter and producing results that can be generally used. This aim can be effectively achieved via particle scale research based on detailed microdynamic information such as the forces acting on and trajectories of individual particles in a considered system. In recent years, such research has been rapidly developed worldwide, mainly as a result of the rapid development of discrete particle simulation technique and computer technology. This paper reviews the work in this area with special reference to the discrete element method and associated theoretical developments. It covers three important aspects: models for the calculation of the particle–particle and 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. 䉷 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 developing interdisciplinary research area with its core being the understanding of the relationships between micro- and macroscopic properties of particulate/granular matter—a state of matter that is widely encountered but poorly understood. Previous studies are largely at a macroscopic or global scale, the resulting information being helpful in developing a broad understanding of a particulate process of particular interest. However, the lack of quantitative fundamental understanding makes it difficult to generate a general method for reliable scale-up, design and control/optimization of processes of different types. The macroscopic behaviour of particulate matter is controlled by the interactions between individual particles as well as interactions with surrounding gas or liquid and wall. Understanding 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 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2006.12.089

is therefore the key leading to truly interdisciplinary research into particulate matter and producing results that can be generally used. This aim can be effectively achieved via particle scale research based on detailed microdynamic information. In recent years, such research has been rapidly developed worldwide, mainly as a result of the rapid development of discrete particle simulation technique and computer technology. Several discrete modelling techniques have been developed, including Monte Carlo method, cellular automata and discrete element method (DEM). DEM simulations can provide dynamic information, such as the trajectories of and transient forces acting on individual particles, which is extremely difficult, if not impossible, to obtain by physical experimentation at this stage of development. Consequently, it has been increasingly used in the past two decades or so. Two types of DEMs are most common: soft-particle and hard-particle approaches. The soft-sphere method originally developed by Cundall and Strack (1979) was the first granular dynamics simulation technique published in the open literature. In such an approach, particles are permitted to suffer minute

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

available until the mid of 2006, and covers the development of simulation techniques and their application to the study of particle packing, particle flow and particle–fluid flow. This article, as part of the review effort, presents a summary of the major theoretical developments in DEM, which are for convenience grouped into three important aspects: models for the calculation of the particle–particle and particle–fluid interaction forces, coupling of DEM with CFD to describe particle–fluid flow, and technique for linking discrete to continuum modelling. Needs for future development are also discussed.

180 160 Number of publications

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140 120 100 80 60 40

2. Governing equations and force models

20 0 1990

1995 Year

2000

2005

Fig. 1. Number of publications related to discrete particle simulation in the recent 20 years, obtained from the Web of Science with the following keywords: 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 elastic, plastic and frictional forces between particles. The motion of particles is described by the well-established Newton’s laws of motion. A characteristic feature of the soft-sphere models is that they are capable of handling multiple particle contacts which are of importance when modelling quasi-static systems. By contrast, in a hard-particle simulation, a sequence of collisions is processed, one collision at a time and being instantaneous; often the forces between particles are not explicitly considered. Therefore, typically, hard-particle method is most useful in rapid granular flows. The two DEMs, particularly the soft-sphere method, have been extensively used to study various phenomena, such as particle packing, transport properties, heaping/piling process, hopper flow, mixing and granulation. DEM has been coupled with computational fluid dynamics (CFD) to describe particle–fluid flows such as fluidization and pneumatic conveying. A survey of the literature indicates that many papers relating to DEM have been published in the past two decades or so, as shown in Fig. 1. The rapid development and application of DEM can also be highlighted by the large number of papers based on DEM at major international conferences, for example, from 29 in 2001 to 80 in 2005 for the International Conference on Powders and Grains, and from 34 in 2002 to 92 in 2006 for the World Congress on Particle Technology. In spite of the large bulk volume, little effort has been made to comprehensively review and summarize the progress made in the past, except for a few relatively focused reviews including, for example, Tsuji (1996) on the work in Japan, Mishra (2003a,b) on tumbling milling processes, Yu (2004) on the work done in his laboratory, Richards et al. (2004) on environmental science, and Bertrand et al. (2005) on mixing of granular materials. To overcome this gap, we have recently reviewed the major work in this area with special reference to the soft-particle model. It is based on the publications from the Web of Science

A particle in a granular flow can have two types of motion: translational and rotational. During its movement, the particle may interact with its neighbouring particles or walls and interact with its surrounding fluid, through which the momentum and energy are exchanged. Strictly speaking, this movement is affected not only by the forces and torques originated from its immediate neighbouring particles and vicinal fluid but also the particles and fluids far away through the propagation of disturbance waves. The complexity of such a process has defied any attempt to model this problem analytically. In DEM approach, it is generally assumed that this problem can be solved by choosing a numerical time step less than a critical value so that during a single time step the disturbance cannot propagate from the particle and fluid farther than its immediate neighbouring particles and vicinal fluid (Cundall and Strack, 1979). Thus, at all times the resultant forces on a particle can be determined exclusively from its interaction with the contacting particles and vicinal fluid for a coarse particle system. For a fine particle system, non-contact forces such as the van der Waals and electrostatic forces should also be included. Based on these considerations, Newton’s second law of motion can be used to describe the motion of individual particles. The governing equations for the translational and rotational motion of particle i with mass mi and moment of inertia Ii can be written as mi

dvi c f g nc Fij + Fik + Fi + Fi , = dt j

Ii

di Mij , = dt

(1)

k

(2)

j

where vi and i are the translational and angular velocities of c and M are the contact force and particle i, respectively, Fij ij nc is the nontorque acting on particle i by particle j or walls, Fik contact force acting on particle i by particle k or other sources, f g Fi is the particle–fluid interaction force on particle i, and Fi is the gravitational force. Fig. 2 schematically shows the typical forces and torques involved in a DEM simulation. Various models have been proposed to calculate these forces and torques, which will be discussed below. Once the forces and torques are known, Eqs. (1) and (2) can be readily solved numerically. Thus, the trajectories, velocities and the transient forces of all particles in a system considered can be determined.

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vj ωj

t

f ij

j

Fijc Fiknc

β m ijr

ωi

k

i vi

f ijn

φ

h

mi g

Fig. 2. Schematic illustration of the forces acting on particle i from contacting particle 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 single point but on a finite area due to the deformation of the particles, which is equivalent to the contact of two rigid bodies allowed to overlap slightly in the DEM. The contact traction distribution over this area can be decomposed into a component in the contact plane (or tangential plane) and one normal to the plane, thus a contact force has two components: normal and tangential. It is very difficult to accurately and generally describe the contact traction distribution over this area and then the total force and torque acting on a particle, as it is related to many geometrical and physical factors such as the shape, material properties and movement state of particles. Alternatively, to be computationally efficient and hence applicable to multiparticle systems, the DEM generally adopts simplified models or equations to determine the forces and torques resulting from the contact between particles. Various approaches have been proposed for this purpose. Generally, linear models are the most intuitive and simple models. The most common linear model is the so-called linear spring–dashpot model proposed by Cundall and Strack (1979), where the spring is used for the elastic deformation while the dashpot accounts for the viscous dissipation. The linear spring model without inclusion of dashpot has also been used by, for example, Di Renzo and Di Maio (2004). More complex and theoretically sound model, Hertz–Mindlin and Deresiewicz model, has also been developed. Hertz (1882) proposed a theory to describe the elastic contact between two spheres in the normal direction. He considered that the relationship between the normal force and normal displacement was nonlinear. Mindlin and Deresiewicz (1953) proposed a general tangential force model. They demonstrated that the force–displacement relationship depends on the whole loading history and instantaneous rate of change of the normal and tangential force or displacement. A complete 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 model is time-consuming for DEM simulations of granular flows often involving a large number of particles, and is therefore not so popular in the application of DEM. Various simplified models based on the Hertz, and Mindlin and Deresiewicz theories have been developed for DEM modelling. For example, Walton and Braun (1986a) and Walton (1993) used a semi-latched spring force–displacement model in the normal direction, and an approximation of the Mindlin and Deresiewicz contact theory for the cases of constant normal force in the tangential direction. Thornton and Yin (1991) proposed a more complex model to simulate the tangential force. While adopting the Hertz theory for the normal force, different from Walton and Braun’s model, their model assumes that the incremental tangential force due to the incremental tangential displacement depends on the variation of the normal force. Both Walton and Braun’s model and Thornton and Yin’s model for tangential force are the direct simplifications of the Mindlin and Deresiewicz theory. A more intuitive model was adopted by Langston et al. (1994). They used a direct force–displacement relation for the tangential force and the Hertz theory for the normal force. Due to its simplicity and intuitiveness, the model has been extensively used to study the dynamic behaviour of granular matter (Langston et al., 1995a,b; Zhou et al., 1999; Zhu and Yu, 2002). More recent advances on contact force incorporating the plastic deformation have also been made by Thornton (1997) and Vu-Quoc and Zhang (1999a,b). However, they need more experimental validation. The above models are often used miscellaneously (Schäfer et al., 1996; Lätzel et al., 2000). Table 1 shows the equations for some commonly used force models for spherical particles, including the linear spring–dashpot model, the simplified Hertz–Mindlin and Deresiewicz model by Langston et al. (1994), and Walton and Braun’s model. The interparticle forces act at the contact point between particles rather than the mass centre of a particle and they will generate a torque causing the particle to rotate. Generally, the torque is contributed by two components of the tangential and asymmetrical 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 is still an active research area (Greenwood et al., 1961; Johnson, 1985; Brillianton and Pöschel, 1998; Kondic, 1999). The rolling friction torque is considered to be negligible in many DEM models. However, it has been shown that the torque plays a significant role in some cases involving the transition between static 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-spherical particles is far more complicated. Two techniques have been


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Table 1 Contact force and torque models Force models

Normal force

Tangential force

References

Linear spring–dashpot model Simplified Hertz–Mindlin and Deresiewicz model

fn = −Kn n nc − Cn (vc · nc )nc

ft = −Kt vct

Cundall and Strack (1979)

Walton and Braun’s model

√ R ∗ (n )3/2 nc fn = − 43 E ∗ ∗ ∗ −Cn (8m E R ∗ n )1/2 ·(vc · nc )nc

⎧ −k n , ˙ 0 1 n c n ⎪ ⎨ (loading) fn = ˙ ⎪ ⎩ −k2 (n − n0 )nc , n < 0 (unloading)

+ Ct (vc × nc ) × nc

ft = −|fn,e |(1 − (1 − |vct |/max )3/2 )ˆvct +2C t (1.5 m ∗ |fn,e | · 1 − |vct |/max /max )1/2 ·(vc × nc ) × nc ⎧ ft −ft∗ 1/3 0 ⎪ vct ⎪ ⎪ ft + kt 1 − fn −ft∗ ⎪ ⎨ if v˙ t in initial direction c ft = ⎪ f + k 0 1 − ft∗ −ft 1/3 vt ⎪ ∗ ⎪ t t c f +f ⎪ n t ⎩ if v˙ ct in opposite direction

Langston et al. (1994, 1995a,b), Zhou et al. (1999) and Zhu and Yu (2002)

Walton and Braun (1986a), and Walton (1993)

where ft = |ft |, fn = |fn |. Torque models Method 1 Method 2

Rolling friction torque mr = −kr r − Cr dr /dt ˆn mr = − min{r |fn |, r |n |}

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

proposed to cope with particles of irregular shapes. One is to model a non-spherical particle as a collection of spherical particles (Gallas and Sokolowski, 1993; Pelessone, 2003; Bertrand et al., 2005). The advantage of the strategy is that it can be used to handle particles with very complex shapes, and only a contact model for spherical particle is required. The other technique considers that particles are of a given shape, such as ellipsoid, polygon and cylinder, and determines whether or not there is a contact between two such neighbouring particles by solving 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

Torque from tangential forces m t = R × ft

References Iwashita and Oda (1998, 2000) Zhou et al. (1999), and Zhu and Yu (2002)

for spherical particles (Favier et al., 2001; Cleary and Sawley, 2002) or its modification (Kohring et al., 1995). This approach is more accurate than the first one, but is computationally more demanding. As discussed above, linear and nonlinear contact force models have been developed for DEM simulations. Theoretically, the more complex nonlinear models simplified based on the Hertz and Mindlin–Deresiewicz theories should be more accurate than the linear model. However, in contrast, the numerical investigations conducted by Di Renzo and Di Maio (2004) showed that the simple linear model sometimes gives better results. This may be because theoretical models are often based on geometrically ideal particles, whereas there are no such perfect particles in practical applications. Selection of proper parameters also plays an important role in generating accurate results. Complicated models may also consume computational time with insignificant gain in DEM simulations. Moreover, as recently demonstrated by Zhu and Yu (2006), most of the force models were developed focused on one or two aspects or based on simplified conditions, their combination in DEM simulation may lead to theoretical or conceptual problems. These issues should be considered in the further development of force models. 2.2. Non-contact forces between particles When fine particles are involved and/or moisture exists, noncontact interparticle forces may affect the packing and flow behaviour of particles significantly. In the past, the interparticle forces were often evaluated by some empirical indexes such as Hausner ratio, angle of repose and shear stress (Hausner, 1972; Svarovsky, 1987). These indexes may partially interpret the behaviour of particles (see, for example, Geldart and Wong, 1984, 1985; Svarovsky, 1987; Yu and Hall, 1994), but general

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Table 2 List of a few typical non-contact forces Cohesive force

Origin

Formula

van der Waals force

Molecular dipole interaction

Fv = −

Electrostatic force

Coulomb force

Liquid bridge force (static)

Surface tension (capillary pressure and contact line force)

Q2 √ h =− 1 − nˆ ij (R 2 +h2 ) 16q0 h2 l F = −[2R sin sin( + ) + R 2 psin2 ]nˆ ij

Ad p nˆ ij 24h2

Fe

1.0E-04

Interparticle Force (N)

Capillary Force

1.0E-05

1.0E-06

Van der Waals Force

Electrostatic

1.0E-07

Weight

1.0E-08 1

10

100

References

1000

Particle Diameter (μm) Fig. 4. Comparison of the magnitude of interparticle forces calculated based ˚ (ii) capillary on: (i) van der Waals forces: A = 6.5 × 10−20 J, h = 1.65 A, force: = 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 be overcome by DEM because such forces can be directly considered. Often the non-contact forces involve a combination of three fundamental forces, i.e. the van der Waals force, capillary force and electrostatic force, which can act concurrently or successively to different extents. Table 2 lists the forces, their origin and equations for their estimation. Their relative importance varies, depending on a range of variables as illustrated in Fig. 4. In this section the mechanisms of these forces and the application in DEM modelling will be briefly discussed. The van der Waals force is the force between molecules having closed shell and consists of several types of interactions (Israelachvili, 1991). The van der Waals force between molecules is proportional to h−6 , where h is the separation distance (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 in DEM simulation to calculate the van der Waals force. This theory is based on the assumption of “pairwise additivity”

Hamaker (1937), and Israelachvili (1991) Krupp (1967) Fisher (1926), Princen (1968), and Lian et al. (1993)

and starts from the interactions between individual atoms (or molecules) and postulates their additive so that the van der Waals attraction between macroscopic bodies can be calculated by integration over all pairs of atoms (de Boer, 1936; Hamaker, 1937). The equation from the Hamaker theory shows the van der Waals force becomes infinite as two particles get into contact (h = 0), which induces a singularity problem in DEM simulation. To solve this problem, a “cut-off” distance is assumed in the calculation of this force, and this distance ranges from 0.165 to 1 nm (Krupp, 1967; Israelachvili, 1991; Yen and Chaki, 1992; Yang et al., 2000). The contribution due to the dispersion interaction to the total van der Waals force decays rapidly as the separation distance between the two interaction bodies becomes larger than a few nanometers, since the time taken for the electromagnetic field to propagate is not the same, with the period of the fluctuating dipole itself at the separation distance larger than a few nanometers (Hough and White, 1980; Israelachvili, 1991). This “retardation effect” should be taken into account for interactions between macroscopic bodies, especially in a liquid medium. A simple and accurate expression of the correction factor for the retardation effect in practical applications is given by Zhang et al. (1999a). Real particles are not rigid and will deform elastically and/or plastically at the contact point, even under zero external load. For plastic deformation, the van der Waals force must include a term for the extended contact area (Visser, 1989; Forsyth et al., 2001). The classical Hertz contact theory is for the elastic deformation of bodies in contact, but neglects the adhesion force due to the van der Waals attraction. The JKR (Johnson et al., 1971) and DMT (Derjaguin et al., 1975) models are two commonly used models developed to improve the Hertz model. The JKR model is derived based on contact mechanics and recognizes that both tensile and compressive interactions contribute to the total contact radius. On the other hand, the DMT model handles the Hertz deformation and adhesion effect separately. It is more straightforward to be implemented in DEM simulation. In particular, the van der Waals force is calculated according to the equation for smooth spheres, as listed in Table 2. The so-called Hamaker constant is the only parameter to be determined for given geometry and materials. Theoretically speaking, this constant depends on many variables related to physical and chemical properties such as particle surface roughness 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 often 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. The electrostatic forces can be catagorized into three types: Coulombic 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 approximated by the classical Coulomb equation. Rumpf (1962) compared the van der Waals attractive force with the electrostatic forces produced by contact potential or by excess charges. The results showed that the van der Waals force is greater by one order of magnitude than the electrostatic force for micro-sized particle. In some DEM simulations, instead of determining explicitly the non-contact forces such as the van der Waals and electrostatic forces, the concept of surface energy is incorporated in the simulation (Subero et al., 1999; Antony, 2000; Moreno et al., 2003). The capillary force is mainly due to the surface tension at solid/liquid/gas interfaces. For the cases shown in Fig. 2, the force due to the reduced hydrostatic pressure in the bridge itself can be given by (Fisher, 1926; Princen, 1968) F = 2R sin sin( + ) + R psin , l

2

2

(3)

where is the liquid surface tension, is the half-filling angle, and is the contact angle. The reduced hydrostatic pressure within the bridge, p, can be given by the Laplace–Young equation, which states that the mean curvature of the meniscus profile is constant and proportional to p. A number of analytical (Orr et al., 1975; Gao, 1997) and numerical (Erle et al., 1971; De Bisschop and Rigole, 1982; Lian et al., 1993) solutions have been proposed. Comparisons between these solutions and the comprehensive set of experimental data (Mason and Clark, 1965) have shown there is a good agreement (Mazzone et al., 1987). Fisher (1926) used a toroidal approximation for the shape of the liquid bridge. Two different methods have been adopted: the neck or gorge method which estimates the force at the neck of the bridge (Hotta et al., 1974), and the contact method which considers the force at the liquid bridge solid contact region (Adams and Perchard, 1985). Both methods show reasonable accuracy theoretically (Orr et al., 1975), experimentally (Cross and Picknett, 1963; Mason and Clark, 1965) and numerically (Lian et al., 1993). However, they may underestimate the force as 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 liquid bridge volume and the separation distance which is easy to implement into DEM simulations, some approximate solution procedures or approximate descriptions of the exact solutions have been proposed (Simons et al., 1994; Weigert and Ripperger, 1999; Willett et al., 2000). The error of these approximations is around 4% when the liquid-to-solid volume ratio is 0.1% but increases with increasing the volume ratio. The accuracy can be greatly improved with a considerably more complex expression which is valid for volume ratio less than 10%

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and gives an error in the force estimate of less than 3% (Willett et al., 2000). To calculate the capillary force between spheres of an unequal size, the Derjaguin approximation (Israelachvili, 1991) is a relatively accurate method for small bridge volumes and for separation distances excluding those at close-contact and near-rupture (Willett et al., 2000). To implement the capillary force in DEM simulation, liquid distribution among particles has to be determined. Muguruma et al. (2000) assumed that liquid can transport among particles and is distributed evenly among all gaps smaller than the rupture distance. On the other hand, Mikami et al. (1998) assumed that liquid is distributed evenly among particles and liquid transport between particles can be neglected if liquid viscosity is sufficiently small. By combining them together, Yang et al. (2003) assumed liquid being distributed evenly and not transferable among particles. Once the particle gap is smaller than the rupture distance, a liquid bridge is formed and the liquid assigned to a particle will be evenly distributed to its liquid bridges. 2.3. Particle–fluid interaction forces The surrounding fluid will interact with particles, generating various particle–fluid interaction forces, in addition to the buoyancy force. For example, the movement of particles is always resisted by stagnant fluid. The particle–fluid interaction force, mainly the drag force, is the driving force for fluidization. Therefore, particle–fluid interaction forces must be properly considered. To date, a number of such forces have been implemented in DEM simulation, including particle–fluid drag force, pressure gradient force, and other unsteady forces such as 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 determine the drag resistance force is well established, described by Newton’s equation (Table 3). The particle–fluid drag coefficient, Cd , is dependent upon Reynold’s number, Re, in addition to liquid properties. There are three regions: the Stoke’s Law region, the transition region, and Newton’s law region. For each region, the drag coefficient is determined by well-established correlations. But for a particulate system, the problem becomes much more complicated. The presence of other particles reduces the space 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 particle configuration, particle–fluid slip velocity and the properties of both particle and fluid. In general, two methods have been used to determine particle–fluid drag force. The first one is based on empirical 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 particles is considered in terms of local porosity, involving the exponent (see Table 3) and related to the flow regimes or particle Reynolds number (see, for example, Di Felice, 1994). The value of the exponent varies in a rather large range (from −3 to 10) (Morgan et al., 1970) and its accurate quantification is important. The other method is based on numerical simulations

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Table 3 Particle–fluid interaction forces Forces

Correlations

References

Drag force

For an isolated particle moving through a gas, Fd = Cd f dp2 |u − v|(u − v)/8 Effect of surrounding particles is described by a voidage function, f (f ): Fd = f (f )Cd f dp2 |u − v|(u − v)/8 Cd = 24(1 + 0.15Re0.687 )/Rep (Rep < 1000) p Cd = 0.44(Rep > 1000) Rep = f dp f |u − v|/f Fd = pf (u − v)/ f

Ergun (1952), and

f (1−f )2 + 1.75(1 − f ) fdp f (p dp )2 p |u−v| f (1−f ) −2.7 3 C ( > 0.8) d f f 4 dp

pf = 150

pf =

|u − v|(f 0.8)

−(+1)

Wen and Yu (1966)

f (f ) = f = 3.7 − 0.65 exp[−(1.5 − log Rep )2 /2]

Di Felice (1994)

F = F0 () + F1 ()Re2p (Rep < 20) F = F0 () + F3 ()Re2p (Rep > 20)

Koch and Sangani (1999), and Koch and Hill (2001)

F 0 ( ) =

1+3(/2)1/2 +(135/64) ln +16.14 1+0.681−8.482 +8.163 3

( < 0.4)

F0 () = 10/(1 − ) ( > 0.4) F1 () = 0.110 + 5.10 × 10−4 e11.6 F3 () = 0.0673 + 0.212 + 0.0232(1 − )5 Pressure gradient force

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

Anderson and Jackson (1967)

Virtual mass force

FV m = Cvm f Vp (u˙ − v˙ )/2 CV m = 2.1 − 0.132/(0.12 + A2c ) Ac = (u − v)2 /(dp d(u − v)/dt) t (u−˙ ˙ v) FBasset = 23 dp2 f f 0 √ dt +

Odar and Hamilton (1964), and Odar (1966)

Basset force Saffman force

Magnus force

t−t

(u−v)0 ) √ t

Reeks and Mckee (1984), and

where (u − v)0 is the initial velocity difference FSaff = 1.61dp2 (f f )1/2 |c |−1/2 [(u − v) × c ] c = ∇ × u 1 FMag = 8 dp2 f 2 ∇ × u − d × (u − v)

Mei et al. (1991) Saffman (1965, 1968)

Rubinow and Keller (1961)

where 21 ∇×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 to the location rotation of the fluid

at a microscale, where the techniques used include the direct numerical simulation (DNS) (Choi and Joseph, 2001) and Lattice–Boltzmann (LB) computation (Zhang et al., 1999b). Although rational, but limited by the current computational capability, to date the numerical studies have been applied only to relatively simple systems. The commonly used correlations for determining the fluid drag force are listed in Table 3. Li and Kuipers (2003) did a systematic study to quantify the difference among these correlations. Their results indicate that these correlations possess similar predictive capability, although their accuracy may differ. Other particle–fluid interaction forces have also been considered, particularly when the fluid involved is liquid rather than gas. These include the pressure gradient force, unsteady force and lift forces (Crowe et al., 1998). In general, the pressure gradient force includes not only the buoyancy force due to gravity but also the acceleration pressure gradient in fluid. There are a few forces in this connection, including the virtual mass force

and the Basset force. The virtual mass force relates to the force required to accelerate the surrounding fluid, and is also called the apparent mass force because it is equivalent to adding a mass to a particle. The Basset force describes the force due to the lagging boundary layer development with changing relative velocity. It accounts for the viscous effects. This term addresses the temporal delay in boundary layer development as the relative velocity changes with time, and is sometimes called the “history” term. According to Hjelmfel and Mockros (1966), the Basset term and virtual mass term become insignificant under some conditions, e.g., for small density ratio ( f / s ∼ 10−3 ). The lift forces, including Saffman lift force and Magnus lift force, on a particle are due to the rotation of the particle. The Saffman lift force is caused by the pressure distribution induced by the resultant velocity gradient. On the other hand, the Magnus force is developed by a pressure differential on the surface of the particle resulting from the velocity differential due to rotation. Equations to calculate these forces are listed in Table 3.


Their implementation in DEM simulation will depend on the particulate 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 in almost all types of particulate processes. Understanding the fundamentals governing the flow and formulating suitable governing equations and constitutive relationships are of paramount importance to the formulation of strategies for process development and control. This necessitates a multiscale approach to understand 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 studies have been done at either atomic/molecular scales related to the thermodynamics and kinetics or large scales related to the macroscopic performance of an operational unit or plant. However, what is missing is the quantitative understanding of microscale phenomena related to the behaviour of particles, droplets and bubbles. Without this information, it is difficult to generate a general method for reliable scale-up, design and control of particulate processes of different types. Therefore, particles scale modelling of particle–fluid flow has been a research focus in the past decade. DEM plays an important role in this development (Yu and Xu, 2003; Yu, 2004). 3.1. Numerical methods In principle, for any particle–fluid flow system, the solution of Newton’s equations of motion for discrete particles and the Navier–Stokes equations for continuum fluid together with boundary and initial conditions will finally determine the solids and fluid mechanics. In practice, however, there are usually a large number of particles. Consequently, this requires a very large number of governing equations to be solved for the motion of each of the particles and the resolution of the fluid field has to be fine enough to resolve the flow of continuum fluid through the pores among closely spaced particles. As a result, depending on the time and length scales of interest, simplifications have to be made when this theoretical approach is followed. The existing approaches to modelling particle flow can be classified into two categories: the continuum approach at a macroscopic level and the discrete approach at a microscopic level. In the continuum approach, the macroscopic behaviour is described by balance equations, e.g., mass and momentum as used in the two fluid model (TFM), closed with constitutive relations together with initial and boundary conditions (see, for example, Gidaspow, 1994). This approach is preferred in process modelling and applied research because of its computational convenience. However, its effective use heavily depends on constitutive or closure relations and the momentum exchange between particles of different type. In the past, different theories have been devised for different materials and for different flow regimes. However, to date, there is no accepted continuum theory applicable to all flow conditions. As a re-

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sult, phenomenological assumptions have to be made to obtain the constitutive relations and boundary conditions, which have very limited application (see, for example, Zhang et al., 1998). The discrete approach is based on the analysis of the motion of individual particles and has the advantage that there is no need for global assumptions on the solids such as steady-state behaviour, uniform constituency, and/or constitutive relations. Various methods have been developed in the past. A major type of discrete approach is based on DEM as discussed above. The method considers a finite number of discrete particles interacting by means of contact and non-contact forces, and every particle in a considered system is described by Newton’s equations of motion. In principle, it is similar to molecular dynamic simulation (MDS) but the forces involved differ because of the difference in time and length scales. The time and length scales for fluid flow can also range from discrete (e.g., MDS, LB, pseudo-particle method (PPM)) to continuum DNS, large eddy simulation (LES), and other conventional CFD techniques including TFM description). In principle, they all can be combined with DEM to describe the coupled particle–fluid flow. Indeed, many of them have been tried, including, for example, LB-DEM (Cook et al., 2004), PPM-DEM ( Ge and Li, 2001, 2003a,b), DNS-DEM (Hu, 1996; Pan et al., 2002), LES-DEM (Zhou et al., 2004a–c), in addition to the CFD-DEM model which, sometimes referred to as the combined discrete and continuum model (CCDM), is most popularly used as discussed below. The so-called smoothed particle hydrodynamics (SPH) (Potapov et al., 2001) and front tracking (FT) (Annaland et al., 2005, 2006) were also attempted. Instead of DEM, other discrete particle models were also attempted, such as direct simulation Monte Carlo (DSMC) (Yuu et al., 1997; Tsuji et al., 1998). Smoothed particle (SP) method was also tried to replace the TFM to eliminate spatial mesh in computation (Yuu et al., 2000b; Sugino and Yuu, 2002). Table 4 lists a few representative combinations of different length scales for fluid and particle phases and their relative merits in different aspects, where for convenience, relative to particle 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 proposed can be further understood with reference to the three popular models: TFM, DNS-DEM, and CFD-DEM, as discussed by Yu and Xu (2003). In TFM, both gas and solid phases are treated as interpenetrating continuum media in a computational cell which is much larger than individual particles but still small compared with the size of process equipment so that the number of governing equations is greatly reduced (Anderson and Jackson, 1967). Since the first numerical simulation of realistic bubbles by Pritchett et al. (1978), TFM has dominated fluidization modelling for decades, as summarized by Gidaspow (1994) and others, for example, Kuipers and van Swaaij (1997) and Arastoopour (2001). However, its effective use heavily depends on constitutive or closure relations for the solid phase and the momentum exchange between phases which are not possible to obtain within its framework; this is particularly true when dealing with differ-

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

Length scale for fluid phase Length scale for particle phase Nature of coupling Example

Closure of equations

Incorporation of distribution effects of dispersed, solid phase Computational effort Suitability for engineering application in relation to process modelling and control Suitability for fundamental research in relation to particle physics

Sub-particle (discrete or continuum) Particle (discrete)

Pseudo-particle (discrete) Particle (discrete)

Computational cell (continuum) Particle (discrete)

Discrete + discrete or continuum + discrete LB-DEM or DNSDEM Yes (but may experience numerical difficulty for systems with strong particle–particle interactions) Yes

Discrete + discrete

Continuum + discrete

Computational cell (continuum) Computational cell (continuum) Continuum + continuum

PPM-DEM

CFD-DEM

TFM

No (difficulty to determine physical properties of a pseudo-particle)

Yes

No (constitutive relation for solid phase and phase interactions not generally available)

Yes

Yes

No

Extremely demanding

Very demanding

Demanding

Acceptable

Extremely difficult

Very difficult

Difficult

Easy

Most acceptable (particle–fluid interaction forces can be determined and used for CCDM)

Acceptable (but only valid for well-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 granular flow is a research area challenging the whole scientific community (see, for example, Jaeger et al., 1996; de Gennes, 1999). In DNS-DEM, the fluid field is resolved at a scale comparable with the particle spacing while particles are treated as discrete moving boundaries (Hu, 1996). One of the key features of the method is that the particle–fluid system is treated implicitly by using a combined weak formulation. DNS has great potential to produce detailed results of hydrodynamic interactions between fluid and particles in a system (Pan et al., 2002). However, one major weakness of this model is its capacity in handling particle collisions. In the earlier development, the particle collisions were not modelled at all; if the gap between two approaching particles was less than a preset small value, the simulation had to stop (Pan et al., 2002). In a recent development, a repulsive body force was introduced into the momentum equation to prevent possible collisions between particles (Glowinski et al., 2000; Singh et al., 2000). Therefore, so far, DNS or DNS-based models have mainly been applied to particle–liquid systems where the hydrodynamic interaction is dominant and particle–particle interaction is nonviolent. This limits its applicability to gas fluidization where particle collisions and interparticle forces are significant. From this point of view, LB-DEM is probably a more promising approach. In CFD-DEM coupling approach, the motion of individual particles is obtained by solving Newton’s equations of motion while 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 equations for fluid phase are the same as those in TFM—a treatise that has been widely accepted in engineering application, its governing equations for solid flow are mainly based on DEM. Therefore, if focused on the solid phase, CFD-DEM has a feature similar to the so-called molecular dynamic simulation (MDS) (Allen and Tildesley, 1987). The main difference between DEM and MDS is that the forces involved differ because of the different length scales concerned. It is also due to the implementation of interparticle forces that distinguishes the CFD-DEM from the previous CFD modelling of gas–solid flow where the interaction among particles is often ignored (see, for example, Crowe et al., 1977; Tanaka et al., 1996). As pointed out by Yu and Xu (2003), at this stage of development, the difficulty in particle–fluid flow modelling is mainly related to solid phase rather than fluid phase. Therefore, the CFD-DEM is attractive because of its superior computational convenience as compared to DNS- or LB-DEM and capability to capture the particle physics as compared to TFM. Therefore, the present review of particle–fluid flow is mainly along with the 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 approach was rationalized by Xu and Yu (1997, 1998). As mentioned above, by this approach, the motion of discrete particles is


described by DEM on the basis of Newton’s laws of motion applied to individual particles and the flow of continuum fluid by the traditional CFD based on the local averaged Navier–Stokes equations. Therefore, for particle phase, the governing equations are the same as Eqs. (1) and (2). For fluid phase, the governing equations comply with the law of conservation of mass and momentum in terms of local-average variables (Anderson and Jackson, 1967) and are the same as those used in TFM. Two formulations, referred to as Model A and Model B, have been proposed (Gidaspow, 1994). Model A assumes that pressure drop shares between the gas and solid phases, and Model B in the gas phase only. Both formulations have been used in CFD-DEM, given by for Model A: jf + ∇ · (f uf ) = 0, jt j( f f uf ) jt

+ ∇ · ( f f uf uf ) = − f ∇p − FA + ∇ · (f ) + f f g

jf + ∇ · (f uf ) = 0, jt jt

vi , (x, y, z)i , i

kc

fpf, i

ui ,(u − v)i , Rep,i

∑ fpf, i

Fpf = V i=1

Continuum Model

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

(4)

(5)

and for Model B:

j( f f uf )

Discrete Model

3387

(6)

+ ∇ · ( 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 and volume of a computational cell, respectively. FA and FB are the volumetric particle–fluid interaction forces for the two models. Note that the definition of the total particle–fluid interaction force 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 s g (s = 1 − f ) (Feng and Yu, 2004a). For monosized particles, there is little difference between Models A and B in the CFD-DEM simulations (Feng and Yu, 2004b; Kafui et al., 2004), which is consistent with the TFM simulation (Bouillard et al., 1989). However, Feng and Yu (2004a) recently demonstrated that for the fluidization of binary mixtures of particles, there is a significant difference between the Model A and Model B simulations. Comparison with the physical experiments conducted under comparable conditions, and the analysis of the numerical scheme to implement a CCDM simulation suggest the Model B formulation is more favoured. Further studies may be necessary in order to clarify fully this important issue. The modelling of the solid flow by DEM is at the individual particle level, whilst the fluid flow by CFD is at the computational cell level. As shown in Fig. 5, their coupling is numerically achieved as follows (Xu and Yu, 1997; Xu et al., 2001). At each time step, DEM will give information, such as the positions and velocities of individual particles, for the

evaluation of porosity and volumetric fluid drag force in a computational cell. CFD will then use these data to determine the fluid flow field which then yields the fluid drag forces acting on individual particles. Incorporation of the resulting forces into DEM will produce information about the motion of individual particles for the next time step. In theory, this coupling method is used by most of the investigators. However, examination of the previous implementations by different investigators showed that different schemes have been employed to couple the two phases modelled at different length scales. According to Feng and Yu (2004a), there are three schemes in the previous CFD-DEM simulation of gas–solid flow in fluidization: Scheme 1: The force from the particles to the gas phase is calculated by a local-average method as used in the TFM, whereas the force from the gas phase to each particle is calculated separately according to individual-particle velocity (Tsuji et al., 1993; Hoomans et al., 1996; Kawaguchi et al., 1998, 2000a,b; Ouyang and Li, 1999a,b; Yuu et al., 2000a,b; Bin et al., 2003; Limtrakul et al., 2003). Scheme 2: The force from the particles to the gas phase is calculated first at a local-average scale as used in scheme 1. This value is then distributed to individual particles according to 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 interaction forces on individual particles in a computational cell are calculated first, and the values are then summed to produce the particle–fluid interaction force at the cell scale (Xu and Yu, 1997, 1998; Hoomans et al., 2000; Helland et al., 2000; Xu et al., 2000, 2001; van Wachem et al., 2001; Kafui et al., 2002; Li and Kuipers, 2002, 2003; Zhou et al., 2002a,b; van Wachem and 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 force of the solid phase acting on the gas phase should be equal to the force of the gas phase acting on the solid phase but in the opposite direction. Scheme 1 does not guarantee that this condition can always be satisfied. Consequently, it is not reasonable. Indeed, this scheme was used only in the early stage of

3388


CFD-DEM development, although it can still be found occasionally. Scheme 2 can satisfy Newton’s third law. However, it uniformly distributes the interaction force among the particles in a computational cell irrespective of the different behaviours of these particles in the cell. This scheme cannot fully represent reality, as the particle–fluid interaction forces for the particles in the cell should differ for nonuniform particle–fluid flow. In addition, in the calculation of the particle–fluid interaction force, a mean particle velocity has to be used. The appropriate method for calculating this mean particle velocity is still an open question, particularly for multisized particle systems. Scheme 3 can overcome the above problems associated with schemes 1 and 2. Indeed, this scheme has been widely accepted since its first introduction by Xu and Yu (1997, 1998). 4. Transition from discrete to continuum By use of a proper averaging procedure, a discrete particle system can be transferred into a corresponding continuum system. Extensive research has been carried out to develop such averaging methods. Various methods have been proposed to derive the balance equations of the continuum system. Earlier work often ignored the effect of the rotational motion of particles. Thus, the resultant balance equations are only for mass and linear momentum. These equations are the same as those in the

classical continuum mechanics, no matter which method is used to derive them. However, recent studies have illustrated that the gradient of particle rotation in some cases such as shear band is very high (Oda and Iwashita, 1999). Therefore, additional quantities should be included in the continuum formulation to describe this gradient. Based on such consideration, an extra equation has been derived to describe the rotation of particles. Therefore, complete balance equations for a continuum system include those of mass, linear momentum and angular momentum. 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 balance equations include mass density , velocity u, angular velocity , stress tensor T and couple stress tensor M. These macroscopic variables can be linked to the microscopic variables in the discrete approaches by means of local averaging. In the past, various averaging methods have been proposed to link the macroscopic variables to the microscopic variables based on different theoretical considerations. They can be classified into: volume, time–volume and weighted time–volume averaging methods. For the volume method, there are two ways

Table 5 Equations for calculation of stress and couple stress Methods

Equations

Volume average Method 1

T=

1 (ui − uj )xij ⊗ fij V i j >i

Method 2

T=

1 (ui rij − uj rj i ) ⊗ fij V i j >i

Method 3

T=

Cp 1 p

V f c ⊗ lpc V p∈V c=1

M=

Availability

References

Valid for quasi-static systems, not sure if satisfying governing equations and valid for other systems

Drescher and de Josselin de Jong (1972)

Rothenburg and Selvadurai (1981), Christoffersen et al. (1981), and Kanatani (1981) Lätzel et al. (2001), and Luding et al. (2001)

Cp 1 p

(lpc × f c ) ⊗ lpc V p∈V V c=1

Time–volume average Method 4

Method 5

T=

T=

1 T VT 0

1 T VT 0

1 T M= VT 0

−

i

−

i

−

mi vi ⊗ vi +

ui Ii vi

⊗ i

Iij ⊗ fij

ds

Valid for rapid flows, not sure if satisfying governing equations and valid for other systems

i j >i

ui mi vi ⊗ vi +

i

+

uij Iij ⊗ fij

ds

i j >i

i j >i

Walton and Braun (1986a,b)

uij Iij ⊗ mij

ds

Zhang and Campbell (1992)


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Table 6 Equations linking microscopic to macroscopic variables (Zhu and Yu, 2002) Macroscopic variablesa Equations linking microscopic to macroscopic variables Mass density

=

Tt

Velocity

u=

Angular velocity

Stress

=

1

Tt

Tt

M=−

1 2

h(¯r, t¯) =

hi Ii vi ⊗ i ds +

i

Tt

1 2

i

gij dij ⊗ fij ds +

Tt

j >i

Tt

i

gib dib ⊗ fib ds

i

gij dij ⊗ (mij − mj i ) ds +

Tt

j >i

gib dib ⊗ mib ds

i

(mij + mj i )(hi + hj ) ds

where hi = h(ri − r, s − t), = a These

hi mi vi ⊗ vi ds +

i

Tt

M =

hi Ii i ds

i

Tt

Rate of supply of internal spin

hi mi vi ds

i

1

T=−

Couple stress

hi mi ds

i

Tt

i

j >i

Tt

i h i Ii

ds, vi = vi − u, gij =

1 0

h(¯ri + rdij − r, s − t) dr, gib =

1 0

g(ri + rdib − r, s − t) dr.

macroscopic quantities depend on the weighting function h(r, s). A weighting function has been recommended (Zhu and Yu, 2002, 2005b) c0 L t 1 1 2 Lt +t¯ 2 Lp +¯r √ , (¯r, t¯) ∈ 2 ¯2 2 2 exp − 2 ln L −t¯ + ln Lp −¯ r 2 2 (Lt −t )Lp (Lp −¯r )

0,

t

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 forces acting at the boundary points of the assembly (Drescher and de Josselin de Jong, 1972). Secondly, the stress tensor is expressed in terms of individual contact forces within the assembly of particles (Rothenburg and Selvadurai, 1981; Christoffersen et al., 1981; Kanatani, 1981). More complex expressions of stress and the micro-mechanical definition of couple stress have also been considered in some recent studies (Oda, 1999; Lätzel et al., 2000, 2001; Tordesillas and Walsh, 2002). For example, the models of Lätzel et al. (2000, 2001) and Luding et al. (2001) included more macroscopic characteristics such as volume fraction and fabric tensor. The volume averaging approaches is mainly developed to describe quasi-static systems. Therefore, the inertial effect is ignored in these approaches. Walton and Braun (1986a,b) proposed an alternative method based on time–volume averaging in their study of a rapid granular flow, which was later improved by Zhang and Campbell (1992) and Campbell (1993a,b) to include a couple stress term. The time–volume averaging has been demonstrated to be applicable to simple shear rapid flows, but its applicability to other granular flows has been questionable (Babic, 1997). It is not clear if the macroscopic properties obtained by use of the volume and time–volume averaging conform with those in the

balance equations (8)–(10) in the continuum approach (Luding et al., 2001). Some commonly used equations for the two approaches are shown in Table 5 (for brevity, only stress and couple stress are included). Babic (1997) first proposed a weighted time–volume averaging 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 shown in Table 6. The concept is shown in Fig. 6. The averaging method has been extended to study the constitutive behaviour of granular materials (Glasser and Goldhirsch, 2001; Goldhirsch and Goldenberg, 2002) and the macro-dynamical behaviour of granular flows (Zhu and Yu, 2003, 2005a,b). The developed approach is suitable for all flow regimes. The selection of the sample size, where the averaging is conducted, is the main problem in the application of this averaging method. To date, this problem has not been properly solved. Nevertheless, the combined approach of discrete approach and averaging method takes into account the discrete nature of granular materials and does not require any global assumption and thus allows a better understanding of the fundamental mechanisms of granular flow. It has been extensively used to investigate the dynamics of granular flows. These studies show that it can be used to analyze the macro-dynamical behaviour of granular flows under different operational

3390


conditions and micro-properties of granular material (Langston et al., 1995a; Potapov and Campbell, 1996, Zhu and Yu, 2003, 2005b; Heyes et al., 2004), to depict the intrinsic characteristics of granular materials such as the constitutive relationship under various flow conditions (Oda and Iwashita, 2000; Alonso-Marroquin and Herrmann, 2005), and to test the continuum theories (Makse and Kurchan, 2002). The approach offers a convenient way to link fundamental understanding generated from DEM-based simulations to engineering application often achieved by continuum modelling.

Probe point

g(r) c(σp)

4πLp3

0

r

Lp

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

5. Concluding remarks The multiscale phenomena associated with particulate matter poses a need for multiscale modelling and analysis. Fig. 7 schematically shows the approaches at different time and length scales. Since the bulk behaviour of particulate matter depends on the collective interactions among individual particles, it is the particle scale modelling and analysis that plays a crucial role in elucidating the underlying fundamentals and linking fundamental to applied research. Indeed, this has been the major research effort in particulate research in the past years, as seen from Fig. 1. The extensive investigations under different packing/flow conditions at either macroor microscopic level by various investigators worldwide, as reviewed by Zhu et al. (2006), clearly indicate that DEMbased discrete particle simulation is an effective way to achieve this goal. We have seen the rapid development in establishing the theories and models for this approach since the pioneer work of Cundall and Strack (1979). In this review, we show this development by highlighting three important aspects mainly related to the modelling of particle or particle–fluid flow. First, the theories underlying the DEM and models for calculating the contact forces between particles are more soundable, although not perfect yet. Secondly, more forces have been implemented in DEM simulation, which makes the DEM model more applicable to particulate research. The most important development in this direction is the so-called CFD-DEM or CCDM approach which can handle the particle–fluid flow widely encountered in process engineering. Thirdly, the theory to link the discrete simulation to continuum modelling is gradually established, generating a new but natural analytical method to depict the fundamentals governing the behaviour of particulate/granular matter. This is an important step for continuum-based process modelling. While DEM or CFD-DEM, as a state-of-the-art simulation technique, is increasingly used to study the physics of particulate and/or multiphase flow related to various industrial problems, as pointed out by Yu (2004), future effort will be made to meet the following research needs (also refer to Fig. 1).

log(Length Scale) (m)

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

-12

-9

-6

-3 0 3 log(Time Scale) (s)

6

9

12

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


• Microscale: To develop a more comprehensive theory and experimental techniques to study and quantify the interaction forces between particles, and between particle and fluid under various conditions, generating a more concrete basis for particle scale simulation. • Macroscale: To develop a general theory to link the discrete and continuum approaches, so that particle scale information, generated from DEM or DEM-based simulation, can be quantified in terms of (macroscopic) governing equations, constitutive relations and boundary conditions that can be implemented in continuum-based process modelling. • Application: To develop more robust models and efficient computer codes so that the capability of particle scale simulation can be extended, say, from two-phase to multiphase and/or from simple spherical to complicated non-spherical particle system, which is important to transfer the present phenomenon simulation to process simulation and hence meet real engineering needs. Notations A c0 Cd Cn Cr Ct CV m dp dij dib E∗ fc fij fn fn,e fpf ,i ft ft∗ c Fij nc Fik f Fi g Fi FA

FB FBasset Fd

Hamaker constant, J normalized constant of weighting function, dimensionless fluid drag coefficient on an isolated particle, dimensionless damping coefficient, dimensionless viscosity coefficient, dimensionless tangential damping coefficient, dimensionless virtual mass force coefficient, dimensionless particle diameter, m branch vector connecting the mass centres of particles i and j, m ray from the mass centre of particle i to j, m reduced Young’s modulus, dimensionless contact force, N interaction force between particle i and j, N normal component of interparticle contact force, N elastic component of normal interaction between particles, N interaction force between fluid and particle i, N tangential component of interparticle contact force, N initially equal to 0 and set to the value of ft whenever v˙ ct reverses its direction, N contact force acting on particle i by particle j, N non-contact force acting on particle i by particle k, N particle–fluid interaction force on particle i, N gravitational force on particle i, N volumetric particle–fluid interaction force in Model A, N m−3 volumetric particle–fluid interaction force in Model B, N m−3 Basset force, N interaction force between isolated particle and fluid, N

Fe Fl FMag Fpf Fp FSaff Fv FV m g h I lpc lij Kn k1 k2 kc kt0 kr Kt Lt Lp m mr mt M Mij nc p p q0 Q rij R R R∗ Re Rep t T T u ui V v vi vct

3391

electrostatic force, N liquid bridge force, N Magnus force, N volumetric particle–fluid interaction force, N m−3 pressure gradient force, N Saffman force, N van der Waals force, N virtual mass force, N gravitational acceleration, m s−2 surface gap between two spheres, m moment of inertia of particle, kg m2 branch vector from the centre of particle p to its contact c, m branch vector connecting the centroids of particle i and j, m normal spring coefficient, dimensionless spring constants for loading, dimensionless spring constants for unloading, dimensionless number of particles in a computational cell, dimensionless initial tangential stiffness, N m−1 rolling stiffness, N m−1 tangential spring coefficient, dimensionless parameter to determine domain in weighting function, s parameter to determine domain in weighting function, m mass of particle, kg torque from rolling friction, N m torque from tangential forces, N m couple stress tensor, Pa torque acting on particle i by particle j, N m unit normal vector at contact pressure, Pa reduced hydrostatic pressure within the bridge, Pa permittivity of vacuum, C2 J−1 m−1 particle charge, C position vector of the contact point between particles i and j, m vector of the mass centre of the sphere to contact plane, m particle radius, m reduced particle radius, dimensionless reynolds number, dimensionless relative Reynolds number around particle, dimensionless time, s stress tensor, Pa duration of the time periodic cell, s velocity, m s−1 equals to 1 if the centroid of particle i is located with V and zero otherwise, dimensionless the averaging volume, m3 translational velocity, m s−1 fluctuation velocity of particle i with respect to the averaged velocity, m s−1 relative tangential displacement at contact, m

3392

vˆ ct Vp V x xij y z


unit vector of vct , m volume of particle, m3 volume of a computational cell, m3 x direction in a coordinate system, m position vector of the contact point between particles i and j, m y direction in a coordinate system, m z direction in a coordinate system, m

Greek letters

pf n max f s r f r r f s p n ˆn p

V i

coefficient defined by Di Felice (1994), dimensionless momentum exchange coefficient, kg m−3 s−1 liquid surface tension, Pa relative normal displacement at contact, m the maximum t when the particles start to slide, m volumetric fraction by fluid, dimensionless volumetric fraction by solid, dimensionless contact angle between two particles, dimensionless relative particle rotation, dimensionless sliding friction coefficient, dimensionless fluid viscosity, Pa s rolling friction coefficient, dimensionless rotational stiffness, N m mass density for solid phase, kg m−3 fluid density, kg m−3 solid density, kg m−3 stress tensor, Pa half-filling angle as indicated inFig. 2, dimensionless solid fraction, dimensionless sphericity, dimensionless angular velocity, s−1 the component of the relative angular velocity in contact plane, s−1 unit vector of n , s−1 weight of the particle’s contribution to the average, dimensionless fluctuation angular velocity of particle i, s−1 average domain, m3 s

Subscripts c d f i ij j k n t

contact damping fluid particle i between particles i and j particle j particle k normal component tangential component

Acknowledgement The authors are grateful to the Australian Research Council for the financial support of their work.

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