Averaging method of particulate systems and its application to particle ...

SPECIAL TOPIC

© 2009

SCIENCE IN CHINA PRESS

Averaging method of particulate systems and its application to particle-fluid flow in a fluidized bed

ARTICLES

Chinese Science Bulletin

ZHU HaiPing†, HOU QinFu, ZHOU ZongYan & YU AiBing Laboratory for Simulation and Modeling of Particulate Systems, School of Materials Science and Engineering, the University of New South Wales, Sydney, NSW 2052, Australia

A particulate system can be described through the discrete approach at the microscopic level or through the continuum approach at the macroscopic level. It is very significant to develop the method to link the two approaches for the development of models allowing a better understanding of the fundamentals of particulate systems. Several averaging methods have been proposed for this purpose in the past, but they mainly focused on cohesionless particle systems. In this work, a more general averaging method is proposed by extending it for cohesionless particle systems. The application of the method to the particle-fluid flow in a gas fluidized bed is studied. The density, velocity and stress of this flow are examined. A detailed discussion has been conducted to understand the dependence of the averaged variables on sample size.

Particulate systems are quite common in nature and industry. Their dynamic behavior is very complicated due to the complex interactions between individual particles and their interactions with surrounding gas or liquid and wall. Understanding the underlying mechanisms in terms of these interactions is key to producing results that can be generally used. In the past, many studies have been done in this area (see ref. [1] for example). However, the development of a general theory to describe particulate systems is still a challenging problem. Generally, two numerical approaches can be employed to describe a particulate system: continuum at a macroscopic level and discrete at a microscopic level. The continuum approach is preferred in process simulation, and has been applied to various particulate systems[2,3]. However, the main limitation of its application is lack of reliable constitutive correlations. Some global assumptions and arbitrary treatments therefore have to be made with this approach[2]. Another shortcoming in the continuum modeling is that the effect of the microstructure is ignored. These difficulties can be overcome by discrete approaches such as the discrete element

method (DEM) for particle flows. For a particle-fluid flow, a typical discrete approach is the so-called CFDDEM (computational fluid dynamics-discrete element method) approach. In this method, the motion of solid flow is described by Newton’s second law of motion whilst the motion of fluid flows by solving the NavierStokes equations, and facilitated by a suitable scheme[1,4]. This method has been recognized as an effective method in the study of the fundamental behavior of particle-fluid flows[1]. With the support of a proper averaging theory, the macroscopic properties of a particulate system can also be investigated by use of the microscopic information generated by the discrete simulation. Extensive research has been carried out to develop an averaging theory to link the microscopic variables in a discrete approach to the macroscopic variables in a continuum approach. In the past, various averaging methods have been proposed based on different theoretical considerations. They can Received March 12, 2009; accepted June 3, 2009; published online August 20, 2009 doi: 10.1007/s11434-009-0500-0 † Corresponding author (email: [email protected]) Supported by the Australian Research Council

Citation: Zhu H P, Hou Q F, Zhou Z Y, et al. Averaging method of particulate systems and its application to particle-fluid flow in a fluidized bed. Chinese Sci Bull, 2009, 54: 4309―4317, doi: 10.1007/s11434-009-0500-0

CONDENSED MATTER PHYSICS

particulate system, gas fluidization, discrete approach, continuum approach, averaging method

be classified into volume[5], time-volume[6] and weighted time-volume averaging[7,8] methods. In theory, the most general approach among these averaging techniques is the weighted time-volume method, which does not include the restrictive assumptions employed in other models and then allows us to examine the nature of macroscopic properties more seriously. This method has been extended to study the constitutive behavior of granular materials[9] and the macro-dynamical behavior of granular flows[1,10]. 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. In addition, the previous studies are mainly focused on the cohesionless particle systems, and thus only contact forces between particles or between particles and walls are considered. In this paper, we propose a more general weighted time-volume method by extending the one for the cohesionless particle systems. In addition to the inter-particle contact forces, the non-contact forces and particle-fluid interaction force are included as well. The application of this method to the particle-fluid flow in a gas fluidized bed is studied.

Aik = (0, fiknc ,0),

Bif = (0, fif ,0) and Gi = (0, fig ,0),

where fijc and mij are the contact force and torque acting on particle i by particle j or walls, fiknc is the non-contact force acting on particle i by particle k or other sources, fif is the particle-fluid interaction force on particle i, and fig is the gravitational force.

Particulate systems can also be considered as a continuum system. Its governing equations can be obtained using the equations of motion for individual particle. Suppose that X (r , t ) is the local average of physical property X i at a point r and a time t, given by

X (r , t ) = ∫ ∑ hi X i ( s )ds,

(2)

Tt i

where hi = h( xi − r , s − t ) , h(r , t ) is the so-called weighting function, being positive in the limited domain

{

}

Ω = (r , t ) r ∈ Ωp ⊂ R3 , t ∈ T = [T0 , T1 ] ⊂ R

and zero

otherwise. xi is the position vector of particle i. s is the time contributing to the averaged quantities, and confined to Tt = [T0 + t , T1 + t ] . The weighting function should

1 Averaging method

satisfy the condition of normalization, i.e. ∫ h(r , t ) d r

A particle in a particulate system can have two types of motion: translational and rotational. During its movement, the particle may interact with its neighboring particles or walls and interact with its surrounding fluid, through which the momentum and energy are exchanged. The resultant forces on a particle can be determined from its interaction with the contacting particles and vicinal fluid for coarse particle systems. 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 of particle i can be written as d X i = ∑ Pij + ∑ Aik + Bif + Gi , (1) dt j k

dt = 1 . From eq. (2), it can be readily obtained that

where X i = (mi , mi vi , I i ωi ) , mi , I i , vi and ωi are the

mass, moment of inertia, translational and angular velocities of the particle, respectively. The terms of the right hand side of eq. (1) are related to the contact force, non-contact force, particle-fluid interaction force and gravity. They are respectively given by Pij = (0, fijc , mij ) , 4310

Ω

X (r , t ) can be written as X (r , t ) = ( ρ , ρ u, ρ ω) , where ρ,

u and ω are respectively the mass density, velocity and angular velocity. Their expressions are listed in Table 1. If L is an element of the system, associated with volume dτ 0 = dxdydz containing point r, then A(r , t ) = X (r, t) × dτ 0 designates the mass, linear momentum and angular momentum of this element. The material derivative DA of A(r , t ) can be expressed as (DX + X ∇ ⋅ u)dτ 0 , where u = dr / dt and D(dτ 0 ) = dτ 0∇ ⋅ u . Assuming that DA = Bdτ 0 (B is a undermined vector), we obtain a balance

equation for X : DX + X ∇ ⋅ u = B . B can be determined from the equations of motion for the particles similar to the treatment in ref. [8]. Therefore we can finally obtain the balance equations for the continuum system corresponding to the discrete system considered: D( X ) + X ∇ ⋅ u = ∇ ⋅ H + P + A + B + G ,

(3)

where H=(0, T, M), P=(0, 0, M′), A=(0, Fnc, 0), B=(0, Fpf, 0) and G=(0, Gp, 0) (T is the stress tensor, M the

www.scichina.com | csb.scichina.com | www.springer.com/scp | www.springerlink.com

SPECIAL TOPIC

Equations linking discrete to continuum variables

a)

Macroscopic variables

Equations linking microscopic to macroscopic variables

ρ=∫

Mass density

∑ hi mi ds

Tt i

u=

Velocity

ω=

Angular velocity

T = − ∫ ∑ hi mi vi '⊗ vi 'ds + ∫

Stress

M = − ∫ ∑ hi Ii vi '⊗ ωi ds + T i

∫ ∑ hi mi vi ds

Tt i

1

λ

∫ ∑ hi Ii ωi ds

Tt i

∑ ∑ gij dij ⊗ fij ds + ∫ ∑ gibdib ⊗ fibds

Tt i j >i

Tt i

Couple stress

1

ρ

Tt i

1 ∑ ∑ g d ⊗ (mij − m ji )ds + ∫ ∑ gibdib ⊗ mibds 2 ∫ i j >i ij ij T T i t

t

Fpf =

Fluid-particle interaction force

Fnc = ∫

Non-contact force

ARTICLES

Table 1

∫

∑ hi fif ds

Tt i

∑∑ hi fikncds

Tt i k

a) λ = ∫ ∑ hi I i ds · v i ' = vi − u is the fluctuant velocity of particle i with respect to the average velocity. gij is the weighting coefficient determined T t

i

by the weighting function, given by gij = ∫ 10 h (ri + rd ij −r , s − t )dr . ri = ri if ri − r ∈ Ωp ; otherwise, ri is the position vector of the point of intersection of vector dij and boundary ∂Ωp of domain Ωp . dij is the part of the branch vector connecting the mass centers of particles i and j

couple stress tensor, M′ the rate of supply of internal spin to particles, Fnc the non-contact force, Fpf the fluid-particle interaction force, and Gp the gravity). They can be respectively given by H = − ∫ ∑ hi vi '⊗ X i ds + Tt

i

1 ∑∑ gij dij ⊗ ( Pij − Pji ) ds 2 ∫ i j >i Tt

+ ∫ ∑ gib d ib ⊗ Bi ds, Tt

(4)

i

P=

1 ∑∑ (hi + h j )( Pij + Pji ) ds, 2 ∫ i j >i

(5)

Tt

A = ∫ ∑∑ hi Aiknc ds, Tt

i

B = ∫ ∑ hi Bif ds, Tt

(7)

i

G = ∫ ∑ hi Gi ds. Tt

(6)

k

(8)

i

Eq. (3) gives the governing equations of mass, linear momentum and angular momentum of the particulate system. Compared with the equations for the cohesionless particle systems[8], the present system has two extra equa-

tions for the non-contact force and fluid-particle interaction force (eqs. (6) and (7)). According to the above equations, the macroscopic properties such as mass, velocity and stress of systems can be determined quantitatively based on the information at the particle scale. Therefore, supported by discrete approaches, the averaging method developed can be used to analyze the macro-dynamical behavior of particulate systems under different operational conditions and micro-properties of granular material, to depict the intrinsic characteristics of granular materials such as the constitutive relationship under various flow conditions, or to test the continuum theories. This method is suitable for the entire considered domain and all flow regimes, and the macroscopic properties obtained in terms of this approach conform with those in the governing equations in the continuum approach. Table 1 lists the detailed expressions to link the discrete properties to continuum properties. The application of this method to particle-fluid flows is discussed in the following section. Its application to the particle systems in which non-contact force is included will be considered in our other work.

Zhu H P et al. Chinese Science Bulletin | December 2009 | vol. 54 | no. 23

4311

CONDENSED MATTER PHYSICS

within domain Ωp in which the quantities are averaged. f i b and mib are the contact force and torque on particle i by walls.

2 Application to the particle-fluid flow in a gas fluidized bed Gas fluidization is observed when gas continuously flows upward through a bed of particles at an appropriate flow rate. It exhibits complex and intriguing flow patterns. The mutual interaction between the discrete particles and continuum gas provides an ideal environment for rapid heat and mass transfer, good mixing of solids and fast chemical reaction inside a fluidized bed. These features are very desirable for many industrial applications. There have been extensive research activities in the area of gas fluidization over the past few decades[11,12]. Although a large amount of data have been accumulated from both laboratory experiment and plant practice, a comprehensive interpretation of these data is difficult, if not impossible, as the information obtained from an experiment is usually incomplete due to the limitation of measurement technique. To date, information on the transient forces acting on individual particles during fluidization is still unavailable. These forces are believed to be the key factors responsible for the complex flow phenomena in a fluidized bed. Information about the flow mechanisms is expected to obtain with the aid of mathematical modelling. 2.1 Simulation method and conditions

In this work, a CFD-DEM approach is used to simulate the gas-solid flow in a fluidized bed. In this approach, particle phase and fluid phase are described by different models. For particle phase, the motion of every particle in a considered system, which can undergo translational and rotational motions, is described by Newton’s laws of motion. For fluid phase, the continuum fluid field is calculated from the continuity and Navier-Stokes equations based on the local mean variables over a computational cell. For discrete particle simulations, explicit time integration is used to solve the translational and rotational motions of particles. The conventional SIMPLE method[13] is used to solve the equations for the fluid phase in the continuum model. The combination of DEM and CFD is numerically achieved as follows. At each time step, DEM simulations give information of Table 2 Physical and geometrical parameters used in simulations

individual particles such as the positions and velocities. From these, the porosity and volumetric fluid-particle interaction force in each computational cell are calculated. CFD is then used to determine the information of gas flow field, from which the fluid drag forces acting on individual particles are calculated. Incorporation of the resulting forces into DEM finally produces information about the motion of the individual particles for the next time step. The details on the CFD-DEM approach and the computational methodology of coupling schemes can be seen elsewhere[1,14]. The geometry used in the present simulation is a twodimensional slot model, as shown in Figure 1. The model thickness is six particle diameters with wall boundary conditions in the front and rear direction. For such a geometry, two-dimensional CFD and three-dimensional DEM are employed in this work as used elsewhere[4]. The bed is first packed with particles of diameter 4.0×10−3 m, and then fluidized by uniform gas introduced from the bed bottom. Table 2 lists the parameters used in the simulations. In the simulation domain shown in Figure 1, the selected locations for recording their macroscopic properties include three lines: vertical central line of the bed x = 0, and horizontal lines at z=0.1 and 0.2 m. At each selected location, its macroscopic properties such as velocity, density and stress are calculated using the averaging method. 2.2 Solid flow pattern

In gas fluidization, different flow patterns can be observed by varying gas superficial velocity at the inlet. Generally, at a low gas superficial velocity, particles in the bed remain stationary, and gas passes through the voids among particles. This state is identified as a fixed or packed bed. When the interaction force between particles and gas just counterbalances the total weight of particles, the bed starts fluidization, and the gas velocity at this point is referred to as the minimum fluidization velocity (Umf). With further increase in U (>Umf), the gas-solid flow behaves quite differently, with gas bubbles formed and particles moving in the bed rigorously.

a)

Number of particles, Particle diameter, Particle density, Young’s modulus, Poisson’s ratio, dp ρ E v N Solid phase 2500 kg/m3 30000 1.0×108 N/m2 0.3 4.0×10−3 m Gas phase

Sliding friction coefficient, μs 0.3

Restitution coefficient 0.8

Gas velocity

Viscosity, μf

Density, ρf

Bed width

Bed height

Cell width, ∆x

Cell height, ∆y

Time step, ∆t

4.0 m/s

1.8×10−5 kg/(m·s)

1.2 kg/m3

0.32 m

2.4 m

1.6×10−2 m

1.6×10−2 m

2.0×10−5 s

a) The wall properties are the same as those of particles 4312

www.scichina.com | csb.scichina.com | www.springer.com/scp | www.springerlink.com

SPECIAL TOPIC ARTICLES The initialized particle bed (a) and the location lines for the average of properties (b).

Figure 2 shows a typical case in a fluidized bed when gas superficial velocity is 4.0 m/s. Two stages can be identified from the variation of bed pressure drop (Figure 2(a)): the start-up stage and stable fluidization stage. The maximum bed pressure drop at the start-up stage as the initial response to the gas flow is much higher than that at the stable stage. After the start-up stage, the bed pressure drop fluctuates around a mean value corresponding to the bed weight per unit area, as shown in Figure 2(a). Figure 2(b) shows the snapshots of simulated solid flow patterns from the start-up to the stable fluidization. It can be seen that, at the start-up stage, the high void space transmits from the bottom to the top part due to the uniform gas introduction at the bottom inlet. The particles above the void space layers are lifted, while the particles beneath them have negative velocities and descend due to the gravity. Once the lifted particles fall back to the bed, stable fluidization is established rapidly which can be reflected by the variation of pressure drop in Figure 2(a). At this stage, the gas-solid flow is featured by the formations of gas bubbles and particle clusters, as shown in Figure 2(b). 2.3 Macroscopic analysis of solid flow

The microscopic quantities of particles obtained by the CFD-DEM simulations can be used to calculate the macroscopic quantities of the solid flow such as velocity, density and stress using the equations listed in Table 1. These macroscopic quantities depend on the weighting function h(r,s). Therefore, the selection of a proper weighting function is important in applying the averaging method. In our previous work, this issue has been discussed and a weighting function for particle flow has been recommended[10]. The function is also suitable for the present particle-fluid flow, and can be given by

cLt ⎧ 1 ⎪ 2 2 2 2 ⎪ 2π 2π ( Lt − t ) Lp ( Lp − r ) ⎪ ⎡ 1⎛ Lp + r ⎞ ⎤ ⎨ L +t + ln 2 h(r , t ) = ⎪× exp ⎢ − ⎜ ln 2 t ⎟ ⎥ , (r , t ) ∈ Ω , (9) Lt − t Lp − r ⎠⎟ ⎥⎦ ⎢⎣ 2 ⎝⎜ ⎪ ⎪0, others, ⎩

where c is the normalized constant of the weighting function, Lt and Lp are two parameters controlling domain Ω. The specification of the sample size (namely, Lt and Lp) is a significant part of the application of the averaging method. Figure 3 shows the distributions of the velocity, density and stress on a vertical line x=0 at t = 0.93 s for different values of Lp. It can be observed that for different Lp, the trends of the distributions of these variables are similar. The velocity distributions can be divided into four sections: A, when z is from 0 to 0.05 m; B, from 0.05 to 0.09 m; C, from 0.09 to 0.16 m; and D, from 0.16 to 0.2 m. In section A, the particles have very low velocities. In section B, the particles move downwards, and the magnitude of the particle velocity increases with the increase of the height. In section C, the particles move downwards, but the magnitude of the velocity decreases to zero with the increase of the height. In section D, the particles move upwards, and the magnitude of the velocity increases slightly with the bed height. These features are in consistence with the particle flow pattern shown in Figure 2. The density decreases with the height in section B and increases in sections C and D if Lp is chosen as 2.0 or 3.0 d. The section with large negative velocity has a small density. The smallest density corresponds to the largest velocity. Such a density distribution has also been identified in the work of Anderson et al.[15]. The stresses increase with the bed height in section A, decrease in section B, have small magnitudes in section

Zhu H P et al. Chinese Science Bulletin | December 2009 | vol. 54 | no. 23

4313

CONDENSED MATTER PHYSICS

Figure 1

Figure 2 Variation of the dimensionless gas pressure drop with time (a) and snapshots of particle flow pattern with contour of vertical velocity when gas superficial velocity is 4.0 m/s (b). Pressure drop shown is a ratio to the bed weight per unit area.

Figure 3 Distributions of the macroscopic properties on the vertical line x=0 at t = 0.93 s for L t = 2.0 d g and different values of Lp (the unit of

Lp is d, where d is the particle diameter). (a) Velocity; (b) density; (c) vertical normal stress; (d) horizontal normal stress; (e) shear stress.

C, and increase in section D. Overall, the section with high stresses has a high density (or low porosity). However, the values of the considered properties are 4314

different for different sample sizes. It can be seen from Figure 3 that these variables fluctuate along z more seriously when Lp = 1.0 d. The change of the properties when

www.scichina.com | csb.scichina.com | www.springer.com/scp | www.springerlink.com

tion, we can choose Lp=2.0 d and Lt = 2.0 d g for the present system. Such selection can ensure that the averaged quantities smoothly vary in a considered domain and retain the local properties of these quantities as largely as possible. The distributions of the variables of particles would vary with time. Figure 7 shows the variation of the solid pressure with time. It can be seen from Figure 7(a) that the distributions of the solid pressure along a horizontal line are different at different instants of time. The values of the solid pressure fluctuate largely. Such fluctuation can also be seen in the variation of the solid pressures at a point shown in Figure 7(b). The reason to cause large

Figure 4

Lt = 2.0 d g respectively. The gas-solid flow behaviors in fluidized beds are very complicated, and would be affected by gas and solid properties and operational conditions. This paper only studies a simple case to demonstrate the applicability of the averaging method to particle-fluid flows. A more comprehensive study is necessary in order to fully understand the flow system.

SPECIAL TOPIC ARTICLES

fluctuation of pressure is because the local structures of the solid flow at a specific point or height are different at different times, as shown in Figure 2(b). These results indicate that according to the present average technique, the local properties of the solid flow can be retained if the values of Lp and Lt are chosen as Lp=2.0d and

3 Conclusions An averaging method for general particulate systems has been developed to obtain the macroscopic quantities of particle phase in the continuum description from the microscopic quantities generated by the discrete simulation. The approach offers a convenient way to link fundamental understanding generated from DEM-based simulations to engineering application often achieved by continuum modeling. The application of the method to

Distributions of the macroscopic properties on the vertical line x=0 at t = 0.93 s for L p = 2.0 d and different values of Lt (the units

of Lt are d g ). (a) Velocity; (b) density; (c) vertical normal stress; (d) horizontal normal stress; (e) shear stress.

Zhu H P et al. Chinese Science Bulletin | December 2009 | vol. 54 | no. 23

4315

CONDENSED MATTER PHYSICS

Lp increases from 2.0 to 3.0 d is much smaller than that when Lp increases from 1.0 to 2.0 d. Similar to Lp, Lt also affects the value of the properties, as shown in Figure 4. The influence of Lp and Lt can be seen more clearly in the distributions of solid mean pressure along two horizontal lines z = 0.1 and 0.2 m, as shown in Figures 5 and 6. The solid mean pressure is the average of the two normal stresses. These results suggest that if Lp and Lt are too small, the resulting macroscopic quantities fluctuate significantly, giving unreasonable average results. On the other hand, if they are too large, the local properties of these quantities may be eliminated. To balance the fric-

Figure 5

Distributions of the solid mean pressure on two horizontal lines at t = 0.93 s for L t = 2.0 d g and different values of Lp (the unit of

Lp is d). (a) z = 0.1 m; (b) z = 0.2 m.

Figure 6

is

Distributions of the solid mean pressure on two horizontal lines at t = 0.93 s for L p = 2.0 d and different values of Lt (the unit of Lt

d g ). (a) z=0.1 m; (b) z=0.2 m.

Figure 7 (a) Distributions of the solid pressure on a horizontal line z = 0.2 m at different instants of time; (b) variation of the solid mean pressure

with time at two points P1 (−0.12, 0.2) and P2 (0.12, 0.2). L p = 2.0d and L t = 2.0 d g are used when calculating the solid pressure.

particle-fluid flows is demonstrated in the calculation of density, velocity and stress of the flow in a gas fluidized bed. The specification of sample size (controlled by parameters Lp and Lt in eq. (9) for the specific weighting function utilized in this work) is a significant part of the application of the averaging method. The results suggest that if the sample size is too small, the resulting macroscopic quantities fluctuate significantly, giving unrea-

4316

sonable average results. On the other hand, if it is too large, the local properties of these quantities may be eliminated. To balance the contradiction, Lp=2.0d and Lt = 2.0 d g are suggested for the present system. The particle-fluid flow in fluidized bed is very important, but complicated. Further studies are being conducted in order to develop a better understanding of this flow system.

www.scichina.com | csb.scichina.com | www.springer.com/scp | www.springerlink.com

3 4

5

6

7 8

10

11 12 13 14

15

Rev E, 2002, 66: 021302 Glasser B J, Goldhirsch I. Scale dependence, correlations, and fluctuations of stresses in rapid granular flows. Phys Fluids, 2001, 13: 407―420 Zhu H P, Yu A B. Micromechanic modeling and analysis of unsteady-state granular flow in a cylindrical hopper. J Eng Math, 2005, 52: 307―320 Davidson J F, Cliff R, Harrison D. Fluidization. 2nd ed. London: Academic Press, 1985 Bi H T, Grace J R. Flow regime diagrams for gas-solid fluidization and upward transport. Int J Multiph Flow, 1995, 21: 1229―1236 Patankar S V. Numerical Heat Transfer and Fluid Flow. New York: Hemisphere Publishing Corporation, 1980 Xu B H, Yu A B. Numerical simulation of the gas-solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics. Chem Eng Sci, 1997, 52: 2785―2809 Anderson K, Sundaresan S, Jackson R. Instabilities and the formation of bubbles in fluidized beds. J Fluid Mech, 1995, 303: 327―366

SPECIAL TOPIC

9

ARTICLES

2

Zhu H P, Zhou Z Y, Yang R Y, et al. Discrete particle simulation of particulate systems: A review of major applications and findings. Chem Eng Sci, 2008, 63: 5728―5770 Zhang S J, Yu A B, Zulli P, et al. Modelling of the solids flow in a blast furnace. ISIJ Int, 1998, 38: 1311―1319 Zaimi S A, Akiyama T, Guillot J B, et al. Sophisticated multi-phase multi-flow modeling of the blast furnace. ISIJ Int, 2000, 40: 322―331 Zhou Z Y, Zhu H P, Yu A B, et al. Discrete particle simulation of gas-solid flow in a blast furnace. Comput Chem Eng, 2008, 32: 1760―1772 Walton O R, Braun R L. Antigranulocytes-temperature, and stress calculations for shearing assemblies of inelastic, frictional disks. J Rheol, 1986, 30: 949―980 Zhang Y, Campbell C S. The interface between fluid-like and solid-like behaviour in two-dimensional granular flows. J Fluid Mech, 1992, 237: 541―568 Babic M. Average balance equations for granular materials. Int J Eng Sci, 1997, 35: 523―548 Zhu H P, Yu A B. Averaging method of granular materials. Phys

CONDENSED MATTER PHYSICS

1

Zhu H P et al. Chinese Science Bulletin | December 2009 | vol. 54 | no. 23

4317