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2011-Discrete particle simulation of gas fluidization of ellipsoidal particles

Discrete particle simulation of gas?uidization of ellipsoidal particles

Z.Y.Zhou a,D.Pinson b,R.P.Zou a,A.B.Yu a,n

a Laboratory for Simulation and Modelling of Particulate Systems,School of Materials Science and Engineering,The University of New South Wales,Sydney,NSW2052,Australia

b BlueScope Steel Research,P.O.Box202,Port Kembla,NSW2505,Australia

a r t i c l e i n f o

Article history:

Received25February2011

Received in revised form

6August2011

Accepted24August2011

Available online1September2011

Keywords:

Fluidization

Ellipsoid

Computational?uid dynamics

Discrete element method

Granular materials

Simulation

a b s t r a c t

Fluidization is widely used in industries and has been extensively studied,either experimentally or

theoretically,in the past decades.In recent years,a coupled simulation approach of discrete element

method(DEM)and computational?uid dynamics(CFD)has been successfully developed to study the

gas–solid?ow and heat transfer in?uidization at a particle scale.However,to date,such studies mainly

deal with spherical particles.The effect of particle shape on?uidization is recognized but not properly

quanti?ed.In this paper,the CFD–DEM approach is extended to consider the?uidization of ellipsoidal

particles.In the simulation,particles used are either oblate or prolate,with aspect ratios varying from

very?at(aspect ratio?0.25)to elongated(aspect ratio?3.5),representing cylinder-type and disk-type

shaped particles,respectively.The commonly used correlations to determine the?uid drag force acting

on a non-spherical particle are compared?rst.Then the model is veri?ed in terms of solid?ow patterns.

The effect of aspect ratio on the?ow pattern,the relationship between pressure drop and gas

super?cial velocity,and microscopic parameters such as coordination number,particle orientation and

force structure are investigated.It is shown that particle shape affects bed permeability and the

minimum?uidization velocity signi?cantly.The coordination number generally increases with aspect

ratio deviating from 1.0.The analysis of particle orientations shows that the bed structures for

ellipsoids are not random as that for spheres.Oblate particles prefer facing upward or downward while

prolate particles prefer horizontal orientation.Spheres have the largest particle–particle contact force

and?uid drag force under the comparable conditions.With aspect ratio deviating from1.0,particle–

particle interaction and?uid drag become relatively weak.The proposed model shows a promising

method in examining the effect of particle shape on different?ow behaviour in gas?uidization.

1.Introduction

Fluidization is widely used in industries and as a typical gas–

solid two-phase?ow system,has been extensively studied,both

experimentally and theoretically,in the past.Generally speaking,

two approaches are available for the modelling of solid phase:

continuum at a macroscopic level and discrete at a microscopic

level.Continuum modelling,based on local average principles,

has been widely used in gas?uidization(for example,Anderson

and Jackson,1967;Gidaspow,1994;Enwald et al.,1996).How-

ever,its effective use depends on reliable constitutive or closure

relations which are generally not available yet.Alternatively,

discrete simulation,based on the analysis of the motion of

individual particles,can overcome this dif?culty.In this connec-

tion,the so called combined approach of computational?uid

dynamics(CFD)and discrete element method(DEM)has been

increasingly used in the study of gas–solid?ow in?uidization,as

recently reviewed by Zhu et al.(2007,2008).This approach can

generate detailed dynamic information such as the trajectories of

and transient forces on individual particles,which is extremely

dif?cult,if not impossible,to obtain experimentally.Such infor-

mation is important to depict the particle–particle and particle–

?uid interactions and hence understand the fundamentals of

particle–?uid?ow under different conditions.

To date,the particles dealt with in the CFD–DEM modelling of

?uidization are spherical particles,which cannot fully represent

the reality.In practice,most particles are non-spherical,of either

regular or irregular shapes.Particle shape is one of the most

important particle properties,and affects the packing/?ow struc-

tures that are critical to transport properties such as permeability

related to pore connection and thermal conductivity related to

particle connection.For example,studies on how particle shape

affects porosity(1-packing fraction)have been done by some

researchers(for example,Yu and Standish,1993;Zou and

Yu,1996;Yu et al.,1996;Donev et al.,2004;Guises et al.,2009;

Jia et al.,2010).Generally speaking,porosity will decrease to a

minimum and then increase when particles become more non-

spherical.Correspondingly,this will affect the bed permeability

Contents lists available at SciVerse ScienceDirect

journal homepage:https://www.doczj.com/doc/4c7512278.html,/locate/ces

Chemical Engineering Science

doi:10.1016/j.ces.2011.08.041

n Corresponding author.Tel.:t61293854429;fax:t61293855956.

E-mail address:a.yu@https://www.doczj.com/doc/4c7512278.html,.au(A.B.Yu).

Chemical Engineering Science66(2011)6128–6145

signi?cantly,as re?ected by the Ergun equation(1952).In the past,some work has also been done experimentally to investigate the effect of particle shape on?ow behaviour.For example, Liu and Litster(1991)investigated how particle shape affects the minimum spouting velocity and fountain height in a spouted bed,revealing that the predicted fountain heights are much lower than the experimental values if particle shape is ignored.Dolejs and Machac(1995)presented a method for estimating the pressure drop for the?ow of Newtonian?uids through?xed beds of spherical and non-spherical particles.Liu et al.(2008) observed that non-spherical particles give poor?uidizing quality as compared to spherical particles.With the same volume-equivalent diameter,non-spherical particles have a lower mini-mum?uidization velocity than spheres.Escudie et al.(2006) investigated the effect of particle shape on segregation in liquid-?uidized beds,revealing that segregation can occur according to shape.Clearly,experimental studies have provided some useful data to understand how particle shape affects the?ow behaviour

at a macroscopic scale.However,the results are often not so quantitative and lack of microscopic details.

DEM has recently been applied to study the granular?ow of non-spherical particles at a particle scale(for example,Vu-Quoc et al.,2000;Cleary and Sawley,2002;Langston et al.,2004; Lia et al.,2004;Fraige et al.,2008).In particular,the shapes considered can be regular or irregular.Due to the dif?culty and complexity in representing the irregular shapes and heavy com-putational requirement,to date,such studies are largely limited to regular shape.Particularly,ellipsoidal particles are commonly used because of its diversity in representing particle shapes varying from oblate to prolate ellipsoids.Recently,attempts have also been made to extend the CFD–DEM approach to modelling the?uidization of ellipsoidal particles(for example,Zhou et al., 2009a;Hilton et al.,2010).Some important aspects are discussed, such as the drag force,local and overall porosity and pressure drop.However,these studies are largely preliminary,without much detailed microscopic information.There are many issues to study here.For example,particle orientation is an important aspect,which probably represents the major difference between spherical and ellipsoidal particles.To date,it is not clear how aspect ratio affects particle orientation and its related parameters such as coordination number and forces.

In this work,the CFD–DEM approach is extended to describe the?uidization of ellipsoidal particles.In the simulation,particles used are either oblate or prolate,with aspect ratio varying from 0.25to3.5.The effect of aspect ratio on dynamic behaviour such as solid?ow pattern,the relationship between pressure drop and gas super?cial velocity,particle contacts and orientation and force structures are investigated in details.

2.Model description

2.1.DEM for ellipsoidal particles

Techniques for DEM modelling of non-spherical particles have been reported in the literature,as,for example,reviewed by Dziugys and Peters(2001).In particular,a number of investiga-tors have contributed to the modelling of ellipsoidal particles (Rothenburg and Bathurst,1991;Ting,1992;Ting et al.,1993; Lin and Ng,1995,1997;Vu-Quoc et al.,2000;Dziugys and Peters, 2001).The present DEM model is largely developed on this basis. For completeness,a brief description of the DEM method used is given below.

As originally proposed by Cundall and Strack(1979),a particle can have two types of motion:translational and rotational. Accordingly,the governing equations for particle i with mass m i and moment of inertia I i can be written as

m i

d v i

?f pf,it

X k i

j?1

ef c,ijtf d,ijTtm i ge1T

and

I i

d x i

X k i

j?1

eM t,ijtM n,ijtM r,ijTe2T

where v i and x i are the translational and angular velocities of the particle,respectively,and k i is the number of particles interacting with the particle.As shown in Fig.1,the forces involved are: particle–?uid interaction force f pf,i,the gravitational force m i g, and inter-particle forces between particles,which include elastic force f c,ij and viscous damping force f d,ij.The torques acting on particle i by particle j include:M t,ij generated by the tangential force,M r,ij commonly known as the rolling friction torque,and also the torque M n,ij generated by the normal force when the normal force does not pass through the particle centre.

Equations used to calculate the interaction forces and torques between two spheres have been well-established in the literature (Zhu et al.,2007).In this work,we simply extend the non-linear model to ellipsoids,and those equations are listed in Table1.This approach was also used by other investigators(Langston and Tuzun,1994,1995;Zhou et al.,1999;Zhu and Yu,2002).It should be noted that an additional torque is introduced,which is caused by the normal force.This is because the normal direction of contact force does not necessarily pass through the centre of an ellipsoid,which,together with the tangential forces and rolling torque,governs the rotational motion of the particle.Another parameter is the so called reduced radius R n in the calculation of the contact forces between particles i and j.For spheres, R n?R i R j=eR itR jT,But for ellipsoidal particles,R n?1=e2

?????????

A0B0

Twhere R i and R j are radii of particle i and j,respectively,where A0and B0can be obtained by solving the following two equations (Dziugys and Peters,2001):

2eA0tB0T?

e3T4eA0àB0T2?

R i

R j

t2cose2bT

R i

R j

e4Twhere1/R i and1/R0i are two curvatures of the contact point on the surface of ellipsoid i(the same concept is applied to particle j), and b is the angle between the normal sections whose radii of curvatures are1/R i and1/R0i(or1/R j and1/R0j for particle j

). Fig.1.Two-dimensional illustration of the forces acting particle i in contact with particle j.

Z.Y.Zhou et al./Chemical Engineering Science66(2011)6128–61456129

The method to determine the curvature in any direction at any point on the surface of an ellipsoid is well described by Harris (2006),and hence used in our work.2.2.CFD for ?uid phase

The continuum ?uid is calculated from the continuity and Navier–Stokes equations based on the local mean variables over a computational cell.In the CFD–DEM modelling,there are different model formulations as recently discussed by Zhou et al.(2010).Generally,three sets of governing equations exist for CFD–DEM model:type I,type II and type III,with the last two corresponding to the so called model A and model B (Bouillard et al.,1989;Gidaspow,1994).Each type of model has its own advantages and disadvantages.Type I is,however,the most rational,and is employed in the present work.Thus,the governing equations can be written as @e f

@t tr U ee f u T?0e5T

@er f e f u T@t

tr U er f e f uu T?àr p àF pf tr U s tr f e f g

e6T

where u ,r f and p are the ?uid velocity,density and pressure,

respectively.(F pf ?eP k c

i ?1f pf ,i T=D V )is the volumetric particle–?uid interaction force in a computational CFD cell of volume D V ;s and e f are the ?uid viscous stress tensor and local porosity,

which are given as s ?m e ?er u Tter u Tà1

,and e f ?1àeP k c

i ?1V p ,i T=D V ,respectively,where V p,i is the volume of particle i (or part of the volume if the particle is not fully in the cell),and k c is the number of particles in the computational cell.m e is the ?uid effective viscosity determined by the standard k-e turbulent model (Launder and Spalding,1974),which has been used in our previous work (Zhang et al.,1998;Zhou et al.,2009b ).2.3.Particle–?uid interaction force for non-spherical particles According to Crowe et al.(1998),particle–?uid interaction force f pf is the sum of all types of particle–?uid interaction forces acting on individual particles by ?uid.In gas ?uidization,the dominant forces are the drag force f d ,pressure gradient force f r p ,and viscous force f r s due to the ?uid shear stress or deviatoric stress tensor.So the total particle–?uid interaction force on a particle can be written as f p f ,i ?f d ,i tf r p ,i tf r s ,i

e7T

It should be noted that ellipsoidal particles can rotate by the torque caused by the non-uniform gas ?ow.As pointed out by Hilton et al.(2010),due to the approximately parallel ?ow of

?uid,the rotation motion of particles caused by ?uid can be neglected.For simplicity,it is thus not considered in the present work.

Many correlations have been proposed to calculate the ?uid drag for spherical particles (for example,Crowe et al.,1998;Zhu et al.,2007).Among those force models,the scheme proposed by Di Felice (1994)is one of the most popular ones,because it is based on individual particles.It has been widely used in our previous studies (for example,Xu and Yu,1997;Xu et al.,2000;Feng and Yu,2004,2007;Zhou et al.,2009b ,2010).In this work,this scheme is extended to ellipsoidal particles,so that f d ,i ?0:5?C D r f A ?e 2f 9u i àv i 9eu i àv i Te àw

f ,i

e8T

where w ?3:7à0:65exp ?àe1:5àlog 10Re i T2=2 ,A ?is the cross-sec-tional area perpendicular to the ?uid ?ow,Re i is the relative Reynolds number,which is de?ned as Re p ,i ?r f d v e f 9u i àv i 9=m f ,where d v is the equivalent diameter de?ned as the diameter of a sphere with the same volume as the ellipsoidal particle.C D is the drag coef?cient,which can be determined by different correlations.

Various efforts have been made to determine the ?uid drag coef?cient C D on non-spherical particles.The Ergun equation (1952)considered the effect of particle shape using the concept of sphericity (j ).But it may have a large deviation when applied to non-spherical particles as demonstrated by some researchers (Wu,2001;Nemec and Levec,2005;Ozahi et al.,2008).Some attempts have been made to modify the Ergun equation,as reviewed by Ganser (1993)and Loth (2008).In particular,Haider and Levenspiel (1989)established a four-parameter correlation for C D ,but ignored the effect of orientation.Ganser (1993)proposed a drag correlation using correction factors (Stokes factor and Newton’s factor),and claimed that it is general and accurate to determine C D .More recently,Holzer and Sommerfeld (2008)proposed a new correlation considering both the shape and the orientation of a non-spherical particle.Table 2lists these correlations whose applications are examined in this work.

It should be noted that the correlations listed in Table 2are developed based on macroscopic observations.In recent years,the lattice Boltzmann method is increasingly used aiming to derive equations to determine the particle–?uid drag for spherical particles (for example,Van der Hoef et al.,2005;Beetstra et al.,2006;Leboreiro et al.,2008;Sarkar et al.,2009;Yin and Sundaresan,2009;Holloway et al.,2010).Attempts have also been made for ellipsoidal particles (for example,Holzer and Sornmerfeld,2009;Guo et al.,2011).To date,however,it is still a challenging area to formulate,at a particle level,reliable equations to determine the ?uid drag on spherical particles with multi-sized system,or non-spherical particles with irregular or

Table 1

Components of particle–particle interaction forces and torques acting on particle i .Forces

Equations

Normal elastic force (f cn,ij )à4=3E n ?????R n p d 3=2

n n

Normal damping force (f dn,ij )àc n e8m ij E n ??????????

R n d n p T1=2v n ,ij

Tangential elastic force (f ct,ij )àm s 9f cn ,ij 9e1àe1àd t =d t ,max T3=2T^d

t Tangential damping force (f dt,ij )àc t e6m s m ij 9f cn ,ij 9??????????????????????????

1àd t =d t ,max p =d t ,max T1=2v t ,ij

Coulomb friction force (f t,ij )àm s

9f cn ,ij

9^d

t Torque by tangential forces (M t,ij )R c ,ij ef ct ,ij tf dt ,ij T

Torque by normal force (M r,ij )R c ,ij ef cn ,ij tf dn ,ij TRolling friction torque (M r,ij )

m r ,ij

9f n ,ij

9_x n

where 1=m ij ?1=m i t1=m j ,E n ?E =2e1àn 2T,_x n ij ?x n ij =9x n ij 9,^d t ?d t =d t ,d t ,max ?m s e2àv T=2e1àv Td n ,v ij ?v j àv i tx j R c ,ji àx i R c ,ij ,v n ,ij ?ev ij :n T:n ,

v t ,ij ?ev ij :n T:n .Note that tangential forces (f ct,ij tf dt,ij )should be replaced by f t,ij when d t Z d t ,max .

Z.Y.Zhou et al./Chemical Engineering Science 66(2011)6128–6145

6130

regular shapes.The present work has to be based on those more generally acceptable in problem solving.

2.4.Solution technique

The explicit time integration method is widely used to solve the translational and rotational motions of a system of discrete particles in the DEM simulations(Cundall and Strack,1979). Although it is established for spheres,such a method can also be extended to ellipsoids.The dif?culties in association with the extension mainly lie in two aspects:particle–particle detection and particle orientation.The methods to overcome the dif?culties are described as follows.

The detection of particle contacts for non-spherical particles is much more complicated than spherical particles.Various analy-tical methods have been proposed to detect the contacts between ellipses or ellipsoids,including the so called intersection algo-rithm(Rothenburg and Bathurst,1991),geometric potential algorithm(Ting,1992;Lin and Ng,1995),and common normal algorithm(Lin and Ng,1995).As noted by Dziugys and Peters (2001),the geometric potential algorithm is more reliable.There-fore,it is used in the present work.It should be pointed out that the computational time for contact detection is huge.This is largely because the algorithm used to determine one contact point involves the numerical solution of a sixth-order polynomial equation(Lin and Ng,1995).For a system with a large number of particles,millions of contact points are required to be determined at each time step, which signi?cantly extends the simulation time.

Particle orientation is another parameter that must be con-

sidered for non-spherical particles.It is generally described by three Euler angles(f,y,c)(Goldstein,1980).Brie?y,at each time step,for the convenience to determine the inertia tensor I i of an ellipsoid,the rotational equation expressed by Eq.(2)in the space-?xed coordinate system(x,y,z)is converted to the body-?xed coordinate system(x0,y0,z0),which is a moving Cartesian coordinate system?xed with the particle and whose axes are superposed by the principal axes of inertia.Thus,in this con-verted coordinate system,the angular velocities o0i of particles can be calculated as used for spheres;they are then used to determine the new three Euler angles on the basis of the so-called quaternion method.More details about the method can be found elsewhere(for example,Goldstein,1980;Dziugys and Peters, 2001).

The modelling of solid?ow by DEM is at the individual particle level,whilst the?uid?ow by CFD is at the computational cell level.The CFD–DEM coupling methodology for spherical particles has been well documented elsewhere(Xu and Yu,1997;Xu et al., 2000;Zhu et al.,2007).It is also used by other investigators (see,for example,Hoomans et al.,2000;Kafui et al.,2002;Li et al., 2004;Di Renzo and Di Maio,2007).Such a methodology is extended to non-spherical particles,and hence only brie?y described here.At each time step,DEM will give information such as the positions and velocities of individual particles.From these,the porosity and volumetric?uid–particle interaction force in each computational cell are calculated.CFD will then use the information to determine the gas?ow?eld,from which the?uid drag forces acting on individual particles are calculated.Incor-poration of the resulting forces into DEM will produce informa-tion about the motion of the individual particles for the next time step.

3.Simulation conditions

The bed geometry used in this work is a slot model with periodic boundary conditions applied to the front and rear direction.For such a geometry,two-dimensional CFD and three-

Table2

Typical correlations to determine the drag force on non-spherical particles. Force models Equations

Ergun(1952)D P

?k1

U g

ed vT

e1àe fT2

tk2

j v

e1àe fT

where both k1and k2are functions of particle sphericity j.

Haider and Levenspiel (1989)C D?

e1tA Re BTt

where A,B,C and D are functions of particle sphericity j.

Ganser(1993)C D

e1t0:1118eRe K1K2T0:6567Tt

0:4305

where K1and K2are Stokes correction factor and Newton’s correction factor,respectively.

Holzer and Sommerfeld (2008)C D?

??????j

161

????j

??????

jt0:42?10

0:4eàlog jT0:2

j is the sphericity of an ellipsoid de?ned as the ratio of surface area of a sphere being equivalent volume of the ellipsoid to the surface area of the ellipsoid,j?is crosswise sphericity,and j99is lengthwise sphericity.

Table3

Various parameters used in the CFD–DEM

simulation.

Variable Values

Bed geometry:

Bed width(x)600mm

Bed thickness(y)60mm

Bed height(z)3600mm

CFD cells(x,z)15?90cells

CFD cell size(D x?D z)20mm

Particle properties:

Particle density,r p2500kg/m3

Particle–particle sliding

friction,m s

0.4

Particle–wall sliding friction,

m s

0.4

Particle–particle rolling

friction,m r

0.1mm

Particle–wall rolling friction,

m r

0.1mm

Young’s modulus,E1?108Pa

Poisson ratio,n0.3

Time step,D t10à6s

Gas properties

Density 1.2kg/m3

Viscosity 1.8?10à5Pa s

Z.Y.Zhou et al./Chemical Engineering Science66(2011)6128–61456131

dimensional DEM are employed as used elsewhere (for example,Feng and Yu,2004;Zhou et al.,2010).The bed is generated under the so-called poured packing condition (Zhang et al.,2001).Ellipsoids with the same size are poured from a certain height,with randomly generated initial velocities and orientations.Dur-ing this process,ellipsoids may collide with each other or with walls,and bounce back and forth.The dynamic process proceeds until all ellipsoids reach their stable positions.The bed is then ?uidized by gas uniformly introduced from the bottom.Table 3lists the parameters including the physical properties of particles and ?uid used in the simulation.Note that the bed geometry and particle properties used for testing the drag force models and model validation purpose are slightly different from those in Table 3,and will be described respectively for each case reported.

The particle shapes considered are spheroids,with its aspect ratio varying from 0.25to 3.5.Here,as shown in Fig.2,the aspect ratio is de?ned as the ratio of diameter 2a in the polar direction to the diameter 2b (?2c )of the equatorial plane.OA is the vector from the particle centre to the polar apex.It can represent how the spheroid is oriented.For this purpose,during the DEM simulation,OA is traced all the time.

In the present work,nine cases corresponding to nine aspect ratios of ellipsoids are selected.The formed bed properties for the nine cases are listed in Table 4.The properties of the initial beds for different shaped particles are mainly re?ected by porosity (1-packing density),coordination number and orientations.Gen-erally speaking,it can be seen that a bed with spherical particles

has the highest porosity and the least particle–particle contact number.Beds with prolate/oblate particles have a lower porosity and higher particle–particle contact number,indicating a

denser

Fig.2.Illustration of (a),a prolate spheroid with aspect ratio Z ?2;and (b)an oblate spheroid with aspect ratio Z ?0.5.The vector OA is used to represent particle orientation (O,particle mass centre;A,polar apex).Table 4

Particle and bed properties used in the simulation cases.Cases

Particle shape

Size (mm),(2a ,2b ?2c ),(d v ?10mm)a Aspect ratio b (Z ?a/b )Sphericity c (j )

Number of particles Porosity

Average CN

Case 1Oblate 3.969,15.87,15.870.250.69630,0000.3858.10Case 2Oblate 6.3,1.26,1.26

0.50.91230,0000.3378.42Case 3Oblate 8.255,11.01,11.010.750.98530,0000.3377.73Case 4Spherical 10,10,10

1.0 1.030,0000.382 6.29Case 5Prolate 13.10,8.736,8.736 1.50.96930,0000.3258.31Case 6Prolate 15.87,7.937,7.937

2.00.91230,0000.3278.97Case 7Prolate 18.42,7.368,7.368 2.50.85030,0000.3399.16Case 8Prolate 2.08,6.934,6.934

3.00.79330,0000.3559.11Case

9Prolate 23.05,6.586,6.586

3.5

0.74230,000

0.3728.92

a is the principal radius in the polar direction,

b and

c are principal radii in the equatorial plane.In the present work,b ?c.

d v is th

e equivalent volume diameter,set to 10mm in this work.So that all the particles with different aspect ratios have the same volume.

For a prolate spheroid,aspect ratio Z 41;for a sphere,Z ?1;for an oblate spheroid,Z o 1.c

See Table 2for its de?nition.

Gas superficial velocity (m/s)

P r e s s u r e d r o p , Δp /(L U g )

2000

4000

6000

8000Fig.3.The relationship between pressure drop D p =LU g and gas super?cial velocity for different drag correlations.

Z.Y.Zhou et al./Chemical Engineering Science 66(2011)6128–6145

6132

packing structure for ellipsoidal particles,which is consistent with these reported (for example,see Zou and Yu,1996;Donev et al.,2004;Jia et al.,2010).

4.Results and discussion

4.1.Analysis of the drag force correlations

There are various correlations proposed to determine the particle–?uid drag coef?cient,with the typical ones listed in Table 2.In order to quantitatively test these correlations,CFD–DEM simulations are performed to determine the bed pressure drop and compare with the measurements.Wu (2001)measured the bed pressure drop using particles with different shapes,and then modi?ed the Ergun equation on this basis.The particles used in his experimental work vary from disk-shaped to cylinder-shaped,and from mono-sized to multi-sized particles.One of the cases is for a kind of oblate particles made of plastics (2a ?2b ?2c ?1.682mm ?5.5mm ?5.5mm,r p ?950kg/m 3,and j ?0.764).The measured data is shown in Fig.3by solid squares ?tted by a solid line.It should be noted that the vertical axis in the ?gure represents D p =LU g ,which can be equal to K 1U g tK 2according to the Ergun equation.Here K 1?k 2r f e1àe f T=ed v e 3f T?5945:2and K 2?k 1m f

e1àe f T2

=ed v 2e 3f T?4369:7.Hence k 1and k 2in the Ergun equation can be determined,and are,respectively,102.93and 2.5(Wu,2001).This correlation will be used in the CFD–DEM simulation as shown below.This also illustrates that to be accurate,the original Ergun equation has to be modi?ed for non-spherical particles.

In the present CFD–DEM simulation,the above experimental case is ?rst considered.For convenience,instead of the cylinder

container in the experiment,the geometry used in the simulation is a 2D slot model with periodic boundary conditions applied in the front and rear direction:240mm in length,30mm in width and 480mm in height.Totally 30,000particles and 20?40CFD cells are used.The ?uid drag coef?cient C D acting on individual plastic particles is calculated by the three correlations listed in Table 2.The obtained pressure drop D p =LU g against gas super?cial velocity is also shown in Fig.3.It can be observed that there is no signi?cant difference between the correlation of Holzer and Sommerfeld (2008)and that of Ganser (1993),but the correlation of Haider and Levenspiel (1989)signi?cantly under-estimates the pressure drop for this case.We also used the modi?ed Ergun equation (k 1?102.93and k 2?2.5)to estimate the pressure drop at a macroscopic level for the bed generated by DEM,showing consistent results with the correlation of Holzer and Sommerfeld,or Ganser.Thus,it is considered that all the three correlations except for that of Haider and Levenspiel can be used in the CFD–DEM modelling as demonstrated for the present case.However,it is not clear if such a conclusion is applicable to other cases with particles of different shapes.

On the other hand,it is noted that to date,a general and reliable correlation to determine the ?uid drag on non-spherical particles is not available yet.Hence,we have to choose a proper one from those in the existing literature for the present CFD–DEM model.It is noted that the Ergun equation or its modi?ed version is based on the macroscopic observation,and does not consider particle orientation.Moreover,ellipsoids of different aspect ratios will lead to different k 1and k 2;such quanti?cation is not yet available.The correlation of Holzer and Sommerfeld (2008)considers the effects of both shape and orientation,and is hence more preferable.It is also simpler

than

https://www.doczj.com/doc/4c7512278.html,parison of snapshots of solid ?ow patterns observed in the experiments (the top ?gures)and CFD–DEM simulations (the bottom ?gures)when the gas super?cial velocity is 3.0m/s.

Z.Y.Zhou et al./Chemical Engineering Science 66(2011)6128–61456133

the correlation of Ganser.Thus,in the present work,the correla-tion of Holzer and Sommerfeld,although not perfect,is used in all the later simulations.

4.2.Simulated macroscopic?ow behaviour

To verify the applicability of the proposed CFD–DEM approach to?uidization,experiments are carried out using a small2D slot model,with bed width of200mm,thickness of50mm and height of900mm.Wall boundary conditions are applied in each direc-tion.The experimental materials used are commercial Smarties s Chocolate Candies:2a?6.53mm,2b?2c?14.67mm;aspect ratio?0.445;and density?1349kg/m3.

Fig.4shows some snapshots observed in the experiments. Once?uidized,?ow patterns vary vigorously.The?gure just shows the typical ones at one single gas velocity.It can be seen that voids may occur on the left or right side,and particles show different orientations during?uidization.The?gure also shows the simulated results.Qualitatively,the?ow patterns observed from the experiments can all be reasonably reproduced by the CFD–DEM modelling.Note that the times for taking the snapshots are different,so quantitative comparison is not adequate.Also, there are various uncertainties in such a quantitative comparison because there are some variables beyond our control,such as particle properties,force models and so on.Nonetheless,Fig.4 clearly demonstrates that the proposed CFD–DEM model is reasonable and can capture the?ow features of ellipsoidal ?uidization.

The CFD–DEM simulations are further carried out on the basis of conditions listed in Tables2and3.Fig.5shows the early response

Fig.5.Snapshots of solid?ow patterns observed in the CFD–DEM simulations for small and large aspect ratios:(a)aspect ratio?0.25,and gas super?cial velocity U/U mf?2.0;and(b)aspect ratio?3.5,and gas super?cial velocity U/U mf?2.0.

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particle bed to the gas introduction for two cases with aspect ratio ?0.25and 3.5.The colour of particles in the ?gure represents the particle vertical velocity.It can be observed that particles are initially lifted up by gas with positive velocity.The lifted particles then descend due to the gravity with negative velocities.After the initiation of ?uidization,different cavities or bubbles are formed producing different complicated ?ow patterns.

The macroscopic behaviour of the bed can be re?ected by the bed pressure drop,which here is the pressure difference between the bed bottom inlet and the bed top outlet at the bed centre.Its temporal variation can indicate the state of bed or ?ow regimes to some degree.Fig.6shows the variation of the pressure drop with time for the cases in Fig.5but with different gas super?cial velocities.It can be seen that when aspect ratio is 0.25,the corresponding minimum ?uidization velocity is around 1.1m/s (Fig.6(a)).Below this value,the pressure drop is a constant.This is also the case when aspect ratio is 3.5(Fig.6(b)).The predicted minimum ?uidization velocity is around 1.2m/s.The observed features are consistent with the general understanding of ?uidization.

Fig.7shows the relationship between pressure drop and gas super?cial velocity.The pressure drop shown here is dimension-less,de?ned by S D p =emgN T,where S is the cross-sectional area

Fig.6.Variation of bed pressure drop with time at different gas super?cial velocities:(a)oblate spheroids with aspect ratio 0.25;and (b)prolate spheroids with aspect ratio 3.5.

0.00.10.20.30.40.50.60.70.80.91.01.11.2D i m e n s i o n l e s s p r e s s u r e d r o

Gas superficial velocity (m/s)

D i m e n s i o n l e s s p r e s s u r e d r o p

Gas superficial velocity (m/s)0.00.10.20.30.40.50.60.70.80.91.01.11.2Fig.7.The relationship between dimensionless pressure drop and gas super?cial velocities for particles with different aspect ratios:(a)oblate spheroids;and (b)prolate spheroids.

Aspect ratio

M i n i m u m f l

u i d i z a t i o n v e l o c i t y (m /s )

1.11.21.31.41.51.61.71.81.9

2.02.1Fig.8.Effect of aspect ratio on the minimum ?uidization velocity.

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the bed,mgN is the bed weight,which is a constant for all the cases.For comparison,the pressure drop determined by the Ergun equation is also calculated for the case of spherical particles.As shown in Fig.7(a),the prediction by the CFD–DEM is consistent with that calculated by the Ergun equation for spherical particles, further con?rming the validity of the proposed model.The?gure also shows cases for oblate particles with aspect ratios varying from0.25to0.75.It can be observed that,in the?xed bed,under the same gas super?cial velocity,the bed with spherical particles has the lowest pressure drop,indicating the highest bed perme-ability.With decreasing aspect ratio from1.0,the bed perme-ability becomes worse,although the bed porosity is slightly higher than spherical particles for the case of aspect ratio?0.25. This indicates that the bed permeability is determined by not only the bed porosity but also particle shape(particle sphericity), which can be clearly seen in the Ergun equation.When the bed is?uidized,the bed pressure drops for the?ve cases are similar,?uctuating around the bed weight.Fig.7(b)shows the case for prolate particles with aspect ratios ranging from 1.0to 3.5. Interestingly,the relationships are similar when aspect ratios are larger than1.5.They have the similar bed permeability,but much less than spherical particles.

Fig.7also shows that the minimum?uidization velocities for different aspect ratios are different.The minimum?uidization velocity corresponds to the velocity at which the maximum pressure drop occurs before the bed is?uidized.The variation of the minimum?uidization velocity against aspect ratio is shown in Fig.8.It can be seen that when aspect ratio decreases from1.0, the minimum?uidization velocity decreases.Such a trend is qualitatively consistent with that observed in the literature (Liu et al.,2008).When aspect ratio increases from 1.0,the minimum?uidization velocity deceases?rst and then increases. The lowest minimum?uidization velocity occurs at aspect ratio around1.75,which indicates the lowest bed permeability.

4.3.Analysis of some microscopic behaviour

The so-called microscopic analysis is a particle scale analysis on the basis of the trajectories or velocities of particles and their corresponding transient forces.Such microscopic information is dif?cult to obtain experimentally but can be readily generated from the CFD–DEM simulation.Its analysis is of importance to elucidating the mechanisms of particle–?uid?ow as

demonstrated

Fig.9.Spatial distributions of some typical results about?ow structures when aspect ratio is0.25(top)at time4.29s and3.5(bottom)at time3.64s:(a)and(e)?ow pattern with coordination number;(b)and(f)?uid?ow?eld and porosity distribution;(c)and(g)particle velocity?eld;and(d)and(h)particle orientation.

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by Zhu et al.(2008).Porosity and coordination number (the number of particles in contact with a given particle)have been widely used to examine the ?ow structure of particles in the study of,for example,particle packing (Liu et al.,1999;Yang et al.,2000)and granular ?ow (Zhu and Yu,2004;Yang et al.,2003;Zhou et al.,2004).As an example,Fig.9shows a snapshot of solid ?ow pattern with the microscopic properties for cases 1and 9.Here the microscopic information includes the spatial distribution of coordination numbers (Fig.9(a)and (e)),porosity distribution and ?uid ?ow ?eld (Fig.9(b)and (f)),particle velocity ?eld (Fig.9(c)and (g)),and particle orientation (Fig.9(d)and (h)),which is here represented by the vectors OA as indicated in Fig.2.The detailed analysis of such microscopic information has been carried out,in order to understand how particle shape affects the ?ow behaviour of ellipsoids in a ?uidized bed,as given below.

Coordination number (CN)is the number of contacts for a given particle.It varies with the de?nition of critical separation less than which two particles are de?ned in contact.Ideally,the critical separation is zero to represent a real contact between particles.But in practical application,it is often slightly larger than zero.This treatment does not affect the analysis as long as the analysis is consistent.In this work,the critical separation is set to 1%d v ,as used by Yang et al.(2000).As an example,Fig.9(a)and (e)shows the CN spatial distribution in ?uidized beds.It can be observed that large CN mainly exists in dense regions while small CN in the regions with voids.This is consistent with the general understanding from the ?uidization of spheres (Kunii and Levenspiel,1991).

To illustrate the effect of aspect ratio on CN,Fig.10shows the variations of bed averaged CN,i.e.,the average CN for all particles in a bed,with time in ?uidized beds for prolate and oblate spheroids,respectively.It can be seen that CN ?uctuates around a certain value,as a result of the appearance and disappearance of voids.Overall,spheres have the lowest CN among the nine cases at U /U mf ?2.0.The averaged value of CN over a certain time for different aspect ratios is further shown in Fig.11,together with the CN in ?xed or packed beds shown in Table 3.Clearly,the trends are qualitatively comparable.However,minor difference can also be identi?ed.For example,oblate spheroids with aspect ratio 0.5have a higher CN in ?xed beds but it is not the case for ?uidized beds.This is because that the ?ow patterns are not certain in ?uidized bed,which may affect the CN signi?cantly.But

generally speaking,when aspect ratio decreases from 1.0for oblate spheroids,or increases from 1.0for prolate spheroids,CN increases.Spheres have the smallest CN in all the cases.Such a feature indicates that the large number of particle–particle con-tacts for prolate/oblate spheroids may lead to the large particle–particle heat transfer.This may be signi?cant for the thermal behaviour in ?uidized beds when the particle–particle conduction is important,which happens to particles with high thermal conductivity (Zhou et al.,2009b ).Further studies should be carried out on how particle shape affects the ?ow and thermal behaviour in a ?uidized bed.

Orientation is one of the most important properties for non-spherical particles.It is known that there is no preferred orienta-tion for spheres,because they are homogeneous in all directions.But this is not the case for non-spherical particles,particularly for very platy or elongated particles.The examination of particle orientation in a ?uidized bed would provide some useful informa-tion about its structure.In the present DEM simulation,the initial

0Time (s)

2345678A v e r a g e C N

0.20.50.751.0

2345678A v e r a g e C N

1.01.5

2.5

3.5

51015200Time (s)

5101520

Fig.10.Variation of bed averaged CN with time for different aspect ratios when gas super?cial velocity U /U mf ?2.0:(a)oblate spheroids;and (b)prolate spheroids.

Aspect ratio

B e d -a v e r a g e d

C N

0.0

0.5

1.0

1.5

2.0 2.5

3.0

3.5

4.0

3.04.05.06.07.08.09.010.0Fixed bed (no gas)Fluidized bed (U/U mf =2.0)

Fig.11.Bed averaged CN for different aspect ratios respectively for ?xed beds when U /U mf ?0,and ?uidized beds when U /U mf ?2.0.

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orientation of a particle is randomly assigned,then its orientation in a packing and then ?uidized process is traced by the vector OA .As an example,Fig.9(d)and (h)shows the spatial distributions of OA .To be more quantitative,the concept of longitude as used in the world map is used here.In particular,OA longitude a is de?ned against an equatorial plane.The equatorial planes can be the horizontal X –O –Y plane,or the vertical X –O –Z or Y –O –Z planes.OA longitude can vary from 01to 3601.01is de?ned at the positive X or Y axis,and then increases in the anticlockwise direction.An example of OA longitude a against the horizontal X –O –Y plane is shown in the inset ?gure in Fig.12(a),where OA 00is the projected vector of OA on the X –O –Y plane.

Fig.12(a)shows the orientation information for spheres at different times in a ?uidized bed.The horizontal axis is OA longitude a ,and the vertical axis is the number of particles countered with similar OA longitudes (D a ?31).As expected,the distribution of a on the horizontal X –O –Y plane is uniform.Note that distributions of a on the vertical X –O –Z and Y –O –Z planes are not shown here due to their similarity to Fig.12(a).It shows that the orientation for spheres is isotropic at any time during the ?uidization process.For further quanti?cation,longitude a vary-ing from 01to 3601can be transformed to an orientation angle less than 901.For example,the orientation angle is calculated as 1801àa if a A (901,1801),or a à1801if a A (1801,2701),or 3601àa if a A (2701,3601).Then the variations of the bed-averaged orientation angles against the time on different equatorial planes can be plotted,and shown in Fig.12(b).As expected,the orientation angles slightly ?uctuate around 451because of the isotropic nature of orientation of spheres.

However,it is not the case for oblate and prolate particles.Fig.13shows the distribution of OA longitude for oblate particles when aspect ratio is 0.25.It can be seen from Fig.13(a)that the OA longitude distribution on the X –O –Y plane is relatively uni-form before ?uidization.This is because OA can theoretically point to any direction during the formation of a packed bed.Due to the small thickness of the bed,OA longitude distribution could be slightly non-uniform as illustrated in the ?gure.However,when the bed becomes ?uidized,the longitude changes signi?-cantly,and ?nally ?uctuates at a W shaped distribution.OA prefers pointing to 01and 1801,i.e.the X axis.Particles do not tend to be along the Y axis.Such a feature is mainly caused by the bed geometry as mentioned above.Fig.13(b)and (c)shows the longitude distribution on the vertical X –O –Z and Y –O –Z planes,respectively.It can be seen that the distribution is non-uniform at the beginning (t ?0s),where most of the particles are at 901and 2701,less particles at 01and 1801.It indicates that oblate particles prefer facing upward or downward,forming an ordered structure to some degree.It is consistent with the general understanding that such a structure is stable for platy particles,with a minimum potential energy of the system.Such a structure usually generates a relatively large resistance to ?uid ?ow,giving a large pressure drop and low permeability,as demonstrated in Fig.7.The lower the aspect ratio,the more signi?cant the effect.But when the bed becomes ?uidized by gas,the degree of non-uniformity is reduced.This is because there exist strong particle–particle collisions,which will affect the particle orientation.Fig.13(d)further shows the variations of bed-averaged orientation angles,which ?uctuate around a certain value.This is mainly caused by the periodic ?ow patterns formed in ?uidized beds,as the case for other parameters,e.g.the bed pressure drop or CN.

Fig.14shows the distribution of orientations of prolate particles when aspect ratio is 3.5.The OA longitude distribution on the horizontal X –O –Y plane is shown in Fig.14(a).Its distribution is similar to the oblate particles shown in Fig.13(a),indicating that in ?uidized beds,OA prefers pointing to the X axis at 01or 901.Such a distribution is mainly caused by the bed with a small thickness.Fig.14(b)and (c)shows the OA longitude distribution respectively on the vertical X –O –Z and Y –O –Z planes.It can be observed that the distribution is non-uniform but the degree of non-uniformity is reduced when the bed is ?uidized.Most of particles are at 01and 1801,less particle at 901and 2701.It indicates that prolate particles prefer oriented horizontally.This is because prolate particles at high aspect ratios,e.g.3.5in this case,can obtain the stable structure for such an orientation.This is particularly true for particles,which are not ?uidized.Such a feature is consistent with the general understanding of stability of elongated particles.Fig.14(d)illustrates the variations of bed-averaged orientation angles against time,revealing that particle orientations change more signi?cantly than oblate particles shown in Fig.13(d).

To be more quantitative and general,the relationship between bed-averaged orientation angle and aspect ratio is established and shown in Fig.15for both ?xed beds and ?uidized beds.In ?xed beds,orientation angles on the horizontal X –O –Y plane are all at 451,independent of aspect ratios.This is because particles can point to any direction on this plane.With aspect ratio increasing from 0.25to 3.5,orientation angles on the vertical X –O –Z and Y –O –Z planes decrease gradually.However,once

beds

Time (s)

O r i e n t a t i o n a n g l e (d e g r e e )

4041424344454647484950Fig.12.(a)Distributions of particle orientations for spheres by OA longitude a on the horizontal X –O –Y plane;and (b)variation of bed averaged orientation angle on different planes.

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are ?uidized,the variation of orientation angles with aspect ratio changes as shown in Fig.15(b).The angles on the X –O –Y plane ?rst increase and then decrease with aspect ratio but all less than 451,indicating the effect of bed geometry and gas ?ow.The orientations for both oblate and prolate particles are re-constructed and their OA longitude prefers pointing to the X axis.The variation of orientation angle on the X –O –Z plane is consis-tent with that in ?xed beds,indicating a strong non-uniformity,particularly for low (e.g.0.25)and high (e.g.3.5)aspect ratios.However,the trend of orientation angle on the Y –O –Z plane is quite different from the ?xed bed,particularly for prolate parti-cles.When aspect ratio is between 1.0and 2.5,the orientation angle is almost a constant around 451,which indicates a more uniform distribution of orientations.With further increase of aspect ratio,the orientation angle decreases a little.The features shown in this ?gure suggests that strong gas ?ow and geometry used in this work cause signi?cant changes in particle orientation.In the ?uidized bed,inter-particle forces cause particles rotating signi?cantly,and hence their orientations change correspond-ingly.In the dense regions,as seen in Fig.9(a)and (e),the orientations of particles behave similar to that in the ?xed bed.Thus,the orientation of particles in a ?uidized bed is complicated,affected by gas ?ow rate and bed geometry.

The ?ow behaviour of particles in gas ?uidization is controlled by the interactions between particles,between particles and gas,and between particles and wall,in addition to the gravitational force.The analysis of these interaction forces is therefore the key to developing a better understanding of the underlying mechan-isms.CFD–DEM approach has shown its unique advantages for such research (Zhu et al.,2008).Here we apply such analysis to ellipsoidal particles.Fig.16shows the spatial distributions of those interaction forces for aspect ratios 0.25and 3.5.The drag force acting on individual particles varies spatially,largely depending on the motion direction of particles.The vertical drag force is always positive,dragging particles moving upward (Fig.16(a)and (d)),as a results of the continuous up?ow of gas.The horizontal drag force can be positive or negative,pushing particles rightward or leftward (Fig.16(b)and (e))depending on the location of varying voids.Generally,the drag force in a dense region is larger than that in a loose region due to the difference in porosity and permeability.Fig.16(c)and (f)shows the spatial distributions of the normal contact particles between particles.It can be observed that large contact forces are found mainly in the region where particles are crowded.

To quantify the effect of aspect ratio on force structures,the probability density distributions of the magnitudes of normal contact forces and ?uid drag force in ?uidized beds are plotted in Fig.17when U /U mf ?2.0.It can be seen that for the normal contact forces,aspect ratio does not affect the distribution much (Fig.17(a)and (b)).But it affects the drag force distributions (Fig.17(c)and (d)).It can be observed that there are two peaks when aspect ratio is close to 1.0,e.g.0.75,1.0and 1.50.If aspect

Longitude at the horizontal X-O-Y plane

N u m b e r o f p a r t i c l e s

Longitude at the vertical X-O-Z plane

N u m b e r o f p a r t i c l e s

0200

400

600

8001000

Longitude at the vertical Y-O-Z plane

N u m b e r o f p a r t i c l e

Time (s)

O r i e n t a t i o n a n g l e (d e g r e e )

Fig.13.Distributions of particle orientations (OA longitudes on different planes)for oblate particles when aspect ratio is 0.25:(a),horizontal X –O –Y plane;(b),vertical Y –O –Z plane;(c),vertical Y –O –Z plane;and (d),variation of bed averaged orientation angles on different planes.

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ratio further decreases from 0.75or increases from 1.5,the two peaks gradually disappear.For example,only one peak can be observed for aspect ratios of 0.25and 3.5.The inter-particle contact forces can be time-bed averaged

de?ned by 1=et 1àt 0TR t 1t 0ee1=N c TP

9f n ,ij 9Tdt ,where N c is the total number of contacts in the bed,f n is the normal contact force

Longitude at the horizontal X-O-Y plane

N u m b e r o f p a r t i c l e s

Longitude at the vertical X-O-Z plane

N u m b e r o f p a r t i c

l e s

100

200

300

400

500600

Longitude at the vertical Y-O-Z plane

N u m b e r o f p a r t i c

l e s

100

200

300

400

500600

Time (s)

O r i e n t a t i o n a n g l e (d e g r e e )

2025

35404550Fig.14.Distributions of particle orientations (OA longitude on different planes)for prolate particles when aspect ratio is 3.5:(a),horizontal X –O –Y plane;(b),vertical X –O –Z plane;(c),vertical Y –O –Z plane;and (d),variation of bed averaged orientation angle on different planes.

0.5

1.52

2.53

3.5

Aspect ratio

2025303540455055606570

7580O r i e n t a t i o n a n g l e (d e g r e e )

On X-O-Y plane On X-O-Z plane On Y-O-Z plane

Aspect ratio

303540455055606570O r i e n t a t i o n a n g l e (d e g r e e )

On X-O-Y plane On X-O-Z plane On Y-O-Z plane

Fig.15.Bed averaged orientation angles for different aspect ratios for:(a),?xed beds when U /U mf ?0;and (b),?uidized beds when U /U mf ?2.0.

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between particle i (A (1,N ))and particle j (A (1,N )),where N is the number of particles in the bed and i o j ,(t 1àt 0)is the time duration for average.Fig.18(a)shows the variation of time-bed averaged normal contact force against aspect ratio.It can be seen that spheres have the strongest particle–particle interaction force.This could be explained by the different R n calculated at the contact point for different shaped particles.The curvature at the contact point for prolate or oblate particles can be very large,corresponding to a small radius.However,it is interesting to note that for oblate particles,there is a minimal averaged normal contact force at aspect ratio 0.5.For prolate particles,the minimum occurs at aspect ratio 1.5.Such a feature is quite different from the CN variation with aspect ratio shown in Fig.11.It indicates that in ?uidized beds,spheres have the smallest CN but the strongest interaction for a given U /U mf .The ?gure also shows that prolate particles have weaker inter-particle interaction than oblate particles under comparable conditions (U /U mf ?2.0).

The time-bed averaged ?uid drag force for different aspect ratios is also examined and shown in Fig.18(b).It shows the similar feature to the effect of aspect ratio on the normal contact force.Under the comparable conditions of U /U mf ?2.0,spheres have the largest ?uid drag force.For oblate particles,the drag force is the smallest when aspect ratio is around 0.75.For prolate

particles,the smallest occurs at aspect ratio around 2.0.Interest-ingly,the trend shown is similar to the bed porosity variation against aspect ratio shown in Table 4or those reported in the literature (Donev et al.,2004;Guises et al.,2009;Jia et al.,2010).But it should be noted that the drag force is signi?cantly affected by ?uid velocity,local porosity and the cross-sectional area A ?as indicated in Eq.(8).As the drag force increases with the increase of porosity,the dominant factors for the effect of aspect ratio on the drag force should be the ?uid velocity and cross-sectional area A ?,which is closely related to particle orientation.This also illustrates that particle orientation in gas ?uidization of ellipsoids is an obvious feature different from spheres.

5.Conclusions

The CFD–DEM approach is extended in this work to study gas ?uidization of ellipsoidal particles,and shown to be promising to generate particle scale information to develop better understand-ing of this complicated ?ow system.The following conclusions can be drawn from the current study:

Particle shape affects the bed permeability.The bed perme-ability is worsened signi?cantly for oblate particles.

Prolate

Fig.16.Snapshots of solid ?ow pattern with different forces when aspect ratio is 0.25(top)and 3.5(bottom):(a)and (d),drag force in the vertical direction;(b)and (e),drag force in the horizontal direction;and (c)and (f),particle–particle contact force.

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Particle-pa rticle contact force (log10(f n ))

P r o b a b i l i t y d e n s i t y

Fluid drag (log10(f d ))Fluid drag (log10(f d ))

P r o b a b i l i t y d e n i s t y

-4.0

-3.5

-3.0-2.5-2.0-1.5

0.00.20.40.60.81.01.21.41.6

1.8

2.00.250.500.751.0

-4.0

-3.5-3.0-2.5-2.0-1.5

0.00.2

0.40.60.81.01.21.41.6

1.8

2.0 1.01.52.5

3.5

Fig.17.Probability density distributions of the magnitudes of the normal contact forces between particles (top)and ?uid drag on particles (bottom)in ?uidized beds when U /U mf ?2.0:(a)and (c),oblate spheroids;and (b)and (d),prolate

spheroids.

Aspect ratio

T i m e -a v e r a g e d n o r m a l c o n t a c t f o r c e (N )

0.03

0.040.050.060.070.08

0.09Aspect ratio

T i m e -a v e r a g e d d r a g f o r c e o n a p a r t i c l e (N )

0.0030

0.0032

0.0034

0.0036

0.0038

Fig.18.Effect of aspect ratio on the time-bed averaged forces under the condition of U /U mf ?2.0:(a),particle–particle normal contact force;and (b)?uid drag force.

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particles do not affect the bed permeability much for the cases considered in the present work.For oblate particle,the mini-mum?uidization velocity decreases with aspect ratio decreas-ing from 1.0.But for prolate particles,the minimum ?uidization velocity decreases?rst then increases.

Spheres have the smallest coordination number in both?xed and?uidized beds.With aspect ratio deviating from1.0,the coordination number generally increases.

The analysis of particle orientations shows that the bed structures for ellipsoids are not random as that for spheres.

To obtain their stable structure,oblate particles prefer facing upward or downward while prolate particles prefer horizontal orientation.With the bed becoming?uidized,the non-uni-formity of the distribution of particle orientations is reduced. Spheres have the largest particle–particle contact force under the comparable conditions.With aspect ratio deviating from

1.0,particle–particle interaction becomes relatively weak.The

minimum exists at aspect ratio0.5for oblate particles and

1.5for prolate particles.The effect of aspect ratio on the drag

force shows the similar feature.It indicates that the particle orientation is one of the dominant factors in affecting the drag force for the gas?uidization of ellipsoids.

Finally,it should be pointed out that the proposed CFD–DEM model is not perfect and requires further developments.For example,particle–particle interaction force models used for ellipsoids are simply extended from spheres.The particle–?uid drag force is determined based on the available correlations in the literature.Their accuracy and applicability are questionable. Moreover,to be general,there may be a need to consider other particle–?uid interaction forces.Nonetheless,the results clearly indicate that this model can capture the key packing and?ow features of ellipsoidal particles.While offering a solid basis for further developments,the model can be used to describe?ow and related behaviour in?uidization.

As clearly demonstrated in this study,this technique can provide detailed dynamic information at a particle scale,includ-ing not only the velocity but also the transient local structures and forces of various types.Previous studies have a dif?culty in obtaining information about the last two.Consequently,there have been problems in probing the underlying mechanics and solving practical problems reliably.In fact,it has been known for years that particle shape plays a signi?cant role in governing the behaviour of particles.Its quanti?cation is however dif?cult, particularly at a microscopic level.Here by the use of an extended CFD–DEM model,we have examined some of the key dynamics of ellipsoids in?uidization.The resulting information should be useful.For example,the structural results about CN and particle orientation can be used in the determination of transport proper-ties such as permeability related to pore connection and thermal conductivity related to particle connection.The force results are useful in understanding why particle shape matters in the?ow of ellipsoidal particles;they can also be used in assessing the possible breakage of particles.Practical particles are often not spherical,so there is a need to extend the CFD–DEM technique to describe the behaviour of such particles more generally.We believe such a development is important not only to fundamental research but also practical problem solving.

Nomenclature

a,b,c Principal semi-diameters of an ellipsoid,m

A0,B0De?ned in Eqs.(3)and(4),m

A?Cross-sectional area of an ellipsoid perpendicular to the ?uid?ow,m2C D Fluid drag coef?cient,dimensionless

c n Normal damping coef?cient,dimensionless

c t Tangential damping coef?cient,dimensionless

d ij Distanc

e between the centres o

f particle i and j,m

d p Particl

e diameter,m

d v Equivalent volum

e diameter,m

E Young’s modulus,Pa

f c,ij Particle–particle contact force,N

f d,ij Particle–particle dampin

g force,N

f d,i Fluid dra

g force on particle i,N

F pf Volumetric?uid–particle interaction force,N/m3

f pf,i Particle–?uid interaction force on particle i,N

g Gravitational acceleration,m/s2

I Moment of inertia of particle,kg m2

k Turbulence kinetic energy,m2/s2

k1,k2Coef?cients in the Ergun equation,dimensionless

K1Stokes correction factor,dimensionless

K2Newton’s correction factor,dimensionless

k c Number of particles in a computational cell,dimensionless k i Number of particles in interaction with particle i, dimensionless

L Bed height,m

M n Torque generated by the normal force,Nám

M r Rolling torque,Nám

M t Tangential torque,Nám

m Mass of particles,kg

N Number of particles in a bed,dimensionless

N c Number of particle–particle contacts in a bed, dimensionless

p Fluid pressure,Pa

R Vector from the mass centre of the particle to the contact point,m

R Particle radius,m

Re Reynolds number,dimensionless

S Cross-sectional area of a bed,m2

t Time,s

u Fluid velocity,m/s

U Gas super?cial velocity,m/s

U mf Minimum?uidization velocity,m/s

v Particle translational velocity,m/s

V Volume of particle,m3

D V Volume of a computational cell,m3

x X direction in a coordinate system,m

D x Width of a computational cell,m

z Z direction in a coordinate system,m

D z Height of a computation cell,m

Greek

a OA longitude on different equatorial planes,degree

w Empirical coef?cient de?ned in Table1,dimensionless d t,max Maximum d t when sliding starts,m

d n Relativ

e normal displacement at contact,m

d t Relativ

e tangential displacement at contact,m

Unit vector of d t,dimensionless

e f Porosity,dimensionless

e Dissipation rate o

f turbulent kinetic energy,m2/s3

f,y,c Euler angles,degree

Z Aspect ratio,dimensionless

j Particle sphericity,dimensionless

m f Fluid molecular viscosity,kg/(m s)

m e Fluid effective viscosity,kg/(m s)

m r Rolling friction coef?cient,m

m s Sliding friction coef?cient,dimensionless

Z.Y.Zhou et al./Chemical Engineering Science66(2011)6128–61456143

n Poisson ratio,dimensionless

r Density,kg/m3

s Stress tensor,Pa

x Angular velocity,1/s

^x Unit vector of x,dimensionless

Subscripts

c Contact

d Damping or drag

f Fluid

i Particle i

ij Between particles i and j

j Particle j

n Normal component

p Particle

r Rolling

t Tangential component

w Wall

Acknowledgements

The authors are grateful to the Australian Research Council (ARC)and BlueScope Steel Research for the?nancial support of this work,and the NCI National Facility for the support in computation.

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