Prediction of particle motion in a two-dimensional bubbling fluidized bed using discrete hard-sphere model

Chemical Engineering Science 60 (2005) 3217–3231

www.elsevier.com/locate/ces

Prediction of particlemotion in a two-dimensional bubbling fluidized bedusing discrete hard-spheremodel

Huilin Lua,∗, ShuyanWanga,Yunhua Zhaoa, LiuYanga, Dimitri Gidaspowb, Jiamin Dingc

aDepartment of Power Engineering, Harbin Institute of Technology, Harbin, 150001, ChinabDepartment of Chemical and Environmental Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA

cDepartment of HYDA, IBM, Rochester, MN 55901, USA

Received 2 October 2004; received in revised form 4 November 2004; accepted 19 January 2005Available online 19 March 2005

Abstract

The solids motion in a gas–solid fluidized bed was investigated using a discrete hard-sphere model. Detailed collision between particlesand a nearest list method are presented. The turbulent viscosity of gas phase was predicted by subgrid scale (SGS) model. The interactionbetween gas and particles phases was governed by Newton’s third law. The distributions of concentration, velocity and granular temperatureof particles are obtained. The radial distribution function is calculated from the simulated spatio-temporal particle distribution. The normaland shear stresses of particles are predicted from the simulated instantaneous particle velocity. The pressure and viscosity of particles areobtained from both the kinetic theory of granular flow and the calculated stresses of particles. For elastic particles the individual lateraland vertical particle velocity distribution functions are isotropic and Maxwellian. The observed anisotropy becomes more pronounced withincreasing degree of inelasticity of the particles.� 2005 Elsevier Ltd. All rights reserved.

Keywords:Discrete hard-sphere approach; Kinetic theory of granular flow; Reynolds stress

1. Introduction

The hydrodynamics of bubbling fluidized bed technologyhas widespread applications in the petroleum, chemical,and energy industries. Understanding of the fundamentalphenomena including the interaction between the particlesis needed to improve the design and performance of flu-idized beds. Broadly speaking the simulation approachesof two-phase flow in bubbling fluidized bed can be classi-fied into Euler–Lagrange discrete particle trajectory modeland Euler–Euler two-fluids model. Several attempts havebeen made to simulate bubble and particle motion in flu-idization using the Eulerian–Lagrangian approach. In thismethod, the fluid phase is treated as a continuum, while theparticles are traced individually by solving the Newtonian

∗ Corresponding author. Tel.: +86010045186412258.E-mail address:[email protected](H. Lu).

0009-2509/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.01.010

equation of motion. The mechanism of particle-to-particlecollisions can be described by soft- or hard-sphere mod-els (Crowe et al., 1997). In the soft-sphere model, theNewtonian equation of motion of individual particles wasintegrated. The collisions between particles and betweenparticle and wall were simulated using Hooke’s linearsprings and dash pots (Cundall and Strack, 1979). Thismodel has been used for investigating inter-particle forceeffect on fluidization characteristics (Kuwagi et al., 2000;Rhodes et al., 2001), mixing and segregation characteristics(Kaneko et al., 1999; Limtrakul et al., 2003), fluid dynam-ics (Rong et al., 1999), particle residence time (Wang and(Rhodes, 2003), and minimum fluidization velocity (Kafui(et al., 2002) in the bubbling fluidized bed.Kawaguchi et al.(1998)simulated particle motions in the spouted bed usingthe discrete element method (DEM). The DEM simulationhas been used to investigate the mechanism of agglomer-ation in a fluidized bed of cohesive fine particles (Mikami(et al., 1998; Kuwagi and Hoiro, 2002). On the other hand,

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3218 H. Lu et al. / Chemical Engineering Science 60 (2005) 3217–3231

the hard-sphere models are successfully used to modelparticle–particle and particle–wall collisions. The bubbleformation and motions of particles in bubbling fluidizedbed are investigated (e.g.,Hoomans et al., 1996; Xu andYu, 1997; Ouyang and Li, 1999; Helland et al., 2000; vanWachem et al., 2001). Goldschmidt et al. (2002)studiedthe particle velocity distributions and collision character-istics from dynamic discrete particle simulations in densegas-fluidized beds. They found that for elastic particlesthe individual particle velocity function was an isotropicMaxwellian, and an anisotropic Maxwellian particle veloc-ity distribution was observed for highly inelastic and roughparticles. Recently,Tsuji et al. (1998)andYuu et al. (2001)studied air and particle motions in a three-dimensional tur-bulent fluidized bed using the direct Monte Carlo simulationmethod (DMCS).On the other hand Euler–Euler two-fluids models con-

sider all phases to be continuous and fully interpenetrat-ing. The equations employed are a generalization of theNavier–Stokes equations for interacting continua. Owing tothe continuum representation of the particle phases, Eule-rian models require additional closure laws to describe therheology of particles. In most recent continuummodels con-stitutive equations according to the kinetic theory of gran-ular flow are incorporated. This provides explicit closuresthat take energy dissipation due to non-ideal particle–particlecollisions into account by means of the coefficient of resti-tution. The models predicted well the bubble formation andthe distribution of time-averaged solids concentration in bub-bling fluidized beds (e.g.,Ding and Gidaspow, 1990; Lyczkowsky et al., 1993; Kuipiers et al., 1993; Schmidt andRenz, 1999; Peirano et al., 2002). An isotropy of the parti-cle velocity distribution is assumed in the kinetic theory ofgranular flow, which makes dense gas–solid fluidized bedsquestionable for two reasons: (1) The net action of all ex-ternal forces in vertical direction will disturb isotropy ofthe particle velocity distribution, if particles are significantlyaccelerated between successive collisions. (2) It seems un-likely that all impact angles are of equal likelihood, whichcauses collisional anisotropy leading to anisotropic velocitydistributions of particles (Goldschmidt et al., 2002).The purpose of this paper is to simulate a detailed pro-

cess of motion of particles in a bubbling fluidized bed usingthe hard-sphere discrete particle model. The particle motionconsists of both collision and free flight steps. The particleinteraction is described as instantaneous, binary, and inelas-tic collision with friction. The particle motion is controllednot only by gas-phase flow determining the flight step butalso by action from its neighboring particles. The interac-tion forces between gas and particle obey Newton’s thirdlaw. The gas-phase turbulent flow is modeled by a simplesubgrid scale (SGS) model. The flow mechanism in a two-dimensional group-D particle bubbling fluidized bed is dis-cussed. The particle pressure is calculated based on boththe kinetic theory of granular flow and the normal stressesobtained from the simulated instantaneous particle velocity

distributions. The radial distribution function is predictedbased on the simulated spatio-temporal particle distribu-tions. The simulation results are compared to the isotropicMaxwellian particle velocity distribution for the kinetic the-ory of granular flow. An agreement with the kinetic the-ory is obtained for elastic particles. The individual particlevelocity distribution function is isotropic and Maxwellian.However, for inelastic particles an anisotropic Maxwellianvelocity distribution is obtained. The observed anisotropybecomes more pronounced with increasing degree of inelas-ticity of the particles.

2. Eulerian–Lagrangian gas–solid flow model

2.1. Continuity and momentum equation for gas phase

The Eulerian–Lagrangian method computes the Navier–Stokes equation for the gas phase and the motion of individ-ual particles by the Newtonian equations of motion. For thegas phase, we write the equations of conservation of massand momentum as

�

�t(�g�g)+ ∇ · (�g�gvg)= 0, (1)

�

�t(�g�gvg)+ ∇ · (�g�gvgvg)

= −∇p + ∇ · �g + �g�gg−Np(v)∑

i=1

fi . (2)

The coupling term∑Np(v)

i=1 fi between the gas and the parti-cle phase is estimated as the sum of the drag on each particlewithin the corresponding fluid control volume. The stresstensor of gas phase can be represented as

�g = �g[∇vg + (∇vg)T] − 23�g(∇ · vg)I . (3)

The gas phase turbulence is modeled using a simple subgridscale (SGS) model. The model was first used and proposedby Deardorff (1971)for channel turbulence flow. The SGSmodel simulates the local Reynolds stresses arising from theaveraging process over the finite-difference grid by aboutthe crudest of methods, that involve an eddy coefficient withmagnitude limited in some way by the size of the averagingdomain. This domain is considered to be the grid volume ina detailed numerical integration. Then the eddy coefficientbecomes a “subgrid scale” coefficient.

�g = �lam,g + �g(Ct�)2√

Sg : Sg, (4)

where�=(�x �y)1/2 andSg= 12[∇ ·ug+∇ ·uTg ]. Deardorff

(1971)suggestedCt be in the range of 0.1–0.2. In this studyCt = 0.1 was used in the simulations.

2.2. Equations for the solid phase

Analysis of particle’s collision in this study is basedon collision dynamics with the following assumptions: (1)

H. Lu et al. / Chemical Engineering Science 60 (2005) 3217–3231 3219

Fig. 1. Two particles before (a) and after (b) impact.

particles are spherical and quasi-rigid, (2) collisions are bi-nary and instantaneous with a contact point, (3) interactionforces are impulsive and all other infinite forces are negligi-ble during collision, (4) particle motion is two dimensionalwith the particle mass center moving in one plane, (5) boththe restitution and the friction coefficients are constant in asimulation.We considered two particles of diameterd1 andd2 with

massesm1 andm2, respectively. The positions of the spherecenters are described by the vectorsr1 and r2. The unitnormaln at the contact point is defined by

n=r1 − r2

|r1 − r2|. (5)

Fig. 1 shows the situation of two particles before andafter collision. For a binary collision of these spheres thefollowing equations can be derived by applying Newton’ssecond and third laws (e.g.,Hoomans et al., 1996; Hellandet al., 2000; van Wachem et al., 2001):

m1(v1 − v1,0)= J , (6)

m2(v2 − v2,0)= −J , (7)

m1d21

4(�1 − �1,0)= n× J , (8)

m2d22

4(�2 − �2,0)= −n× J , (9)

whereJ is the impulse force. From Eqs. (6)–(9) it is clearthat the post-collision velocities of both particles can becalculated when the impulse is known. The relative velocityat the contact point is

v12 = (v1,c − v2,c). (10)

The tangential unit vector is as follows:

t =v12,0 − n(v12,0 · n)|v12,0 − n(v12,0 · n)|

. (11)

The coefficients of (normal) restitutione, (dynamic) friction� and tangential restitution�t are defined as

v12 · n= −e(v12,0 · n), (12)

|n× J | = −�(n · J ), (13)

n× v12 = −�t (n× v12,0). (14)

The definitions of restitution coefficients in Eqs. (12)–(14)hold for spherical particles but not for non-spherical parti-cles where energy inconsistencies may be resulted in. De-pendence of these coefficients on particle size and impactvelocity is not taken into account in this model. If a colli-sion occurred at a normal impact velocity less than a thresh-old value (typically 10−4m/s), the collision is assumed tobe perfectly elastic, ore= 1.0. The threshold value was in-troduced for computational convenience. It does not have asignificant effect on the simulation results. The normal com-ponent of the impulse vector is

Jn = −(1+ e)m1m2

(m1 +m2)(v12,0 · n). (15)

For the tangential component, two types of collisions can bedistinguished, namely sticking and sliding collisions. If thetangential component of the relative velocity is sufficientlyhigh in comparison to the coefficient of friction and tangen-tial restitution that gross sliding occurs throughout the wholeduration of the contact, the collision is of the sliding type.The non-sliding collisions are of the sticking type. When�tis equal to zero the tangential component of the relative ve-locity becomes zero during a sticking collision. When�t isgreater than zero in such a collision, reversal of the tangen-tial component of the relative velocity will occur. These twotypes of collisions can be determined from

�<(1+ �t )m1m2

3.5(m1 +m2)Jn(v12,0 · t) sliding, (16)

��(1+ �t )m1m2

3.5(m1 +m2)Jn(v12,0 · t) sticking. (17)

For collisions of the sticking type, the tangential impulse is

Jt = −(1+ �t )m1m2

3.5(m1 +m2)(v12,0 · t). (18)

For collisions of the sliding type, the tangential impulse isgiven by

Jt = −�tJn. (19)

The total impulse vector is then simply obtained by additionof

J = Jnn+ Jt t . (20)

The post-collision velocities of particles 1 and 2 can becalculated from Eqs. (6) and (7). For particle to wall colli-sion, the mass of wall is assumed as infinitely large and thevelocity vectors are all set equal to zero.

2.3. Motion of single particle

The motion of individual particle is completely deter-mined by Newton’s second law of motion. Magnus force,Saffman force, Basset, and the unsteady force are neglecteddue to the high ratio of particle density to gas density. The


particle motion in suspension is governed by gravity anddrag force

mpdvpdt

=mpg + fi , (21)

wherefi is the drag force on a particle involving the effectof neighboring particles. If the particles are more or lessuniformly distributed throughout the fluid, the restriction ofthe flow spaces between the particles in denser zones resultsin steeper velocity gradients of the gas phase, thus greatershearing stresses and an increase in resistance of the gasflow. When numerically evaluating the effective drag force,these additional forces, and the possible resulting modifica-tions in force equation by the so-called “void function” or“correction function”, the drag forcefi is then expressed as

fi = 18�d

2Cd�g�2g|ug − ui |(ug − ui)�−4.7

g , (22)

whereCd is the drag coefficient.Schiller and Naumann(1935)give the drag coefficient on a single sphere:

Cd ={

24(1+ 0.15Re0.687p )/Rep, Rep <1000,0.44, Rep�1000.

(23)

The rate of angular momentum change of a spherical par-ticle interacting with a viscous fluid may be calculated as(Lun and Liu, 1997)

mid2i

10

d�idt

=�sd

2i

64

(

6.45

Re�+

32.1

Re�

)

|�i |�i , (24)

where the spin Reynolds number is defined asRe� =d2�g|�|/(4�g).

2.4. Sequence of binary collisions of particles

The collision time of a pair of particles is defined as thetime interval until collision. It can be calculated from theinitial position and velocity vectors of both particles (Allenand Tildesley, 1980; Bird, 1994):

t12

=−r12 · v12 −

√

(r12 · v12)2 − v212(r212 − 0.25(d1 + d2)

2)

v212,

(25)

wheret12 is the collision time andr12= 0.5(d1 + d2) is therelative position of particles. If(r12 · v12)>0, the particleswill move away from each other and they will not collide.For the case of a collision between particle and wall, thecollision time depends simply on the distance to the walland on the normal velocity component toward the wall.

2.5. Calculation of porosity

For each cell of the computational domain, porosity can becalculated on the basis of the area occupied by the particlesin the cell. The porosity is obtained by subtracting the sum

of the particle volumes from the volume of a fluid cell.When a particle overlaps one neighboring cell or more, thevolume fraction included in a particular fluid cell is takeninto account to calculate the porosity of the cell. A “2D”porosity in the cell is defined as

�g,2D = 1−∑Ni=1 si

�x �y, (26)

where�x and�y are the width of the computational cellands is the surface area of the particle located in the cell.However, the above-defined “2D” porosity is inconsistentwith the applied empiricism in the calculation of the dragforce exerted on a particle and of the interfacial friction,since the relevant correlations are from 3D systems. In or-der to be more consistent, we used a 3D porosity�g,3D as(Ouyang and Li, 1999)

�g,3D = 1−2

√

�√3(1− �g,2D)

3/2. (27)

2.6. Neighbor list

In the hard-sphere approach, a sequence of binary colli-sions is processed one collision at a time. A collision list hasto be compiled and a corresponding collision time needs tobe stored. This requires a substantial CPU time. In order toreduce the required CPU time, a neighbor list approach isused: For each particle, a list of neighbor particles is storedand only particles in this list are checked for possible colli-sion. In molecular dynamic simulations, the nearest neigh-bor list is constructed by defining a cutoff distanceRlistwhich gives a circular area in the two-dimensional system(or a spherical space in the three-dimensional system). Thisinvolves the computation of circulars or spheres at cost ofCPU time.Hoomans et al. (1996)used the square neigh-bor list area in the two-dimensional system. All particlesscanned within the square of sizeDlist are listed. The size ofDlist is chosen to be 5 times the particle diameter.Ouyangand Li (1999)used a searching method of particle collisioncombining a staggered gas grid with a particle grid.In this study, we have developed an algorithm for de-

tecting particle binary collision in order to save CPU time.Several fluid control volumes (FCV) are taken to representan elementary volume in the Eulerian coordinate system,shown inFig. 2. Each particle in the FCV represents a par-ticle control volume (PCV). Every particle within the FCV(particlei) and its adjacent four FCVs are included in a list,so that this collision searching is only limited to the particlei located FCV and its nearest four neighbor FCVs (seeingthe hatched parts inFig. 2). We try to include as many PCVswithin an FCV as possible in order to make the collision listas small as possible. For the first PCV in the collision list,we search for overlapping of the particle with other parti-cles in its own FCV and the nearest four neighbor FCVs.When a first overlapping has been detected, we treat thecollision dynamics for the two particles involved. For the


Fig. 2. Adjacent PCVs included in the collision.

first particle the search for overlapping is terminated. Then,we repeat the searching procedure for the next particles inthe list. For these particles, we only search for overlappingwith the subsequent particles. When all PCVs within the re-spective FCV have undergone the procedure, we repeat thesame treatment for the next PCV until every collision listfor all the PCVs in the calculation domain has been treated.Nevertheless, this method seems to be efficient as no or in-significant fraction of particle overlapping is observed evenin high particle concentrations.

2.7. Grid mapping

For the calculation of the force acting on a suspended par-ticle, local averaged values of pressure, porosity and veloc-ities of gas and particles at the position of the particle (theLagrangian grid) are required. Due to the numerical solu-tion method used, these variables are only known at discretenodes of the computational domain. An area weighted av-eraging technique is used to obtain the local averaged valueQ̄ of a quantityQ(i, j) from the four surrounding computa-tional nodes shown inFig. 3. The local averaged value canbe calculated as follows:

Q̄E→L =Ai,jQi,j + Ai+1,jQi+1,j + Ai,j+1Qi,j+1 + Ai+1,j+1Qi+1,j+1

Dx Dy, (28)

with

Ai,j = �x�y,

Ai+1,j = (Dx − �x)�y,

Ai,j+1 = �x(Dy − �y),

Ai+1,j+1 = (Dx − �x)(DY − �y).

Fig. 3. The principle of area weighted averaging.

The distances�x and �y in this averaging technique arecalculated from the position of the particle in the staggeredgrid.

2.8. Simulation procedure

The particle motion was divided in two steps: collisionstep and free flight step. The former was controlled by dy-namics of particle collision, and the latter was controlled bydynamics of particle motion. The simulation procedures arethe following: (1) To randomly set particles in the bed toobtain initial condition, and then induce gas flow uniformlywith a center jet into the bed; (2) To calculate drag forceacting on a particle. From particle position and velocity,a sequence of collisions was processed using the collisionmodel. At the end of this step, the new particle position andvelocity were determined. The porosity and force acting onthe gas of a cell were obtained. (3) To obtain gas flow fieldby solving Eqs. (1)–(4).

3. Results and discussion

The parameters used are listed inTable 1. The minimumfluidization velocity of particles is 1.77m/s based on theequation proposed byWen and Yu (1966). The calculated

terminal velocity of particles is 16.5m/s. In all simulations,the identical initial conditions were used. A velocity of15m/s in the central cell of the bottom was specified. Otherthan central cells in the bottom, the inlet gas velocity of0.5m/s was specified. The superficial gas velocity of the bedis 2.6m/s. A time step was set to be 10−5 s and the simula-tions were continued for 10 s.


Table 1Parameters used in simulations

Particles Bed

Diameter,d (mm) 4.0 Width (mm) 150Density (kg/m3) 2700 Height (mm) 650Coefficient of restitution 0.9, 1.0 Cell width (mm) 10Dynamic friction coefficient 0.1 Cell height (mm) 20Coefficient of tangential restitution 0.3 Numberx-cell 15Number of particles 900, 1200 Numbery-cell 25

time=1.0 s time= 2.0 s time=3.0 s time=4.0 s time=5.0 s time=6.0 s

Fig. 4. Simulations of bubble and particles motions in a bubbling fluidizedbed with coefficient of restitutione = 0.9 at the superficial gas velocityof 2.6m/s.


Fig. 5. Simulations of bubble and particles motions in a bubbling fluidizedbed with coefficient of restitutione = 1.0 at the superficial gas velocityof 2.6m/s.

Snapshots of the simulated particle motion and bubble for-mation are shown inFig. 4with coefficient of restitution of0.9 at the superficial gas velocity and the number of particlesof 2.6m/s and 1200. A bubble is initiated at the orifice andgradually grown in size through the bed. The bubble wakewith particles is observed. For elastic particles(e=1.0), thebubble growth is quite different from that of inelastic parti-cles. After 0.5 s, no bubbles are found for fluidizing elasticparticles in the same bed, shown inFig. 5. The bed heightremains relatively constant in time. This indicates that themotions of particles and bubbles in the bed are related to


Fig. 6. Simulations of instantaneous particle velocity in a bubbling flu-idized bed with coefficient of restitutione = 0.9.

the momentum transfer and energy dissipation by collisions.This means that the simulation is sensitive to the coeffi-cients of restitution. Similar conclusions are also found inthe published literature (e.g.,Hoomans et al., 1996; Hellandet al., 2000) in two-dimensional fluidized bed simulations.For quantitative comparison, three-dimensional simulationhas to be carried out, which calls for a huge computationcapacity, and therefore has not been realized.Fig. 6 shows the instantaneous 2D solids velocity vector

plots for inelastic particles in the bubbling fluidized bedat the superficial gas velocity of 2.6m/s. A characteristicfeature of the flow is the oscillating motion of particles. Itis observed that particles move upward with a high velocitywhen bubbles go through the bed. The particles in the gapsbetween the bubbles have a high velocity because of theirstrong interaction with the particles on the bubbles. Suchparticle movement results in solids mixing in the bed. Thehistograms of inelastic particle vertical and lateral velocitieswith the total particle number of 900 and 1200 are shownin Figs. 7 and 8 at the two locationsr/R of 0.1364 and0.5, respectively. The mean values of particle velocity andstandard deviation (SD) are also indicated in the figures.The results fitted by Gaussian distribution are also indicated.From these figures, we see that the standard deviations ofvertical and lateral velocity of particles are not the same. Thestandard deviations of the vertical component of velocity arealways larger than those of the lateral component. First fourmoments of simulated data can be calculated. These firstfour moments of particle velocity can give the characteristicof particle flows in the bed. The third moment, skewness,and the fourth moment, kurtosis, are defined as

�skew=1

N

N∑

i=1

(vi − vm)3

v3rms, (29)

�kurt =1

N

N∑

i=1

(vi − vm)4

v4rms, (30)


-1.0 -0.6 -0.2 0.2 0.60

400

800

1200

1600

N=1200N=900Gauss fit

(a)

r/R = 0.1364y = 3.0 cmMean = -0.3059 m/sσ = 0.1933 m/sµskew = 0.00497

µkurt = 2.356Frequ

ency (/)

Vertical particle velocity (m/s)

-1.0 -0.6 -0.2 0.2 0.60

400

800

1200

1600

N=1200N=900

µskew = 0.00015

µkurt = 2.1386

Gauss fit

(b)

r/R = 0.1364y = 3.0 cmMean = 0.1568 m/sσ = 0.0769 m/s

Frequ

ency (/)

Lateral particle velocity (m/s)

-1.0 -0.6 -0.2 0.2 0.60

400

800

1200

1600

N=1200

N=900

(c)

r/R = 0.5000y = 3.0 cmMean = 0.0026 m/sσ = 0.1705 m/sµskew = 0.7984

µkurt = 3.7137

Gauss fit

Frequen

cy (/)


-1.0 -0.6 -0.2 0.2 0.60

400

800

1200

1600

N=1200N=900

µskew = 0.4716

µkurt = 2.6359

Gauss fit

(d)

r/R = 0.5000y = 3.0 cmMean = -0.0417 m/sσ = 0.10166 m/s

Frequency (/)


Fig. 7. Histogram of particles velocity at the height of 3.0 cm,e = 0.9.

-1.0 -0.6 -0.2 0.2 0.60

400

800

1200

1600

N = 1200

N = 900

Gauss fit

(a) (b)

(c) (d)

r/R = 0.1364y = 3.0 cmMean = -0.0774 m/sSD = 0.2030 m/sµskew = 0.1115

µkurt = 2.9787

Frequen

cy (/)


-1.0 -0.6 -0.2 0.2 0.60

400

800

1200

1600

N = 1200N = 900

N = 1200N = 900

N = 1200N = 900

Gauss fit

r/R = 0.1364y = 3.0 cmMean = 0.0401 m/sσ = 0.1803 m/sµskew = 0.00114

µkurt = 3.2437

Frequen

cy (/)


-1.0 -0.6 -0.2 0.2 0.60

400

800

1200

1600

Gauss fit

r/R = 0.5000y = 3.0 cmMean = -0004 m/sσ = 0.2154 m/sµskew = 0.268

µkurt = 3.6811

Frequency (/)


-1.0 -0.6 -0.2 0.2 0.60

400

800

1200

1600

Gauss fit

r/R = 0.5000y = 3.0 cmMean = -0.0058 m/sσ = 0.1983 m/sµskew = 0.4314

µkurt = 3.2741

Frequen

cy (/)


Fig. 8. Histogram of particles velocity at the height of 3.0 cm,e = 1.0.

wherevi and vm are the computed instantaneous particlevelocity and mean particle velocity, respectively.vrms isthe root-mean-square velocity andN is the number of datapoints. For a Gaussian distribution, the skewness is zero and

the kurtosis is 3. From figures we see that the values of kur-tosis are close to 3.0 for elastic particles. These results in-dicated that for elastic particles the differences between theGaussian distribution and the simulated velocity distribution


0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

(a)

Granu

lar tempe

rature (cm

/s)

2

Particle concentration

ug = 2.6 m/s

e = 0.9

0.0 0.1 0.2 0.3 0.4 0.5 0.60

50

100

150

ug = 2.6 m/s

e = 1.0

Garnular temperature (cm

/s)

2

Particle concentration(b)

Fig. 9. Distribution of granular temperature with two different coefficients of restitution of 0.9 and 1.0.

in the vertical and lateral directions are too small to be no-ticeable. For particles with realistic collision parameters thevertical and lateral velocity distributions of particles showa slight deviation from the Gaussian distributions. Increas-ingly anisotropic flow behavior for decreasing coefficientsof restitution was observed in computer simulations of ho-mogeneous shear flows of granular media (Campbell andGong, 1986; Walton and Braun, 1986;Lun et al., 1994) andbubbling fluidized bed (Goldschmidt et al., 2002). It can beseen that, even for inelastic particles, the fluctuating particlemotion in the performed hard-sphere discrete particle sim-ulations can be closely approximated by a Gaussian veloc-ity distribution. Though the particle velocity distributions invertical and lateral directions are Gaussian, significant dif-ferences are observed between the standard deviations cal-culated for different directions. The more inelastic the par-ticles, the stronger the anisotropy of the particle velocitydistribution. The large particle velocity fluctuations are ob-served in the vertical direction, whereas the fluctuations inthe lateral are small.

3.1. Distribution of granular temperature

In the Eulerian–Eulerian gas–solids flow model from thekinetic theory for granular flow, the granular temperature isused to predicate particle viscosity and pressure. The granu-lar temperature can be defined as a measure of particle fluc-tuations. The granular temperature of particles,�, may becalculated from the two-dimensional simulations (Gidaspowand Huilin, 1996; Rhode et al., 1999):

� = (132z + 2

32x), (31)

where the standard deviation of particle velocities is givenby

=

√

1

(N − 1)

∑N

i=1(vi − vm)

2. (32)

Fig. 9a and b shows the computed granular temperature as afunction of particle concentration in the bed at the superficialgas velocity of 2.6m/s. At the coefficient of restitution of0.9, the computed granular temperature decreases with theincrease in particle concentration but approaches zero whenparticle concentration is near zero. A similar trend was alsofound byNeri et al. (2000)andHuilin et al. (2003)by us-ing the Eulerian–Eulerian two-fluid model with the kinetictheory of granular flow. For elastic particles, the granulartemperature increases with the decrease in particle concen-trations. Hence, the coefficient of restitution of particles af-fects the simulated flow behavior of particles in the bubblingfluidized beds.

3.2. Radial distribution function determination

Fig. 10shows a typical spatio-temporal particle distribu-tion. The simulations store the coordinates of each particlein the picture. From the spatial distribution of particles, theradial distribution functiong(r) can be calculated as follows(Gidaspow and Huilin, 1998):

�N = 2�rN

AREAg(r)�r, (33)

whereN is the total particle number in the AREA and�Nis the particle number in the computing area.Fig. 11givesthe concept of the radial distribution function calculation.Thus, the radial distribution function is

g(r)=�N × AREA

2�r�rN(34)

and the local density of particles is

Local density of particles=(

N

AREA

)

g(r), (35)


Fig. 10. A small portion of a picture of instantaneous particle distributionin fluidized bed.

r

r

r

Fig. 11. Calculation of radial distribution function.

wherer =√

(xi − xj )2 + (yi − yj )

2. To prevent two parti-cles from being at the same location, forr�d (particle size),g(r)= 0. As r → ∞, local density= local density× g(r).Hence,g(r) → 1.0.The typical instantaneous radial distribution function pro-

files in the fluidized bed are shown inFig. 12at the superfi-cial gas velocity of 2.6m/s. The radial distribution functionis the highest when the distancer equals particle diameter,oscillating decreases, and then goes to unity. From these fig-ures, the radial distribution functiong0 at contactr = d wasdetermined.Fig. 13shows the distributions of radial distri-bution function at contact and concentration of particles asa function of time. The local concentration of particles and

the radial distribution function are oscillations with time.The radial distribution function could be related with particleconcentrations.Fig. 14shows the comparison of simulatedradial distribution function at contact versus particle con-centrations with results calculated from the Bagnold equa-tion (Bagnold, 1954) and Carnahan and Starling’s equation(Carnahan and Starling, 1969). Qualitatively, the radial dis-tribution function increases with increase in concentrationof particles. We see that the Carnahan and Starling equa-tion is not applicable to the dense flow of particles as thequantity of the radial distribution function at contact doesnot become infinite as particle concentration approaches themaximum packing value.Goldschmidt et al. (2002)simu-lated flow of particles in the fluidized bed using a 3D hard-sphere discrete particle model. The total number of colli-sions of particles was obtained from simulations. The valueof the radial distribution function can be calculated fromthe equation of particle collision frequency that was derivedfrom the kinetic theory of granular flow (Gidaspow, 1994).The radial distribution functions obtained byGoldschmidtet al. (2002)are shown inFig. 14. Simulated results indi-cated that the values of radial distribution function at con-tact increase with the increase in the granular temperatureand concentration of particles (Goldschmidt et al., 2002).Goldschmidt et al. (2002)pointed out that the radial dis-tribution function should, except from the solids concentra-tion, also depend upon the granular temperature at the pointof contact. To obtain a more generally applicable relation ofthe radial distribution function many more simulations arerequired.

3.3. Solids pressure

From the kinetic theory of granular flow, the solids pres-sure consisting of kinetic portion and collisional part can becalculated with the assumption of Maxwellian distributionof particle velocity as follows (Ding and Gidaspow, 1990;Gidaspow, 1994):

ps = �s�s[1+ 2(1+ e)�sg0]�. (36)

Once particle concentration, granular temperature, and ra-dial distribution function at contact are calculated, the solidspressure can be obtained from Eq. (36).Fig. 15a and b showsthe profiles of particle pressure calculated from Eq. (36) asa function of particle concentration at the superficial gas ve-locity of 2.6m/s. For inelastic particles, the particle pressureincreases with the increase in particle concentration at thelow concentration, and decreases with the increase in par-ticle concentration in the higher concentration of particles.The pressure of particles will becomes zero at the fixed bedwhere the particle cannot move. From Eq. (36), it is clearthat there are competing effects of particle concentration andgranular temperature on solids phase pressure when particleconcentration is high enough. For elastic particles, the solidspressure increases with the increase in particle concentra-


0 2 4 6 8 100

5

10

15

20

25

30

Radial distribution function

r/d(a) (b) (c)

(d) (e) (f )

ug = 2.6 m/s

computed location: x/R = 0.5, y/H = 0.3computation time t = 1.0 s

0 2 4 6 8 100

5

10

15

20

25

30

ug = 2.6 m/s


Rad

ial distribution function

r/d

0 2 4 6 8 100

5

10

15

20

25

30

ug = 2.6 m/s


Rad


r/d

0 2 4 6 8 100

5

10

15

20

25

30

ug = 2.6 m/s



r/d

0 2 4 6 8 100

5

10

15

20

25

30

ug = 2.6 m/s



r/d

0 2 4 6 8 100

5

10

15

20

25

30

ug = 2.6 m/s



r/d

Fig. 12. Calculated instantaneous radial distribution function in fluidized bed.

1 2 3 4 5 6 7 801020304050607080900.0

0.2

0.4

0.6

0.8

1.0

Rad


Time (seconds)(a)

ug = 2.6 m/s

computed location: x/R = 0.5, y/H = 0.3Con

centration of particles

1 2 3 4 5 6 7 801020304050607080900.0

0.2

0.4

0.6

0.8

1.0

Rad


Time (seconds)(b)

ug = 2.60 m/s

computed location: x/R = 0.5, y/H = 0.3Con

centration of particles

Fig. 13. Calculated instantaneous radial distribution function at contact and concentration of particles.

tion. When particle concentration reaches 0.3 and above, asshown inFig. 15a, the solids pressure significantly oscil-lates as a function of particle concentrations due to strongeffect of granular temperature on solids pressure. At higherparticle concentration, as soon as not in packed condition,more collisions between particles are expected and thereforemore energy loss is expected. This causes the granular tem-

perature to decrease as the particle concentration increases.For elastic particles, however, there is no such energy lossand a clear trend of solids pressure increase with the particleconcentration is found, as shown inFig. 15b.The oscillating velocity of particles can be obtained

from the instantaneous and mean velocity of particles. Thenormal stress components of particles can be calculated as


0.3 0.4 0.5 0.60

10

20

30

40

50

60

Rad

ial distribution function g o


u = 2.6 m/sg

e = 0.9 Calculations Bagnold equation Carnahan and Straling equatuon Data from Goldschmidt et al. (2002):

(θ= 2.47x10-3 m2/s2) Data from Goldschmidt et al. (2002):

(θ = 7.81x10-4 m2/s2)

Fig. 14. Calculated radial distribution function as a function of concen-tration of particles.

follows:

Ps,ii = �s(u′iu

′i), (37)

u′iu

′i =

1

T

∫ t+T

t

u′iu

′i(t)dt , (38)

whereu′i is the oscillating velocity of particles,i is x or

y direction, andT is the time interval.Fig. 16 shows thenormal stress components of particles as a function of parti-cles concentration at the superficial gas velocity of 2.6m/s.The vertical and lateral normal stresses of particles,ps,yyandps,xx , increase, reach a maximum, and then decreasewith the increase in particle concentration. We see that thevertical normal stress of particles is larger than the lateralcomponent. The values of vertical normal stress of parti-cles are 10–20 times of the lateral component. The particle

0.0 0.1 0.2 0.3 0.4 0.5 0.60

50

100

150

200

(a)

Particle pressure (Pa)


ug = 2.6 m/s

e = 0.9

0.0 0.1 0.2 0.3 0.4 0.5 0.60

50

100

150

200

ug = 2.6 m/s

e = 1.0

Particle pressure (Pa)

Particle concentration(b)

Fig. 15. Profiles of particle pressure predicted from the kinetic theory of granular flow with two different coefficients of restitution of 0.9 and 1.0.

pressure based on the normal stress of particles is expressedas

ps = (13ps,zz + 23ps,xx). (39)

Fig. 17shows the pressure of particles as a function of par-ticle concentrations at the superficial gas velocity of 2.6m/s.We see that the pressure of particles increases with theincrease in particle concentration, reaches a maxima, andthen goes to zero at the particle packing. Comparing withFig. 15, the calculated particle pressure from Eq. (39) is dif-ferent from that calculated from Eq. (36) based on the ki-netic theory of granular flow. This disagreement is probablydue to the assumption of isotropic flow of particles in thekinetic theory of granular flow.

3.4. Particle viscosity

The solids viscosity consisting of kinetic portion andcollisional part may be written as (Gidaspow, 1994)

�s =4

5�2s�sdg0(1+ e)

√

�

�

+10�sd

√��

96(1+ e)�sg0

[

1+4

5g0�s(1+ e)

]2

. (40)

The solids viscosities for both inelastic(e=0.9) and elastic(e=1.0) particles are shown inFig. 18a and b as a functionof particle concentrations at the superficial gas velocity of2.6m/s. We see that the solids viscosity increases with theparticle concentration in general with some oscillations athigher particle concentration for inelastic particles but notfor elastic particles. As discussed before such oscillationsare due to the granular temperature dependence on particleconcentration at collision. The viscosity oscillation is notso significant as for the solids pressure because the granulartemperature effect is square rooted as shown in Eq. (40).


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

200

400

600

800

1000

1200

(a)

Normal stress p s, yy=

ε s(u

yuy) (cm

/s)2


ug = 2.6 m/s

e = 0.9 Vertical component of normal stresses

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

10

20

30

40

50

60

(b)

Normal stress p s, xx= es(u

xux) (cm

/s)2


ug= 2.6 m/s

e = 0.9 lateral component of normal stresses

Fig. 16. Distribution of normal stresses of particles at the coefficients of restitution of 0.9.

The shear stress of particles can be calculated as follows:

�s,ij = �s�s(u′iu

′j ). (41)

Fig. 19 shows the shear stress of particles as a functionof particles concentrations at the superficial gas velocity of2.6m/s. The large shear stress of particles contributes tothe motions of bubbles in the bed. The laminar type shearviscosity of particles can be expressed as follows:

�s,laminar=�s,ij

(dvs/dx)=

�s�s(u′iu

′j )

(dvs/dx), (42)

where(dvs/dx) is the shear rate of particles.Fig. 20showsthe laminar-type shear viscosity of particles as a functionof particle concentrations at the superficial gas velocity of2.6m/s.We see that the laminar-type shear viscosity of parti-cles increases with the particle concentration in general withsome oscillations at higher particle concentration. We seethat the mean value of laminar-type shear viscosity of par-ticles is about 0.39Pas in the particle concentration rangeof 0.36–0.62 in the bed. This value is smaller than that ofthe particle viscosity calculated from the kinetic theory ofgranular flow.

4. Conclusions

A discrete particle model of a gas–solids bubbling flu-idized bed has been proposed and the 2D motion of theindividual spherical particles was directly calculated. Thesearching list approach of fluid volume–particle volume wasdeveloped for the processes of one collision at a time. Bub-ble formation, growth and eruption were predicated in thebubbling fluidized bed with a jet. The distributions of con-centration, velocities, and granular temperature of particleswere presented.The results obtained for elastic particles showed an excel-

lent agreement with the kinetic theory of granular flow. Thedistributions of individual particle velocity are found to be

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

50

100

150

200

Particle pressure P s (Pa)


ug= 2.6 m/s

e = 0.9 Particle pressure

Fig. 17. Profile of particle pressure at the coefficients of restitution of 0.9.

closed to Gaussian distributions. However, for inelastic par-ticles an anisotropy of the particle velocity distribution wasobserved. The vertical standard deviation of particle veloc-ity is much larger than the lateral standard deviation. Theformation of bubbles seemed to disturb spatial homogeneityand resulted in collisional anisotropy.The normal stress and shear stress of particles are calcu-

lated from instantaneous particle velocity. The vertical nor-mal stress of particles is larger than the lateral normal stress.Results show that the particle pressure calculated from thenormal stresses is the same magnitude as the values calcu-lated from the kinetic theory of granular flow. However, adifferent trend is observed due to the anisotropy flow of par-ticles. The laminar-type shear viscosity of particles calcu-lated from shear stress is smaller than that calculated fromthe kinetic theory of granular flow.To describe the observed vertical and lateral standard de-

viation differences, the model of kinetic theory of granu-lar flow developed by Jenkins and Richman based on thenon-Gaussian distribution of particle velocity (Jenkins and


0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

2.0

4.0

6.0

8.0

10.0

(a)

Particle viscosity (Pa s)


ug = 2.60 m/s

e = 0.9

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

2.0

4.0

6.0

8.0

10.0

(b)

Particle viscosity (Pa s)


ug = 2.60 m/s

e = 1.0

Fig. 18. Profiles of particle viscosity predicted from the kinetic theory of granular flow with two different coefficients of restitution of 0.9 and 1.0.

0.0 0.4-50

-30

-10

10

30

50

She

ar stress τ xy= εsρ

s(u'yu' x) (cm

/s)2


ug= 2.6 m/s

e = 0.9 Shear stress

0.60.2

Fig. 19. Shear stress of particles as a function of particle concentrations.

0.0 0.2 0.4 0.60.0

1.0

2.0ug= 2.6 m/s

e = 0.9laminar type viscosity of particles

Laminar viscosity of particles (Pa.s)


Fig. 20. Laminar viscosity of particles as a function of particle concen-trations.

Richman, 1998) may be utilized. Its applicability seems tobe an interesting topic for future research.The hard-sphere discrete particle model can give de-

tailed information of particle phase since it computes themotion of every individual particle, taking collisions, andexternal forces acting on the particles directly into ac-count. Therefore, the discrete particle model can be appliedas a valuable research tool to verify and develop closurelaws for the continuum models. A limitation for the futureapplicability of hard-sphere models seems to arise fromthe significant increase in computation times, since accu-rate detection of the collision point requires much morecomputational effort. Though more and more detailed de-scriptions of the particle collisions and the gas-phase flowfield might result in better discrete particle models, thepresented sampling technique (or its refinements based onnew insights) will be a helpful tool to validate and cal-ibrate constitutive theories for application in continuummodels.

Notation

Cd drag coefficientd particle diametere coefficient of restitutionf drag forceg gravityg0 radial distribution functionI unit vectorJ impulsemean mean valuem particle massn particle number density, normal unit vectorN number of data, particle numberp fluid pressureps solid pressure


r position, distance from wallsR half-width of bedRe Reynolds numbers local area of a particleSD standard deviationt time, tangential unit vectort12 collisional timeu gas velocityug superficial gas velocityu′ oscillating velocity of particlev particle velocityv1, v2 velocity of particles 1 and 2 after collisionv10, v20 velocity of particles 1 and 2 before collisionv12 impact velocity at the point of contact

Greek letters

� porosity�s solid concentration� granular temperature�g gas viscosity�s solid viscosity� dynamic friction coefficient�t coefficient of tangential restitution�skew skewness�kurt kurtosis�g gas density�s particle density standard deviation�g gas stress tensor�s shear stress of particles�1,�2 angular velocity of particles 1 and 2 after col-

lision�10,�20 angular velocity of particles 1 and 2 before

collision

Subscripts

g gas phasek cell indexs particlesx lateral directiony vertical direction

Acknowledgements

This work was supported by the National Science Foun-dation in China through Grant No. 50376013, and the co-operative project by NSFC-Petro China Company Limitedunder Grant No. 20490200.

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Prediction of particle motion in a two-dimensional bubbling fluidized bed using discrete hard-sphere model