Chemical Engineering Science 60 (2005) 4567–4580

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The effects of particle and gas properties on the fluidizationof GeldartA particles

M. Ye, M.A. van der Hoef, J.A.M. Kuipers∗

Fundamentals of Chemical Reaction Engineering, Faculty of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede,The Netherlands

Received 17 November 2004; received in revised form 8 March 2005; accepted 8 March 2005

Abstract

We report on 3D computer simulations based on the soft-sphere discrete particle model (DPM) of Geldart A particles in a 3D gas-fluidizedbed. The effects of particle and gas properties on the fluidization behavior of Geldart A particles are studied, with focus on the predictionsof Umf andUmb, which are compared with the classical empirical correlations due to Abrahamsen and Geldart [1980. Powder Technology26, 35–46]. It is found that the predicted minimum fluidization velocities are consistent with the correlation given by Abrahamsen andGeldart for all cases that we studied. The overshoot of the pressure drop near the minimum fluidization point is shown to be influenced byboth particle–wall friction and the interparticle van der Waals forces. A qualitative agreement between the correlation and the simulationdata forUmb has been found for different particle–wall friction coefficients, interparticle van der Waals forces, particle densities, particlesizes, and gas densities. For fine particles with a diameterdp <40�m, a deviation has been found between theUmb from simulationand the correlation. This may be due to the fact that the interparticle van der Waals forces are not incorporated in the simulations, whereit is expected that they play an important role in this size range. The simulation results obtained for different gas viscosities, however,display a different trend when compared with the correlation. We found that with an increasing gas shear viscosity theUmb experiences aminimum point near 2.0× 10−5 Pa s, while in the correlation the minimum bubbling velocity decreases monotonously for increasing�g .� 2005 Elsevier Ltd. All rights reserved.

Keywords:Discrete particle simulation; Geldart A particles; Fluidized bed; Fluidization

1. Introduction

Geldart A particles are defined asaeratableparticles,which normally have a small particle size (dp <130�m)and low particle density (<1400 kg/m3). This kind of par-ticles can be easily fluidized at ambient conditions (Geldart,1973). The enormous relevance of the fluidization proper-ties of Geldart A particles for industrial applications haslong been recognized in chemical reaction engineering, inparticular in the context of fluidized bed reactors contain-ing FCC powders. A typical property of Geldart A particlesis that they display an interval of non-bubbling expansion

∗ Corresponding author. Tel.: +31 53 489 3000; fax: +31 53 489 2479.E-mail address:j.a.m.kuipers@ct.utwente.nl(J.A.M. Kuipers).

0009-2509/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.03.017

(homogeneous fluidization) between the minimum fluidiza-tion velocityUmf and the minimum bubbling velocityUmb,which is absent in the fluidization of large particles (Gel-dart B and D particles). It is precisely this homogeneousfluidization which is responsible for many unique featuresdisplayed by these reactors. Notwithstanding the intenseexperimental research that has been conducted in the past30 years (Geldart, 1973; Abrahamsen and Geldart, 1980;Tsinontides and Jackson, 1993; Menon and Durian, 1997;Cody et al., 1999; Valverde et al., 2001), there is still noconsensus on the precise mechanism underlying the homo-geneous fluidization. Consequently, there exists currentlyno comprehensive theoretical approach, which is capableof describing both the homogeneous fluidizationand bub-bling behavior on the basis of gas and particle properties.Foscolo and Gibilaro (1984)suggested that the fluid–particle

4568 M. Ye et al. / Chemical Engineering Science 60 (2005) 4567–4580

interaction is the dominant factor that controls the stabil-ity of the homogeneous fluidization regime. On the otherhand,Rietema and Piepers (1990)andRietema et al. (1993)proposed that the interparticle forces are responsible for thehomogeneous fluidization behavior of small particles. Al-though both viewpoints are partially supported by some ex-periments (Geldart, 1973; Abrahamsen and Geldart, 1980;Tsinontides and Jackson, 1993; Menon and Durian, 1997;Cody et al., 1999; Valverde et al., 2001) and theoretical work(Koch and Sangani, 1999; Buyevich, 1999; Buyevich andKapbasov, 1999; Sergeev et al., 2004), a complete hydro-dynamical description, based on either of them, is still notsufficient to model dense gas–solid flows involving GeldartA particles. This significantly limits the use of state-of-the-art CFD techniques in the design and scale-up of fluidizedbed reactors with Geldart A particles.

Clearly, a detailed study of the particle–particle interac-tions and particle–fluid interaction at a more fundamentallevel is highly desirable. Discrete particle models (DPM)can play a valuable role in such studies. DPM has beenwidely used in the study of gas-fluidized beds, for example,the hard-sphere approach byHoomans et al. (1996), Ouyangand Li (1998), andZhou et al. (2002), and the soft-sphereapproach byTsuji et al. (1993), Xu andYu (1997), Mikami etal. (1998), andKafui et al. (2002). The idea of discrete par-ticle simulation is to track the motion of each particle in thesystem by solving Newton’s equations of motion. In DPMthe details of the particle–particle (and particle–wall) colli-sions, including friction, can be readily incorporated. Fur-thermore, because of thetwo-way coupling, discrete particlesimulations allows to study the influence of particle prop-erties on the bed dynamics or vice versa (Li and Kuipers,2003).

Recently, several attempts have been made (Kobayashi etal., 2002; Xu et al., 2002; Ye et al., 2004a) to study thefluidization behavior of Geldart A particles by use of 2Ddiscrete particle simulations.Kobayashi et al. (2002)stud-ied the effect of both the lubrication forces and the van derWaals forces on the relationship between pressure drop andthe gas velocity for Geldart A particles. They showed theexistence of a non-bubbling (homogeneous) regime, whereit was found that both the cohesive and lubrication forcesaffected the profile of pressure drop for a decreasing gasvelocity, but not for an increasing gas velocity.Xu et al.(2002)investigated the force structure in the homogeneousfluidization regime of Geldart A particles, where they foundthat the van der Waals forces acting on the particles are bal-anced by the contact forces. They also reported void struc-tures during the “homogeneous” fluidization. In a previous2D DPM study, we observed many of the typical featuresof Geldart A particles in gas-fluidized beds, such as the ho-mogeneous expansion, gross particle circulation in the ab-sence of bubbles, fast bubbles at fluidization velocities be-yondUmb (Ye et al., 2004a), and void structures (Ye et al.,2004b). An analysis of the velocity fluctuation of GeldartA particles suggests that homogeneous fluidization actually

represents a transition phase resulting from the competitionbetween three kinds of basic interactions: the fluid–particleinteraction, the particle–particle collisions (and particle–wallcollisions) and the interparticle van der Waals forces (Ye etal., 2004a,b). However, these DPM simulations were basedon 2D geometries, and focused on the influence of cohesiveforces on the flow patterns or flow structures. No modelingwork has been carried out so far which studies the effect ofthe properties of both the particulate phase and gas phaseon fluidization of Geldart A particles, although the classi-cal empirical correlations (Abrahamsen and Geldart, 1980)have been proposed more than two decades ago. The mainpurpose of this paper is, for the first time, to make a com-prehensive comparison with the well-known empirical cor-relation byAbrahamsen and Geldart, 1980(in particular forUmf andUmb.), using a full 3D soft-sphere DPM to modelthe fluidization of Geldart A particles. In Section 2 the dis-crete particle model is briefly described. The details of thesimulation procedure are discussed in Section 3, which isfollowed by a presentation of the simulation results. The pa-per ends with conclusions and a discussion.

2. Discrete particle model

In the discrete particle model, the gas-phase hydrody-namics is described by the volume-averaged Navier–stokesequations, following the approach ofKuipers et al. (1992).

�(ε�g)

�t+ (∇ · ε�gu) = 0, (1)

�(ε�gu)

�t+ (∇ · ε�guu)

= −ε∇p − Sp − ∇ · (ε�) + ε�gg. (2)

No energy equations are considered in our model. This canbe justified since we are studying the fluidization behaviorat ambient conditions where it is anticipated that heat effectsare small, so that the gas and particle flows can be safely as-sumed as isothermal. The gas flow is treated as compressibleas the local gas pressure and density might be locally differ-ent. The gas phase flow field is computed on a Eulerian grid(with computational cell volumeV ) using the well-knownSIMPLE algorithm (Patankar, 1980). The gas phase density�g is calculated via the equation of state of an ideal gas law:

�g = pMg

RT, (3)

whereR is the universal gas constant(8.314 J/mol K), Tthe temperature, andMg the molar mass of the gas. Theequation of state of the ideal gas can be applied for mostgases at ambient temperature and pressure. The couplingwith the particulate phase is included by means of a sourcetermSp, which is formally defined as

Sp = 1

V

∫ ∑Fdrag,a�(r − ra)dV , (4)

M. Ye et al. / Chemical Engineering Science 60 (2005) 4567–4580 4569

whereFdrag,a is the drag force acting on particlea, and theintegration is performed over the volume of the computa-tional cellV . To solve the gas-phase hydrodynamical equa-tions, there are two types of boundary conditions that canin principle be used to account for the sidewalls: no-slipand free-slip boundary conditions. For the free-slip bound-ary conditions, the normal components of gas velocity nearthe sidewalls are zero and the normal gradient of the tan-gential components vanishes. In the case of no-slip bound-ary conditions, both the normal and tangential componentsof gas velocity at the side-walls vanish (Kuipers, 1990). Inthis research, we use both no-slip and free-slip boundaryconditions for the sidewalls. As we will see in the followingsections, in our particular system, the free-slip wall bound-ary conditions predict smaller minimum bubbling velocities(compared to no-slip boundary conditions), which are in bet-ter agrement with the values calculated from the empiricalcorrelations (Abrahamsen and Geldart, 1980). The minimumfluidization velocity, on the other hand, is hardly affected bythe type of wall boundary condition.

The particulate phase is described by the Newtonian equa-tions of motion for each individual particle in the system.The equations of motion for a single particlea are given by

ma

d2radt2

= Fc,a + Fvdw,a + Fdrag,a − Va∇p + mag, (5)

Iad�a

dt= Ta . (6)

The first and second term on the RHS of Eq. (5) are thetotal contact force and the van der Waals force exerted byneighboring particles, respectively.

The contact force between two particles (or a particle anda wall) is obtained from a soft-sphere model proposed earlierby Cundall and Strack (1979). In that model, a linear-springand a dashpot are used to formulate the normal contact force,while a linear-spring, a dashpot and a slider are used tocompute the tangential contact force.

The interaction of particlea with the surrounding fluidfollows from a drag forceFdrag,a , which depends on therelative velocity of the two phases, and can be written as

Fdrag,a = 3��gε2dp(u− vp)f (ε). (7)

In Eq. (7), the effect of the neighboring particles on thedrag force experienced by particlea is included via theso-called porosity functionf (ε), which depends on the lo-cal porosityε and the particle Reynolds numberRep. Thelocal porosity, which is calculated for each fluid cell, isdetermined on a scale that is much smaller than the compu-tational domain, and can thus reflect the effect of local struc-tures in the fluidized bed. Many attempts have been madeto obtain accurate drag force correlation from either exper-iments (Ergun, 1952; Wen and Yu, 1966) or lattice Boltz-mann (LB) simulations (Koch and Hill, 2001; Van der Hoefet al., 2004, 2005). One of the most widely used correlationsis the Ergun–Wen–Yu model, where the well-known Ergun

equation (Ergun, 1952) is employed for porosities lower than0.8, and the Wen and Yu correlation (Wen and Yu, 1966) forporosities higher than 0.8. In terms of the porosity function,the Ergun–Wen–Yu drag model can be written as

f (ε) = 150(1 − ε)

18ε3 + 1.75Rep18ε3 ε <0.8 (8)

and

f (ε) = (1 + 0.15Re0.687p )ε−4.65 ε�0.8 (9)

for Rep <1000. The Ergun–Wen–Yu correlation is based onthe measurement of both pressure drop for stationary bedsand terminal velocities of more dilute assemblies of spheres.Basically this correlation accounts for the gross effect ofthe fluid flow, and is suited to two-phase hydrodynamics atmacro-scale. The LB simulations can capture the details ofthe flow field around each particle (Koch and Hill, 2001; Vander Hoef et al., 2004, 2005), and the drag law based on LBsimulations is expected to give a more accurate representa-tion of the drag force acting on a single particle in an as-sembly, at least for model conditions (static, homogeneoussystems). The effect of mobility and heterogeneity on theLB-based drag laws are unclear at present. For this reason,we have used the Ergun–Wen–Yu correlation in this work.Although the applicability of that relation to the systems westudy is also questionable, in this way one can make contactwith previous modeling results, where the Ergun–Wen–Yucorrelation has been used almost exclusively.

Note that the drag force relation Eqs. (8) and (9) are basedon theaveragerelative velocity of the particles to the fluidvelocity in the drag force relation. In the simulations, how-ever, we use the individual velocity of each particle relativeto the fluid velocity. This seems the most straightforwardway to take the mobility of the particles into account inthe drag model, although the validity of this should still betested by LB methods. The individual particle velocity is inany case much smaller than the fluid velocity for the sys-tems we study, so the effect is expected to be small for thisparticular application.

To calculate the interparticle van der Waals force betweentwo spheres, we adopt the Hamaker expression (Chu, 1967;Israelachvili, 1991):

Fvdw,ab(S) = A

3

2rarb(S + ra + rb)

[S(S + 2ra + 2rb)]2

×[

S(S + 2ra + 2rb)

(S + ra + rb)2 − (ra − rb)

2 − 1

]2

. (10)

Note that Eq. (10) exhibits an apparent numerical singu-larity if the intersurface distanceS between two particlesapproaches zero. In the present model, we define a cut-off(maximal) value of the van der Waals force between twospheres to avoid such a numerical singularity when two par-ticles approach, and start to compress. In practice, an equiv-alent cut-off valueS0 for the intersurface distance is usedinstead for the interparticle force (Seville et al., 2000).

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3. Simulation procedures

3.1. Fluidizing the system

In the simulations, the superficial gas velocityU0 is setto increase linearly in time

U0 = Kt . (11)

Here the slopeK is chosen in such a way that the gas velocityU0 increases slowly, thereby avoiding sudden changes of theflow field inside the fluidized bed. This procedure was foundto be more efficient, compared to the “step-wise” proceduresadopted byRhodes et al. (2001)andYe et al. (2004a). Inthe step-wise procedure, the gas velocity is increased stepby step, and for each gas velocity a sufficiently long com-puting time is required to ensure that the bed reaches a fi-nal dynamical equilibrium since a sudden change of the gasflow will lead to large fluctuations in the flow conditions.We stress, however, that the linear-increase approach differsfrom the common experimental procedure, and could causesome systematic errors when compared to the experimentalcorrelations. Nevertheless, this linear-increase procedure isexpected to be useful for investigating the origin of bubblingfluidization.

Preliminary simulations showed that the larger the slopeK the higher the predicted minimum bubbling pointUmb un-der the same conditions. A smallerK was found to predictsaUmb closer to that of the step-wise procedure, which, how-ever, requires a longer computing time. InFig. 1, we presentthe typical results of pressure drop and bed height obtainedfrom the simulations for two differentK values and alsofor the step-wise procedure. The valueK = 0.03 m/s2 wasfound to be optimal in the sense that this yields a reason-able speed-up compared to the step-wise method, where thedeviation in the predicted pressure drop and bed height areminimal (again compared to the results from the step-wisemethod).

Some input parameters that have been used in the simu-lations are listed inTable 1. Other parameters not indicatedhere will be specified for the individual simulations.

3.2. The determination of minimum bubbling point

One of the most important parameters that characterizethe fluidization behavior of Geldart A particles is the mini-mum bubbling pointUmb, which is generally defined as theinstant at which the first obvious bubble appears (Geldart,1973). However, such a definition is qualitative, and it provesdifficult to describe the minimum bubbling point in a morequantitative way. It has been found that the change of thespatial fluctuation of local porosities is the most outstandingobservation (Kobayashi et al., 2002; Ye et al., 2004a), al-though a temporal fluctuation of pressure drop and granulartemperature can also be observed near the transition fromhomogeneous fluidization to bubbling fluidization. The typ-

Fig. 1. The pressure drop and bed height obtained from the simulationsfor: K = 0.02 (Dashed line);K = 0.03 (Solid line); and the step-wiseprocedure (Squares). Simulations are carried out with no-slip boundaryconditions for the sidewalls. Other parameters not listed inTable 1are:particle diameterdp = 75�m; Hamaker constantA = 1.0 × 10−22 J;particle density�p=1290 kg/m3; gas shear viscosity�g=1.8×10−5 Pa s;

gas molar massMg = 2.88 × 10−2 kg/mol; size of the fluidized bed12.0× 3.0× 1.2 mm; initial bed heightH0 = 3.68 mm; and particle–wallfriction coefficient�f = 0.2.

ical fluctuation of local porosities with respect to the gasvelocity is shown inFig. 2. Here the fluctuation of localporosities is calculated by

�ε =√

1

Nsub−1

[∑Nsub

k=1ε2k−

1

Nsub

(∑Nsub

k=1εk

∑Nsub

k=1εk

)](12)

with εk is the local porosity in subdomaink. As can be seenfrom Fig. 2, there are two clear transitions occurring for thefluctuation of local porosities with increasing gas velocity.These two transition points are very close to the minimumfluidization point (0.0034 m/s) and minimum bubbling point(0.0082 m/s) determined from the visualization of the sim-ulation results. The window of homogeneous expansion inthe discrete particle simulations can thus be determined bythe transition points of porosity fluctuation. In this paper, theminimum bubbling point is determined by visual inspection.

M. Ye et al. / Chemical Engineering Science 60 (2005) 4567–4580 4571

Table 1Parameters used in the simulations

Parameters Value

Particle number, 36 000Normal restitution coefficient,en 0.9Tangential restitution coefficient,et 0.9Friction coefficient between particles,�f 0.2Normal spring stiffness,kn 7 or 3.5 N/mTangential spring stiffness,kt 2 or 1 N/mCFD time step, 1.0 × 10−5 or 2.0 × 10−5 sParticle dynamics time step, 1.0×10−6 or 2.0 × 10−6 sMinimum interparticle distance,S0 0.4 nmNumber of cells 48× 12× 5Gas temperature,T 293 KGas constant,R 8.314 J/(mol K)

Umf Umb

Fig. 2. The spatial fluctuation of the local porosity. Simulations arecarried out with free-slip boundary conditions for the sidewalls. Otherparameters not listed inTable 1are: particle diameterdp=75�m; HamakerconstantA = 0.0; particle density�p = 1495 kg/m3; gas shear viscosity

�g = 1.8× 10−5 Pa s; gas molar massMg = 2.88× 10−2 kg/mol; size ofthe fluidized bed 12.0 × 3.0 × 1.2 mm; initial bed heightH0 = 3.68 mm;and particle–wall friction coefficient�f = 0.2.

We compare our results forUmb with the correlation de-rived byAbrahamsen and Geldart (1980), which is given by

Umb = 2.07dp�0.06g

�0.347g

exp(0.176W45), (13)

whereW45 is the weight fraction of particles having a di-ameter less than 45�m.

3.3. The determination of minimum fluidization point

The determination of the minimum fluidization velocityUmf is straightforward. The pressure drop�p0 across thebed will just support the weight of particles at the minimumfluidization point, hence the following relation should hold:

�p0

H0= ε0�gg + (1 − ε0)�pg. (14)

Therefore, in the simulation the minimum fluidization pointis determined as the first instant at which the pressure dropacross the bed equals�p0. The empirical minimum fluidiza-tion pointUmf is given by (Abrahamsen and Geldart, 1980)

Umf = 9.0 × 10−4d1.8p [(�p − �g)g]0.934

�0.066g �0.87

g

. (15)

Note that Eq. (15) is a purely empirical correlation directlyobtained from a fit of experimental data of fine particles(Abrahamsen and Geldart, 1980). On the other hand, sev-eral correlations forUmf based on drag force models werealso derived from the balance of the gravitational force anddrag force acting on the bed of particles (Abrahamsen andGeldart, 1980). However, the latter correlations are valid formore general type of particles, and are not necessarily accu-rate for GeldartA particles. In fact, according toAbrahamsenand Geldart (1980), Eq. (15) gave the best predictions forfine powders used in 48 different solid–gas systems. There-fore, we prefer to to use Eq. (15) for comparison with theUmf found in our simulations.

4. Simulation results

4.1. The effect of the sidewalls

Since our simulations have been carried out in a relativelysmall fluidized bed, it is essential to check first the effectof the sidewalls on both the solid and gas phase inside thefluidized bed.

To investigate the effect on the solid phase, several sim-ulations have been carried out with different particle–wallfriction coefficients under no-slip boundary conditions. Theresults are given inFig. 3. It is shown that the predicted min-imum fluidization velocitiesUmf agree well with the valuescalculated from the correlation given by Eq. (15). By con-trast, the minimum bubbling velocities are overestimated inthe simulation, compared to correlation (13). However, theinfluence of the particle–wall friction on the minimum bub-bling point is negligible. InFig. 4 we show the pressuredrop observed in the simulation for different particle–wallfriction coefficients�f . Note that in the initial state the lo-cal gas pressure has not been assigned a uniform value,the real pressure drop is not zero at zero superficial veloc-ity. From Fig. 4 it is found that the overshoot of pressuredrop near the minimum fluidization point obviously dependson the particle–wall friction coefficients. The bigger theparticle–wall friction coefficient, the higher the overshootof the pressure drop cross the bed. Without interparticle vander Waals forces (A= 0), the overshoot is nearly zero for azero particle–wall friction coefficient. These findings are inaccordance with the recent experimental results obtained byLoezos et al. (2002).

To establish the effect of wall boundary conditions on thegas phase inside the fluidized bed, we mutually compare

4572 M. Ye et al. / Chemical Engineering Science 60 (2005) 4567–4580

Fig. 3. The effect of particle–wall friction onUmf andUmb. The symbolsindicate the simulation results forUmb (circles) andUmf (triangles). Thelines correspond to the prediction forUmb from Eq. (13) (dashed line)andUmf from Eq. (15) (solid line). Simulations carried out with no-slipboundary conditions for the sidewalls. Other parameters not listed inTable 1 are: particle diameterdp = 75�m; Hamaker constantA= 1.0× 10−22 J; particle density�p = 1495 kg/m3; gas shear viscosity

�g = 1.8× 10−5 Pa s; gas molar massMg = 2.88× 10−2 kg/mol; size ofthe fluidized bed 12.0 × 3.0 × 1.2 mm; initial bed heightH0 = 3.68 mm.

the simulation results with no-slip and free-slip boundaryconditions. InFig. 5we show the fluctuation of local porosityobtained in the simulations. In both cases the first transitionoccurs at nearly the same gas velocity, which implies thatthe minimum fluidization point is not influenced by the wallboundary conditions. This is not surprising since the onsetof fluidization only depends on the weight of particles insidethe bed. As can be seen, however, in the case of no-slipboundary conditions the second transition occurs at a muchlater stage (Umb=0.0128 m/s), compared to the results from

(a) (b) (c)

Fig. 4. The effect of particle–wall friction on the pressure drop. Hamaker constantA = 0. Other simulation conditions are the same as inFig. 3. The

particle–wall friction coefficient�(wp)f

is (a) 0.3; (b) 0.2; (c) 0.0.

Fig. 5. The effect of the wall boundary conditions on the fluctuation ofthe local porosity. Circles: free-slip condition; crosses: no-slip conditions.Simulations are carried out under the same conditions as inFig. 2.

free-slip boundary condition (Umb = 0.0082 m/s). Accord-ing to the correlation byAbrahamsen and Geldart (1980),a Umb = 0.0070 is expected for the specified system indi-cated inFig. 5. Thus the simulation results forUmb are 17%and 82% over-predicted under free-slip and no-slip bound-ary conditions, respectively. On the other hand, the pressuredrop and bed height do not show large deviations for thesetwo different wall boundary conditions, as can be seen inFig. 6.

In fact, we also carried out some parallel simulations witheither no-slip or free-slip boundary conditions for some otherconditions. The general qualitative agreements have beenfound, except that higher values forUmb are observed incase of no-slip boundary conditions.

M. Ye et al. / Chemical Engineering Science 60 (2005) 4567–4580 4573

Fig. 6. The effect of the wall boundary conditions on pressure drop andbed height. Solid line: no-slip conditions; dotted line: free-slip conditions.The simulations are carried out under the same conditions as inFig. 2.

4.2. The effect of interparticle van der Waals force

In Fig. 7, we show the effect of the strength of the vander Waals forces on the minimum bubbling point and theminimum fluidization point. The strength of the van derWaals forces can be quantified by a granular Bond numberBo, which is defined as the ratio of the interparticle van derWaals force between two identical spheres to the weight of asingle particle. The simulations have been conducted underfree-slip boundary conditions. It is found that the influenceof interparticle van der Waals forces onUmf is negligible,and that the predicted minimum fluidization velocitiesUmf

again agree well with the value obtained from Eq. (15).On the other hand, the predictedUmb increases with anincreasing Bond numberBo, as shown inFig. 7. This seemsto imply that the cohesive interactions between particles willdelay the minimum bubbling point in the gas-fluidized bed,which is in accordance with previous experimental work(Rosensweig, 1979). In case of relatively strong cohesiveforces, e.g.,Bo�10, the bed behaves effectively as GeldartC particles, where obvious bubbles have not been identified,not even at a very high gas velocity.

Granular Bond number Bo, [-]

Um

f &

Um

b, [m

/s]

Fig. 7. The effect of the interparticle van der Waals forces onUmf andUmb. The symbols indicate the simulation results forUmb (circles) andUmf (triangles). The solid line corresponds to the prediction forUmf fromEq. (15). Simulations carried out under free-slip boundary conditions.Other parameters not listed inTable 1are: particle diameterdp = 75�m;particle density�p=1495 kg/m3; gas shear viscosity�g=1.8×10−5 Pa s;

gas molar massMg = 2.88 × 10−2 kg/mol; size of the fluidized bed12.0 × 3.0 × 1.2 mm; initial bed heightH0 = 3.68 mm; the particle–wallfriction coefficient�f = 0.2.

In Fig. 8 the profiles of pressure drop for differentHamaker constants are shown. It is found that the overshootis also affected by the interparticle van der Waals force: thestronger the interparticle van der Waals force, the higherthe overshoot of the pressure drop near the minimum flu-idization point. So on the basis of our discrete particle sim-ulations, we conclude that the generation of the overshootof pressure drop in the fluidization of Geldart A particlesis due to both the particle–wall friction and the interparti-cle van der Waals forces. This confirms the conclusion ofRietema and Piepers (1990).

4.3. The effects of particle density

In Fig. 9we show the simulation results ofUmf andUmb

for different particle densities. Again the predicted minimumfluidization pointsUmf agree well with the correlation givenby Eq. (15). When the particle density gets higher,Umf

increases rapidly. By contrast, only a weak dependence ofUmb on the particle density is found. The predictedUmb

changes slightly from 0.0082 to 0.0094 m/s by increasing theparticle density from 900 to 4195 kg/m3. Hence the windowof homogeneous fluidization is decreased for heavy particles,but this is mainly due to the increase inUmf . At a very highparticle density (�p=4195 kg/m3),Umf almost equalsUmb,which indicates a transition from Geldart A to B fluidizationbehavior. This transition is clearly shown inFig. 9. Notethat the correlation ofAbrahamsen and Geldart (1980), asshown in Eq. (13), does not include any information aboutthe particle density, which suggests a negligible effect of

4574 M. Ye et al. / Chemical Engineering Science 60 (2005) 4567–4580

(a) (b) (c)

Fig. 8. The effect of the interparticle van der Waals forces on the pressure drop. Simulation conditions are the same as inFig. 7. The Hamaker constantA is (a) 10−21J; (b) A = 10−22 J; (c) 0.

particle density onUmb. Our simulation results support thisconclusion.

4.4. The effects of particle size

In Fig. 10the results ofUmf andUmb for different parti-cle diameters are shown. The predicted minimum fluidiza-tion pointsUmf agree well with the correlation given byEq. (15). A general qualitative agreement is found forUmb

when the particle diameterdp is bigger than 40�m. Thevalues ofUmb are typically over-predicted by 15–25% incomparison with the correlation byAbrahamsen and Gel-dart (1980). For particles having a diameterdp = 180�m, atransition of Geldart A to B fluidization behavior can be dis-tinguished. For fine particles with a diameterdp <40�m,the simulation results clearly deviate from the correlation.For instance, for a particle diameterdp = 37.5�m, a muchlowerUmb = 0.0022 m/s is obtained, compared to the pre-diction from the correlation (Umb = 0.006m/s). Note thatthe interparticle van der Waals forces are absent in this sim-ulation, i.e., the Hamaker constantA=0. As was mentionedpreviously, the incorporation of interparticle van der Waalsforces can delay the minimum bubbling point and extendthe interval of homogeneous fluidization. For fine particleswith a diameterdp less than 40�m, the interparticle van derWaals forces may become more stronger, and will make ashift of Umb to a higher value that are normally observed inthe experiments. Thus it can be argued that for fine particlesthe interparticle van der Waals forces are playing a criticalrole in the formation of homogeneous fluidization.

4.5. The effects of the gas density

To change the gas density, we can change either the mo-lar massMg or gas pressurep, as shown in Eq. (3). Herewe will only consider the change of the gas molar mass,and the effect of the gas pressure on the fluidization of Gel-dart A particles will be the subject of a future research. Weshould mention, however, that the gas density is not uniforminside the fluidized bed since the gas pressure is spatiallyheterogeneous. Since the pressure drop across the fluidizedbed is quite small compared to the absolute gas pressure,the gas density can be determined solely by the inlet gaspressure. We change the molar massMg of the gas phasefrom 1.44×10−2 to 5.04×10−2 kg/mol. As can be seen inFig. 11, the influence of the gas density on the porosityfluctuation is negligible (except in the bubbling regime),which implies that the effect of gas density on bothUmf

andUmb are negligible. Incidentally, in the correlations ofAbrahamsen and Geldart (1980), as shown in Eqs. (15) and(13), only a weak dependence ofUmb andUmf on gas den-sity is found, i.e.,Umb ∼ �0.06

g andUmf ∼ �−0.066g .

4.6. The effect of the gas viscosity

Figs.12and13showUmf andUmb for different gas shearviscosities. For simplicity, the interparticle van der Waalsforces are switched off by setting the Hamaker constantA=0. Again, the minimum fluidization velocities agree wellwith the values calculated from Eq. (15).Umf experiences acontinuous decrease as�g increases. The minimum bubbling

M. Ye et al. / Chemical Engineering Science 60 (2005) 4567–4580 4575

500 1000 1500 2000 2500 3000 3500 4000 45000.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

0.011

Um

f and

Um

b, [m

/s]

1000 2000 3000 40001.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Particle density �p, [kg/m3]

Particle density �p, [kg/m3]

Um

b /U

mf,

[-]

Fig. 9. The effect of particle density onUmf and Umb. The symbolsindicate the simulation results forUmb (circles) andUmf (triangles). Thelines correspond to the prediction forUmb from Eq. (13) (dashed line)and Umf from Eq. (15) (solid line). Simulations are carried out underfree-slip boundary conditions. Other parameters not listed inTable 1are:particle diameterdp = 75�m; gas shear viscosity�g = 1.8 × 10−5 Pa s;

gas molar massMg = 2.88 × 10−2 kg/mol; size of the fluidized bed12.0×3.0×1.2 mm; initial bed heightH0=3.68 mm; and the particle–wallfriction coefficient�f = 0.2.

velocities, however, manifest a systematic deviation from theempirical correlation byAbrahamsen and Geldart (1980). Asillustrated in theFig. 13, Umb first drops and subsequentlyincreasesfor an increasing shear viscosity, passing a mini-mum point at�g =2.0×10−5Pa s. This is clearly in contra-diction with Eq. (13), where the minimum bubbling velocitydecreases monotonously for increasing�g. At present, wehave no explanation why a minimum inUmb is observed inour simulations. It is worthwhile to mention, however, thatthe correlation ofAbrahamsen and Geldart (1980)was actu-ally obtained from gas shear viscosities ranging from 0.9 to2.0×10−5 Pa s. In this regime the predictedUmb also expe-riences a continuous decrease with an increasing gas shearviscosity. It would therefore be interesting to perform exper-iments with gas shear viscosities larger than 2.0×10−5 Pa s,where our simulations predict anincreasein Umb.

0 40 80 120 160 200

0.000

0.004

0.008

0.012

0.016

0.020

0.024

Particle diameter dp, [m]

Particle diameter dp, [m]

Um

f and

Um

b, [m

/s]

0 40 80 120 160 2001

2

3

4

Um

b /U

mf,

[-]

Fig. 10. As inFig. 9, but now for varying particle diameter. The particledensity is equal to�p = 1495 kg/m3.

Fig. 11. The effect of gas density onUmf andUmb. The inlet gas densitiesused in the simulations are 0.5990 (squares), 1.1979 (crosses), 1.4974 (tri-angles) and 2.2461 (circles) kg/m3. The simulations are carried out underfree-slip boundary conditions. Other parameters not listed inTable 1are:particle diameterdp=75�m; particle density�p=1495 kg/m3; gas shear

viscosity�g=1.8×10−5 Pa s; size of the fluidized bed 12.0×3.0×1.2 mm;initial bed heightH0 =3.68 mm; and the particle–wall friction coefficient�f = 0.2.

4576 M. Ye et al. / Chemical Engineering Science 60 (2005) 4567–4580

Fig. 12. The effect of gas shear viscosity onUmf . The triangles denotethe simulation results while solid line represents Eq. (15). The simulationsare carried out under free-slip boundary conditions. Other parametersnot listed inTable 1are: particle diameterdp = 75�m; particle density�p=1495 kg/m3; size of the fluidized bed 12.0×3.0×1.2 mm; initial bedheightH0 = 3.68 mm; and the particle–wall friction coefficient�f = 0.2.

Fig. 13. The effect of gas shear viscosity onUmb. The circles represent thesimulation results while dashed line represents Eq. (13). The simulationsare carried out under the same conditions as inFig. 12.

To further compare with the correlation ofAbrahamsenand Geldart (1980), in Fig. 14 we showUmb vs �g on alog–log scale. The best linear fit to our data has a slope−0.267, compared to the power−0.347 in Eq. (13). If weonly include data up to 1.8 × 10−5Pa s, which correspondsto the shear viscosity of air under normal condition (T =293K), a slope of−0.354 is obtained.

4.7. The analysis of dimensionless numbers

The analysis of the independent dimensionless parame-ters is of theoretical interest in the study of fluidization ofGeldart A particles (Sundaresan, 2003). Of particular inter-est are Froude numberFr = U2/gdp, Reynolds numberRe=�f Udp/�f , and Stokes numberSt=�pudp/�f . In this

Fig. 14. Same asFig. 13, but now in a log–log scale, and where only thefirst 8 points ofFig. 13 are shown. The solid line is the best linear fit toall 8 data points, the dashed line for the first 5 data points.

work, some typical values aredp=75�m,�p=1495 kg/m3,and�f = 1.8× 10−5 Pa s. The terminal settling velocity forthis kind of particle isUt = 0.254 m/s, and thus the corre-sponding dimensionless numbers areFrt =Ut/gdp =87.9,Stt =�pUtdp/�f =1583.2, andRet =�f Utdp/�f =1.269,respectively.

In Fig. 15we show the dimensionless numbers based onthe minimum bubbling velocityUmb. The results are ob-tained for systems with different particle and gas proper-ties. It is interesting to note that for the Froude number aminimum value can be observed, if we neglect the pointthat corresponds to a particle sizedp = 37.5�m. In fact, fordp = 37.5�m a very lowUmb has been obtained, which isprobably due to the fact that we have not included the cohe-sive forces in the simulation for this type of particles. Sincethe cohesive forces are relatively large for particles sizes be-low 40�m, they are expected to give rise to a largerUmb

for such a fine powder. For particles with a diameter largerthan 50�m, a minimum valueFrm = 0.09 can be distin-guished, which means that the bed will manifest homoge-neous fluidization ifFrm�0.09. In an early experimentalstudy,Wilhelm and Kwauk (1948)found that a fluidized bedwould display homogeneous fluidization ifFrm � 1. Weargue, however, that the Froude number is still too roughto offer a criterion for the transition from homogeneous flu-idization to bubbling fluidization. The typical Stokes num-ber and Reynolds number areStm = O(101 ∼ 102) andRem = O(10−2 ∼ 10−1).

5. Discussion and conclusions

In this research, computer simulations based on the soft-sphere DPM have been used to investigate the fluidizationbehavior of Geldart A particles. The simulations have beencarried out in a 3D fluidized bed, with interparticle van der

M. Ye et al. / Chemical Engineering Science 60 (2005) 4567–4580 4577

0.00

0.04

0.08

0.12

0.16

0.20

0.24

Fr m

b

1-�mb

0

20

40

60

80

100

120

140

160

St m

b

0.30 0.35 0.40 0.45 0.50 0.55 0.60

1-�mb

0.30 0.35 0.40 0.45 0.50 0.55 0.60

1-�mb

0.30 0.35 0.40 0.45 0.50 0.55 0.60

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

Re m

b

Fig. 15. The dimensionless numbers with respect to the solid volumefraction at the minimum bubbling point for particle size larger than 40�m.The results are obtained fromFig. 9 (squares),Fig. 10 (crosses), andFig. 12 (triangles).

Waals forces that follow from the Hamaker theory. We firststudied the effects of the sidewalls on the hydrodynamicsinside fluidized beds. It has been found that the generation

of the overshoot of the pressure drop near the minimum flu-idization point is affected by both the particle–wall frictionand the interparticle van der Waals forces, which confirm theexperimental results byLoezos et al. (2002)and Rietemaand Piepers (1990).

For all cases we studied in this research, the predictedUmf was found to be in good agreement with the corre-lation by Abrahamsen and Geldart (1980). The minimumbubbling velocityUmb, in general, shows a qualitative agree-ment with the correlation. First, the wall boundary condi-tions are found to have influences on the predictedUmb. Thefree-slip boundary conditions predicts a lowerUmb than theno-slip boundary conditions in our small-scale simulations.The predictedUmb under free-slip boundary conditions isfound 15–25% higher than value calculated by the correla-tion, while under no-slip boundary conditions this accountsto more than 80%. Second, the action of interparticle vander Waals is found to delay the origin of bubbles and ex-tend the interval of homogeneous fluidization. The higherthe granular Bond number, the higherUmb, until a transi-tion to Geldart C behavior where noUmb can be determined.Third, the particle density and gas density are shown to havea weak effect onUmb. For heavy particles, the window ofhomogeneous fluidization is decreased mainly due to the in-crease inUmf , where a transition from Geldart A to B be-havior (no homogeneous fluidization) is found at a densityof 4195 kg/m3 for dp = 75�m. Also, it has been found thatthe particle size has a strong effect onUmb. The predictedUmb with different particle diameter agrees with the corre-lation except for fine particles with a diameterdp <40�m.This may due to the fact that we turn off the interparticlevan der Waals forces in the simulations. It can be arguedthat for larger particles with a diameterdp >40�m the in-terparticle van der Waals forces may have a negligible ef-fect on the formation of homogeneous fluidization. For fineparticles, however, a proper incorporation of interparticlevan der Waals forces is highly desired. One of the prob-lems is that there are currently no reliable estimates for themagnitude of the cohesive forces, since it has proved notpossible to directly measure these forces in experiments.Eq. (15) might implicitly reflect the effect of cohesive inter-action since for finer particles the cohesive forces might giverise to some clustering of the particles which would affect thedrag force. However, we are not expecting an apparent in-crease ofUmf for slightly cohesive particles as the clusteringeffect will be minimal, where the onset of fluidization is onlydue to the balance of gravity and gas–particle interaction(drag force) for homogeneous systems. Finally, the effect ofgas viscosity has been examined. The minimum bubblingvelocities predicted with different gas viscosity, however,manifest a systematic deviation from the empirical correla-tion byAbrahamsen and Geldart (1980). We found that withan increasing gas shear viscosity theUmb experiences a min-imum point near 2.0×10−5 Pa s, while in the correlation byAbrahamsen and Geldart (1980)the minimum bubbling ve-locity decreases monotonously for increasing�g. If we fit the

4578 M. Ye et al. / Chemical Engineering Science 60 (2005) 4567–4580

Fig. 16. The snapshots of the simulation results of the homogeneous fluidization of Geldart A particles. The far left graph shows the fluidized bed in3D. Graphs 1–5 show the cross sections of the bed (cutting through the width direction are shown. The simulation conditions are the same as inFig. 2.

Fig. 17. As inFig. 16, but for the bubbling fluidization.

data up to 2.0× 10−5 Pa s, a power of−0.267 has been ob-tained, which is not far away from the value (−0.347) givenby Eq. (13). Clearly, a more elaborate study of the effectof viscosity onUmb is required, both from experiment andsimulation.

With respect to the latter, however, we note that it is anon-trivial task to determine the minimum bubbling velocityUmb, since there is no unique, quantitative formalism torelateUmb to parameters that can be directly measured inthe discrete particle simulations. This is directly related tothe fact that the minimum bubbling point is rather looselydefined, namely, as the instant at which “the first obviousbubble” appears in the fluidized bed. To illustrate this point,in Figs. 16 and 17, we show some typical snapshots forboth homogeneous fluidization and bubbling fluidization. Ascan be seen, even during the homogeneous fluidization, wecan still find some void structures. It would be extremelydifficult to define a formalism (i.e., a computer code) whichcould discriminate the voids and cavities of homogeneousfluidization from the first obvious bubble, just on the basisof the particle coordinates.

Notation

A Hamaker constant, Jd particle diameter, mF,F force, NFr Froude number,−g,g gravitational acceleration, m/s2

H bed height, mI moment of inertia, kg m2

K Slop, m/s2

m mass, kgn normal unit vector, dimensionlessNsub number of sub-domains, dimensionlessNpart number of particles, dimensionlessp gas pressure, Par particle radius, mr particle position, displacement, mRe Reynolds number, dimensionlessS Intersurface distance between spheres, mSt Stokes number, dimensionlesst time, s

M. Ye et al. / Chemical Engineering Science 60 (2005) 4567–4580 4579

T torque, N mu local gas velocity, m/sU superficial gas velocity, m/sv particle velocity, m/sV volume, m3

W the weight fraction of fine particles, dimensionless

Greek letters

ε porosity, dimensionless�f coefficient of friction, dimensionless�g gas shear viscosity, Pa s� density, kg/s3

� viscous stress tensor, kg/m s2

� angular velocity, 1/s

Subscripts

0 initial statea, b particle indexc contact forcedrag drag forceg gas phasemf minimum fluidization pointmb minimum bubbling pointp particlephaset terminal setting velocityvdw van der Waals force

Superscripts

n normal directiont tangential direction

Acknowledgements

This work is part of the research program of the Stichtingvoor Fundamenteel Onderzoek der Materie (FOM), which isfinancially supported by the Nederlandse Organisatie voorWetenschappelijk Onderzoek (NWO).

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