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Available online at www.sciencedirect.com

Particuology 6 (2008) 475–482

DEM simulation of particle percolation in a packed bed

Mahbubur Rahman a,∗, Haiping Zhu a, Aibing Yu a, John Bridgwater b

a Lab for Simulation and Modeling of Particulate Systems, School of Materials Science and Engineering, The University of New South Wales,Sydney, NSW 2052, Australia

b Department of Chemical Engineering, University of Cambridge, Cambridge CB2 3RA, UK

Received 10 March 2008; accepted 15 July 2008

bstract

The phenomenon of spontaneous particle percolation under gravity is investigated by means of the discrete element method. Percolation behaviorsuch as percolation velocity, residence time distribution and radial dispersion are examined under various conditions. It is shown that the verticalelocity of a percolating particle moving down through a packing of larger particles decreases with increasing the restitution coefficient betweenarticles and diameter ratio of the percolating to packing particles. With the increase of the restitution coefficient, the residence time and radialispersion of the percolating particles increase. The packing height affects the residence time and radial dispersion. But, the effect can be eliminatedn the analysis of the residence time and radial dispersion when they are normalized by the average residence time and the product of the packing

eight and packing particle diameter, respectively. In addition, the percolation velocity is shown to be related to the vertical acceleration of theercolating particle when an extra constant vertical force is applied. Increasing the feeding rate of percolating particles decreases the dispersionoefficient.

© 2008 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All

ial di

pi(m((r(efRDtcHp

eywords: Discrete element method; Particle percolation; Residence time; Rad

. Introduction

Percolation phenomena are very common in nature and engi-eering, for example, in the flow of liquid in a porous medium,pread of disease in a population, movement of fires through for-st, metal–atom dispersion in insulators, and mixing of particlesf different size (Kesten, 1982; Rosato, Strandburg, Prinz, &wendsen, 1987; Stauffer, 1985; Weil, 1983). The spontaneous

nter-particle percolation is one of the main issues in this area,hich is widely encountered in industries. A specific example

rises in iron making blast furnace, where small sintered parti-les are placed upon much larger coke particles. When subject tooth gravity and gas flow, the smaller components pass throughhe larger. The physics of spontaneous inter-particle percolationlso provides one means of assessing the effectiveness of theacking models (Bridgwater, Sharpe, & Stocker, 1969).

The inter-particle percolation has been investigated by meansf various physical and numerical experiments in the past.ridgwater et al. (1969) and Bridgwater and Ingram (1971)

∗ Corresponding author.E-mail address: mahbub@materials.unsw.edu.au (M. Rahman).

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674-2001/$ – see inside back cover © 2008 Chinese Society of Particuology and Institute of Process

oi:10.1016/j.partic.2008.07.016

rights reserved.

spersion

erformed the experiments to study the percolation velocity, res-dence time and radial distance distribution. Moore and Masliyah1973) developed computer programs to describe the downwardotion of small spheres through an ordered bed of spheres. Li

1998) and Richard, Oger, Lemaitre, Samson, and Medvedev1999) used Monte Carlo method to analyze various propertieselevant to percolation. Asmar, Langston, Matchett, and Walters2002) used the discrete element method (DEM) to study theffect of the density ratio, diameter ratio, initial velocity andriction coefficient on settling behavior of percolating particles.ecently, Lomine and Oger (2006) performed experiments andEM simulations to analyze transport properties of particles

hrough porous structure, and found that the number of per-olating particles affected the relevant percolation behaviors.owever, to fully understand the percolation phenomenon, theercolation behaviors such as percolation velocity, residenceime and radial dispersion should be studied under various cir-umstances. Previous work has a lack of complete investigationn this aspect although some of these properties have been ana-

yzed under some specific conditions.

This work attempts to study spontaneous inter-particle per-olation by means of DEM simulation. The percolation velocity,esidence time and radial dispersion of the percolating particles

Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

476 M. Rahman et al. / Particuol

Nomenclature

c damping coefficientd percolating particle diameter (m)D packing particle diameter (m)E Young’s modulus (kg/(m/s2))Er radial dispersion coefficientf inter-particle forces (N)fe extra forceg gravitational acceleration (m/s2)H packing heightI moment of inertia of particle (kg m2)m torque (N m)m mass of particle (kg)N number of percolating particles having centers

within radius r at time tN0 total number of percolating particlesNt number of percolating particles deposited at the

bottom in an interval of t = √(gD)

R vector from the mass centre of the particle to thecontact point (m)

r radius describing dispersion (m)t time (s)T residence time (s)tmean average of residence time (s)�t time for averaging (s)v particle translational velocity (m/s)v percolating particle velocity (m/s)

Greek lettersδt vector of the accumulated tangential displace-

ment between particles i and jν Poisson ratioμ friction coefficientω particle angular velocity (s−1)ω̂ unit vector defined by ω̂ = ω/|ω|ρ density (kg/m3)α restitution coefficient

Subscriptsc contactd dampingi particle iij between particles i and jj particle jn normal componentr rolling frictions slidingt tangential component

aapp

ip

2

ptcwi

m

I

wlTgpFcptrafti2

c

sbi

Ft(adppuloaoThe initial positions of the percolating particles for these cases

re analyzed. The effects of the properties of the percolating

nd packing particles such as the restitution coefficient betweenarticles, diameter ratio of percolating to packing particles andacking height on the percolation behaviors, are discussed. The

apt

ogy 6 (2008) 475–482

nfluence of the extra force and feeding rate on the particleercolation is also considered.

. Simulation method and conditions

The DEM model adopted in this work has been used in ourrevious work (Zhu & Yu, 2003). In the DEM simulation, a par-icle has two types of motion: translational and rotational, whichan be described by Newton’s law of motion. These equations,hich are based on the forces and torques originated from its

nteraction with neighboring particles, can be given by

i

dvi

dt=

∑j

fij + mig, (1)

i

dωi

dt=

∑j

mij, (2)

here mi and Ii, vi and ωi are the mass, moment of inertia, trans-ational velocity and angular velocity of the particle respectively.he forces involved are the gravitational force, mig (g is theravitational acceleration), and the interaction force fij betweenarticles i and j, which depends on the deformation of particle i.or simplicity, our present study is limited to dry spherical parti-les whose inter-particle force only includes elastic and dampingarts. The torque acting on particle i by particle j, mij, stems fromhe contact force and causes particle i to rotate or slow down theelative rotations between particle i and particle j. Wall is treateds a particle of infinite size. The equations used to calculate theorce and torque are listed in Table 1. They are the same ashose used by Zhu and Yu (2003). The restitution coefficient, α,s related to the damping coefficient, c, given by (Asmar et al.,002):

= − ln α√π2 + ln2 α

. (3)

Therefore, the trajectories, velocities, angular velocities, tran-ient forces and torques of all particles in a system can be tracedy solving Eqs. (1) and (2). The computational procedure useds similar to that in our previous work (Zhu & Yu, 2003).

The geometry of the model used in this work is shown inig. 1. A cylindrical container is used to form a packing of par-

icles with a diameter of D. In each simulation, the particleswhich are referred to as packing particles here) are first gener-ted randomly in the container. After these particles are settledown under gravity to form a packing, small particles are thenut on the top of the packed bed. These percolating particlesass through the packed bed towards the bottom of the columnnder gravity. Three cases are considered: (1) only one perco-ating particle for analyzing the percolation velocity, (2) a layerf 171 percolating particles for determining the residence timend radial dispersion, and (3) same as Case (2) but more layersf percolating particles for quantifying the effect of feeding rate.

re given in different ways. For the first case, the percolatingarticle is initially set to be at the centerline of the column. Forhe second case, one layer of percolating particles is generated

M. Rahman et al. / Particuology 6 (2008) 475–482 477

Table 1Components of forces and torque acting on particle i.

Forces and torques Symbols Equations

Normal elastic force Fcn,ij − 43 E∗√R∗δ2/3

n n

Normal damping force Fdn,ij −cn(8mijE∗√R∗δn)

1/2Vn,ij

Tangential elastic force Fct,ij −μs|Fcn,ij |(1 − (1 − δt/δt,max)2/3)δ̂t (δt < δt,max)

Tangential damping force Fdt,ij −ct(6μsmij |Fcn,ij |√

1 − |vt |/δt,max/δt,max)1/2

Vt,ij (δt < δt,max)

Coulomb friction force Ft,ij −μs|Fcn,ij |δ̂t (δt ≥ δt,max)

Torque by tangential forces mt,ij Rij × (Fct,ij + Fdt,ij)

Rolling friction torque mn,ij μr,ij |Fn,ij |ωnt,ij

Where 1R∗ = 1

|Ri| + 1|Rj | , E∗ = E

2(1−v2), ωt,ij = ωt,ij

|ωt,ij | , δ̂t = δt|δt | , δt,max = μs

2−v2(1−v) δn,

Note that tangential forces (Fct,ij + Fdt,ij) should be replaced by Ft,ij when δt ≥ δt,max.

Fig. 1. Geometry of the model used in this work.

rtiwpecpchw

3

3

pp&ccarat

TP

P

PPPYPRSRT

N(t

Vij = Vj − V i + ωj × Rj − ωi × Ri, Vn,ij = (Vij · n)n, Vt,ij = (Vij × n) × n

andomly at the centerline of the column in a circle with a diame-er of 5D. For the third case, percolating particles are generatedn the same way as Case (2) but number of layers is changedhile total feeding time is kept constant. The percolating andacking particles are assumed to have the same material prop-rties, including Young’s modulus, Poisson’s ratio, restitutionoefficient, and sliding and rolling friction coefficients. Threearameters, restitution coefficient between particles, α, parti-le diameter ratio of percolating to packing, d/D, and packingeight, H, are considered in this work. Their effects are examinedhen other parameters are constant as listed in Table 2.

. Results and discussion

.1. Percolation velocity

It has been suggested that the vertical velocity of a percolatingarticle moving down through a random packed bed of largerarticles is a constant (Bridgwater & Ingram, 1971; Lomine

Oger, 2006; Moore & Masliyah, 1973). This has also beenonfirmed in the present DEM simulations. In general, the per-olation velocity is dependent on a number of variables, such

s the sizes and densities of percolating and packing particles,estitution coefficient and friction coefficient between particles,nd void fraction of packing. Here, we would like to quan-ify this dependence, but only focus on the two main properties

able 2arameters used in the present DEM simulations.

arameter Value

ercolating particle diameter, d 0.001–0.068D (0.068D)acking particle number 1600–5000 (5000)acking height, H 9–30D (30D)oung’s modulus 50,000 g�Dρ/6oisson’s ratio 0.3estitution coefficient, α 0.3–0.9 (0.8)liding friction coefficient 0.5olling friction coefficient 0.001Dime step 10−5√(gD)

ote: (1) g (=9.81 m/s2) is the magnitude of g, and ρ is the particle density and2) the values in brackets are the base values used to study the effect of one ofhe three considered variables α, d/D, H.

478 M. Rahman et al. / Particuology 6 (2008) 475–482

FrH

fp

tTodddtv

ibcfmvcvc

Fna

Fc

pstcmtci

3

i1iaS

ig. 2. The variation of the dimensionless percolation velocity on the diameteratio of percolating to packing particles for different restitution coefficients when

= 30D.

or brevity, namely the diameter ratio of percolating to packingarticles and restitution coefficient.

Fig. 2 shows the variation of the percolation velocity againsthe particle diameter ratio for different restitution coefficients.he percolation velocity is normalized by

√gD. It can be

bserved that the velocity decreases with increasing the particleiameter ratio. The variation of the velocity is higher when theiameter ratio is larger. The trends of the variations are similar forifferent restitution coefficients. However, increasing the resti-ution coefficient can reduce the magnitude of the percolationelocity.

Fig. 3 shows the effect of the collision number of the percolat-ng particle per unit height (i.e., the average number of collisionsetween percolating and bed particles per unit height) on the per-olation velocity. From microscopic point of view, increasingrequency of collision leads to an impedance of the downwardotion of the percolating particle, hence reduce its percolation

elocity, as shown in Fig. 3. The smaller particle diameter ratioauses fewer collisions, and hence results in higher percolationelocity. The higher restitution coefficient leads to that moreollisions are necessary to remove the energy of the percolating

ig. 3. The variation of the dimensionless percolation velocity on the collisionumber per unit height for different restitution coefficients when d/D = 0.068nd H = 30D.

srafilbeh

hvdiathtthdto

ig. 4. Statistic distributions of residence time for different restitution coeffi-ients when d/D = 0.068 and H = 30D.

article to pass down one particular surface into the next inter-tice, and therefore causes lower velocity. On the other hand,he velocity is also related to the intensity of collisions. Higherollision intensity leads to a larger impedance of the downwardotion of the percolating particle, and hence reduce the percola-

ion velocity. This is the reason why we cannot obtain a generalorrelation between the collision number and percolation veloc-ty as shown in Fig. 3.

.2. Residence time distribution

Residence time distribution is an important characteristicn the previous experimental studies (Bridgwater & Ingram,971). It has been obtained that the residence time distributions roughly similar to a normal distribution although there may beslight difference for some percolating particles such as lead.imilar trend can also be observed in the present simulations, ashown in Fig. 4. Furthermore, it can be seen that with increasingestitution coefficient, the distribution curve shifts to the right,nd becomes wider. It is because the higher the restitution coef-cient, the lower the particle percolation velocity, and then the

onger the time for the particles to reach the bottom of the packeded. Similar feature has also been observed for different diam-ter ratio of percolating to packing particles, but is not shownere for brevity.

The experimental studies of Bridgwater and Ingram (1971)ave suggested that the cumulative residence time distributionaries little with the particle properties, when plotted againstimensionless time T/tmean. However, the variation with pack-ng height has not been seriously considered in the previous worklthough the packing height is also an important factor affectinghe residence time as shown in Fig. 5(a). The higher the packingeight, the longer the residence time. Therefore, it is meaningfulo consider if the dimensionless time distribution is dependent onhe packing height. It can be seen from Fig. 5(b) that the height

as limited effect on the normalized cumulative residence timeistributions, which indicates that the description of the percola-ion behavior can be simplified when the cumulative distributionf the dimensionless residence time is considered.

M. Rahman et al. / Particuology 6 (2008) 475–482 479

F denceh

3

ddte

wnpTLubTdctT

Fw

litvtp

ppdF

3

ptpt

ig. 5. Cumulative distributions of: (a) residence time, and (b) normalized resieights when d/D = 0.068 and α = 0.8.

.3. Radial dispersion

The process of radial dispersion is a random walk one. Radialispersion is generally affected by some parameters such as sizeistribution of particles and restitution coefficient between par-icles. It has been found in the experimental work of Bridgwatert al. (1969) that the radial dispersion can be described by

r2

4Ert= ln

(N0

N0 − N

), (4)

here Er is the radial dispersion coefficient, N0 is the totalumber of the percolating particles, and N is the number ofercolating particles having centers within radius r at time t.his correlation has been confirmed by the numerical study ofi (1998) using Monte Carlo method, and the present DEM sim-lations as shown in Fig. 6. However, there is a large deflectionetween the Monte Carlo simulations and experimental results.he experiments of Bridgwater et al. (1969) suggested that the

ispersion coefficient increases slowly with restitution coeffi-ient. But, the Monte Carlo simulations of Li (1998) showedhat it increases sharply when increasing restitution coefficient.he reason why there is such a deflection is not clear. This prob-

ig. 6. Plot of r2 against ln[N0/(N0 − N)] for various restitution coefficientshen d/D = 0.068 and H = 30D.

ftt(

Fd

time, T/tmean (tmean is the average of the residence time) for different packing

em can be overcome by the present DEM simulations, as shownn Fig. 7. Both the DEM simulations and experiments show thathe dispersion coefficient varies slowly. The difference of thealues of the dispersion coefficient between the DEM simula-ions with experimental results can be attributed to the effect ofarticle properties.

Fig. 8 shows the dependence of the dispersion distribution onacking height. It can be seen from Fig. 8(a) that the higher theacking height, the more off-centre the dispersion distance. Theispersion distribution can be normalized by HD, as shown inig. 8(b).

.4. Effect of extra force

If the effect of other sources such as gas is considered, thearticles should be subject to extra forces. Such forces will con-ribute the acceleration of particles, and hence influence thearticle percolation velocity. For brevity, we assume that the par-icles are only affected by a constant force in vertical direction,

e, except contact forces and gravity. Fig. 9 shows the effect ofhe force on the particle percolation velocity. It can be observedhat increasing the magnitude of the force in downward directionnegative) can increase the percolation velocity, while increas-

ig. 7. Dispersion coefficient for various restitution coefficients when/D = 0.068 and H = 30D.

480 M. Rahman et al. / Particuology 6 (2008) 475–482

)] for different percolating heights when d/D = 0.068 and α = 0.8.

ira

werto

cfthgiFitv

Fe

F

w

3

Fig. 8. Plot of (a) r2, and (b) r2/(HD) against ln[N0/(N0 − N

ng the magnitude of the force in upward direction (positive)educes the velocity. Further, the percolation velocity and forcere found to have a relationship, given by

v√gD

= k

√1 − fe

mg, (5)

here m is the mass of the percolating particle, and k is a param-ter related to particle properties such as the particle diameteratio and restitution coefficient. This indicates that the percola-ion velocity is related to the vertical acceleration (=(g − fe/m))f the percolating particle.

We also simulated the case with one layer of percolating parti-les under the effect of extra force. Fig. 10 shows the cumulativerequency distribution of residence time. It can be observed thathe residence time is affected by the vertical acceleration. Forigh vertical acceleration applied in the opposite direction of theravity, the residence time is longer. It is because the percolat-ng particles take longer time to reach the bottom of container.ig. 11 shows the radial dispersion distribution of these percolat-

ng particles under different vertical accelerations. It is observedhat the radial dispersion coefficient increases with increasingertical acceleration.

ig. 9. Relationship between the dimensionless percolation velocity and thextra force when d/D = 0.068, α = 0.8, and H = 30D.

lwit

F

d

ig. 10. Cumulative distribution of residence time with different extra forces

hen d/D = 0.068, α = 0.8, and H = 30D. Va =√

1 − fe/mg.

.5. Effect of feeding rate

If several layers of percolating particles are fed, the perco-ating particles may act on with each other besides interacting

ith packing particles, which will affect the percolation behav-

ors. To understand such feature, Fig. 12 shows the residenceime distributions for the cases with one, two, three, four, five

ig. 11. Plot of r2 against ln[N0/(N0 − N)] for different extra forces when

/D = 0.068 and H = 30D. Va =√

1 − fe/mg.

M. Rahman et al. / Particuol

Fd

apbfitwtawcmbiotmciopbfe

p

Fd

bccftoc

4

bbbc

•

•

•

•

•

ig. 12. Statistic distributions of residence time for different feeding rates when/D = 0.068 and H = 30D.

nd six layers of percolating particles. For all cases, percolatingarticles are fed batch by batch within a fixed time, and eachatch has 171 particles. So, the feeding rate is different for dif-erent number of layers of percolating particles. From Fig. 12,t can be seen that the frequency distribution of the residenceime is dependent on the feeding rate, especially, for the casesith fewer batches. When only single batch of particle is fed,

he distribution has one distinctive high peak. If more batchesre fed, the distribution curves shift to the right, and becomeider, which indicates that the mean residence time for these

ases is higher. This is because more batches of particles needore time to feed, and hence take longer time to reach the bed

ottom. Another reason is that the interactions between percolat-ng particles of different layers reduces the percolation velocityf the particles. In particular, for the case with two batches,he distribution fluctuates largely and has more peaks. The two

ain peaks correspond to the two layers of percolating parti-les, respectively. The curves corresponding to the two peaksntersect with Nt/N0 = 0.07, which indicates that some particlesf the two batches collide during the percolation process. Morearticles of different batches interact for the cases with moreatches, leading to relatively smooth distribution. Furthermore,or the cases with four or more batches, the percolating particles

xhibit similar distributions.

Fig. 13 shows the radial dispersion for different batches ofercolating particles. It can be seen that increasing the num-

ig. 13. Plot of r2 against ln[N0/(N0 − N)] for different feeding rates when/D = 0.068 and α = 0.8.

A

B

R

A

B

B

KL

L

ogy 6 (2008) 475–482 481

er of layers of percolating particles decreases the dispersionoefficient. This indicates that more percolating particles arelose to the center of the packed bed bottom with increasing theeeding rate. It is because as the feeding rate of percolating par-icle is increased, more percolating particles interact with eachther, move cohesively and reach the bottom of the packed bedollectively.

. Conclusions

The spontaneous inter-particle percolation phenomenon haseen investigated by means of DEM simulation. Percolationehaviors such as percolation velocity, residence time distri-ution and radial dispersion have been examined under variousonditions. The results can be summarized as follows:

The percolation behavior is complicated. The percolationvelocity increases with decreasing restitution coefficient andparticle diameter ratio. The velocity is related to not onlythe collision number of the percolating particle, but also thecollision intensity.With the increase of the restitution coefficient, the residencetime of the percolating particles increase. The dispersion coef-ficient increases slowly with restitution coefficient, which isconsistent with the result obtained by the previous experi-ments.The packing height affects the residence time and radial dis-persion. But, the effect can be eliminated in the analysis ofthe residence time and radial dispersion when they are nor-malized by the average residence time and the product of thepacking height and packing particle diameter, respectively.The percolation velocity is shown to be related to the verticalacceleration of the percolating particle when an extra constantvertical force is applied.The feeding rate affects the percolation behaviors. Increasingthe number of batches of percolating particles decreases thedispersion coefficient.

cknowledgments

The authors are grateful to Australian Research Council andlueScope Steel Research for the financial support of this work.

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