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Available online at www.sciencedirect.com

Particuology 6 (2008) 557–571

A numerical investigation into the solids phase chromatography usinga combined continuous and discrete approach

Wei Yang a, Fang Yang b, Yulong Ding a,b,∗a Institute of Particle Science & Engineering, University of Leeds, Leeds LS2 9JT, UKb On visiting from University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

Received 8 May 2008; accepted 15 July 2008

bstract

Solids phase chromatography for particle classification is based on different retention times of particles with different properties when they arelutriated through a confined geometry. This work aims at a fundamental understanding of such a technology by using the combined continuousnd discrete method. A packed bed is employed as the model confined geometry. The numerical method is compared first with experimentalbservations, followed by a parametric analysis of the effects on the flow hydrodynamics and solids behaviour of various parameters including theumber of injected particles, the superficial gas velocity, the contact stiffness and the diameter ratio of the packed column to the packed particles.he results show that the modelling captures some important features of the flow of an injected pulse of fine particles in a packed bed. An increase

n the number of injected particles or the superficial gas velocity reduces the retention time, whereas the contact stiffness does not show much

ffect over the range of 5 × 102 to 5 × 104 N/m. It is also found that the effect on the retention time of the diameter ratio of the packed column tohe packed particles seems complex showing a non-monotonous dependence.

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

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eywords: Solids phase chromatography; Gas–solid two-phase flows; Packed b

. Background

Solids phase chromatography (SPC) is a technology pro-osed for particle classification in the dry form under the processonditions (Ding, 2006). SPC uses a confined geometry, calledeparation channel, through which particles to be fractionatedre pneumatically conveyed by a carrying gas under controlledumidity and temperature. The separation channel can be emptyr structurally packed and the surface of the channel and packingay be coated with anti-electrostatic materials. Various detec-

ors can be used to measure evolution of pressure along thehannel, particle velocity and retention time. This work con-iders packed beds as the separation channels. Fundamentally,lassification using such separation channels relies on different

engths of retention time of particles with different properties dueo interactions among the carrying gas, particles to be classified,eparation channel wall and packing. Our recent experiments,

∗ Corresponding author. Tel.: +44 113 343 2747; fax: +44 113 343 2405.E-mail address: y.ding@leeds.ac.uk (Y. Ding).

htNM1Ga

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

oi:10.1016/j.partic.2008.07.007

ombined continuous and discrete method; Modelling

sing packed beds as the separation channels, showed that, givenpacked column and a superficial gas velocity, the retention

ime of particles depended on the size, density, and shape ofarticles, with larger, spherical or lighter particles passing moreuickly through the channel (Yang & Ding, in press). We cannotxplain the experimental observations, particularly why largearticles elutriate faster than smaller particles. This forms onef the motivations of this work.

There are various industrial processes involving aas–particle mixture flowing through packed beds. Exam-les include deep bed filtration of particle laden flue gases,ulverised coal injection into blast furnaces, and sorptionnhanced reaction processes. As a results, a number studiesave been carried out over the past few decades on theydrodynamics of a steady-state flow of gas–solid mixtureshrough packed beds in a continuous manner (Chen, Akiyama,ogami, & Yagi, 1994; Hidaka, Matsumoto, Kusakabe, &

orooka, 1999; Shibata, Shimizu, Inaba, Takahashi, & Yagi,

991; Song, Wang, Jin, & Tanaka, 1995; Wang, Ding, &hadiri, 2004). These studies are different from the SPC wherepulse of gas–solid mixture is injected into the packed bed. The

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

558 W. Yang et al. / Particuolog

Nomenclature

a radius of the circular area of contactCD drag coefficientdp diameter of packed particledp,i diameter of particle iDc diameter of packed columne coefficient of restitutionE elastic modulusfn normal contact forcefp,ij interaction forces between particle i and jft tangential contact forceFgp gas–particle interaction forceg gravitational accelerationG shear modulusIi moment of inertia of particle ik contact stiffnesski number of particles in contact with particle ikn normal contact stiffnessL length of packed columnmi mass of the particle in̂ unit vector in the normal directionpg gas pressureP normal forceR radius of sphereRep particle Reynolds numberRc radius of packed columnr radial positionSn particle displacement in normal directionSt particle displacement in tangential directiont̂ unit vector in tangential directiont timeTp,ij intraparticle torqueUg superficial gas velocityvg velocity vector of gasvi velocity of particle ivp velocity vector of particlevr relative velocity<. . .> calculation of an average value

Greek lettersα relative approach of the centres of the two con-

tiguous spheresε porosityη coefficient of dampingμfric coefficient of frictionμg viscosity of gasν Poisson’s ratioρg density of gasτg viscosity stress tensor of gas

sum

etontapstrtAcipswwostswcanaooAppSvbeWiddvbtoat

2

sN

ωi rotational velocity of particle i

econd motivation of this work is therefore to understand thensteady-state solids motion and hydrodynamics of gas–solidixtures flowing through the packed column.

(wst

y 6 (2008) 557–571

Packed beds have a complex interior structure. Several mod-lling approaches have been employed in the past to predicthe flow field in the pores of the packing with different degreesf success. The most sophisticated models are based on directumerical simulation where the packed particles are treated ashe solid boundaries (Manz, Gladden, & Warren, 1999). Thispproach uses very fine meshes to characterise the flow in theores and hence is capable of addressing some important issuesuch as predicting non-parallel flows. Direct numerical simula-ion is advantageous in revealing details of the flow fields butequires very expensive computational powers—a major limi-ation for applications in simulating real industrial processes.nother shortcoming is probably the difficulties in handling the

ollisions between particles. As a consequence, direct numer-cal simulation is rarely used in simulating gas–solid flows inacked beds. A more conventional (and most popular) way ofimulating gas–solid flows in packed beds is the so-called net-ork modelling (Martins, Laranjeira, Lopes, & Dias, 2007),here the local packing structure is represented by a networkf regular geometries, usually spheres and cylinders or con-tricted tubes. For example, Kozeny theory treats the pores inhe packed bed as an ensemble of channels of various cross-ections (Fan & Zhu, 1998), whereas the constricted tube modelas used by Pendse, Chiang, and Tien (1983) who found suc-

ess in improving the prediction of axial dispersion. Imdakmnd Sahimi (1991) combined the Monte Carlo method with theetwork model to study particle transportation in porous mediand obtained quantitative agreement with experimental data. Anbvious disadvantage of the network modelling is its inabilityf providing detailed information at the single particle level.nother popular approach of modelling gas–solid flows in theacked beds is the two-fluid model where both gas and the sus-ended particle phases are treated as interpenetrating continua.uch an approach uses separate mass and momentum conser-ation equations for each phase and hydrodynamic interactionsetween the two phases are through the drag coefficient; see forxample Dong, Zhang, Pinson, Yu, and Zulli, 2004 and Li, Ding,en, and He (2006). This approach requires the use of empir-

cal relationships for both the drag coefficient and the porosityistribution. This work employs the combined continuous andiscrete method (CCDM)—a technique that is capable of pro-iding detailed single particle level information hard to obtainy using other approaches (Xu & Yu, 1997). This also formshe third motivation of this work—to interpret the experimentalbservations by using the CCDM modelling results. Addition-lly, the CCDM could also provide information for improvinghe SPC.

. Model formulation and numerical solution

The combined continuous and discrete particle method con-iders the fluid phase as a continuum with the locally averagedavier–Stokes equations governing the momentum transport

Anderson & Jackson, 1967), and the particle phase as discreteith the motion of individual particles governed by Newton’s

econd law of motion. The two phases are coupled throughhe Newton’s third law of motion. The concept of the dis-

uolog

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R

Ts

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f

w

eiRwaaCsraBy using the model proposed by Cundall and Strack (1979), thecontact force is expressed by

fp = fn + ft, (12)

W. Yang et al. / Partic

rete element method was originally developed by Cundall andtrack (1979) and has been extensively explored and used forodelling various particle systems, particularly gas–solid and

ranular flow systems over the past two decades; see for exam-le Xu and Yu (1997), Mikami, Kamiya, and Horio (1998) andoomans, Kuipers, and van Swaaij (2000) on gas fluidisation,suji, Tanaka, and Ishida (1992), Di Maio and Di Renzo (2004)n pneumatic conveying, Foo, Sheng, and Briscoe (2004) ongglomeration, Moreno, Ghadiri, and Antony (2003), Morrisonnd Cleary (2004) on particle breakage, Hsiau and Yang (2005)n shearing granular materials, and Zou et al. (2003) and Sobolevnd Amirjanov (2004) on particle packing. In the following,overning equations for both the fluid and particle phases andhe coupling equation between the two phases are given firstn Section 2.1, whereas the numerical schemes and simulationonditions are detailed in Section 2.2.

.1. Governing equations

The governing equations for the gas phase include the massnd momentum balances given, respectively by

∂ρgε

∂t= ∇ · (ρgεvg), (1)

∂ρgεvg

∂t+ ∇ · (ρgεvgvg) = −ε∇pg − ε∇ · �g + ρgεg + fgp,

(2)

here ρg, vg, pg, and τg are the density, velocity vector, pressurend viscosity stress tensor, respectively, ε is the porosity, and gs the gravitational acceleration. fgp in Eq. (2) is the gas–particlenteraction force expressed by:

gp = β · (〈vp〉 − vg

), (3)

here 〈vp〉 is particle velocity vector averaged in a computa-ional cell, and β is the interaction term given by either the Ergunquation (Ergun, 1952), Eq. (4), or the Wen and Yu correlationWen & Yu, 1966), Eq. (5):

= 150(1 − ε)2

ε〈dp〉 μg + 1.75(1 − ε)ρg

〈dp〉∣∣〈vp〉 − vg

∣∣ (ε ≤ 0.8),

(4)

= 3

4ρgCD

(1 − ε)ε−1.7

〈dp〉∣∣〈vp〉 − vg

∣∣ (ε > 0.8), (5)

ith 〈dp〉 the average diameter of particles in the cell, μg the gasiscosity, and CD the drag coefficient for single particles giveny

D = 24

Rep(1 + 0.15Re0.687

p ) (Rep ≤ 1000), (6)

D = 0.44 (Rep > 1000), (7)

y 6 (2008) 557–571 559

here Rep is the particle Reynolds number defined by

ep = ρgε∣∣〈vp〉 − vg

∣∣ 〈dp〉μg

. (8)

he velocity of individual particles is determined by Newton’second law of motion:

i

dvi

dt= mig + ff,i +

ki∑j=1

fp,ij, (9)

here mi and vi are, respectively the mass and velocity of particle, fp,ij is the interaction forces between particles i and j, and ki

s the number of particles in contact with particle i. Interactionetween particles may cause particle to rotate and the rotationalotion is calculated by

i

d�i

dt=

ki∑j=1

Tp,ij, (10)

ith Ii the moment of inertia, ωi the rotational velocity of particleand Tp,ij the torque. ff,i in Eq. (9) is the force exerted by theuid on the particle and can be expressed by

f,i = πd3p,i

6

(β

1 − ε(vg − vi) − ∇Pg

), (11)

here dp,i is the diameter of particle i.When the long-range interparticle forces (e.g. van der Waals,

lectrostatic, liquid bridge) are negligible, the most importantnterparticle force is the direct contact force due to collisions (Dienzo & Di Maio, 2004). There are several approaches to dealith the contact mechanics; see for example Johnson, Kendall,

nd Roberts (1971), Cundall and Strack (1979) and Thorntonnd Yin (1991). The linear spring-dashpot model proposed byundall and Strack (1979) was employed in this work. Fig. 1

hows a schematic diagram of the approach. Here the spring rep-esents the elastic contribution to the contact while the dashpotccounts for the energy dissipation by the plastic deformation.

Fig. 1. Models of contact forces: (a) normal force; (b) tangential force.

560 W. Yang et al. / Particuology 6 (2008) 557–571

Table 1Simulation conditions.

Length of the column 20 mmDiameter of the column 9.5, 14.3, 19 mmDiameter of packed particles 2 mm spheresFriction coefficient of walls 0.3Friction coefficient of particles 0.2, 0.5, 0.8Density of particles 2500 kg/m3

No. of particles 5250–52,500Particle sizes 75–90, 106–125 �mRestitution coefficient 0.9Contact stiffness 5 × 102, 5 × 103, 5 × 104, 5 × 106

Density of gas 1.205 kg/m3

VS

wt

f

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f

wi

2

idttt2Fnbfipia

mflctnasTi

pSceava

pdOamtwt

c(t

T

wY

iscosity of gas 1.8 × 10−5 Pa superficial gas velocity 1.5, 1.7, 1.9 m/s

here fn and ft are, respectively the normal and tangential con-act forces given by

n = −kSn − ηvn, ft = −kSt − ηvt, vn

= (vr · n)n, vt = vr − vn, (13)

here Sn and St are displacements in the normal and the tan-ential directions, respectively, k is the contact stiffness, η is theamping coefficient, and n is the unit vector in the normal direc-ion. If sliding takes place, Eq. (13) for ft should be replacedy

t = −μfric |fn| t, (14)

here μfric is the coefficient of friction, and t is the unit vectorn the tangential direction.

.2. Numerical method and simulation conditions

A 20-mm long vertically oriented column was consideredn this work. The majority of simulations were on a 9.5-mmiameter column, the one used in our experiments. Two addi-ional column diameters, 14.3 and 19 mm, were also simulatedo examine the effect of diameter ratio of the packed columno packed particles. The separation channels were packed withmm spheres, which gave an average packing porosity of 0.45.ine particles of two sieve cuts, 75–90 and 106–125 �m (withormal size distributions in each sieve cut), were carried upwardy an airflow from the bottom of the packed column. Thesene particles were generated just below the entrance of theacked column. Table 1 shows the simulation conditions. Thenlet velocity and outlet pressure of the carrying gas were givennd the non-slip conditions were assumed at the column wall.

The simulations were performed in the Itasca PFC3D environ-ent. In such an environment, the fluid scheme is invoked at eachuid timestep between the fluid motion and particle dynamicsalculations. The fluid scheme is executed when the accumula-ive mechanical time equals or exceeds the predicted time for theext fluid step. The pressure and velocity of fluid in each cell

re calculated by applying the semi-implicit method for pres-ure linked equations (SIMPLE) algorithm (Patankar, 1980).he fluid–particle interaction forces are calculated during the

teration. Fig. 2 shows some cells used in the fluid scheme. The

flaoc

Fig. 2. Scalar cell and staggered cell.

orosity and pressure are defined at the centre of scalar cells.taggered cells, which are shifted a half size relative to the scalarells in each direction, are used for discretising the momentumquations, so that the fluid velocity in each direction is definedt the boundary of the scalar cells. The parameters (diameter andelocity) for particles used for calculating interaction forces areveraged over each staggered cell.

It is computationally prohibitive to check all possible pairs ofarticles within each timestep. Contact detection in this work isone by mapping particles into rectangular cells (Cundall, 1988).nly particles contained within the same cell are recognised

s neighbours and detected for contact. As individual particlesove during the course of a simulation, they are remapped and

ested for contact with new neighbours. Contact is said to occurhen the centre-to-centre distance between two particles is equal

o or less than the summation of the radii of the two particles.Timestep for particles is determined by the trade-off between

omputation time and numerical stability. Cundall and Strack1979) proposed the following relation for the mechanicalimestep determination.

imestep < 2

√m

k, (15)

here m is the mass of the particle and k is the stiffness. Xu andu (1997) used a drop test to evaluate timestep in simulations of

uidised beds and found the smallest timestep occurred whenfree falling particle bounds back to the same height as the

riginal height. Timestep is also affected by the smallest parti-le size (dpmin). Kafui, Thornton, and Adams (2002) calculated

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W. Yang et al. / Partic

he critical timestep based on the Rayleigh wave speed of forceransmission on the surface of elastic bodies and proposed theollowing relations for critical timestep determination.

ritical Timstep = πdpmin

2λ

√ρp

G, (16)

ith λ = 0.1631� + 0.8766, ν the Poisson ratio, G the Young’sodulus and ρp the particle density. In this work, the method

roposed by Cundall and Strack (1979) was used. The fluidimestep is generally greater than the mechanical timestep forarticles. PFC3D chooses the minimum value from the followinghree values for the fluid timestep: (a) 100 times the mechani-al timestep; (b) half of the minimum value over all fluid cellsefined by the size of the fluid cell divided by the magnitudef the fluid velocity in each direction; and (c) half of the min-mum value over all fluid cells defined by the size of the fluidell divided by the magnitude of the averaged particle velocityn each direction. If the fluid timestep calculated by this pro-edure is smaller than the mechanical timestep, the mechanicalimestep is chosen for the fluid timestep.

Plastic deformation occurs when the relative impact veloc-ty between two colliding spheres is higher than the criticalield velocity (Fan & Zhu, 1998). In the model of Cundall andtrack (1979), a damping coefficient was introduced to take intoccount kinetic energy loss in an inelastic collision. However,t is difficult to determine this coefficient theoretically due tonvolvement of multi-collision dynamics (Xu & Yu, 1997). Tsuji,awaguchi, and Tanaka (1993) proposed a simplified relationetween the damping coefficient and the coefficient of restitutione):

= 2

√mk

2 . (17)

(π/ln(e)) + 1

In this work, the drop test proposed by Xu and Yu (1997) wassed.

ttsb

ig. 3. Schematic illustration of the procedure of making a packed column: (a) generavity; (e) removal of extra particles and placing a mesh at the outlet to prevent part

y 6 (2008) 557–571 561

. Results and discussion

The first step of the work was to compare the modellingesults with experimental observations on a qualitative basis.he work was then focused on understanding the mechanismsf particle classification in packed columns. This was done byxamining the effects of various factors on the hydrodynamics ofas–solid flow and particle dynamics. These factors included theumber of fine particles, the superficial gas velocity, the mate-ial property (contact stiffness), and the size ratio of the packedolumn to the packed particles. The results are presented andiscussed in the following. In this section, except for otherwisendicated, the friction coefficient is taken as 0.5 for particles and.3 for the wall, the stiff coefficient as 5 × 103, the number ofarticles as 5250 and particle size as 106–125 �m sieve cut.

.1. Preparation of packed columns

The column was packed by the free-fall method, as shown inig. 3. First, packed particles of 2 mm spheres were generated

n a hopper above the column. The initial velocities of thesearticles were set to zero. Driven by the gravity, these particlesell downward into the column. The packing process was termi-ated when the ratio of maximum unbalanced forces to contactorces was equal to or below 0.001. Then, these particles wereermanently bonded to one another so that there was no rela-ive motion among the packed particles and between the packedarticles and the column during the rest of the simulation.

.2. Comparison between experiments and simulations

Experiments were carried out using a column of the sameiameter and packed with the same sized particles as used inhe modelling. However, the column was 2 m long, much longerhan that used in the modelling. The amount of fine particles for

he classification experiments was about 2 g, which are far morehan we could handle using the discrete method within a rea-onable period of time. As a consequence, the comparison wille on a qualitative basis. Despite the difference in the length

ration of particles in a hopper above the column; (b–d) free fall of particles byicles from escaping.

5 uology 6 (2008) 557–571

obct

sar

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•

•

•

mp

3

5a

Fig. 4. Pressure responses: qualitative comparison between experiments andsimulations at superficial gas velocity of 1.5 m/s (2 g of fine particles in experi-m(si

62 W. Yang et al. / Partic

f the column and the amount of fine particles, pressure distri-ution was measured along the column in experiments, whichould provide a means for comparison as will be discussed inhe following.

Fig. 4(a) and (b) shows the measured and calculated pres-ure responses after fine particles are injected at the inlet andposition some distance from the inlet in the packed column,

espectively. Several observations can be made:

Both experimental measurements and modelling results showa rapid increase in the pressure at the inlet when fine particlesare injected. Such rapid increase in the pressure is due to thelocal decrease in the bed porosity as a result of injection offine particles, and energy consumed for transporting the fineparticles and for overcoming the energy loss due to collisionsbetween fine particles, fine particles and packed particles andfine particles and the wall; see more discussion on this later.The pressure quickly reaches a maximum and then declinesslowly to approach the pressure level before the injection;see the upper curves in Fig. 4(a) and (b). A similar pressureresponse is observed at a downstream position with a timedelay (lower curves in Fig. 4(a) and (b)). Note that the timedelay is of the order of the distance divided by the superficialgas velocity.Both the modelling and experiments show that the pressureresponse peaks at the downstream positions are lower than thatat the inlet, which is likely due to the damping of the packedbed on the pressure signal propagation and also dispersion offine particles so that the decrease in the local porosity is lowerthan that at the inlet; see later for more discussion.Both the experiments and modelling show that the smaller par-ticles (75–90 �m) give a higher pressure peak and narrowerpressure distribution than the larger particles (106–125 �m),though the predicted pressure distributions are sharper andnarrower than the measured ones.There are small-scale fluctuations in the predicted outlet pres-sure, which does not occur in the measured pressure becausethe data acquisition rate in the experiments is much lowerthan in the modelling. The fluctuation disappears in predictedpressure drop between the inlet and outlet, similar to the exper-imental observation. This is shown in Fig. 4(c) where thepressure drop is normalised by the total particle weight perunit cross-sectional area (wp/A) of the packed column. It isinteresting to note that the elutriation time ratio of the exper-iment to that of prediction is about 20, which compares wellwith the length ratio of the column used in the experiments tothat used in the simulation.

The above observations suggest that modelling captures someain features of the flow hydrodynamics of a pulse of fine

articles through a packed bed.

.3. Effect of the injected number of fine particles

Numerical simulations were performed on the injection of250, 26,250 and 52,500 particles of sieve cut 106–125 �m atsuperficial gas velocity 1.9 m/s. Fig. 5 shows the time evolu-

tafib

ents, 5250 of 106–125 �m and 13,718 of 75–90 �m particles in simulations).a) Measured pressure response at the inlet and 0.4 m from inlet (P1 pressureensor); (b) calculated pressure responses at the inlet and outlet (20 mm from thenlet); (c) normalised pressure drop for the injection of 106–125 �m particles.

ions of the porosity and pressure drop where the porosity is the

verage of the packed column. It can be seen that injection ofne particles results in a reduction in the average porosity of theed as expected and the extent of the reduction increases with

W. Yang et al. / Particuology 6 (2008) 557–571 563

Fig. 5. Effect of the number of injected fine particles: column with 20 mm lengthand 9.5 mm diameter, superficial gas velocity at 1.9 m/s and injected particleswp

ipiTdpi5ppdtpbaTocl

Ffig

tttRzcwpp(poDpfio

owtmanslpp

tladof injected particles increases from 5250 to 26,250, but further

ith a sieve cut of 106–125 �m. (a) Average porosity of the packed column; (b)ressure drop across the packed column.

ncreasing number of injected particles; an injection of 52,500articles causes a decrease in the porosity by nearly eight timesn comparison to the injection of 5250 particles; see Fig. 5(a).he porosity decrease corresponds to the increase in the pressurerop across column as shown in Fig. 5(b); an injection of 52,500articles leads to a peak pressure drop of around 20 kPa, whichs about 2.5 times the peak pressure drop caused by injecting250 particles. It is interesting to note that the time at which theeak pressure drop occurs (Fig. 5b) differs from that the peakorosity occurs (Fig. 5a)—the peak time for the pressure dropecreases with increasing number of injected particles, whereashat for the porosity is little dependent on the number of injectedarticles. Such a difference is likely due to the porosity dataeing averaged across the whole volume of the packed columnnd hence they do not reflect the true dynamics of the system.he pressure drop data, on the other hand, reflect the dynamics

f the system. An increase in the number of injected particlesauses a decrease in the local porosity, hence an increase in theocal gas velocity and a fast pressure response.

iis

ig. 6. Effect of number of injected fine particles on the radial occupancy of thene particles: column with 20 mm length and 9.5 mm diameter at a superficialas velocity of 1.9 m/s, injected particles of a sieve cut of 106–125 �m.

Fig. 6 shows the radial occupancy distribution of the fine par-icles in the packed column, where the occupancy is defined ashe time ratio of fine particles spending in an annular voxel tohe total time that the particles spend in the column. In Fig. 6,c is the column radius, r is the radial co-ordinate, and the hori-ontal scale is normalised by the packed particle diameter. Onean see that the fine particles have the highest occupancy at theall (Rc − r)/dp = 0, where the porosity is the highest. The occu-ancy falls quickly to a minimum at (Rc − r)/dp ∼0.5, i.e. halfarticle diameter away from the wall. With increasing value ofRc − r)/dp, a periodical change of the occupancy with the radialosition is observed, similar to the radial porosity distributionbserved experimentally; see for example de Klerk (2003) anding et al. (2005). Fig. 6 also shows that the number of injectedarticles has a small effect on the radial occupancy distribution ofne particles except for the wall region where a greater numberf injected fine particles give a slightly lower occupancy.

Fig. 7 illustrates the effect of injected fine particle numbern the mean velocity and the kinetic energy of the particleshere the particle velocity refers to the translational velocity and

he kinetic energy accounts for both translational and rotationalotions of the particles. One can see that both the mean velocity

nd kinetic energy of the fine particles increase with increasingumber of injected fine particles. This is consistent with the pres-ure drop increase and porosity decrease as discussed above—aarger number of fine particles leads to a reduction in the localorosity, which gives a higher local gas velocity, hence the higherarticle velocity and kinetic energy.

The effect of injected fine particle number on their break-hrough behaviour is shown in Fig. 8, which reveals that aarger number of injected particles give a shorter retention timend hence a faster breakthrough. Fig. 8 also suggests that theecrease in the retention time is significant when the number

ncrease in the number of particles gives a very small effect. Themplication of this observation is that the SPC for particle clas-ification should be operated with a moderate amount of fine

564 W. Yang et al. / Particuology 6 (2008) 557–571

Fig. 7. Effect of injected fine particle number on the mean velocity and kineticenergy of particles for a superficial gas velocity of 1.9 m/s and injected particlesof a sieve cut of 106–125 �m. (a) Mean translational particle velocity; (b) kineticenergy.

Fig. 8. Effect of the number of injected particles on the breakthrough behaviourof fine particles at a superficial gas velocity of 1.9 m/s and injected particles ofa sieve cut of 106–125 �m.

Fi

pe

rptptacatptbrtsttcinfrpl

tofnttogtt

ig. 9. Normalised contacts for a superficial gas velocity of 1.9 m/s, 5250njected particles of a sieve cut of 106–125 �m.

articles. As the span between 5250 and 26,250 is large, thexact range of particle number needs further investigation.

Illustrated in Fig. 9 are normalised contact numbers withespect to the total number of particles for an injection of 5250articles at superficial gas velocity 1.9 m/s. It can be seen that, athe very beginning of the injection, collisions among the injectedarticles dominate. Once the particles enter the packed bed, con-acts between the injected and the packed particles arise quicklynd soon overtake the collisions among the injected particles. Inontrast, the number of contacts between the injected particlesnd the wall are much lower than that among the injected par-icles in the initial stage and that between the injected and theacked particles after the initial stage. It is therefore concludedhat the interactions among the injected particles and interactionsetween the injected and packed particles mainly determine theetention time. Fig. 10 compares the normalised particle con-acts with different numbers of injected fine particles. One canee that an increase in the injected particle number increaseshe number of contacts among the injected particles, whereashe number of contacts between the injected and packed parti-les increases very little when the number of injected particlesncreases from 5250 to 26,250 and little change to the contactumber occurs when the number of injected particles increasesrom 26,250 to 52,500. This seems to indicate some sort of satu-ation in terms of contact between the injected particles and theacked particles when particle concentration exceeds a certainimit—similar to the adsorption process.

A particle may slide upon contact with another particle orhe wall. This can cause kinetic energy loss usually in the formf heat. Fig. 11 shows the mean accumulative frictional workor three different numbers of injected particles, calculated byormalising the accumulative frictional work with respect to theotal number of injected particles. The mean accumulative fric-ional work is seen to vary significantly for the different numbers

f injected particles. A larger number of injected particles give areater amount of mean frictional work. As the number of con-acts between the injected and packed particles is less sensitiveo the injected particle number (Fig. 10), the difference in the

W. Yang et al. / Particuology 6 (2008) 557–571 565

Fig. 10. Effect of injected fine particle number on particle collisions for a superfi-cial gas velocity of 1.9 m/s and injected particles of a sieve cut of 106–125 �m. (a)Collisions among injected particles; (b) collisions between injected and packedparticles.

Fig. 11. Effect of injected particle number on the mean accumulative frictionalwork for a superficial gas velocity of 1.9 m/s and particles of the sieve cut of106–125 �m.

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ig. 12. Effect of superficial gas velocity on the time evolutions of mean particleelocity and kinetic energy for 5250 particles of 106–125 �m sieve cut. (a) Meanarticle velocity; (b) kinetic energy.

ean frictional work as shown in Fig. 11 is most likely to be dueo the increased interactions among the injected particles. Thessociated energy loss is reflected in the pressure drop increases discussed above (Figs. 4 and 5).

.4. Effect of superficial gas velocity

Fig. 12 plots the time evolutions of the mean velocity and theotal kinetic energy of the injected particles for three superficialas velocities. One can see that an increase in the superficial gaselocity from 1.5 to 1.9 m/s (∼1.27 times) leads to an increase inhe peak mean particle velocity from ∼0.58 to ∼0.75 m/s (∼1.29imes) and the corresponding peak kinetic energy increase isbout 1.58 times. Note that the slightly higher extent of thencrease in the mean particle velocity than that in the superficialas velocity increase does not violate the first law of thermo-ynamics as the peak values are used. If there were no kinetic

nergy loss, the increase in the mean particle velocity would havemplied a kinetic energy increase of ∼1.66 times. The actualower value of 1.58 times is due to kinetic energy loss to otherorms of energy loss such as heat due to friction.

566 W. Yang et al. / Particuology 6 (2008) 557–571

Fig. 13. Effect of superficial gas velocity on the breakthrough behaviour for5n

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Fig. 14. Effect of superficial gas velocity on the frictional energy loss for 5250particles of 106–125 �m sieve cut.

250 particles of 106–125 �m sieve cut. (a) Normal breakthrough curves; (b)ormalised breakthrough curves with respect to (L/Ug).

The increased mean particle velocity also implies a fast par-icle elutriation and hence a shorter retention time. This ishown in Fig. 13(a) where particle breakthrough curves areeen to shift towards the left when the superficial gas velocitys increased. It is interesting to note that all the breakthroughurves collapse into a single curve (Fig. 13b) if the elutria-ion time is scaled with respect to L/Ug where L is the columnength.

Increasing the superficial gas velocity increases the accu-ulative frictional loss; see Fig. 14. However, the normalised

umber of contacts seems to be insensitive to the change ofhe superficial gas velocity in the range of 1.5–1.9 m/s as illus-rated in Fig. 15. This is likely to be due to a shorter retentionime of fine particles at a higher superficial gas velocity. Its also found that the superficial gas velocity imposes little

ffect on the radial distribution of particle occupancy (Fig. 16),hich is in agreement with the experimental observations bying et al. (2005) using the positron emission particle tracking

echnique.

Fig. 15. Effect of superficial gas velocity on particle collisions for 5250 particlesof 106–125 �m sieve cut. (a) Contacts between injected and packed particles;(b) contacts among injected particles.

W. Yang et al. / Particuology 6 (2008) 557–571 567

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ig. 16. Effect of superficial gas velocity on radial particle occupancy distribu-ion for 5250 particles of 106–125 �m sieve cut.

.5. Effect of contact stiffness

As discussed above, collisions among injected fine particlesnd that between injected and packed particles dominate at dif-erent stages of fine particle motion through the packed bed. Theontact stiffness is expected to impose a significant effect on theetention time of fine particles. This was investigated by usingrange of stiffness values between 5 × 102 and 5 × 106 N/m in

he simulation with 5250 particles of 106–125 �m sieve cut at auperficial gas velocity of 1.9 m/s. Fig. 17 shows the simulatedreakthrough curves for the different values of the contact stiff-ess. It can be seen that the breakthrough curves for the contacttiffness ranging from 5 × 102 to 5 × 104 N/m are almost over-apping and are in qualitative agreement with the experimentalbservations as shown earlier, whereas the breakthrough curveor the contact stiffness of 5 × 106 N/m is very much different

rom the experimental observations. By looking at the detailedarticle motion, it was found that the use of contact stiffness of× 106 N/m leads to rebounding of the injected particles at the

nlet of the packed column upon injection.

ig. 17. Effect of contact stiffness on fine particle breakthrough behaviour for250 particles at a superficial gas velocity of 1.9 m/s.

1

mttakecinnaedFtbcTe

ig. 18. Effect of the contact stiffness on the mean gas velocities and pressurerop for 5250 particles of 106–125 �m sieve cut at a superficial gas velocity of.9 m/s. (a) Mean gas velocity; (b) pressure drop.

Fig. 18 illustrates the effect of the contact stiffness on theean gas velocity and pressure drop. One can see that both

he mean gas velocity and the pressure drop depend little onhe value of the contact stiffness over the range of 5 × 102

nd 5 × 104 N/m. This is also true for the mean velocity andinetic energy of the fine particles as shown in Fig. 19. How-ver, the change in the contact stiffness changes the number ofontacts between the injected and packed particles as shownn Fig. 20, which indicates that the normalised accumulativeumber of contacts increases with increasing contact stiff-ess. This suggests that particles of higher contact stiffnessre more heavily intercepted by the packed particles. How-ver, as shown in Fig. 17, the breakthrough behaviour is littleependent on the stiffness value over the range considered inig. 20, other factors must have played a role to compensate

he interception of particles. The frictional energy loss could

e a main factor as shown in Fig. 21, where increasing theontact stiffness decreases the accumulative frictional work.his implies that higher contact stiffness results in less kineticnergy loss of particles, because the frictional force is directly

568 W. Yang et al. / Particuology 6 (2008) 557–571

Fig. 19. Effect of contact stiffness on the mean particle velocity and the totalkinetic energy for 5250 particles of 106–125 �m sieve cut at a superficial gasvelocity of 1.9 m/s. (a) Mean particle velocity; (b) total kinetic energy.

Fig. 20. Effect of contact stiffness on the accumulative number of contactsbetween the injected and the packed particles for 5250 particles of 106–125 �msieve cut at a superficial gas velocity of 1.9 m/s.

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ig. 21. Effect of contact stiffness on the accumulative frictional work for 5250articles of 106–125 �m sieve cut at a superficial gas velocity of 1.9 m/s.

elated to the contact force, which is proportional to the contacttiffness.

.6. Effect of the ratio of packed column diameter to packedarticle diameter

The diameter ratio of the packed column to the packed par-icles has a strong impact on the packing structure (de Klerk,003). To investigate such an effect, packed column diameteras changed to give diameter ratios of 4.75, 7.13 and 9.50 and the

orresponding injected fine particle numbers were 5250, 11,813nd 21,000, respectively to keep the same volume fraction. Inhese simulations, fine particles of 106–125 �m sieve cut weresed; the superficial gas velocity was 1.9 m/s and the contacttiffness was taken as 5 × 103 N/m.

Fig. 22 shows the breakthrough curves for the different diam-ter ratios. One can see that the diameter ratio of 7.13 gives theastest breakthrough suggesting that the effect of the diameteratio is complex. On one hand, an increase in the diameter ratio

ig. 22. Effect of diameter ratio of the packed column to the packed particlesn the breakthrough curves for 5250 particles of 106–125 �m sieve cut at auperficial gas velocity of 1.9 m/s.

W. Yang et al. / Particuology 6 (2008) 557–571 569

Fig. 23. Effect of diameter ratio of the packed column to the packed particlesosv

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Fig. 24. Effect of diameter ratio of the packed column to the packed particles onthe mean particle velocity for particles of 106–125 �m sieve cut at a superficialgas velocity of 1.9 m/s.

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n the pressure drop and mean gas velocity for 5250 particles of 106–125 �mieve cut at a superficial gas velocity of 1.9 m/s. (a) Pressure drop; (b) mean gaselocity.

rom 4.75 to 9.5 leads to a reduction in the peak porosity from.428 to 0.396 (not shown in figures). This should lead to anncrease in the breakthrough time. However, the increase in theiameter ratio also leads to an increase in the mean gas veloc-ty (and hence an increase in the pressure drop) as shown inig. 23. The consequence of this is that the injected particlesould have a greater velocity (Fig. 24). The combination of the

bove two factors seems to provide an explanation to the fastestreakthrough for the diameter ratio of 7.13. The above analy-es also suggest existence of an optimal size ratio for a shortestreakthrough time.

The diameter ratio of the packed column to the packed parti-les has an effect on the radial distribution of particle occupancy.ig. 25 shows the results. Increasing the diameter ratio leads todecreased peak height of the oscillatory dependence of radial

ccupancy distribution. This is because an increase in the diame-er ratio provides more paths for the injected particles and hencearticles are more widely and evenly distributed in the radialirection. The results shown in Fig. 25 also reveal that a greater

•

ig. 25. Effect of diameter ratio of the packed column to the packed particles onhe radial distribution of particle occupancy for 5250 particles of 106–125 �mieve cut at a superficial gas velocity of 1.9 m/s.

iameter ratio gives a reduced particle occupancy at the wallnd hence an increased occupancy in the centre region of theolumn.

. Concluding remarks

This work aims at a fundamental understanding of the SPCor particle classification by using the CCDM. The followingonclusions are obtained:

The numerical simulations capture some important featuresof the flow hydrodynamics and solids behaviour of injectedfine particles flowing through a packed bed. The simulated

time evolutions of the pressure and pressure drop agree atleast qualitatively with the experimental observations.Injection of fine particles into a packed bed reduces the aver-age bed porosity, leading to an increased local gas velocity

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70 W. Yang et al. / Partic

and hence an increased mean particle velocity and a reducedparticle retention time.The radial particle occupancy changes periodically with radialposition with the highest occupancy occurring at the wallwhere the porosity is the highest. Such an observation is littleaffected by the number of injected particles and the super-ficial gas velocity. However, the effect of diameter ratio ofthe packed column to the packed particles is considerable; anincrease in the diameter ratio leads to a decreased peak heightof occupancy distribution, indicating a more evenly particledistribution across the column cross-section.Collisions among the injected particles and that between theinjected and packed particles are dominant in determining theretention time of the injected particles, with the former domi-nates the initial stage of the injection and the latter dominatesthe rest of the elutriation process. An increase in the number ofinjected particles increases the number of contacts among theinjected particles, but this does not affect much the number ofcontacts between the injected and packed particles, indicating‘saturation’ of the contact similar to the adsorption process.The contact stiffness over the range of 5 × 102 to 5 × 104 N/mimposes some effects on the number of contacts between parti-cles. However, such a parameter has little influence on particleretention time, mean gas velocity, pressure drop and meanparticle velocity.

The results reported in this paper are for one sieve cut, focus-ng on the effects of the number of fine particles, the superficialas velocity, the material properties (contact stiffness), and theize ratio of the packed column to the packed particles. Furtherork is underway investigating numerically the effects of parti-

le size and surface energy, which is expected to provide cluess for why large particles elutriate faster than small ones.

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