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Particuology 7 (2009) 233–237

Contents lists available at ScienceDirect

Particuology

journa l homepage: www.e lsev ier .com/ locate /par t ic

as–solid interaction force from direct numerical simulation (DNS) of binaryystems with extreme diameter ratios

. Sarkar, S.H.L. Kriebitzsch, M.A. van der Hoef ∗, J.A.M. Kuipersepartment of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

r t i c l e i n f o

rticle history:eceived 5 November 2008ccepted 27 February 2009

eywords:

a b s t r a c t

Fluid–particle systems as commonly encountered in chemical, metallurgical and petroleum industriesare mostly polydisperse in nature. However, the relations used to describe fluid–particle interactions areoriginally derived from monodisperse systems, with ad hoc modifications to account for polydispersity. Inprevious work it was shown that for bidisperse systems with moderate diameter ratios of 1:2 to 1:4, this

idisperseluid–particle interaction forceattice Boltzmann simulationirect numerical simulation

approach leads to discrepancies, and a correction factor is needed. In this work we demonstrate that thiscorrection factor also holds for more extreme diameter ratios of 1:5, 1:7 and 1:10, although the force on thelarge particles is slightly overestimated when using the correction factor. The main origin of the correctionis that the void surrounding the large particles becomes less in case of a bidisperse mixture, as comparedto a monodisperse system with the same volume fraction. We further investigated this discrepancy bycalculating the volume per particle by means of Voronoi tessellation.

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btstnmitswfflfautsis

© 2009 Chinese So

. Introduction

Flow through an assembly of spheres like packed beds oruidized beds are widely encountered in chemical engineering

ndustry. In order to describe the interaction between the fluid andarticles, mainly empirical relations are used, which can be roughlylassified into two types. The first type of relation (Eq. (1)) is basedn the expression of the drag force on a single particle, Fd (0, Re)nd the effect of neighbouring particles are taken into account byultiplying with the void fraction ε to some power −n, viz.,

d(ε, Re) = Fd(0, Re)ε−n. (1)

he Wen and Yu (1966) equation is a prototypical example of such aelation. The second class of relations – of which the famous Ergun1952) equation can be viewed as the archetype – is based on thexpression of drag force in the limit of Stokes flow, Fd (ε, 0) where aerm linear in the Reynolds number is added to account for inertialffects, viz.,

d(ε, Re) = Fd(ε, 0) + ˛(ε)Re. (2)

part from the empirical approaches, there have also been effortso derive an expression for drag force from a purely theoretical

∗ Corresponding author. Tel.: +31 53 489 2953; fax: +31 53 489 2882.E-mail address: m.a.vanderhoef@tnw.utwente.nl (M.A. van der Hoef).

ranec(

e

674-2001/$ – see front matter © 2009 Chinese Society of Particuology and Institute of Process Eoi:10.1016/j.partic.2009.02.002

of Particuology and Institute of Process Engineering, Chinese Academy ofSciences. Published by Elsevier B.V. All rights reserved.

asis, such as by Kaneda (1986). However, the applicability ofhese expressions to systems relevant for chemical engineeringcience is very limited, since they only hold for very dilute sys-ems at low Reynolds number. With the recent advance of directumerical simulations (DNS) methods of fluid–solid flow, a thirdethod of estimating the drag force has become available, which

n some sense fills the void between the empirical approaches andhe purely theoretical approaches. One of the first elaborate DNStudies – using the lattice Boltzmann method – of gas–solid flowas by Hill, Koch, and Ladd (2001). However, their studies were

ocused more towards the fundamental understanding of inertialow in particulate systems rather than constructing a drag relation

or practical use, although they provided a correlation for Re < 2nd 40 < Re < 120. Later Benyahia, Syamlal, and O’Brien (2006) havesed the work of Hill et al. to construct an explicit expression forhe gas–particle drag for the entire Reynolds number range. At theame time, Beetstra, van der Hoef, and Kuipers (2007a) derivedndependently a similar expression, also from lattice Boltzmannimulations. Since both expressions are found to give very similaresult, it is safe to conclude that they currently represent the mostccurate estimate of the gas–solid drag force, at least for homoge-eous, monodisperse, static arrays of spheres. Note that the Ergun

quation gives a markedly different result, which is further dis-ussed in Hill et al. (2001) and van der Hoef, Beetstra, and Kuipers2005).

In practical applications, systems are often polydisperse, theffect of which has – until recently – been neglected when applying

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

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34 S. Sarkar et al. / Parti

he drag force correlations. That is, the gas–solid force on a sin-le particle was calculated from the particle’s slip velocity andiameter, and local void fraction, without taking into considera-ion the (variation in) diameters of the particles in the immediateeighbourhood. In other words, the same drag force correlationsmonodisperse) are used for polydisperse systems, where the diam-ter that appears in the monodisperse drag relation (such as thergun equation) is simply replaced by the individual diameter of aarticle. In two of our recent publications (Beetstra et al., 2007a;an der Hoef et al., 2005) it was shown that an extra correctionactor – which depends on the local degree of polydispersity in theeighbourhood of the particle for which the drag is to be evaluatedis required to get good agreement with the data from DNS simu-

ations for binary systems. It was verified for bidisperse systems ateynolds numbers up to 500, solid volume fractions ranging from.1 to 0.65, diameter ratios (d2/d1) in the range 1.65–4, and massatios (�2 = �1) ranging from 1/9 to 20, where d1(d2), �1 (�2) arehe diameter and solid volume fraction of the small (large) par-icles respectively. Recently, this correlation has also been testedor general polydisperse systems (Sarkar, van der Hoef, & Kuipers,008, 2009).

In this study we want to test whether this correlation also holdsor bidisperse systems with more extreme diameter ratios, namely:5, 1:7 and 1:9. These were not included in our previous workBeetstra et al., 2007a; van der Hoef et al., 2005), because of thearge system sizes that are required for such a study, and conse-uently long simulation times. For that reason, we have also limitedurselves in the present study to a single solid volume fraction (0.4)nd 4 different Reynolds numbers (1, 10, 100, 500). For the DNS sim-lations we used the lattice Boltzmann method, the data of whichre compared with the prediction from our previous work on binaryystems.

. Fluid–particle interaction relation for bidisperse systems

The total force Ff→i that a fluid exerts on a particle i is the com-ined effect of drag force (Fd,i) and buoyancy force:

f →i = Fd,i − Vi∇P. (3)

he drag force follows from the solid–fluid interaction at the surfacef the solid sphere if there is a relative velocity between the parti-le and the fluid medium, whereas the buoyancy force is due to thetatic pressure gradient �P. In literature, both Ff→i and Fd,i are usedo describe the fluid–solid interaction. For monodisperse systemshe choice is arbitrary, since the two forces differ by a simple factorf the porosity; for bidisperse systems, however, the two forces can-ot be linked by such a simple relation (see Beetstra, van der Hoef,Kuipers, 2007b; Sarkar et al., 2009). In this work, we consider the

otal interaction force Ff→i rather than the drag force, where it isost convenient to use the force in its non-dimensionalized form:

toti = Ff →i

3��diU, (4)

here 3��diU is the Stokes drag, with U the superficial velocity, dihe diameter of the particle, and � the viscosity.

Recently, van der Hoef et al. (2005) and Beetstra et al. (2007a)ound that for bidisperse systems, the interaction force on a par-icle of type i can be well represented by the interaction force Ftot

s calculated from a monodisperse correlation, when the averageeynolds number 〈Re〉 is used, and the force is multiplied by aorrection factor fi, viz.:

toti = fiF

tot(〈Re〉, ε) (5)

5wtgt

y 7 (2009) 233–237

ith

i = εdi⟨d⟩ + (1 − ε)

d2i⟨

d⟩2

, (6)

Re⟩

=�U

⟨d⟩

�, (7)

here the average diameter⟨

d⟩

of the polydisperse system isefined as

d⟩

= N1d31 + N2d3

2

N1d21 + N2d2

2

=[

�1

�d1+ �2

�d2

]−1

, (8)

here � = 1 − ε the overall solid volume fraction, and �i =i�d3

i/(6V) the individual solid volume fraction, in which Ni is the

umber of particles of species i, and V the volume of the system. Eq.6) has been derived on the basis of the Carman–Kozeny analysisor porous systems, and reflects the fact that in a binary mixturehe smaller (larger) particles will experience a smaller (larger) nor-

alized gas–solid force when compared to a monodisperse system.his is because the local porosity around a small (large) particle isarger (smaller) compared to the average porosity. This is furtheriscussed in Section 5.

In principle the correction factor is not coupled to any particu-ar Ftot, that is, one could take any correlation for a monodisperseystem which one expects to be the most accurate for the system atand. For comparison with the polydisperse DNS data it is logicalo use a correlation for Ftot that is derived from DNS data of similarbut monodisperse) systems, as given by Beetstra et al. (2007a):

totmono = 10�

(1 − �)3+ (1 − �)(1 + 1.5�1/2)

+ 0.413Re

24(1 − �)3

{(1 − �)−1+3�(1 − �)+8.4Re−0.343

1 + 103�Re−((1+4�)/2)

}. (9)

or polydisperse systems this correlation was found to give the bestgreement with the DNS data reported by Sarkar et al. (2009).

. Simulation technique

The binary systems contained 512 particles in a periodic domain,here for all cases the number of small particles was set to 496ith a diameter of 6.4 grid sizes. The size of the 16 large parti-

les was chosen such that the desired diameter ratio (1:5, 1:7 or:10) was obtained, and the volume of the domain was made suchhat the solid volume fraction equals 0.4. Initially, all the particlesre given positions according to a BCC lattice. Then of 16 parti-les – picked at random – the diameter is increased to that of thearge particles. Subsequently a Monte Carlo procedure is applied,

here particles are randomly displaced – which is accepted whenhis does not result in any overlap – until a random configurations achieved. A fluid is set to flow past the static array, where theow field is calculated via the lattice Boltzmann method. Periodicoundary conditions are applied at the system boundary, whereast the fluid–particle interface no slip condition is employed. Totalorce Fi→f (=−Ff→i) which particle i exerts on the fluid phase isalculated from the change in fluid momentum per unit timehich is required to maintain the stick boundary condition at thearticle–fluid interface. In order to improve statistics, for each setf parameters (d2 = d1, Re) we performed simulations for typically

different random configurations, and averaged over the results,here the RMS deviation is used to estimate the error bars. Note

hat since the box size has to have an integer value in units of therid size, the actual volume fraction �sim cannot be exactly equalo the desired volume fraction � = 0.4. Although the difference is

S. Sarkar et al. / Particuology 7 (2009) 233–237 235

Fig. 1. Comparison of simulated individual normalized total force with predicted data as a function of the individual diameter over the average diameter (di/⟨

d⟩

) for

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4

E�

s

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idisperse systems with a solid volume fraction 0.4. The symbols are the results fromashed lines represent the prediction from the monodisperse correlation where thre: (i) 〈Re〉 = 1, (ii) 〈Re〉 = 10, (iii) 〈Re〉 = 100, and (iv) 〈Re〉 = 500.

sually small (less than 2%), the effect on the gas–solid force cane appreciable, and larger than the error margin. To this end, theesult of the simulation is extrapolated to the desired solid volumeraction 0.4 by second-order Taylor series expansion, assuming aunctional form of total interaction force. Further details of the sim-lation procedure can be found in van der Hoef et al. (2005) andeetstra et al. (2007a).

. Results and discussions

In Fig. 1, simulation data are compared with the prediction fromq. (5) in combination with relation (9), for a solids volume fraction= 0.4 and average Reynolds number 〈Re〉 = 1, 10, 100 and 500.

For the small particles good agreement is observed between theimulated total force and our drag force correlation for all values

tstTh

ce Boltzmann simulations, the solid lines represent the prediction from Eq. (5). Thevidual Reynolds number is used, and no correction factor. The Reynolds numbers

f 〈Re〉. However, for the large diameters our correlation overpre-icts the DNS data by about 10% for 〈Re〉 = 1 and 10, and about0% for 〈Re〉 = 100 and 500. This discrepancy is inherent to howhe correction factor has been constructed. When derived from the-ry using the Carman–Kozeny approximation, the correction factorontains one unknown parameter, which is then chosen such thatn the limit of the diameter ratio approaching zero, the force on themall particles takes the proper form, namely that of a monodis-erse system in a reduced volume. It is to our knowledge not knownhat the force on the large particle will be in the same limit (i.e.

he force on a particle in suspension of a fluid with infinitesimalmall particles), but it will be incompatible with the result fromhe Carman–Kozeny approach as obtained by the above procedure.he dashed line in Fig. 1 represents the prediction when the “adoc” procedure is followed, that is, the binary gas–solid interaction

2 cuology 7 (2009) 233–237

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Table 1Average value of ı for the small and large particles, calculated fromthe average Voronoi volumes Vvor

1 and Vvor2 via Vvor

i= �(di + ıi)

3/6.

Ratio ı1 ı2

1:2 2.491 2.5101:3 3.151 3.1181:4 4.128 3.9321:5 5.456 5.0211:6 6.939 6.1551:7 8.579 7.3001:8 10.242 8.6231:9 11.977 9.9341:10 13.851 11.395

rVatotififVficw

iVoronoi tessellation, and from the Patwardhan–Tien approach bysolving Eq. (11) exactly.

We find very good agreement between the two results. Note thatthe local free volume of each individual particle (not species) canstill vary substantially due to the random position of the particle,

36 S. Sarkar et al. / Parti

orce is estimated from the monodisperse relation (9), where theiameter (which appears via Re) is simply replaced by the individ-al diameter. For most cases, it can be seen that the prediction usinghe correction factor (Eq. (5)) is much closer to the DNS data thenhen using the ad hoc approach, apart for the larger particles at

arge Reynolds numbers, where, as said, the correction factor over-redicts the data, and the ad hoc prediction is in remarkable goodgreement with the DNS data. One explanation is that the massatio is very extreme in these simulations, so that the large particlesompletely dominate the system. This is discussed in more detailn the next section. Interestingly, for moderate diameter ratios theorrection factor underpredicted the simulation data, which was

emedied by adding a term 0.064d3i/⟨

d⟩3

to the correction factorvan der Hoef et al., 2005), whereas for the more extreme diam-ter ratios presented in this work, it turns out that the correctionactor overpredicts the simulation data. We have not attempted to

mprove on relation (9) by including a term proportional to d3i/⟨

d⟩3

s we did for the moderate ratios. For such a procedure to be mean-ngful we should have far more data over a wider range of solidolume fractions.

. Predicting the gas–solid force using the concept of aocal porosity

Let us consider the non-normalized gas–solid force Ff→i on aarticle of type i in a binary system, which can be written asf →i = 3��diU × F tot

i, that is, as the Stokes–Einstein force, multi-

lied by a term to take into account the effect of the neighbouringarticles, which depends on the solid volume fraction. Let us forimplicity consider the case in the limit 〈Re〉 → 0. In a simple pic-ure, one may expect that the force on a small and a large particlen a binary system only differs because the term 3��diU is differ-nt, and that the term to correct for the presence of other particlestoti

is the same for large and small particles, and could be calcu-ated as for a monodisperse suspension at the same average solidsolume fraction (F(�)). The DNS data shows that this is not true:toti

is larger for the large particles, and smaller for the small parti-les. The origin of this is that the local volume fraction of the largearticles is smaller, and that of the small particle larger, comparedo the average solid volume fraction. Patwardhan and Tien (1985)ave used this insight to construct an alternative improved corre-

ation, where a monodisperse relation is used, only with differentffective volume fractions �(1) and �(2) for the particle species, viz.:

toti (�) = F(�(i)) with �(i) = �d3

i/6

V loci

, (10)

ith V loci

the average local free volume of a particle i. Patwardhannd Tien (1985) approximate V loc

iby �(di + ı)3/6, where ı is then

btained from the constraint that the sum of all free volumes shoulde equal to the total volume Vtot:

1(

d1 + ı)3 + N2

(d2 + ı

)3 = 6Vtot

�. (11)

he main difficulty lies in evaluating ı from this expression.atwardhan and Tien (1985) approximate ı by ı = (�1d1/� +2d2/�)(�−1/3 − 1), however, also an exact solution of (11) can bebtained, by solving a cubic equation.

In van der Hoef et al. (2005) indirect evidence was given thathe origin of the failure of the ad hoc approach indeed can be traced

ack to the different local volume fractions of the large and smallarticles, since the gas–solid force as evaluated from (10) was inery good agreement with the lattice Boltzmann data, at least when(i) was evaluated from the exact solution to Eq. (11). We now want

o test whether this picture still holds for more extreme diameter

Fcbf

In the approach of Patwardhan and Tien (1985), it is assumed thatı1 = ı2 = ı, i.e. the free volume can be estimated by a single parameterı. This assumption holds reasonably well if the particle ratio is not tooextreme.

atios. We first want to test if the approximate evaluation of theloci

by the Patwardhan–Tien approach is correct, by evaluating thectual free volume around the particles directly by use of a Voronoiessellation. In this, the computational domain is divided into a setf volume V̂vor

isuch that each point within the volume is closer to

he surface of particle i than to the surface of another particle j /= in the system. In this way a unique tessellation is obtained. As arst test of the Patwardhan–Tien approach, we have calculated ıi

rom V loci

= �(di + ıi)3/6, where V loc

iwas estimated by the average

oronoi volumes of the particles of type 1 and 2 (see Table 1). Wend that the two values for ı are reasonably close, proving that theoncept of estimating the free volume by a single parameter ı holdsell.

In Fig. 2 we show that average effective volume fraction of thendividual species for all diameter ratios, as calculated from the

ig. 2. Comparison of average solid volume fraction of the particle species (�(i))alculated from the Voronoi tessellation and from the Patwardhan–Tien method forinary systems with different diameter ratios and with the average solid volumeraction being 0.4 (1:2 *, 1:3 �, 1:4 �, 1:5 ©, 1:6 �, 1:7 ♦, 1:8 +, 1:9 �, 1:10 ×).

S. Sarkar et al. / Particuolog

Table 2Prediction for the normalized gas–solid interaction force Fi in the limit of Re → 0.F(�) is the ad hoc approach, F(�(i)) the approach with the use of a local volumefraction, fiF(�) is the approach of van der Hoef et al. (2005), and Fsim

ithe DNS result.

Ratio Small sphere Large sphere

F(�) F(�(1)) f1F(�) Fsim1 F(�) F(�(2)) f2F(�) Fsim

2

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S

S

:5 19.69 3.972 5.246 4.152 19.69 146.11 46.46 41.79:7 19.69 2.300 2.893 2.489 19.69 131.92 35.45 30.63:10 19.69 1.569 1.624 1.754 19.69 103.49 27.63 23.88

hich will naturally give rise to a fluctuation in the gas–solid forceompared to the prediction from a correlation using average val-es for the volume fraction. This is discussed in more detail inriebitzsch, van der Hoef, and Kuipers (2008).

Now that we have a reliable estimate of the local free volume, wean calculate the gas–solid force by use of Eq. (10), that is, where theffect of bidispersity only enters via �(1) and �(2), for which we usehe values calculated from a Voronoi tessellation. Table 2 compareshe gas–solid force calculated this way with the actual simulationata, and the prediction calculated from Eqs. (5) and (9).

It can be seen that for the small spheres the use of a local vol-me fraction works well; however, for the large spheres it results inhuge overestimation of the gas–solid force, where the predictionith the use of a correction factor fiF(�) yields reasonable agree-ent, as can be seen in Fig. 1. The conclusion is that for the systems

nvestigated in this work, where not only the diameter ratio, butlso the mass ratio is more extreme than in van der Hoef et al.2005), the local volume fraction does not provide sufficient infor-

ation to calculate the gas–solid interaction for all cases. Note thator the most extreme diameter ratio (1:10) the prediction from thed hoc approach (F2 = F(�)) is closer to the simulation data thanor the more moderate ratios (see also Fig. 1d, where the ad hocrediction for 1:7 and 1:10 is very close to the DNS data). This iso because we keep the number of spheres fixed while increasinghe diameter ratio, which means that the volume of the solid parti-les completely dominates the system as d2/d1 increases, so that �2pproaches �. The force F2 on the large particles then approacheshat of a monodisperse system at �2 ≈ �, i.e. F2 ≈ F(�). This alsoxplains why the use of a local volume fraction does not work well.he situation would be different when the mass fractions of themall and large particles were similar; however, to achieve equalass ratio with 16 large particles and a diameter ratio of 1:10, weould require 16 000 small particles, which is currently not feasibleith DNS.

. Conclusions

In this paper, we have presented results for the fluid–particlenteraction force from lattice Boltzmann simulations of a bidisperse

v

W

y 7 (2009) 233–237 237

as–particle system, with more extreme diameter ratios (1:5, 1:7nd 1:10). Simulated data are compared with the prediction fromhe relation proposed by van der Hoef et al. (2005), using the

onodisperse relation of Beetstra et al. (2007a). It is seen that theelation proposed by van der Hoef et al. (2005) could reasonablyell predict the simulation data, even for extreme diameter ratios.

he overall conclusion is that the correction factor as given by Eq.9) yields reasonably accurate results over a wide range of diameteratios (1:2 to 1:10), where it slightly overpredicts for intermediateatio 1:4, and underpredicts for the most extreme ratios 1:7 and:10.

cknowledgements

We thank Prof. A.J.C. Ladd for allowing us to use his lattice Boltz-ann code (SUSP3D) to get the flow field through the assembly

f spheres. This project is funded by the Nederlandse Organisatieoor Wetenschappelijk Onderzoek (Netherlands Organization forcientific Research, NWO).

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eetstra, R., van der Hoef, M. A., & Kuipers, J. A. M. (2007b). Erratum to “Drag forceof intermediate Reynolds number flow past mono- and bidisperse arrays ofspheres” [AIChE Journal 53 (2007) 489–501]. AIChE Journal, 53, 3020.

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arkar, S., van der Hoef, M. A., & Kuipers, J. A. M. (2008, June). Fluid–particle interac-tion force for polydisperse systems from lattice Boltzmann simulations. In 6thinternational conference on CFD in oil & gas, metallurgical and process industries,SINTEF/NTNU (CFD08-89) Trondheim, Norway.

arkar, S., van der Hoef, M. A., & Kuipers, J. A. M. (2009). Fluid–particle interac-tion from lattice Boltzmann simulations for flow through polydisperse randomarrays of spheres. Chemical Engineering Science, 64, 2683–2691.

an der Hoef, M. A., Beetstra, R., & Kuipers, J. A. M. (2005). Lattice Boltzmann sim-ulations of low Reynolds number flow past mono- and bidisperse arrays ofspheres: results for the permeability and drag force. Journal of Fluid Mechanics,528, 233–254.

en, C. Y., & Yu, Y. H. (1966). Mechanics of fluidization. Chemical Engineering ProgressSymposium Series, 62, 100–111.