COMPUTATIONAL FLUID DYNAMICS FOR DENSE GAS-SOLID FLUIDIZED BEDS: A MULTI-SCALE MODELING STRATEGY

CHINA PARTICUOLOGY Vol. 3, Nos. 1- -2, 69-77, 2005COMPUTATIONAL FLUID DYNAMICS FOR DENSE GAS-SOLIDFLUIDIZED BEDS: A MULTI-SCALE MODELING STRATEGYM. A. van der Hoef, M. van Sint Annaland and J. A. M. Kuipers*Department of Science & Technology, University of Twente, 7500 AE Enschede, The Netherands"Author to whom correspondence should be addressed. E-mail: j.a.m.kuipers@utwente.nlproduction of fuels, frilizers and base chemicals. The scale-up of these processes is often problematic and is related tothe intrinsic complexities of these flows which are unfortunately not yet flly understood despite signifcant efrts made inboth academic and industial research laboratories. In dense gas-particle flows both (fectve) fuid-paricle and (issipative) paricle-particle interactions need to be accounted for because these phenomena to a large extent govern theprevailing flow phenomena, i.e. the formation and evolution of heterogeneous structures. These structures have signif-cant impact on the quality of the gas solid contact and as a direct consequence thereof strongly affect the performance ofthe process. Due to the inherent complexity of dense gas particles flows, we have adopted a multi-scale modeling ap-proach in which both fuid-particle and particle-particle interactions can be properly accounted for. The idea is essentiallythat fundamental models, taking into account the relevant details of fluid-particle (attice Boltzmann model) and parti-cle- particle (discrete particle model) interactions, are used to develop closurelaws to feed continum models which canbe used to compute the flow structures on a much larger (industrial) scale. Our multi-scale approach (see Fig. 1) involvesthe lttie Boltzmann model, the discrete particle model, the continuum model based on the kinetic theory of granular flow,and the discrete bubble model. In this paper we give an overview of the multi-scale modeling strategy, accompanied byilustrative computational results for bubble formatin. In aditin, areas which need substantial further atetion will behighlighted.Keywords dense gas- solid flow, gas-luidized beds, multi-scale modelling1. Introductionelementary physical principles such as drag, friction andDense gas-particle flows are frequently encountered in adissipation (i) based on this insight, develop models withvariety of industrially important gas-solid contactors, ofpredictive capabilties for dense gas-particle flows en-countered in engineering scale equipment. To this end, wewhich the gas-fuidized bed can be mentioned as a veryconsider gas solid flows at four distinctive levels of mod-important example. Due to their favorable mass and heattransfer characteristics, gas-fluidized beds are often ap-At the most detailed level of description the gas flow fieldplied in the chemical, petrochemical, metallurgical, envi-is modeled at scales smaller than the size of the solid par-ronmental and energy industries in large scale operations tcles. The iteraction of the gas phase with the solid phaseinvoling i.e. coating, granulation, drying, and synthesis of is incorporated by imposing "stick* boundary conditins atfuels and base chemicals (Kuni & Levenspiel, 1991). Lackthe surface of the solid particles. This model thus alows usof understanding of the fundamentals of dense gas-particleto measure the effective momentum exchange betweenflows, and in particular of the effects of gas-particle dragthe two phases, which can be used in the higher scaleand particle-particle interations (Kuipers et al, 1998;models. In our model, the flow field between sphericalKuipers & van Swaaij, 1998), has led to severe dificultiesparticles is solved by the latice Bolzmann model (Succi,in the scaleup of these nustily important gas-soid 2001; Ladd & Verberg, 200) although in principle othercontactors (van Swaij, 1990). To arrive at a better under-methods (such as standard computational fluid dynamics)standing of these complicated systems in which bothcould be used as well. At the intermediate level of descrip-gas-particle and priclpartcle iteractions play a domi- tion the Aow feld is modeled at a scale larger than the sizenant role, computer models have become an indispensa-of the particles, where a grid cell typically containsble tool. However, the prime difculty with modeling0(10)~0(10) particles, which are assumed to be perfectgas-fluidized beds is the large separation of scales: thespheres (diameter d), This model consists of two parts: alargest flow structures can be of the order of meters; yetLagrangian code for updating the positions and velocitiesthese structures are found to be directly infuenced byof the solid particles from Newton's law, and a Euleriandetails of the particle-particle collisions, which take placecode for Jndatina the local aas densitv and velocity fromon the scale of millimeters or less. Therefore, we havethe Navi中国煤化工et al, 1996). The :adopted a muti-level modeling strategy (see Fig. 1), withadvantag_Jel (DPM) is that itthe prime goal to () obtain a fundamental insight in thecan accMHCN M H Gie-partitle itera-complex dynamic behavior of dense gas-particle fluidized tions in a realistic manner, for system sizes of about 0(10")suspensions; that is, to gain an understanding based on particles, which is sufcienty large to llow for a direct70CHINA PARTICUOLOGY Vol. 3, Nos. 1-2, 2005Larger geometryLattice BoltzmannDiscrete ParticleContinuumDiscrete BubbleModelFluid-particleParticle-particleLarge scale motioninteractionIndustrial sizeBubble behavior→Larger scale phenomenaFig. 1 Mutilevel modeling scheme for dense gas-fluidized beds.comparison with laboratory scale experiments. As a logicalHence, in this model both the gas-phase and the solidconsequence of this approach a closure law for the effec-phase are treated on an equal footing, and for both phasestive momentum exchange has to be specified, which canan Eulerian code is used to describe the time evolutionbe obtained from the aforementioned lttice Boltzmann (see Kuipers et al, 1992; Gidaspow, 1994, amongst oth-simulations. Note that in chemical engineering, to dateers). The information obtained in the two smaller-scalemainly empirical relations are used for the friction ceffi-models is then included in the continuum models via thecient β (defined by (1) and (2)), such as the Ergun (1952)kinetic theory of granular flow. The advantage of this modelcorrelation for porosities ε<0.8:is that it can predict the flow behavior of gas-solid flows atlife-size scales, and these models are therefore widelyβd2= 150(1-E)尸- +1.75Re ,(1)used in commercial fluid flow simulators of industrial scaleμequipment. Finally, at the largest scale, the (larger) bub-and the Wen and Yu (1966) equation for porositiesbles that are present in gas-solid fluidized beds are con-ε>0.8:sidered as discrete objects, similar to the solid particles inthe DPM model. This model is an adapted version of the= EcRe(1- e)e2s,discrete bubble model for gas-liquid bubble columns. Wewant to stress that this model, as outlined in section 5, hasCg=24(1+ 0.15Re°r)Re .Re <103been developed quite recently, and the results should beconsidered as very preliminary. In this paper we will give0.44Re>10(2)an overview of these four levels of modeling as they areemployed in our research group. In the following sectionswhere μ is the viscosity of the gas phase, Re is the particlewe will describe each of these models in more detail.Reynolds number, and C the drag coefficient, for which中国煤化工the expression of Schiller and Nauman (1935) is used. At2.an even larger scale a continuum description is employediYHCNMH Cal (LBM)for the solid phase, i.e. the solid phase is not described byThe lttie Boltzmann model (L .BM) originates from theindividual particles, but by a local density and velocity field. lttice-gas cellular automata (LGCA) models (Frisch et al,72CHINA PARTICUOLOGY Vol. 3, Nos. 1-2, 2005diamele lor delails see van der HOet et al. (00) IFi9g.3 3. Discrete Particle Model (DPM)we present some LBM results for a binary mixture at fniteReynolds numbers. In this figure the individual drag forceThe discrete particle model is one level higher in theF dided by the drag force F() of a monodisperse system muli-scale hierarchy. The most impotant diterencne withat the same solids volume fraction中, is plotted as a func~the ltite Boltzmann model is that in this model the size oftion of the coretion factor (1- p)y, +Y that we derived,he particles is smaller than the grid size that is used tosolve the equations of motion of the gas phase. Thiswhere F(中) is our best ft to LBM siulaton data for mono- means that for the itrctont with the gas phase, the par.disperse systems:ticles are simply point sources and sinks of momentum,F(Q)=10_0 +(1-0)/[1+1.501.(9where the finite volume of the particles only comes in via(1-0)an average gas fraction in the drag force relations. A sec-As can be seen from Fig. 3, we find excellnt agreementond (technical) dfference with the LB model is that thebetween our data and theory. It should be noted here that evolution of the gas phase now fllws from a finite difter-the assumption F=F(0), which is curently used in ltrature ence scheme of the Navier-Stokes equation, rather thancan lead to dfferences with the LBM simulation data up tothe Boltzmann equation. A complete description of thea factor of 5. This finding indicates that one should be cau-method can be found in Hoomans et al. (1996), however,tius with relying on ad hoc mrdications of drag laws for we will briely discuss some of the basic elements here.monodisperse systems to extend their "alidity" toThe discrete particle model consists of two parts: a La-polydisperse systems. In additin, this result highlights thegrangian part for updating the positins and velocites ofusefulness of microscopic simulation methods, becausethe solid particles, and an Eulerian part ftor updating thethe experimental determination of the individual ffctite local gas density and velocity. In the Lagrangian part, thedrag force in a dense assembly would be extremely diiequation of motion of each particle i(velocity的, mass mcult.volume V) is given by Newton's law业。+V(u-V)-VVρ+加+户， (10) .dt(1-2)where the RHS represents the total force acting on theparticle. This includes extemal forces (the gravitationalforce mg), interaction forces with the gas phase (dragforce - B(-附) and pressure force Vp ), and fnally theparticle-particle forces印and particle-wall forces户which represents the momentum exchange during coll-ions, and possible long-range atractions between thparticles, and particles and walls, respectively. There are,in principle, two ways to calculate the trajectories of thesolid particles from Newton's law. In a time-driven nu-merical simulatin, the new position (1+ dr) and velocity可(t+d1) are calculated from the values at time t, via a。LBMdata'standard integration scheme for ODE's. Such type of5 --- Currently usedsimulation is in principle suitable for any type of interactionforce between the particles. In an event driven simulation,the interactions between the particles are considered in-stantaneous (Cllisions), and the systems evolves directly告。(4ree flight) from nearest collision event to next-nearestcollision event, etc. This method is eficient for low-densitysystems, however it is not suitable for dense packed sys-tems, or systems with long-range forces. In the Eulerianpart of the code, the evolution of the gas phase is deter-mined by the volume-averaged Navier-Stokes equations:中国煤化工(11)(1-0)y+r?Fig.3 Example of particle configuration generated with a Monte CarloMTHCNM.HGr- 5+g，(12)euton 山procedure for a binary system (upper) and dimensionless dragforce computed for small and large particles from LBM (lower).where ? is the usual stress tensor, which includes thevan der Hoet, van Sint Annaland & Kuipers: Computational Fluid Dynamics for Dense Gas Solid Fluidized Beds 73cofficient of shear viscosity. Note that there is a full anisotropy of the distribution in case of non-ideal particles.two-way coupling with the Lagrangian part, i.e., the reac- A possible explanation is the formation of dense particletion from drag and pressure forces on the solid particles isclusters in the case of inelastic ollions, which may dis-included in the momentum equation for the gas phaseviaa turb the spatial homogeneity and thereby causing colli-source term s:sional anisotropy. Analysis (Jenkins & Savage, 1983) ofthe normal and tangential component of the impact velocitys=HZ(-V)8(7 -F)dV .(13) indeed showed that, in dense gas-fluidized beds, not allEquations (11) and (12) are solved with a semi-implicitimpact angles occur with the same frequency.method for pressure linked equations (SIMPLE-algorithm), .with a time step that is in general an order of magnitudelarger than the time step used to update the particle posi-tions and velocities. The strength of the DP model is that it8allows to study the effect of the particle-particle interactionson the fluidization behavior. In the most detailed model of一Gaussdescription, the interparticle contact forces includes normal. Maxwelland tangential repulsive forces (modeled by linear springs),and dissipative forces (modeled by“dash pots"), and tan-gential friction forces (Walton, 1993). A DPM simulationstudy by Hoomans et al. (1996) showed that the hetero-geneous flow structures in dense gas-fluidized beds areC/(ms)partly due to the ollisional energy dissipation. More re-cently, Li and Kuipers (2003) demonstrated that such flowstructures are also strongly infuenced by the degree of020 -010 000 0.10 020 030 040non-linearity of the particle drag with respect to the gas frac-tion e. Bokkers et al. (2004a; 2004b) studied the effect of theclosures for gas particle drag on the bubble-induced mixingin a pseudo 2D gasfluidized bed and found that the bestagreement between theory and experiment was obtained in+ Gausscase the LBM-generated drag closures reported by Hill et al.+ Maxwell(2001a; 2001b) were used in their DPM simulations.One of the great advantages of discrete particle simula-tions is that it alows to study properties of the system thatare very dificult to obtain via experimentation. A particu-larly important example is the velocity distribution of theC(ms')particles, i.e. the probability of finding a particle with a ve-locity component Va with at x,y,Z. It would be extremely-0.10 -0000 05 0.10 015 020ificult to obtain reliable estimates for the velocity distribu-tion from experiments; yet, this function is of great rele- Fig.4 DPM simulation data for the normalized particle velocity dis-vance for the validity of the higher scale to-fluid modeltribution t(Gx), fCy), f(C2) and (C, compared to a Gaus-(see next section) derived from the kinetic theory, where itsianMaxwellan distribution. Upper graph: ideal particles;lower graph: non-ideal particles.is assumed that the velocity distribution is both isotropicand nearly Gaussian. The discrete particle simulations are4. Two-Fluid Model (TFM)an ideal tool for testing this assumption, since it is relativelystraightforward to measure the velocity distribution as allThe maximum number of particles that can be simulatedparticle velocities are known at any moment in time.with the DP model, as described in the previous section, isWe have studied the velocity disribution for two cases: tyicaly less than a mllion, whereas the number of pari-in Fig.4 we show the result for a fuidized bed of ideal (i.e. cles that are present in an industrial size fluidized bed canperfectly smooth and elastic) and non-ideal (i.e. rough and be two to three orders of magnitude higher. Since both theinelastic) particles. The system contained 25000 particles CPU time and the required memory scales linear with theof 2.5 mm diameter, where the gas velocity is set to 1.5 number of particles, it is obvious that DPM simultions oftimes the minimum fluidization velocity. Details of the industrial中国煤花nd the capabiilyy ofsampling procedure for obtaining the velocity distributions commer_es within the fore-can be found in Goldschmidt et al. (2002). Figure 4 shows seeablepe of model is usedthat for both ideal and non-ideal particles, the velocity dis- for simujY HCNMHGthelnr’tributions do not deviate significantly from a Gaussian and solid phase consisting of individual, distinguishable parti-Maxwellian distribution. However, Fig.4 reveals a clear cles is abandoned. This so-called two-fluid continuum74CHINA PARTICUOLOGY Vol. 3, Nos. 1-2, 2005model (TFM) describes both the gas phase and the solidswhere C。represents the particle fluctuation velocity andphase as fully inter-penetrating continua, using a set ofthe brackets indicate ensemble -averaging. The time evolu-generalized Navier-Stokes equations (Kuipers et al, 1992;tion of the granular temperature itself is given by: .Gidaspow, 1994). That is, the time evolution of the gasphase is sil governed by (11) and (12); for the solid phase,文(e。p.0)+V. (,p,0v)|=the discrete particle part (10) is now replaced by a set of(18)continuum equations of the same form as (11) and (12):(p.i +e):vv-V (,g)-3p0-r.&(<。p2)+V epv=0,(14)with g。 the kinetic energy fux, and r the dissipation of(esp.0)+V e.p.W= ε.Vp-Vp.-V .e.+S+e.p.g，kinetic energy due to inelastic particle ollions. In equa-tions (14)-(18) there are still a number of unknown quanti-(15)ties (pressure, stress tensor, energy fux), which must bewith p。， v and e。=1-ε the local density, velocity, andexpressed in terms of the basic hydrodynamic variablesvolume fraction of the solid phase, respetively. In this(density, velocity, granular temperature), in order to get adescription the source term s is slightly different fromclosed set of equations. The derivation of such constitutiveequations follows from the KTGF, and can be found in the(13), namelybooks by Chapman and Cowling (1970) and GidaspowS=β(u-D). .(16)(1994) and the papers by Jenkins and Savage (1983) andObviously, the numerical scheme for updating the solidDing and Gidaspow (1990). In this work, the constitutivephase is now completely analogous to (and synchronous equations developed by Nieuwland et al. (1996) have beenwith) that of the gas phase. Since the concept of particlesused for the particle phase heology.has disappeared completely in such a modeling, the effectIn Fig.5 we show the simulated bubble formation for aof particle particle interactions can only be included indi-pseudo two-dimensional (2D) bed (bed geometry:rectly, via an effective solids pressure and effective solids0.57 mx0.015mx1.0m (wX dx h)) operated with a centralviscosity. A description which alws for a slightly morejet (diameter 0.015m) at a velocity of 40 times the incipientdetailed description of particle-particle interactions followsfluidization velocity. The bed contains ballotini with a parti-from the kinetic theory of granular flow (KTGF); suchcle diameter and density of 500μm and P。 =2660kg.mstheory expresses the diagonal and off-diagonal elementsrespectively. Clearly, a very complex bubble pattern resultsof the solids stress tensor (i.e. the solids pressure andfrom the jet operation where the size and the shape of thesolids shear rate) as a function of the granular temperatureformed bubbles continuously change. It can also cleartly befor a monodisperse particle system, defined as:seen that bubble coalescence occurs leading to a rapido=(% qg),(17)increase in the bubble size.10000s1=200005to 3.0000s1=4.0000st= 5.0000s1-6.000051=7.000051-8.0000S-9.00005t-10.0000s中国煤化工HCNMHGFig.5 Computed bubble formatin in a pseudo 2D gas-fluidized bed with a central jet. Bed material: bllotini with df= 500μmand PAs=2 660 kg-ms. Jet velocity: 10.0 m-s' (40um).van der Hoef, van Sint Annaland & Kuipers: Computational Fluid Dynamics for Dense Gas-Solid Fuidized Beds 755. Discrete Bubble Model (DBM)the behaviour of bubbles in fluidized beds can be readilyAlthough the two fluid model can simulate fluidized bedsincorporated in the force balance of the bubbles. In thisrespect, one can think of the rise velocity, and the tendencyat ife-size scales, the largest scale industial fuidized bed of ring bubbles to be drawn towards the center of the bed,reactors (diameter 5 meters, height 16 meters) are stillfrom the mutual interaction of bubbles and from wall effectsbeyond its capabilties. However, it is possible to introduce(Kobayashi et al, 2000). Coalescence, which is an highlyyet another upscaling by considering the bubbles as dis. prevalent phenomenon in fuidized beds, can also be easilycrete entities, as observed in the DPM and TFM models ofincluded in the DBM, since all the bubbles are trackedgasfluidized beds. This is the so-called discrete bubbleindividually.model, which has been succestully applied in the field ofWith the DBM, two preliminary calculations have beengas-liquid bubble columns (Delnoij et al., 1997). The ideaperformed for industrial scale gas-phase polymerizationto apply this model to describe the large scale solids cir-reactors, in which we want to demonstrate the effect of theculation that prevails in gas-solid reactors is new, however.superficial gas velocities, set to 0.1 m-s”and 0.3 m-s '. TheIn this paper, we want to show some first results of thegeometry of the fluidized bed was 1.0 mx3.0 mx1.0 mdiscrete bubble model applied to gas-solid systems, which(wxhxd). The emulsion phase has a density ofinvolves some slight modifications of the equivalent model400kg.m3 and the apparent viscosity was set to 1.0Pas.for gas-liquid systems. To this end the emulsion phase isThe density of the bubble phase was 25 kg.m?. The bub-modeled as a continuum, like the liquid in a gas-iquidbles were injected via 49 nozzles positioned equally dis-bubble column, and the larger bubbles are treated as dis-tributed in a square in the middle of the column.crete bubbles. Note that granular systems have no surtaceIn Fig. 6 snapshots are shown of the bubbles that rise intension, so in that respect there is a pronounced diferencethe fluidized bed with a superficial velocity of 0.1 m-s ' andwith the bubbles present in gas-liquid bubble columns. For0.3ms , respectively. It is clearly shown that the bubbleinstance, the gas will be free to flow through a bubble in thegas-soid systems, which is not the Ccase for gas-iqid holdup, is much larger with a superficial vlocity ofsystems. As far as the numerical part is concerned, the0.3m.s'. However, the number of bubbles in this caseDBM strongly resembles the discrete particle model as might be too large, since calscence has not been takenoutined in section 3, since it is also of the Euler-Lagrangeinto account in these simulations. In Fig. 6 in adition time-type with the emulsion phase described by the vol-averaged plots are shown of the emulsion velocity aterume-averaged Navier-Stokes equations:(ep)+V .epu=0，(19)at(epu)+V epuu= -eVp-V.ετ-S+epg， (20)whereas the discrete bubbles are tracked individually ac-cording to Newton's second law of motion:matd=户(21)where F is the sum of difterent forces acting on a singlebubble:后=后+后+后+F+w.(22)As in the DPM model, the total force on the bubble hascontributions from gravity (后), pressure gradients (后)and drag from the interaction with emulsion phase (后)For the drag force on a single bubble (diameter o6), thecorrelations for the drag force on a single sphere are used,only with a modified drag coeficient Ca, such that it yieldsthe Davies-Taylor relation Vr =0.711/90。for the rise ve-locity of a single bubble. Note that in (21), there are twoforces present which are not found in the DPM, namely thelit force片and the virtual mass force的The lit force中国煤化工is neglected in this application, whereas the virtual massFig.; (ef) computed from theMHCNMH(the time average vectorforce cofficient is set to 0.5. An advantage of this ap-plots ot the emusion pnase (ight) atter 100 s of simulation; top:proach to model large scale fluidized bed reactors is thatu6=0.1 m-s', d6=0.04 m; bottom: uc=0.3ms ', d=0.04 m.CHINA PARTICUOLOGY Vol. 3, Nos. 1-2, 2005100s of simulation. The large convection pattems, upfow granular flows. Although the continuum models have beenin the middle, and downflow along the wall, and the efect studied extensively in the literature (e.g. Kuipers et al,of the superficial gas velocity, are clearly demonstrated.1992; Gidaspow, 1994), these models still lack the capa-Future work will be focused on implementation of closure bilty of describing quantitatively particle mixing and seg-equations in the force balance, like empirical relations for regation rates in multi-disperse fluidized beds. An impor-bubble rise velocities and the interaction between bubbles.tant improvement in the modeling of lf-size fluidized bedsThe model can be augmented with energy balances tocould be made if direct quantitative information from thestudy temperature profiles in combination with the largediscrete particle simulations could find its way in the con-circulation patterms.tinuum models. In particular, it would be of great interest tofind improved expressions for the solid pressure and the6. Summary and Outlooksolid viscosity, as they are used in the two fluid model,however, it is a non-trivial task to extract direct data on theIn this paper we have presented an overview of thesolid viscosity and pressure in a DPM simulation. A verymulti-scale methods that we use to study gas-solid fluid-simple, indirect method for obtaining the viscosity is toized beds. The key idea is that the methods at the smaller,monitor the decay of the velocity of a large spherical in-more detailed scale can provide qualitative and quantita-truder in the fluidized bed. The viscosity of the bed followstive information which can be used in the higher scalethen directly from the Stokes-Einstein formula for the dragmodels. A typical example of such qultive iformatin is force. Very preiminary resuts - obtained from data of athe insight (from the DPM simulations) that inelastic cll-high velocity impact- were in reasonable agreement withsions and nonlinear drag can lead to heterogeneous flowthe experimental values for the viscosity. More elaboratestructures. Even more important, however, is the quantita-simulations of these systems are currently underway.tive information that the smaller scale models can provide.Finally, the discrete bubble model applied to gas-solidA typical example of this is the drag force relation obtainedsystems seems to be a promising new approach for de-from the LBM simulations, which finds its direct use in bothscribing the large scale motion in life-size chemical reac-the DPM and TFM simulations. We should note here thattors. Essential for this model to be successful is that relialthough the new drag force relations seem to give resultsable information with regard to rise velocities and mutualat the DPM/TFM level which compare better with the ex-interaction of the bubbles is incorporated, which can beperimental findings, these relations are still far from optimal.obtained from the lower scale simulations. In particular, theIn particular, it should be bome in mind that these dragTFM and DPM simulations will be used to guide the for-force relations are derived for static, unbounded, homo-mulation of aditional rules to properly describe the coa-geneous arrays of mono-disperse spheres. Yet, at thelescence of bubbles, which is at present not incorporatedDPM/TFM level these relations are applied to systemsin the model. This will be the subject of future research.which are, even locall, inhomogeneous and non-static;furthermore, rather ad hoc modifications are used to allowAcknowledgementfor polydispersity. In future work, we want to focus on de-veloping drag force relations for systems which deviateThe lttie Boltzmann simulations have been performed with thefrom the ideal conditions, where the parameters which SUSP3D code developed by Anthony Ladd. We woud like towould quantity such deviation may be tivial to deine thank him for making his code ailable, and Albert Bokkers for(polydispersity: width of the size distribution; moving parti-performing the DPM, TFM and DBM simulations.cles: granular temperature) or not so trivial (inhomogenei-ties). Our attice Boltzmann results for the drag force inReferencesbinary systems (van der Hoef et al., 2005) revealed sig-Bokkers, G. A.. van Sint Annaland, M. & Kuipers, J. A. M. (2004a).niflcant deviations with the ad hoc modifications of theMixing and segregation in a bidispersed gas-solid fluidized bed:monodisperse drag force relations, in which it is assumeda numerical and experimental study. Powder Technol, 140that the drag force scales linearly with the particle diameter.176- 186.At present, only qualitative information from the DPMBokkers, G. A., van Sint Annaland, M. & Kuipers, J. A. M. (2004b).simulations is obtained, such as the aforementioned het-Comparison of continuum models using the kinetic theory oferogeneous flow structures, which is caused by dsspativegranular flow with discrete particle models and experiments:extent of particle mixing induced by bubbles. Proceedings offorces.Fluidization X7 (p.187 -194), May 9- 14, 2004, Naples, ltaly.Another example is the functional form of the velocityChapman, S. & Cowling, T. G. (1970). The Mathematical Theorydistibution. It was found that that dissipative interactionof Non-uniform Gases (Trial mode edition). Cambridge: Cam-forces cause an anisotropy in the distribution, although thefunctional form remains close to Gaussian for all three Delno中国煤化工j,w. P. M. (997. Com- .directions (Goldschmidt et al, 2002). 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