Gas-solid fluidized beds are almost always polydisperse in industrial\ud
application. However, to describe the fluid-particle interaction\ud
force in models for large-scale gas-solid flow, relations\ud
are used which have been derived for monodisperse system, for\ud
which ad-hoc modifications are made to account for polydispersity.\ud
Recently it was shown, on the basis of detailed lattice\ud
Boltzmann simulations, that for bidisperse systems these\ud
modifications predict a drag force which can be factors different\ud
from the true drag force. In this work fluid-particle interaction\ud
forces for polydisperse system are studied by means of\ud
lattice Boltzmann simulation, using a grid that is typically an\ud
order of magnitude smaller than the sphere diameter. Two different\ud
lognormal size distributions are considered for this study.\ud
The systems consist of polydisperse random arrays of spheres\ud
in the diameter range of 8-24 grid spacing and 8-40 grid spacing,\ud
a solid volume fraction of 0.5 and 0.3 and Reynolds number\ud
0.1 to 500. The data confirms the observations made for bidisperse\ud
systems, namely that an extra correction factor for the\ud
drag force is required to adequately capture the effect of polydispersity.\ud
It was found that the correction factor derived by van\ud
der Hoef et al (J. Fluid Mech. 528 (2005) 233) on the basis of\ud
bidisperse simulation data, applies also to general polydisperse\ud
system

Hoef, M.A. van der

Kuipers, J.A.M.

Sarkar, S.

English

Gas-solid fluidized beds are almost always polydisperse in industrial application. However, to describe the fluid-particle interaction force in models for large-scale gas-solid flow, relations are used which have been derived for monodisperse system, for which ad-hoc modifications are made to account for polydispersity. Recently it was shown, on the basis of detailed lattice Boltzmann simulations, that for bidisperse systems these
modifications predict a drag force which can be factors different from the true drag force. In this work fluid-particle interaction forces for polydisperse system are studied by means of
lattice Boltzmann simulation, using a grid that is typically an order of magnitude smaller than the sphere diameter. Two different lognormal size distributions are considered for this study. The systems consist of polydisperse random arrays of spheres in the diameter range of 8-24 grid spacing and 8-40 grid spacing, a solid volume fraction of 0.5 and 0.3 and Reynolds number 0.1 to 500. The data confirms the observations made for bidisperse
systems, namely that an extra correction factor for the drag force is required to adequately capture the effect of polydispersity. It was found that the correction factor derived by van
der Hoef et al (J. Fluid Mech. 528 (2005) 233) on the basis of bidisperse simulation data, applies also to general polydisperse systems

Sarkar, S

Hoef, MA Martin  van der

Kuipers, JAM Hans

Repository TU/e

Fluid-particle interaction force for polydisperse systems from lattice boltzmann simulations

Gas-solid fluidized beds are almost always polydisperse in industrial application. However, to describe the fluid-particle interaction force in models for large-scale gas-solid flow, relations are used which have been derived for monodisperse system, for which ad-hoc modifications are made to account for polydispersity. Recently it was shown, on the basis of detailed lattice Boltzmann simulations, that for bidisperse systems these modifications predict a drag force which can be factors different from the true drag force. In this work fluid-particle interaction forces for polydisperse system are studied by means of lattice Boltzmann simulation, using a grid that is typically an order of magnitude smaller than the sphere diameter. Two different lognormal size distributions are considered for this study. The systems consist of polydisperse random arrays of spheres in the diameter range of 8-24 grid spacing and 8-40 grid spacing, a solid volume fraction of 0.5 and 0.3 and Reynolds number 0.1 to 500. The data confirms the observations made for bidisperse systems, namely that an extra correction factor for the drag force is required to adequately capture the effect of polydispersity. It was found that the correction factor derived by van der Hoef et al (J. Fluid Mech. 528 (2005) 233) on the basis of bidisperse simulation data, applies also to general polydisperse systems

Hoef, van der, M.A.

NARCIS 

Pure OAI Repository

Gas-solid fluidized beds are almost always polydisperse in industrial application. However, to describe the fluid-particle interaction force in models for large-scale gas-solid flow, relations are used which have been derived for monodisperse system, for which ad-hoc modifications are made to account for polydispersity. Recently it was shown, on the basis of detailed lattice Boltzmann simulations, that for bidisperse systems these modifications predict a drag force which can be factors different from the true drag force. In this work fluid-particle interaction forces for polydisperse system are studied by means of lattice Boltzmann simulation, using a grid that is typically an order of magnitude smaller than the sphere diameter. Two different lognormal size distributions are considered for this study. The systems consist of polydisperse random arrays of spheres in the diameter range of 8-24 grid spacing and 8-40 grid spacing, a solid volume fraction of 0.5 and 0.3 and Reynolds number 0.1 to 500. The data confirms the observations made for bidisperse systems, namely that an extra correction factor for the drag force is required to adequately capture the effect of polydispersity. It was found that the correction factor derived by van der Hoef et al (J. Fluid Mech. 528 (2005) 233) on the basis of bidisperse simulation data, applies also to general polydisperse system

van der Hoef, Martin Anton

University of Twente Research Information

6th International Conference on CFD in Oil & Gas, Metallurgical and Process IndustriesSINTEF/NTNU, Trondheim, Norway10-12 June 2008CFD08-89FLUID-PARTICLE INTERACTION FORCE FOR POLYDISPERSE SYSTEMS FROMLATTICE BOLTZMANN SIMULATIONSS. Sarkar, M.A. van der Hoef ∗ and J.A.M. KuipersDepartment of Science and Technology, University of Twente, P.O.Box 217, 7500 AE ENSCHEDE, THE NETHERLANDS∗E-mail: m.a.vanderhoef@tnw.utwente.nlABSTRACTGas-solid fluidized beds are almost always polydisperse in in-dustrial application. However, to describe the fluid-particle in-teraction force in models for large-scale gas-solid flow, relationsare used which have been derived for monodisperse system, forwhich ad-hoc modifications are made to account for polydis-persity. Recently it was shown, on the basis of detailed lat-tice Boltzmann simulations, that for bidisperse systems thesemodifications predict a drag force which can be factors differ-ent from the true drag force. In this work fluid-particle inter-action forces for polydisperse system are studied by means oflattice Boltzmann simulation, using a grid that is typically anorder of magnitude smaller than the sphere diameter. Two dif-ferent lognormal size distributions are considered for this study.The systems consist of polydisperse random arrays of spheresin the diameter range of 8-24 grid spacing and 8-40 grid spac-ing, a solid volume fraction of 0.5 and 0.3 and Reynolds number0.1 to 500. The data confirms the observations made for bidis-perse systems, namely that an extra correction factor for thedrag force is required to adequately capture the effect of poly-dispersity. It was found that the correction factor derivedby vander Hoef et al (J. Fluid Mech. 528 (2005) 233) on the basis ofbidisperse simulation data, applies also to general polydispersesystems.Keywords: Polydispersity, Fluid-particle interaction force,Lattice Boltzmann simulationINTRODUCTIONPacked bed or fluidized bed reactors are widely used inthe chemical, metallurgical and petrochemical industries,where in most cases the beds are polydisperse in nature.One of the weakest links in the modeling of gas-fluidizedbeds is our limited understanding of the resistance be-haviour of an assembly of particles to fluid flow, judg-ing from the many different drag force relations avail-able in the literature, even for monodisperse systems. Themost widely used correlation for the gas-particle interac-tion force as a function of the solid volume fractionφ andparticle Reynolds number Re is Ergun’s relation (Ergun(1952)), which for a static bed reads:F totmono=15018φ(1−φ)3+1.7518Re(1−φ)3(1)whereF totmono is the total gas-particle interaction force nor-malized by the Stokes-Einstein force 3πµdU, and Re= ρd|U |/µ, in whichU , µ, andρ are superficial velocity,viscosity, and density of the gas-phase, respectively, andd the diameter of a particle. Ergun’s relation is derivedfrom the pressure drop data over beds of spheres, sandand pulverized coke and is applicable to dense systemslike packed beds, but not to dilute systems (solids frac-tion < 0.2), for which the Wen and Yu (1966) relation isused.F totmono=[1+0.15 Re0.687]ε−4.65 (2)Recently, on the basis DNS type simulations data for flowpast monodisperse and bidisperse arrays of spheres, Beet-stra et al. (2007) showed that the linear scaling of dimen-sionless drag force with the Reynolds number (as is thecase in the Ergun correlation, and many others), is anover simplification. Similarly, in the Wen and Yu rela-tion, the dependence of drag force on void fractionε−nis an oversimplification of the contribution of neighbour-ing particles to the drag force. On the basis of the DNSdata, an new relation for monodisperse systems has beensuggested by Beetstra et al. (2007).Another difficulty with the current class of drag forcerelations is that they are not well defined with respectto homogeneity, sphericity and monodispersity. With re-spect to the latter, the degree of polydispersity is not ex-plicitely included into the drag force correlations. That is,the gas-solid force on a single particle is calculated fromthe particle’s slip velocity and diameter, and local voidfraction, without taking into consideration the (variationin) diameters of the particles in the immediate neighbour-hood. In other words, the same drag force correlations1S. Sarkar, M.A. van der Hoef & J.A.M. Kuipers(monodisperse) are used for polydisperse systems, wherethe diameter that appears in the monodisperse drag re-lation (such as the Ergun equation) is simply replacedby the individual diameter of a particle. In two earlierpublications (van der Hoef et al. (2005); Beetstra et al.(2007)) it was shown that an extra correction factor -which depends on the local degree of polydispersity inthe neighbourhood of the particle for which the drag isto be evaluated - is required to get good agreement withthe data from DNS simulations for binary system. In thiswork, we have extended the DNS simulations to generalpolydisperse systems which have a lognormal size dis-tribution, to test whether this correction factor also ap-plies to such systems. Two different range of diametersare chosen: 8-24 grid spacing and 8-40 grid spacing, fortwo average porosities packing fraction (0.5 and 0.3), andReynolds numbers ranging from 0.1 to 500. Simulationdata are compared with the prediction from the relationby van der Hoef et al. (2005).DRAG RELATION FOR POLYDISPERSE SYSTEMWe consider a fluid (liquid or solid), with superficial ve-locity U is flowing past a static array of polydispersespheres, in which there areNi spheres with diameterdi , i = 1,2,3, ...,m (where m represents different ”Com-ponents” (diameter in this case) of particles). For con-venience we consider only flow in one direction, whichallows us to use scalar notation (instead of a vector no-tation) for the forces and velocities. Assuming all theparticles with same diameter experience the same fluidto particle force, we write the total fluid-particle interac-tion forceFg→s,i on a particle of typei as the sum of thedrag force (Fd,i), which exists if there is relative velocitybetween fluid and particles, and buoyancy force whichappears due to the static pressure gradient (∇P):Fg→s,i = Fd,i −Vi∇P (3)with V i is the volume of the particle of typei. Note thatin literature often bothFg→s andFd are defined as dragforce, which is a matter of taste (for monodisperse sys-tems one can show thatFg→s = Fd/(1−ε). In this pa-per, we will consider the total interaction force ratherthan drag force. It is convenient to consider this force inits dimensioneless form by normalizing it by the Stokes-Einstein drag force 3πµdiU:Ftoti = Fg→s,i/3πµdiU (4)whereµ is the viscosity.In two recent publications (van der Hoef et al. (2005),Beetstra et al. (2007)), it is shown that for bidispersesystems a correction factor (fi ) should be applied to themonodisperse drag force correlationF totmonoto describe thefluid-particle interaction force i.e.Ftoti = fi Ftotmono(〈Re〉,ε) (5)fi = εdi〈d〉+(1−ε)d2i〈d〉2+0.064εd3i〈d〉3(6)where〈d〉 is the Sauter mean diameter defined as〈d〉 =∑i Nid3i∑i Nid2iIn principle the correction factor is not coupled to a par-ticular F totmono, that is, one could take any correlation for amonodisperse system which one expects to be the mostaccurate for the system at hand. For comparison withthe polydisperse DNS data it is logical to use a correla-tion for Ftotmono that is derived from DNS data of similar(but monodisperse) systems, as given by Beetstra et al.(2007):Ftotmono=10φ(1−φ)3+(1−φ)(1+1.5φ12 )+0.413Re24(1−φ)3{(1−φ)−1+3φ(1−φ)+8.4Re−0.3431+103φRe−(1+4φ)2}(7)with φ = 1− ε the solids volume fraction, and〈Re〉 de-fined as〈Re〉 =ρU〈d〉µ(8)It was shown that this equation differs significantly fromthe well-used Ergun and Wen & Yu correlations given insection 1. In this work, our aim is to test whether expres-sion (5) is also valid for general polydisperse systems andfor this reason we have performed number of DNS sim-ulations of fluid flow past random arrays of spheres withten different diameters.It should be stressed that the correction factorfi asgiven above is valid for thetotal gas-solid force. Forthe normalizeddrag force Fd,i/3πµdiU , the correctionfactor is fi = εdi/〈d〉+(1−ε)d2i /〈d〉2 + 0.064εd3i /〈d〉3,where it is understood that in the contribution to thegas-solid force from the pressure gradientVi∇P the in-dividual diameter of the particle is taken. In our earlierpublications (van der Hoef et al. (2005); Beetstra et al.(2007)) we wrongly assumed that both forces had thesame correction factor; this was derived from our as-sumption thatFg→s,i = Fd,i/ε, which is, however, onlytrue for monondisperse systems (for polydisperse sys-tems, the individual forcesFg→s,i andFd,i cannot be re-lated by a simple single relation).SIMULATION RESULTSThe DNS simulations were performed using the latticeBoltzmann method to resolve the fluid flow between thespheres, under the condition of no-slip boundary condi-tions at the surface of the sphere. For details on the simu-lation method we refer to the paper by Ladd (1994a). Thesimulation procedure to measure the drag force is similar2Fluid-particle interaction force for polydisperse systems from lattice Boltzmann simulations / CFD08-89to what we used for binary systems, the details of whichare described in van der Hoef et al. (2005); we just brieflysummarize the method below. A static (random) array ofspheres is set to move with a constant velocity throughthe LB fluid. The change in fluid momentum resultingfrom setting the local fluid velocity equal to the surfacevelocity of a particle ”a” is equal (when taken per unittime), to the total force that the particlea exerts on thefluid; the inverse of this force is then the instantaneousgas-solid interaction forcēFg→s,a on particlea, which ismonitored in time for each particle. Averaging over time(when steady state is reached) and over all particlesa ofthe same typei yields the average gas-solid interactionforceFg→s,i . To improve accuracy, additional averagingover different (typically 5) initial configurations is done.Note that a uniform force density is applied to the fluidwhich counteract the solid-to-fluid force, otherwise thesystem would continue to accelerate.The random arrays of spheres are created by start-ing with a regular array (typically BCC), which is sub-sequently randomized by a Monte Carlo procedure as de-scribed in Beetstra et al. (2007). At the domain boundaryperiodic boundary condition are employed. The config-urations we studied contain 512 and 1000 spheres in to-tal of ten different diameters both with a log-normal anda Gaussian size distribution. The results for the Gaus-sian distributions will be published elsewhere (Sarkar etal. (2008)). In this paper, we will show results from thethe log-normal distributions, for spheres in the range of8-24 and 8-40 lattice spacings (see Figure 1 for the pre-cise number of particles present for each particular diam-eter).8 12 16 20 24 28 32 36 400306090120150180Particle diameter in unit of grid sizeNumber of particles(a)(b)Figure 1: Two different size distribution of particles. log-normal: (a) diameter range of 8-24 for 1000 particles in totaland (b) diameter range of 8-40 for 512 particles.In Figure 2 to 5 simulation data are compared withthe prediction (5) using the monodisperse relation (7) ofBeetstra et al. (2007) at packing fraction 0.3 and 0.5 andReynolds number 0.1, 10.0, 100.0, 500.0. It can be seenthat excellent agreement is found for all the Reynoldsnumbers and packing fractions.0.4 0.6 0.8 1 1.2 1.4 1.501020304050607080di/<d>Normalized total force (F itot)Simulation (0.3)Prediction (0.3)Simulation (0.5)Prediction (0.5)Figure 2: Comparison of simulated individual normalizedtotal force with the prediction (5) as a function of individualdiameter over the average diameter (di/ < d >) for a lognor-mal particle size distribution (type (a)) with a diameter rangeof 8 to 24 grid spacings, at Reynolds number(i)< Re>=0.1.0.4 0.6 0.8 1 1.2 1.4 1.50102030405060708090di/<d>Normalized total force (F itot)Simulation (0.3)Calculated (0.3)Simulation (0.5)Calculated (0.5)Figure 3: As figure 2, but now for Reynolds number< Re>= 10.3S. Sarkar, M.A. van der Hoef & J.A.M. Kuipers0.4 0.6 0.8 1 1.2 1.4 1.5020406080100120140160180200di/<d>Normalized total force (F itot)Simulation (0.3)Calculated (0.3)Simulation (0.5)Calculated (0.5)Figure 4: As figure 2, but now for Reynolds number< Re>= 100.0.4 0.6 0.8 1 1.2 1.4 1.50100200300400500600di/<d>Normalized total force (F itot)Simulation (0.3)Calculated (0.3)Simulation (0.5)Calculated (0.5)Figure 5: As figure 2, but now for Reynolds number< Re>= 500.Simulated data are compared for larger range of diame-ter in Figures 6 and 7 at packing fraction 0.3 and 0.5 andReynolds number 0.1 and 500. Again a very good agree-ment is observed between the simulated data and the pre-diction.0.3 0.6 0.9 1.2 1.5 1.8 2020406080100120di/<d>Normalized total force (F itot)Simulation (0.3)Calculated (0.3)Simulation (0.5)Calculated (0.5)Figure 6: As figure 2, but now for for a lognormal size distri-bution (type (b)) with a diameter range of 8 to 40 grid spacingsand Reynolds number< Re>=0.1.0.3 0.6 0.9 1.2 1.5 1.8 201002003004005006007008009001000di/<d>Normalized total force (F itot)Simulation (0.3)Calculated (0.3)Simulation (0.5)Calculated (0.5)Figure 7: As figure 6, but now for Reynoldsnumber< Re>= 500.CONCLUSIONSIn this paper we have presented the results of the gas-particle interaction force from DNS simulations of fluidflow past static arrays of polydisperse spheres. We con-sidered two different size ranges with a log-normal dis-tribution. We found that the individual drag force is pre-dicted very well by expression (5), in combination withthe monodisperse relation (7) of Beetstra et al. (2007).Note that when the correlation by Ergun is used, with thediameter replaced by the individual diameter of the par-ticle species, the disagreement with the simulation datacan amount to up to 200 %. In a more extended futurepublication (Sarkar et al. (2008)) we will present resultsfor a Gaussian distribution, as well as a detailed compar-ison with predictions using various other monodisperse4Fluid-particle interaction force for polydisperse systems from lattice Boltzmann simulations / CFD08-89relations.ACKNOWLEDGEMENTSWe thank Anthony Ladd for allowing us to use his lat-tice Boltzmann code (SUSP3D). This work is financiallysupported by the ‘Nederlandse Organisatie voor Weten-schappelijk Onderzoek (NWO) (Netherlands Organiza-tion for Scientific Research).REFERENCESBeetstra R., van der Hoef M.A., and Kuipers J.A.M., 2007. Dragforce of intermediate Reynolds number flow past mono- andbidisperse arrays of spheresAICHE Journal, 53: 489.Beetstra R., van der Hoef M.A., and Kuipers J.A.M., in Press.Erratum to ”Drag force of intermediate Reynolds number flowpast mono- and bidisperse arrays of spheres”[AICHE Journal53 (2007) 489-501]AICHE Journal.Ergun S., 1952. Fluid flow through packed columns.Chem. En-gng. Progs., 48: 89.Ladd A. J. C., 1994(a). Numerical simulations of particulatesuspensions via a discretized Boltzmann equation Part I. Theo-retical foundationJ. Fluid Mech., 271: 285.Sarkar S., van der Hoef M.A., and Kuipers J.A.M., 2008. Dragforce from lattice Boltzmann simulations for flow through poly-disperse random arrays of spheressubmitted to Chemical En-gineering Science.van der Hoef M.A., Beetstra R., and Kuipers J.A.M., 2005. Lat-tice Boltzmann simulations of low Reynolds number flow pastmono- and bidisperse arrays of spheres: results for the perme-ability and drag force.J. Fluid Mech., 528: 233.Wen C.Y., and Yu Y.H., 1966. Mechanics of fluidization.Chem.Engng. Prog. Symp. Ser., 62: 100.5

Drag force from lattice Boltzmann simulations for ﬂow through polydisperse random arrays of spheres submitted to Chemical Engineering Science.

Drag force of intermediate Reynolds number ﬂow past mono- and bidisperse arrays of spheres

Erratum to ”Drag force of intermediate Reynolds number ﬂow past mono- and bidisperse arrays of spheres”[AICHE

Fluid ﬂow through packed columns.

Lattice Boltzmann simulations of low Reynolds number ﬂow past mono- and bidisperse arrays of spheres: results for the permeability and drag force.

Mechanics of ﬂuidization.

Numerical simulations of particulate suspensions via a discretized Boltzmann equation Part I. Theoretical foundation

http://purl.utwente.nl/publications/68587

Fluid-particle interaction force for polydisperse systems from lattice boltzmann simulations

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