Hoef2008 Hoef Fluidized Bed Annurev.fluid.40.111406.102130

7/30/2019 Hoef2008 Hoef Fluidized Bed Annurev.fluid.40.111406.102130

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Numerical Simulation ofDense Gas-Solid FluidizedBeds: A MultiscaleModeling Strategy

M.A. van der Hoef, M. van Sint Annaland,N.G. Deen, and J.A.M. Kuipers

Department of Science and Technology, University of Twente, 7500 AE Enschede,The Netherlands; email: [email protected]

Annu. Rev. Fluid Mech. 2008. 40:4770

The Annual Review of Fluid Mechanicsis online atfluid.annualreviews.org

This articles doi:10.1146/annurev.fluid.40.111406.102130

Copyright c 2008 by Annual Reviews.All rights reserved

0066-4189/08/0115-0047$20.00

Key Words

fluidization, direct numerical simulation, discrete element model,

two-fluid model

Abstract

Gas-solid fluidized beds are widely applied in many chemical pro-

cesses involving physical and/or chemical transformations, and for

this reason they are the subject of intense research in chemical engi-neering science. Over the years, researchers have developed a large

number of numerical models of gas-fluidized beds that describe gas-solid flow at different levels of detail. In this review, we discriminate

these models on the basis of whether a Lagrangian or a Eulerian ap-proach is used for the gas and/or particulate flow and subsequently

classify them into five main categories, three of which we discuss inmore detail. Specifically, these are resolved discrete particle mod-

els (also called direct numerical simulations), unresolved discreteparticle models (also called discrete element models), and two-fluid

models. For each of the levels of description, we give the general

equations of motion and indicate how they can be solved numeri-cally by finite-difference techniques, followed by some illustrative

examples of a fluidized bed simulation. Finally, we address someof the challenges ahead in the multiscale modeling of gas-fluidized

beds.

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1. INTRODUCTION

Gas-fluidized beds consist of granular particles (usually with a diameter less than5 mm) that are subject to a gas flow from below, large enough so that the gas drag on

the particles can outbalance gravity, and the particles fluidize. When in the fluidized

state, the moving particles work effectively as a mixer, which results in a uniformtemperature distribution anda high mass transfer rate, both of which arebeneficial for

the efficiency of many physical and chemical processes, such as coating, granulation,drying, and the synthesis of fuels and base chemicals (Kunii & Levenspiel 1991). For

this reason, gas-solid fluidized beds are encountered in many industrial plants in thechemical, petrochemical, metallurgical, environmental, and energy industries. Many

of these operations are on large scales. For instance, the FCC (fluid catalytic cracking)unitwhich is the heart of almost any oil refineryconsists of fluidized beds that

are typically 14 m high and 6 m in diameter, with a circulation rate of up to 1 ton persecond. At present, the design and scale-up of such fluidized bed reactors are mostly

fully empirical processes, owing to limited insight into the fundamentals of densegas-particle flows at such scales, in which the phenomena related to effective gas-

particle interactions (drag forces) and particle-particle interactions (collision forces)

in particular are not well understood (Kuipers & van Swaaij 1998). For this reason,many preliminary tests on pilot-scale model reactors have to be performed, which is

a time-consuming and thus expensive activity.To aid this design process of fluidized bed reactors, computer simulations can

clearly be a useful tool. However, the prime difficulty in modeling life-size fluidizedbeds is the large separation of scales: The largest flow structures can be of the order

of meters, yet these structures can be directly influenced by details of particle-particlecollisions and particle-gas interactions, which take place below the millimeter scale.

Clearly, it will not be possible to have one single simulation method that can coverall length and time scales; instead, one needs a hierarchy of methods, modeling the

gas-solid flow phenomena on different length and time scales, and thus also with

different levels of detail. We can classify these different models most conveniently byconsidering the possible models for the solid phase and the gas phase separately. The

dynamics of each of these phases can be described by (a) considering the phase as acollection of discrete particles that obey Newtons law, which requires a Lagrangian

type of model, or (b) adopting a continuum description of the phase, which is thentypically governed by a Navier-Stokes-type equation, which requires a Eulerian type

of model. Based on these two options for each phase, we categorize the differentmodels available for gas-solid flow in Table 1. A graphical representation of the

models is shown in Figure 1. Loth (2000) has made a similar classification in a moregeneral context for engineering science (including bubbles and droplets).

A useful starting point for the discussion ofTable 1 is to examine the Lagrangian-

Lagrangian (LL) model and the Eulerian-Eulerian model. The LL model is the mostfundamental model, in which both the solid phase and the gas phase are representedby particles, so the solid-gas interaction comprises simply the collisions of the gas par-

ticles with the bigger solid particles. The positions and velocities of both phases are

updated by molecular dynamics (MD) type methods; that is, Newtons law is solved

48 van der Hoef et al.


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Table 1 Classification of the various models used for simulating dense gas-solid flow in the context of

gas-fluidization

Name Gas phase Solid phase Gas-solid coupling Scale

1. Discrete bubble

model

Lagrangian Eulerian Drag closures for bubbles Industrial (10 m)

2. Two-fluid model Eulerian Eulerian Gas-solid drag closures Engineering (1 m)

3. Unresolved discreteparticle model

Eulerian (unresolved) Lagrangian Gas-particle drag closures Laboratory (0.1 m)

4. Resolved discrete

particle model

Eulerian (resolved) Lagrangian Boundary condition at

particle surface

Laboratory (0.01 m)

5. Molecular dynamics Lagrangian Lagrangian Elastic collisions at particle

surface

Mesoscopic (


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Figure 2

Example of a simulationusing aLagrangian-Lagrangianmodel, in which the fluidphase is modeled via anextremely simplified

molecular dynamics model,a lattice-gas cellularautomata model (see van derHoef et al. 1991). Note thecharacteristic double-vortexin the fluid flow field, set upby the large particle, whichis moving to the right.

the large solid particles (Ladd et al. 1988). Figure 2 shows an example of a largeparticle moving in an LGCA fluid. On the other end of the scale is the Eulerian-

Eulerian model, also referred to as the two-fluid model (TFM). This model employsa continuum description for both the solid phase and the gas phase and uses a finite-

difference code to describe the time evolution of both phases (see Gidaspow 1994and Kuipers & van Swaaij 1998, among others). The interaction between the two

phases is incorporated by drag force correlations, which depend on the local relativevelocity of the phases and the local solids volume fraction. Also correlations for the

solids phase pressure and viscosity have to be specified. The drawback of this methodis that it does not adequately model the details of particle-particle and particle-gas

interactions. The latter is taken care of by intermediate Euler-Lagrange models, alsocalled discrete particle models (DPMs), in which the solid phase is represented by the

actual particles, as in the LL models. The update of the solid particles is the same as

in the LL models; however, the interaction with the continuous gas phase is differ-ent. Basically one can decide between two choices for the Euler-Lagrange coupling:

(a) unresolved and (b) resolved. In the unresolved discrete particle model (UDPM),often referred to in the literature as the discrete element model, the Eulerian grid is

at least an order of magnitude larger than the size of the particles, so the particles arereduced to point sources and sinks of momentum, at least with respect to the gas-

particle interactions. Thus for this interaction, one still requires correlations similarto those for the gas-solid interaction in the TFM. In the resolved discrete particle

model (RDPM), often referred to in the literature as the direct numerical simulationmodel, the Eulerian grid is an order of magnitude smallerthanthe size of the particles,

so the flow in between the particles is also computed. The gas-particle interactionis now handled by (stick) boundary conditions at the surface of the solid spheres.

In this case, no correlations are required: Both the particle-particle and particle-gas

interactions are modeled in a realistic way.



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Recently, a fifth level of modeling has been suggested, which could be termeda Lagrangian-Eulerian model. This model treats the gas bubbles (which typically

appear in gas-fluidized beds) as discrete entities, which can collide, coalesce, breakup, shrink, and grow. The Eulerian description is not of the actual solid phase, but

rather of the emulsion phase of gas and particles. This model is still in its earlystages, and we only briefly discuss it in Section 6. Note that not all methods from the

literature can be categorized according to Table 1. For example, the force-couplingmethod (Climent & Maxey 2003) falls between the RDPM and UDPM, whereas the

multiphase particle-in-cell method (Snider 2001)in which the solid phase is treatedas both Eulerian and Lagrangianfalls between the UDPM and TFM. Although

these methods may provide an interesting alternative for modeling gas-fluidized beds,

we do not consider them further here.The basic philosophy behind the multiscale modeling approach is that the smaller-

scale models, which take into account the various interactions (fluid-particle, particle-particle) in detail, are used to develop closure laws that can represent the effective

coarse-grained interactions in larger-scale models. Note that it is not guaranteed thatall relevant correlations between small- and large-scale processes can be captured by

effective interactions. However, experience has shown that in many cases the maincharacteristics of gas-solid flows can be well described by the use of closure relations.

At present, the modeling of the gas fluidization of granules is limited to the threeintermediate models (TFM, UDPM, RDPM); hence in this review we focus on these

models. It is convenient, however, to first briefly discuss the various models for single-phase flow because these are the building blocks of the models for two-phase flow.

2. MODELS FOR SINGLE-PHASE FLOW

2.1. Eulerian Models

In the Eulerian models the phase (gas or solid) is considered as a continuum, which is

characterized by a local mass density(r, t) and a local momentum densityj(r, t). The

local flow velocityu(r, t) is then defined by j = u. The basic equations of motionare the conservation equations for mass and momentum:

t + (u) = 0, t(u) + (uu) = p g, (1)

where gis the gravity constant (=9.81 ms2), p is the hydrostatic pressure, and isthe stress tensor, for which the general form for a Newtonian fluid can be used:

=

23

( u)I + (u + (u)T), (2)

with and the coefficients of bulk and shear viscosity, respectively, and I the unittensor.TherearecurrentlytwomainclassesofmethodsforsolvingEquation1numer-

ically, namely computational fluid dynamics (CFD) methods and lattice Boltzmann(LB) methods.

In CFD methods, Equation 1 is solved via finite-difference, finite-element, orfinite-volume methods on a Eulerian grid. A large number of different discretization

www.annualreviews.org Numerical Simulation of Gas-Fluidized Beds 51


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schemes have been developed in the past, and for an in-depth discussion of CFDmethods in general, we refer the reader to Fletcher (1988a,b). For the TFMs and

DPMs, a computational scheme based on the implicit continuous Eulerian methodis mostly used. The implementation is based on a finite-difference technique, which

uses a staggered grid for stability and defines the scalar variables (pressure and den-sity) at the cell centers, whereas the velocities are defined at the cell faces. One can

use higher-order schemes [Barton, (W)ENO] to construct the mass and momentumfluxes at the cell faces from the information defined at the centers. In its most simple

form, the discretization of Equation 1 comprises a first-order time differencing with afully implicit treatment of the pressure gradient and convective mass flux, and explicit

treatment of momentum convection, viscous interaction, and gravity. The resulting

set of equations is solved iteratively, for instance, by a two-step correction projec-tion method such as the SIMPLE algorithm, in which the pressure is adapted via a

Newton-Rhapson procedure until mass conservation is achieved (for more details,see Kuipers & van Swaaij 1998 and van der Hoef et al. 2006).

In the LB methods (Chen & Doolen 1998, Succi 2001), Equation 1 is not solveddirectly, but rather the Boltzmann equation is solvedwhich one can view as the

fundamental equation underlying Equation 1; in its most simple form, it reads

t f(r,v, t) +v f(r,v, t) = 1

a[ f(r,v, t) feq(, u, T)], (3)

with fthe single-particle distribution function and feq(, u, T) the local equilibriumdistribution function at temperature T. Because frepresents the particle density in

velocity and coordinate space, the zeroth moment

f dvis equal to , and the firstmoment

fvdvis equal toj; taking the zeroth and first moment of Equation 3 yields

Equation 1. If the temporal and spatial variations of and u are sufficiently small,the stress tensor takes the form of Equation 2, with vanishing bulk viscosity and a

shear viscosity a T, with the average density. In the LB method, Equation3 is solved by a finite-difference scheme, in which a single time step consists of a

propagation of fon a grid, followed by a relaxation to equilibrium (Chen & Doolen1998, Succi 2001); in the popular D3Q19 model, the grid is simple cubic, in which

distributions can move to nearest and next-nearest neighbor sites.

Both the CFD and LB methods have proven to be powerful techniques for mod-eling fluid flow at low to moderate Reynolds numbers; both have their relative merits

and weaknesses, which mainly have to do with stability and efficiency (Bernsdorf et al.1999, Succi 2001), and we do not elaborate on this here. We discuss the use of both

models in the numerical simulation of dense gas-solid flow in Sections 35.

2.2. Lagrangian Models for the Solid Phase

In Lagrangian models, the phase (gas or solid) is represented by discrete particles. In

mainstream simulation models of gas-fluidized beds, Langrangian models are onlyused for the solid phase, in which the particles are almost exclusively represented

by perfect spheres, for obvious computational reasons; Lagrangian models of drygranular flow with nonspherical particles have been developed, however (Langston



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et al. 2004, Vu-Quoc et al. 2000). There are two main classes of simulation modelsfor (in)elastic spheres: hard-sphere models and soft-sphere models. Both methods

originate from MD simulation models, initiated by Alder and Wainwright in the1950s and 1960s, and a number of excellent books discussing the details of MD

methods have appeared since (e.g.,Allen & Tildesley1990,Frenkel & Smit 2002).Thetechniques canwith only minor modificationsbe transferred directly to granular

systems. One of the most important modifications is that in granular systems theparticles dissipate energy in collisions and also experience surface friction. Energy

dissipation is usually described via the coefficient of normal restitution e , which isdefined as the factor by which the postcollisional relative velocity of two particles

in the normal direction is reduced as a result of dissipation, compared with a fully

elastic collision. The surface friction is governed by the friction coefficient f andaffects the relative tangential velocity at the contact point, at which both sticking and

sliding can occur. Then one can alsoinclude the dissipation of energy in the tangentialdirection, governed by a coefficient of tangential restitution e t. For more details on

the dynamics of inelastic collisions, we refer the reader to Walton (1992). We nowbriefly discuss the essentials of hard-sphere and soft-sphere methods. A much more

detailed description can be found in van der Hoef et al. (2006) and Deen et al. (2007).In hard-sphere methods, particles interact via instantaneous collisions; in between

these collisions, the particles undergo free flight, during which their velocity is notchanged. The evolution of the systems is thus from collision event to collision event,

and therefore the method is often classified as event-driven. The advantage of the

hard-sphere method is that it is efficient for dilute systems. Moreover, the instanta-neous collision is physically more realistic compared to soft-sphere models, in which

for computational reasons the duration of a collision is much longer than it shouldbe on the basis of the material properties (i.e., the elasticity) of the particles. The

drawback is that the event-driven scheme is inefficient for very dense systems, owingto the extremely high number of collisions; if the particles get jammed, the method

breaks down completely.In the soft-sphere model, Newtons law for each particle i (position ri, mass mi),

miri = Ftoti , Ftoti =

j

Fij mig, (4)

is solved numerically. The inelastic collision process is represented by the force Fij,

which is the force that particle iexperiencesfrom particle jwhenincontact(otherwiseFij = 0). One of the most widely used models for the contact force is the linearspringdashpot model by Cundall & Strack (1979), which is a good compromise

between efficiency and accuracy. Newtons equation can be solved approximately bya (time-driven) integration scheme; in MD for molecular systems, one normally uses

second-order schemes, such as the popular Verlet scheme (Allen & Tildesley 1990).However, for granular systems often a first-order scheme suffices because, owing

to the presence of nonconservative forces (drag, inelastic collisions) in gas-fluidizedbeds, the conservation of energy in the integration is not such a crucial issue as

in molecular systems. In gas-fluidized systems, the equilibrium state follows froma balance between the energy fed to the system to maintain the fluidized state and



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the energy lost in dissipation. A nonconserving integration algorithm may shift thisbalance slightly, but the energy would not drift away as it would in molecular systems.

One of the main advantages of the soft-sphere model is that it works well both fordilute systems (although it is less efficient than the hard-sphere model) and dense

systems. Also one can include other forces such as gas-solid drag and the van derWaals force more naturally in a time-driven scheme (they can simply be added to

Ftot

i ) than in an event-driven scheme, in which the update is not based on forces.

2.3. Lagrangian Models for the Gas Phase

The Lagrangian models for the solid phase described in the previous section above

can also be used to model the gas phase; for many applications, however, this iscomputationally extremely expensive, and also such a detailed description of the fluid

phase is not required to model the essential features. For this reason, researchers havedeveloped a number of simplified MD models of a fluid phase. One such method that

has become popular in recent years is stochastic rotation dynamics (Malevanets &Kapral 1999), also called multiparticle collision dynamics (for an extensive review,

see Padding & Louis 2006). Stochastic rotation dynamics is basically a time-drivenMD method in which the collisions between the particles are replaced by stochas-

tic rotations of the velocity vectors relative to the local flow field. Other simplifiedparticle-based methods of fluid flow are dissipative particle dynamics, direct simula-

tion Monte Carlo, and LGCA. Figure 2 shows an example of an LGCA simulation.An upcoming technique that merits attention is smoothed particle hydrodynamics

(SPH) (Liu & Liu 2003, Monaghan 2005), which is also a particle-based model, yet

it is different from the ones discussed above. SPH is a mesh-free method used tosolve the hydrodynamics equations, by replacing the fluid with a set of particles.

Basically the particles, which move with the flow, represent the interpolation points toevaluatethespatialderivativesofthephysicalquantities.Akernelisusedtosmooththe

particle-based information (which is thus represented at points in space) to obtain thecontinuous hydrodynamics fields. Thereareclose similaritiesbetween MD models for

fluid flow and the SPH model, yet the latter does not contain the thermal fluctuationsinherent to the other particle-based models.

3. RESOLVED DISCRETE PARTICLE MODEL

In this section, we discuss two possible implementations of the RDPM: one in whichthe gas phase is modeled by the CFD method and one in which the gas phase is

modeled by the LB method. The key element is how to implement the stick bound-ary condition, in which the fluid velocity vanishes at the surface of the solid spheres.

In both cases, the solid phase is modeled by the Lagrangian method discussed inSection 2.2. We stress that apart from the two methods discussed below, a num-

ber of other finite-difference/finite-element techniques have appeared to model fullyresolved gas-particle flow at finite Reynolds numbers. These include the arbitrary-

Lagrangian-Eulerian technique (see Hu et al. 2001, and references therein), thedistributed Lagrange-multiplier/fictitious domain (DLM) method (see Sharma &



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Patankar 2005 and references therein), and PHYSALIS (Zhang & Prosperetti 2005).PHYSALIS couples the exact solution of the Stokes equation in the vicinity of the

particles surface (with a no-slip boundary condition) to the finite-difference solutionof the full Navier-Stokes equation further away from the surface.

Stick boundary conditions in a CFD description of the gas phase can be imple-mented efficiently by the immersed boundary (IB) method (Mittal & Iaccarino 2005,

Peskin 2002), in combination with a direct forcing method (Uhlmann 2005). In thismethod, Lagrangian force points are defined, which are uniformly distributed on

the surface of the sphere; the velocity Wm of each force point m is evaluated fromthe linear and angular velocity of the sphere. The interaction between the gas phase

and the solid phase is then controlled by a force densityfsg, which is added to theright-hand side of the momentum conservation in Equation 1:

t(gug) + (gugug) = pg g gg+ fsg, (5)

where the subscriptg refers to gas-phase variables. In the discretization of the mo-mentum equation, fsg is treated explicitly. To discuss the method further, we find itconvenient to only consider the interaction with a single sphere i(i.e., fsg = fig),inwhich extension to systems with more than one sphere is straightforward. The localforce density in the Eulerian frameworkfig is calculated from the force densitiesfm(rm) at the location of Lagrangian force points m by using a distribution function

D: fig(r) =

m D(r rm)fm(rm), where the sum is over all force points rm withinthe range of influence of r. For most cases a distribution based on simple volume

weighing is used. The force densities fm(rm) are calculated from the constraint that ateach force point, the local gas velocity should match the local particle velocity, which

yields

fm =m(Um Wm)

t

3N

12

R2

l2+ 1

, (6)

with

m and Um the initial gas density and velocity, respectively, at the Lagrangianpointm, mapped from the Eulerian grid using the same distribution function D; l3

is the volume of an elementary CFD grid cell. The force on particle i as a result ofthe boundary rules is then Fg i =

Nm fml

3, which can simply be added to Ftoti in

Equation 4.We next discuss fully resolved flow around particles using the LB method. Ladd

(1994a,b) introduced a particularly efficient and simple way to enforce stick boundary

rules on a sphere i in the LB method. First the boundary nodes are identified, whichare defined as the points halfway between any pair of neighboring lattice sites, one of

which is located inside the sphere, whereas the other one is outside the sphere. Fora static particle, the boundary rule is simply that a distribution fmoving such that

it would cross the boundary bounces back at the boundary node, which results in avanishing boundary velocityUb of the fluid at the location of the node b . For moving

and rotating particles, the bounce-back rule is modified such that part of fcan leakthrough, so Ub =Wb when summed over all link directions, where Wb is the localboundary velocity that is calculated from the linear and angular velocity of the sphere.The force on particle i is then Fg i =

Nb pb /t, with pb the change in fluid



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momentum as a result of the boundary rule at node b . The advantage of the bounce-back method is that it is simple to implement. The disadvantage is that the fluid

velocity is matched at the Eulerian points b , which give a stepwise description of thesurface. By contrast, in the IB method the fluid velocity is matched at the Lagrangian

points m, which are located exactly on the sphere. Verberg & Ladd (2001) and Rohdeet al. (2003) have suggested methods that have a smoother description of the particle

boundary in the LB method, and more recently Fen & Michaelides (2005) introducedthe IB method into the LB scheme.

Fully resolved methods such as CFD-IB, LB-bounce-back, PHYSALIS, andDLMcan be used for two different purposes in the multiscale framework of gas-fluidized

beds: (a) to perform a fully resolved simulation of gas-fluidized beds and (b) to obtain

estimates for thegas-solid interaction force Fg i. To our knowledge, no fully resolvedsimulations of gas-fluidized beds have appeared in the literature to date. Researchers

have carried out LB simulations of many particle systems to study suspension flowin Couette systems (Kromkamp et al. 2006) and sedimentation (Nguyen & Ladd

2005), both at low Reynolds numbers. Sedimentation has also been simulated byZhang et al. (2007) for 1024 spheres in liquid using the PHYSALIS method and by

Glowinski et al. (2001) for 6400 discs in liquid using the DLM method. The onlyfully resolved simulation of a fluidized bed has been reported by Pan et al. (2002),

who studied 1204 particles fluidized in water using the DLM approach to resolve thefluid flow around the particles. In Figure 3 we show the results from fully resolved

simulation using the CFD-IB method for 3600 particles in two-dimensions, fluidizedin air at 100 bar. For computational convenience, the particle/gas density ratio was

Figure 3

Snapshots from a fully resolved simulation of a two-dimensional fluidized bed using theimmersed boundary method. The system (width 75 cm) contains 3600 particles with adensity of 1100 kg m3 and a diameter of 1 cm, fluidized at 0.20 m s1; the density of thefluid phase is 100 kg m3. The computational fluid dynamics cells are 0.625 0.625 mm2,so the total number of cells is nearly 3 million. The particles are colored to visualize themixing.



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limited to 11 in this case. Higher-density ratios can be used in principle, but theylead to (excessively) small permissible time steps owing to the explicit treatment of

the fluid-solid interaction.A different use of fully resolved simulations is to obtain information on the effective

gas-solidinteraction force, which is an essential input in thehigher-scalemodels.Suchsimulations have been carried out almost exclusively using the LB method, for static

random arrays of spheres that are monodisperse (Hill et al. 2001a,b, Kandhai et al.2003, Maier et al. 1999) and bidisperse (Beetstra et al. 2007a, van der Hoef et al. 2005)

and for nonstatic arrays (Wylie et al. 2003). For monodisperse static arrays, the LBresults forthe gas-solidinteraction force reported by thevarious authors areconsistent

and show relatively large deviations (up to 100%) with the well-accepted empirical

correlations by Ergun (1952) and Wen & Yu (1966). The following correlation for thegas-solid interaction force on a particle of diameter di represents a good fit (within

8%) to the LB data of monodisperse and polydisperse systems (Beetstra et al. 2007a):

Fg i3 dig(ug vi)

=

gdid + s

d2id2 + 0.064g

d3id3

10 s3g

+ g + 1.5 g1/2s + 0.413 Re243g

1

g + 3sg + 8.4 Re0.343

1 + 103sRe(1+4s)/2

, (7)

with g and s the volume fraction of the gas phase and solid phase, respectively

(g + s = 1), and the Reynolds number Re = gdg|ug vi|/. The last term insquare brackets of Equation 7 represents the gas-solid interaction force for monodis-perse systems, whereas the first term represents the correction for polydispersity, withd = i d3i /(i d2i ). The bar indicates thatFg i represents the average gas-solidinteraction force, which only depends on other average quantities (d, s, g) of acertain domain. Note that Fg i represents the total gas-solid interaction forceincluding the contribution from the pressure gradientwhich is different from the

drag force

Fd,i as it is commonly defined; for monodisperse systems one can derivethatFg i = Fd,i/g. We discuss this point further in Section 4. It should be stressed

that Relation 7 is for homogeneous arrays; ten Cate & Sundaresan (2006) showed thatfor heterogeneous configurations (which are typically encountered in gas-fluidized

beds), there can be large deviations between the domain-averaged interaction forceFg i andthetrue individual interaction force Fg i.Thisisatypicalexampleinwhichthe lack of resolution in the higher-scale models can be the source of discrepancies

compared with the results from detailed-scale simulations. Note that the problem ofunresolved structures is central to the whole multiscale modeling approach.

In this section on resolved DPM we focus on models with a Eulerian descriptionof the gas phase. In the past years, researchers have developed a number of other

numerical models for fully resolved gas-particle flow, in which the gas phase is repre-sented by the particle-based models described in Section 2.3. However, such models

have been applied almost exclusively to simulate colloidal systems, that is, systems atlow Reynolds numbers and small scales, at least compared to gas-fluidized beds. Yet

some of these models could present an interesting alternative to LB or CFD in thefully resolved modeling of gas-fluidized beds. One particular example is the work of



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Ma et al. (2006), who studied the effect of heterogeneity on the drag force in gas-solidsuspensions using the SPH method for the gas phase.

4. UNRESOLVED DISCRETE PARTICLE MODEL

In UDPMs (also termed discrete element models), the size of the Eulerian grid is

typically an order of magnitude larger than the size of the particles, so the gas phasesees the particles only as point sources and sinks of momentum. In the Lagrangiandescription of the solid phase, however, the particles do have a finite volume and

interact via collision rules, as discussed in Section 2.2. Although LB models have

been used (Derksen 2003), at present almost all UDPMs of gas-fluidized beds haveadopted a CFD-type discretization of the gas phasepioneered by Tsuji et al. (1993),

Hoomans et al. (1996), and Xu & Yu (1997)and we focus on such a descriptionhere. Below we briefly discuss the essentials of the UDPM as it is implemented at the

University of Twente; there are some minor differences with UDPMs developed byother research groups, which are discussed by Kafui et al. (2002). A slightly different

version of the model has been developed by Ouyang & Li (1999).

The UDPM closely resembles the CFD-IB method outlined in Section 3, with oneimportant difference: The Lagrangian force points usedto construct the force densityfsg in Equation 5 are now the solid particles in the domain. Also, the force densityfi at the Lagrangian pointi (note that we switch notation from m to i) is not given byEquation 6, but is calculated from a correlation such as Equation 7: fi = Fg i/l3,where l3 is the volume of an elementary CFD grid cell. Importantly, in such a

description the solid particles do not exclude volume for the gas phase in a naturalway. This has to be incorporated artificially in the conservation in Equation 1, by

replacing g bygg and g byg:1

t(gg) + (ggug) = 0, (8)

t(ggug) + (ggugug) = pg (gg) ggg+ fsg. (9)In practice, however, the equation is rewritten is a slightly different form2 becausethe total gas-to-particle interaction force Fg i can be split into a drag force Fd,i anda force from the pressure gradient: Fg i = Fd,i Vipg, with Vi the volume ofparticle i. The Eulerian force density arising from Vipg is equal to spg, so thefinal expression for the momentum conservation equation in the UDPM becomes

t(ggug) + (ggugug) = pg (gg) ggg fd + spg, (10)with fd the Eulerian force density constructed from the force densityFd,i/l3 at theLagrangian points i, for which a similar type of distribution is used as in the IB

method; i.e., fd(r) = i D(r ri)Fd,i(ri)/l3, where the sum is over all force points ri

within the range of influence of r. For monodisperse systems Fd,i = gFg i, whichcan be calculated at each Lagrangian pointi from a correlation such as Equation 7, in

1This is not just an ad hoc modification but has a sound theoretical basis ( Jackson 1997).

2See also Gidaspow 1994: Equation 9 corresponds to his model B and Equation 10 to his model A.



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which an inverse Euler-Lagrange mapping is used to extrapolate the porosityg andfluid velocityug from the nearby Eulerian grid points to the location of particle i. In

chemical engineering, a combination of the empirical correlations by Ergun (1952)and Wen & Yu (1966) for Fg i is currently the most widely used (Gidaspow 1994).Note that the UDPM as described above does not take the effect of gas turbulenceinto account specifically; one of the reasons is that even for moderately high Reynolds

numbers (


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f= 0.0

f= 0.1

e = 1.00 e = 0.99 e = 0.97 e = 0.94 e = 0.90Figure 4

Snapshots from eightdifferent unresolved discreteparticle model simulations,demonstrating the effect ofthe coefficient of normalrestitution e and friction

coefficient fon thefluidization in air of 4-mmglass particles (density2700 kg m3) in aquasi-two-dimensionalsystem (0.15 0.60 m,7200 particles). Thesuperficial fluidizationvelocity was set at 2.5 m s1;for all cases we selected theframes with the highestlevel of heterogeneity. It canbe observed that dissipationinduces heterogeneity,whereas the simulationswith friction show slugflowtype fluidization.

is governed by Equations 8 and 10. For the solid phase, a similar set of equations is

used, which is arrived at by replacing the subscriptgbys and changing the sign of thegas-particle interaction terms fd and spg (whose subscript should not be changedto s).3 Therefore, the equation for the momentum density of the solid phase is

t(ssus) + (susus) = ps + (ss) ssg+ fd spg. (11)

Note thats is the density of the material of which the solid particles are made. Theforce densityfd in Equation 11 can be directly calculated from Fg

i:

fd =s

VpFd =

s

VpgFg i,

3Again this is not an ad hoc procedure, but it has a sound theoretical basis ( Jackson 1997).



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where Vp is the volume of a particle (we assume monodisperse systems for simplicity),and Fg i can be evaluated from a correlation such as Equation 7, with vi replacedbyus. Note that in the literature the gas-solid interaction is usually described by adrag coefficient, which is defined as fd = (ug us). For the solid stress tensor inEquation 11, again one can use the general form (Equation 2) of a Newtonian fluid.

Because the concept of particles has disappeared in the TFM, one can only include

indirectly the effect of particle-particle interactions, via an effective solids pressureps and an effective solids shear and bulk viscosity s and s, for which appropriate

closures should be used. In the early hydrodynamic modelsdeveloped by Anderson& Jackson (1967), Anderson et al. (1995), Tsuo & Gidaspow (1990), and Kuipers et al.

(1992)the viscosity is defined as an empirical constant, and also the dependence of

the solids phase pressure on the solids volume fraction is determined from experi-ments. The advantage of this model is its simplicity; the drawback is that it does not

take into account the underlying characteristics of the solids phase rheology. Anotherclass of models, pioneered by Elghobashi & Abou-Arab (1983), uses a particle turbu-

lent viscosity, derived by extending the concept of turbulence from the gas phase tothe solid phase. The state-of-the-art closures are obtained from the kinetic theory of

granular flow (KTGF), initiated by Jenkins & Savage (1983), Lun et al. (1984), andDing & Gidaspow (1990). The KTGF expresses the solids shear and bulk viscosity,

and the pressure in terms of the solids volume fraction s, the coefficient of normalrestitution e , and the granular temperature = Cp Cp/3, where Cp represents theparticle fluctuation velocity. As an example, we give the expression for the pressure

and shear viscosity (Gidaspow 1994):

ps =

1 + 1 + e2

y(s)

ss,

s =5

12

1

(1 + e)y(s)+ 2

5+ 0.193(1 + e )y(s)

sd

,

where y(s) is the excess compressibility of an elastic hard-sphere system, for whichone can use the expressions by Carnahan & Starling (1969) or Ma & Ahmadi (1986).

The time evolution of the granular temperature itself is given by

3

2

t(ss) + (ssus)

= ps us ss : us (sqs) 3 , (12)

where qs is the kinetic energy flux, and is the dissipation of kinetic energy owing

to inelastic particle collisions.One of the strengths of the TFM combined with the KTGFalthough still under

developmentis that it can describe two-phase flow at relatively large scales, yet itis directly controlled by the physics at the level of particle-particle interactions, such

as the amount of energy dissipated in a collision. To illustrate this, Figure 5 showssome snapshots of a TFM simulation of a gas-fluidized bed for various values of the

coefficient of restitution e . The fluidization behavior as function of e , predicted byTFM, agrees well with results from the DPM model (see Figure 4).

In the past 20 years, researchers have used the TFM to study a wide number ofdifferent systems relevant to chemical engineering science, and a full survey of all



16/26

e = 1.00 e = 0.99 e = 0.97 e = 0.94 e = 0.90

Figure 5

Snapshots from five different two-fluid model simulations, demonstrating the effect of thecoefficient of normal restitution e . The shade of gray indicates the local level ofs. Theclosures for viscosity and pressure are obtained from the kinetic theory of granular flow. Thesystem conditions (bed and particle size, fluidization velocity, etc.) are exactly equal to those ofFigure 4. The linear size of a computational fluid dynamic grid cell is equal to 0.5 mm (seealso Goldschmidt et al. 2001).

past and present developments is beyond the scope of this review. Below we presenta limited list of some selected applications and developments. For example, Sinclair

& Jackson (1989) predicted the core-annular regime for steady developed flow in a

riser, whereas Ding & Gidaspow (1990) simulated a bubbling fluidized bed. Transient

simulations and comparisons to data were done by Samuelsberg & Hjertager (1996),and Nieuwland et al. (1996) investigated a circulating fluidized bed using the KTGF.Detamore et al. (2001) have performed an analysis of the scale-up of circulating

fluidized beds using kinetictheory. Thevalidity of some of theassumptions underlyingthe KTGF has been studied by Goldschmidt et al. (2004) by comparing it directly

with unresolved discrete particle simulations. McKeen & Pugsley (2003) simulatedthe fluidization of fine FCC particlesin a freely bubbling bed, and Hansen et al. (2005)

studied the ozone decomposition in the riser of a circulating fluidized bed. Somerecent general developments of the TFM include the extension to binary systems by

Huilin et al. (2003) and the inclusion of heat and mass transfer by Patil et al. (2003,2006).

6. OUTLOOK

In this review, we discuss the essentials of three general classes of models to the studyof gas-fluidized beds: RDPM, UDPM, and TFM. Within each class, one can use



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different types of techniques for single-phase flow (CFD, LB, or MD) to model thegas and solid phase. Each of these classes will continue to play an important role in

the modeling of gas-solid systems in the foreseeable future, although the system sizesthat can be handled will clearly shift with the advancement of computer resources.

Whereas the first discrete particle simulations of gas-fluidized beds in the mid-1990sused an unresolved gas-particle model to study two-dimensional laboratory-scale sys-

tems (which typically contained less than 10,000 particles), soon such systems will bewithin reach of the fully resolved DPM, whereas the unresolved models can now

handle full three-dimensional laboratory-scale systems containing millions of parti-cles. This does not mean that no further developments are required for those models.

With respect to the RDPM, the accuracy, stability, and efficiency of the method leave

room for improvement. The reliability of the UDPM can be improved by using moreadvanced gas-solid drag relations, which take into account the heterogeneity and mo-

bility of the particles; such expressions can be derived from extensive fully resolvedDPM simulations.

A large number of challenges lie ahead for the two-fluid class of models, whichinclude the simulation of (a) fine powders, (b) polydisperse systems, (c) particles with

friction, and (d) industrial-scale systems. With respect to fine powders, at presentTFMs cannot predict the fluidization properties of Geldart A class powders, at least

not without using some ad hoc scaling of the drag force (McKeen & Pugsley 2003).The problems may well have to do with a lack of resolution (see below). With respect

to polydisperse systems, the current class of TFMs still lacks the capability of de-scribing quantitatively particle mixing and segregation rates in polydisperse fluidized

beds. Recently, the KTGF has been formulated for bidisperse systems (Huilin et al.

2003; M. van Sint Annaland, G.A. Bokkers, M.A. van der Hoef, M.J.V. Goldschmidt& J.A.M. Kuipers, submitted), and the next challenge is to extend it to general poly-

disperse systems. Also, the effect of particle-particle friction is not incorporated inthe current KTGF. A recent simulation study using the DPM showed that particle

friction has a large influence on the mixing behavior when a single bubble is injectedinto the system (M. van Sint Annaland, G.A. Bokkers, M.A. van der Hoef, M.J.V.

Goldschmidt & J.A.M. Kuipers, submitted). It was also found that the effects of lackof friction could not be remedied by using a smaller coefficient of normal restitution

(see also Figure 4), which implies that friction should be taken into account explicitlyin the KTGF at the level of the encounter model. There are also limitations to the

TFM with respect to the system sizes that can be studied. The current class of TFMscan simulate fluidized beds only at engineering scales (height of 12 m), at least for

millimeter-sized particles; large-scale industrial fluidized bed reactors (diameter of

15 m, height of 320m) are stillfar beyond its capabilities.The problem is that owingto computational limits, the total number of CFD cells is restricted, which means that

larger systems can only be studied by using a larger grid size, which cannot capturethe structures formed on the smaller scales (Agrawal et al. 2001, Sundaresan 2000).

Researchers have approached the problem of unresolved structures through variousapproximate schemes (McKeen & Pugsley 2003, Yang et al. 2004). Some very re-

cent initiatives in this direction have been undertaken by Andrews et al. (2005), whoconstruct the filtered two-fluid equations, which take the same form as the normal



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t= 10.0 s t= 12.5 s t= 15.0 s t= 17.5 s

Figure 6

Snapshots from a discrete bubble simulation of a polymerization reactor of 4 m 4 m 8 m,fluidized at 0.25 m s1. The bubbles represent the gas phase; note that the surroundingemulsion phase (density 400 kg m3) is not visible. The initial bubble size (injected uniformlyat the bottom plate) is 8 cm; the maximum bubble size is 80 cm. The system typically contains5000 bubbles.

TFM equations, only with a different set of closure relations, suitable for coarse-gridsimulations (see also van der Hoef et al. 2006). The idea of filtered equations is anal-

ogous to large eddy simulation equations for turbulent flows, except that now thereis the added complication of having to constitute a model for the filtered drag force.

Note that the effective drag coefficient depends on filter size, as does the particlephase stress.

Finally, in this review we do not discuss the two classes of model at both ends ofthe scale inTable 1, the LL and the Lagrangian-Eulerian models, because they have

only just begun to be explored in the context of gas-fluidized beds. With respect to

the LL models, we expect that stochastic rotation dynamics and SPH will receivemuch attention in the near future. With respect to the Lagrangian-Eulerian mod-els, some initiatives have been started at the University of Twente, where a discrete

bubble model for gas-solid flow has been developed, adapted from the equivalent

model for gas-liquid systems, including a simple coalescence and break-up model(Bokkers et al. 2006) for the gas bubbles. Figure 6 shows some snapshots from a

discrete bubble model simulation of an industrial-scale gas-phase polymerizationreactor (4 m 4 m 8 m). This model is still in its early stages, and, in par-ticular, the rules for breakup and coalescence can be much further refined on thebasis of UDPM simulations. The ultimate goal of both the filtered TFM and the

discrete bubble model is to make contact with the phenomenological models of life-

size gas-fluidized beds, that is, to provide closures for the bubble size, phase holdups,velocities, and mass dispersion coefficients in the lateral and axial direction (Kunii &Levenspiel 1991). This will be the final step in the multiscale modeling strategy and

will complete the link that one wants to make between (simple) elementary physicsat the detailed scale and (complex) multiphase flow phenomena at the industrial

scale.



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DISCLOSURE STATEMENT

The authors are not aware of any biases that might be perceived as affecting theobjectivity of this review.

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Annual Review

Fluid Mechanic

Volume 40, 2008

Contents

Flows of Dense Granular Media

Yol Forterre and Olivier Pouliquen p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 1

Magnetohydrodynamic Turbulence at Low Magnetic Reynolds

Number

Bernard Knaepen and Ren Moreau p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 25

Numerical Simulation of Dense Gas-Solid Fluidized Beds:

A Multiscale Modeling StrategyM.A. van der Hoef, M. van Sint Annaland, N.G. Deen, and J.A.M. Kuipers p p p p p p p 47

Tsunami Simulations

Galen R. Gisler p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 71

Sea Ice Rheology

Daniel L. Feltham p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 91

Control of Flow Over a Bluff Body

Haecheon Choi, Woo-Pyung Jeon, and Jinsung Kim p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 113

Effects of Wind on PlantsEmmanuel de Langre p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 141

Density Stratification, Turbulence, but How Much Mixing?

G.N. Ivey, K.B. Winters, and J.R. Koseffp p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 169

Horizontal Convection

Graham O. Hughes and Ross W. Griffiths p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 185

Some Applications of Magnetic Resonance Imaging in Fluid

Mechanics: Complex Flows and Complex Fluids

Daniel Bonn, Stephane Rodts, Maarten Groenink, Salima Rafa,

Noushine Shahidzadeh-Bonn, and Philippe Coussotp p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

209

Mechanics and Prediction of Turbulent Drag Reduction with

Polymer Additives

Christopher M. White and M. Godfrey Mungal p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 235

High-Speed Imaging of Drops and Bubbles

S.T. Thoroddsen, T.G. Etoh, and K. Takehara p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 257

v


26/26

Oceanic Rogue Waves

Kristian Dysthe, Harald E. Krogstad, and Peter Mller p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

Transport and Deposition of Particles in Turbulent and Laminar Flow

Abhijit Guha p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

Modeling Primary Atomization

Mikhael Gorokhovski and Marcus Herrmannp p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

Blood Flow in End-to-Side Anastomoses

Francis Loth, Paul F. Fischer, and Hisham S. Bassiouny p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

Applications of Acoustics and Cavitation to Noninvasive Therapy and

Drug Delivery

Constantin C. Coussios and Ronald A. Roy p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

Indexes

Subject Indexp p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

Cumulative Index of Contributing Authors, Volumes 140 p p p p p p p p p p p p p p p p p p p p p p p p p p

Cumulative Index of Chapter Titles, Volumes 140 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p

Errata

An online log of corrections to Annual Review of Fluid Mechanicsarticles may be

found at http://fluid.annualreviews.org/errata.shtml

Hoef2008 Hoef Fluidized Bed Annurev.fluid.40.111406.102130

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