cfd simulations of gas-liquid-solid flow in fluidized bed ......review cfd simulations of...

Powder Technology 299 (2016) 235–258

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Review

CFD simulations of gas–liquid–solid flow in fluidized bed reactors —A review

Hui Pan a, Xi-Zhong Chen b, Xiao-Fei Liang a, Li-Tao Zhu a, Zheng-Hong Luo a,⁎a Department of Chemical Engineering, School of Chemistry and Chemical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR Chinab University of Chinese Academy of Sciences, Beijing 100490, PR China

⁎ Corresponding author.E-mail address: [email protected] (Z.-H. Luo).

http://dx.doi.org/10.1016/j.powtec.2016.05.0240032-5910/© 2016 Elsevier B.V. All rights reserved.

a b s t r a c t
a r t i c l e i n f o
Article history:Received 25 September 2015Received in revised form 25 April 2016Accepted 15 May 2016Available online 18 May 2016

Gas–liquid–solid fluidized bed reactors (FBRs) are of considerable importance as proved by their widespread useand popularity in various industrial processes, which results in extensive theoretical analyses, experimental in-vestigations and computational fluid dynamics (CFD) studies in the past. Many reviews have been publishedconcerning the theoretical and experimental works of three-phase FBRswhile little effort is on the CFD approachof three-phase fluidization system. This review attempts to summarize and analyze CFD simulations for thesethree-phase fluidization systems from two aspects: the fundamentals and their applications, which are of para-mount importance to the formulation of strategies for scale-up, design and control of three-phase FBRs. The fun-damentals of CFD approach in the three-phase FBRs are focusing on various multi-scale models, such as pseudotwo fluidmodel, three fluidmodel, two fluidmodel + discrete particle method (DPM), CFD+ the front tracking(FT)/front capturing (FC) + DPM etc. and the coupling fundamental theories of interaction forces. With the em-phasis on the hydrodynamics of individual phase atmacroscopic ormicroscopic level, the applications of CFD ap-proach for three-phase fluidization system are analyzed. This review also proposes that future emphasis andchallenges of CFD simulation for three-phase FBRs are to present specific and appropriate closure laws for inter-phase momentum exchange, incorporate chemical reactions transport phenomena properties into numericalmodels, and develop measurement techniques and provide more experimental data for three-phase systems.

© 2016 Elsevier B.V. All rights reserved.

Keywords:ReviewGas–liquid–solid fluidization systemFBRCFDMulti-scale modelCoupling fundamental theory

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2372. Multi-scale models of CFD approach for three-phase FBRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2363. Fundamentals of coupling interactions for three-phase system in FBRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

3.1. Coupling between gas and liquid phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2393.2. Coupling between solid and liquid phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2423.3. Coupling between gas and solid phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2433.4. Analysis for particle–particle collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2433.5. Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

4. Applications for CFD simulations in three-phase FBRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2454.1. Gas hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

4.1.1. Bubble formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2454.1.2. Bubble rise velocity, size and shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2464.1.3. Gas holdup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

4.2. Solid hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2484.2.1. Solid velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2484.2.2. Solid holdup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2494.2.3. Particle entrainment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

4.3. Liquid hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

236

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236 H. Pan et al. / Powder Technology 299 (2016) 235–258

4.4. Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2535. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2546. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

1. Introduction

A phase is defined as any part of a system, which is homogeneousand physically distinct and also possesses a definite boundary. Hence,phases are capable of being mechanically separated from the otherparts of the system. A system composed of two phases is called a two-phase system; if there are three phases it is a three-phase system [1].The three-phase systems are mainly classified as gas–liquid–liquid, liq-uid–liquid–solid and gas–liquid–solid three phase system [2–4] whichis also the emphasized system of this review. The gas–liquid–solidfluid-ized bed reactors (FBRs) can be often classified as three-phase fluidizedbed (TFB), draft tube three-phase fluidized bed (DTFB), external circu-lating three-phase fluidized bed (ETFB), inverse three-phase fluidizedbed (ITFB), and tapered three-phase fluidized bed (TIFB) based on dif-ferent industrial purposes [5], which can be traced back to the work ofBergius from 1912 to 1926 on the direct coal liquefaction or coal hydro-genation [5], and ever since then gas–liquid–solid fluidization systemshave been used extensively in the chemical, petrochemical, refining,pharmaceutical, biochemical, food, environmental and physical indus-tries [5–11]. Industrial applications of gas–liquid–solid FBRs in chemicaland petrochemical processing include hydrogen peroxide production,heavy oil and synthetic crude from coal, oil shale and tar sand, produc-tions of methanol and polyethylene, flue gas desulfurization, etc. Gas–liquid–solid three-phase fluidization systems are also found in manyphysical operations such as dust collection, air flotation, sand filtercleaning, crystallization and air humidification [5,12]. The applicationsof three-phase FBRs to biochemical, pharmaceutical and food industrieslike wastewater treatment, antibiotics production and fermentationhave received a vast number of attentions in recent years [13,14].Three-phase FBRs are of considerable industrial importance as provedby their widespread use and popularity in various industries, which re-sults in intensive traditional theoretical analyses [5,15] and experimen-tal investigations [16–22] on them in the past.

Apart from these theoretical and experimental endeavors, computa-tional fluid dynamics (CFD) has been a powerful tool and increases inpopularity as amethod tomodel flow hydrodynamics inmultiphase re-actors with the rapid advancement in computer hardware and physicalunderstanding during the past decades. Though empirical and experi-mental methods can generate reliable results with varied influencingfactors, they still have their own restrictions: traditional theoreticalanalyses where calculation objects are always simplified are restrictedby the nonlinearities of flow hydrodynamics to get analytical solutionsfor multiphase systems; experiments are restricted by specific reactor,fluid disturbance, human security andmeasurement precision [23]. Fur-thermore, the phenomenon of three-phase fluidization system is rathercomplicated that solid particles are suspended by the gas bubbles or theflow of liquid mixed with gas bubbles, which evolves the interactionsand interphase coupling of mass, momentum, and energy betweenthe individual phases. Due to the limitations, it is difficult to describethe fundamental characteristics of the sophisticated three-phase fluidi-zation system completely through these two methods. On the otherhand, unlike the two methods above, CFD can provide satisfactory nu-merical solutions and thus engineers can test various numerical designsand compare the solutions without realistic experiments to find outsome optimized proposals and provide more understanding of flow hy-drodynamics, heat and mass transfer in three-phase system. Therefore,the budget of realistic experiments could be significantly reduced,

which is desirable in order to help design, scale-up, and optimize ofgas–liquid–solid fluidized bed reactor [24]. Numerical simulations canalso reproducefluid flowvividly and provide deep insight into hydrody-namics of the multiphase flow in reactors. Hence, the CFD approach isconsolidating its status as an efficient and inexpensive tool to predictthe reactors' performances.

Numerous studies have applied the CFD approach to study two-phase flow phenomena [25–50]. For instance, our group [25–45] hasstudied various two-phase flow systems, such as polymerization,MTO, FCC systems. We investigated the multi-scale phenomena ofgas–solidflow in the reactors numerically and proposed a single particlemodel with considering the effect of intraparticle transfer on the flowfield. Moreover, a variety of aspects of two-phase flow hydrodynamicshave been investigated, such as particle growth and aggregation, pres-sure drop, bed expansion etc. We also incorporated chemical reactionsinto CFD model to predict mass and heat transfer in two-phase flow re-actors.Withmore CFD studies on two-phase flow, researchers could ob-tain a deep understanding of two-phase flow phenomena in reactors,which provides adequate information and studying foundation forCFD research of three-phase flow behaviors. Therefore, researchersbegin to put great effort on numerically investigating three-phase flowphenomena in reactors. It is noteworthy that many reviews have beenpublished concerning the numerical works of two-phaseflow in variouskinds of reactors [51,52] as well as the theoretical and experimentalworks of three-phase fluidization system in FBRs [53–60]. However, lit-tle effort is on reviewing the CFD approach for three-phase system inFBRs.

To overcome this gap,we summarize and analyze CFD simulations ofgas–liquid–solid multiphase flow behaviors in three-phase FBRs. Thisreview focuses on the fundamentals and applications of CFD approachfor the three-phase fluidization system published in the literature.With the emphasis on various multi-scale models and the interactionforces between individual phases, the fundamentals of CFD approachin the three-phase FBRs are discussed in Sections 2 and 3. The CFD ap-proach has been applied to simulate many aspects in three-phaseFBRs, of which themost significant one is the hydrodynamics of individ-ual phase at macroscopic or microcosmic levels. Section 4, which is onthe hydrodynamic phenomena of three-phase fluidization system,mainly discusses bubble formulation, bubble shape and size, gas andsolid holdup, particle entrainment etc. And the unique challenges ofCFDmodelingwork for three-phase fluidized in the near future are pre-sented in Section 5.

2. Multi-scale models of CFD approach for three-phase FBRs

In sciences such as physics, geography, astronomy, meteorology andstatistics, the term scale is used for describing or classifying the featuresand changes of objects (or phenomena) from spatial and/or temporalrange. The three-phase flow in FBRs that is inherently unstable and en-compasses nonuniform structures spanning awide range of lengths andtime scales is intrinsically multi-scale in nature, which mainly includesmacro-scale (reactor scale), meso-scale (like particle clusters, streamersor bubble swarms) andmicro-scale (particle or bubble) [61–63]. For ex-ample, Particle–fluid and particle–particle interactions at the micro-scale (1–5 particle diameters dp) result in meso-scale structures (10–100 dp) that affect macro-scale gas–solid flow behavior [63,138]. There

Nomenclature

A cross-sectional area of particle.CD,gl drag coefficient between gas and liquid phases.CD,gs drag coefficient between gas and solid phases.CD,sl drag coefficient between solid and liquid phases.CD,sl' modified drag coefficient between solid and liquid

phases.CD,ls' modified drag coefficient between liquid and a

suspended particle.ds particle diameter.D Initial bubble diameter.Ds Distance between the bubble edge and the symmetric

plane.Dw Distance between the bubble edge and the wall.Dcr Critical distance between two orifices for negligible

interaction.D0 Orifice diameter.en coefficient of normal restitution.FD,gl gas–liquid interphase drag force.FD,sl solid–liquid interphase drag force.FD,ls drag force acting on a suspended particle.Fsl solid–liquid interaction force.Fgs gas–liquid interaction force.Fls the forces acting on an individual particle from liquid.FAM Added mass force.FBA Basset force.Fab contact force between particle–particle.H height.I moment of inertia.J impulse.Jab impulse vector.k spring stiffness.Ksl momentumexchange coefficient between solid and liq-

uid phase.L length.Ll,Lg Magnus force for liquid phase and gas phase,

respectively.Nc number of particles in the controlled volume.mab effective mass.nab normal unit vector.p scalar pressure.ps surface pressure.pv gas phase pressure inside bubbles.r position.R,ID radius of fluidized bed.Re Reynolds number.U mean stream velocity.vl velocity of liquid phase.vg velocity of gas phase.vs velocity of solid phase.vp velocity of particles.vi velocity of particle i.vlr liquid velocity under radial direction.vgr gas velocity under radial direction.vab relative velocity at the contact point.Vp volume of a particle.W width.x position vector.t time.tab tangential unit vector.

Greek lettersβ0 coefficient of tangential restitution.

εl volume fraction of liquid phase.εg volume fraction of gas phase.εs volume fraction of solid phase.κ curvature of free surface.η damping coefficient.μl liquid dynamic viscosity.μ' dynamic friction coefficient.ν kinematic viscosity.ρ density.σ surface tension.τ viscous stress tensor; time.Ω computation domain.ω angular velocity.

Subscripts0 prior to collision.a,b particle indices.l liquid phase.g gas phase.s solid phase.ij cell indices.n normal direction.t tangential direction.

237H. Pan et al. / Powder Technology 299 (2016) 235–258

is no doubt that suchmulti-scale characterwill make numerical simula-tions a real challenge.

At present, there are mainly three different approaches applied inthe CFD technology to simulate multiphase flow systems: the Eulerianmethod, the Lagrangian method, and the direct numerical simulation(DNS)method. For the Eulerian approach, phases are treated as fully in-terpenetrating continuous phases and all the phases have same formconservation equations, in which additional closure relationships likethe stress and viscosity of the solid phase are required. Thus, theEulerian approach provides the macroscopic average quantities thatare of practical values to engineers on the basis of statistical descriptionsof multiphase flow can and hence can simulate device-scale systems orreactors [63]. However, closuremodels and constitutive relations are re-quired in this approach. On the other hand, the Lagrangian approachsolves the Newtonian equations of motion for the object of dispersedphase like each individual particle or bubbles considering the effectsof particle collisions and forces acting by other phases, which cantrack the trajectories of each individual particle and describe the dis-crete character of the solid phase at micro-scale or meso-scale by spec-ifying both a drag force closure and a collision model. The advantage ofthis approach is that it can account for particle-wall and particle–parti-cle interactions in a realistic manner. In the DNS approach, the grid sizeis commonly much smaller than the object size of the dispersed phase,and the moving interface can be represented by implicit or explicitschemes in the computational domain, whichmainly includes themov-ing-grid method, the grid-free method and the fixed-grid method [60].In order to predict interface position andmovement, themost common-ly applied methods in DNS approach are the front tracking [64–68] andfront capturing that belong to the fixed-grid method. The front tracking(FT) technique defines the interface and tracks its location explicitlythrough the Lagrangian makers on a fixed grid, which mainly includesthe maker-and-cell (MAC) method first proposed by Harlow andWelch [69] in 1965 and the immersed boundary method (IBM) intro-duced by Peskin [70] in 1977. The front capturing (FC) technique treatsthemoving and deformable interfaces implicitly by amaker function ona fixed grid, which includes the level-set method put forward by Osherand Sethian [71] in 1988, the volume of fluid (VOF) method presentedby Hirt and Nichols [72] in 1981 and the maker density function(MDF) raised by Kanai and Miyata [73] in 1995. In this approach, no

Fig. 1. Four common multi-scale models applied to simulate gas–liquid–solid FBRs atvarious time and length scales at present. (DPM: discrete particle method; FT: fronttracking; FC: front capturing).


closure models are required and the micro-scale information can becaptured. For example, the flow field around every particle is resolvedand the micro-scale information can be captured through IBM method,which can provide the insight and data to formulate improved correla-tions for higher scales.

As mentioned above, the multiphase flows manifest the multi-scalephenomena (macro-, meso- and micro-scales) in reactors, which pres-ent formidable modeling challenges, especially in three-phase FBRs.Thus, in order tomodel three-phaseflowat various spatial and temporalscales in the reactor under limited computing capacity, researchers al-ways select method for individual phase from three approaches aboveand then combine multi-scale models to describe three-phase system.With the increase of reactor scale, the Lagrangian or DNS approaches re-quire numerous computer resources to simulate flows behaviors. Al-though these two approaches can, in principle, be used to any lengthscale of simulation reactors, it has been almost applied to the experi-mental scale or micro-scale at present [74,75]. Specifically, the compu-tational cost of the Lagrangian or DNS approaches limits the size ofsystems or reactors that can be simulated. However, the DNS and La-grangian approaches required less empirical closure relations and thushave higher accuracy than the Eulerian approach. Hence, in order to ac-curately describe multi-scale flow behaviors in various length sizes ofthree-phase reactors under limited computer resources, we need tocome to a balance point so that the model becomes computationallyfeasible and at the same time we do not lose much information, withthe help of multi-scale modeling approach. Based on different numeri-cal modeling treatments (like the Eulerian, Lagrangian or DNS ap-proach) for individual phases in gas–liquid–solid three phase system,four common multi-scale models are always combined and applied to

Table 1Comparisons among four multi-scale models applied in CFD simulations for three-phase FBRs.

Multi-scalemodels

Applicationsconditions Advantages

Pseudo two fluidmodel [79]

Particles small enough, low particleloading

Less computer resources,reactors can be simulated

Three fluid model[80–82]

High particle loading, large-scale reactors Relative less computationsimulate industrial scale

Two fluid model+ DPM [88–90]

Dilute particle phase, small-scale reactors Particle–wall and particlecan be resolved

CFD + FT/FC +DPM [91–96]

Dilute particle phase, need to know thefluid around every particle or bubble

Can generate insight to inclosures for other models

simulate gas–liquid–solid FBRs at various time and length scales at pres-ent (See Fig. 1) and Table 1 compares these four multi-scale models indetails.

(a). Pseudo two fluid model: This model is based on the Eulerian ap-proach. Solid phase and liquid phase are treated as a pseudo-ho-mogeneous suspension phase with varying density and viscositywhich should be modified based on particle loading [76–79].Hence, three-phase system can be simplified as two-fluid system,which not only reduces numerical cost but also gets enough infor-mation in CFD simulation, especially for the three-phase systemwith small enough particles. For example, Feng et al. [76] investi-gated gas holdup, gas and liquid velocity in a lab-scale gas–liq-uid–TiO2 nanoparticle three-phase bubble columnexperimentally and numerically. In their work, the nanometersolid particles are small enough to behave similar to liquid mole-cules and the fine nanometer solid particles are dispersed in thewater uniformly. Accordingly, the Eulerian approach was used todescribe pseudo-homogeneous liquid–nanoparticle suspensionphase and gas phase in the gas–liquid–nanoparticle three-phasefluidization, which gets a successful prediction of local hydrody-namics of this three-phase system. The model is only suitable forsystems with sufficiently small particle and low particle loading.Moreover, the particles should uniformly dispersed in the liquidphase and have small slip velocity relative to the liquid phase.

(b). Three fluid model: In this model, all three phases are treated asmutually interpenetrating continua and interacting with eachother everywhere in the computational domain within a tripleEulerian approach. The pressure field is assumed to be shared byall three phases, in proportion to their volume fraction. Themotionof eachphase is governedby respectivemass andmomentumcon-servation equations [80–87]. Three fluid model needs the closurerelationships for interphase interaction that play significant rolesin achieving accurate results. Furthermore, solid pressure and vis-cosity are required to calculate by closure models, including thewell-known kinetic theory of granular flow (KTGF) and constantviscosity model (CVM). For example, Panneerselvam et al. [81]simulated the local hydrodynamics of a gas–liquid–solid three-phase FBR using three fluid model. The simulation results showedgood agreement with experimental data for solid phase hydrody-namics in term of mean and turbulent velocities and for gas andliquid phase hydrodynamics in terms of phase velocities and hold-up. Thoughpseudo twofluidmodel and threefluidmodel are bothable to simulate three-phase industrial scale reactor due to theirlittle computational cost, three fluid model get more attentionsthan pseudo two fluid because it is capable of predicting theoverall performance of three-phase systemwithout losing infor-mation of an individual phase. However, the drawback of thismodel is that too many empirical closures for interaction be-tween phases are required and constitutive relation for solidphase is necessary.

DisadvantagesScale (m)(at present)

Operationmodes

industrial-scale Particle information loss, lowaccuracy

~100–101 orlarger

Mode-2-a

al cost, able toreactors

Empirical closure laws andconstitutive relations arerequired

~100–101 Mode-1-a,Mode-2-a

–particle interactions Suffered from computationalintensive

~10−1–100 Mode-1-a

terphase interaction, high accuracy

Not practicable forindustrial-scale reactors atpresent

~10−2–10−1 Mode-1-a


(c). Two fluidmodel+DPM: In order to trace the trajectories of par-ticles in three-phase system, solid phase is tracked using La-grangian approach while gas and liquid phases are modeledthrough two fluid model [88–90]. In this model, gas phase andliquid phase are regarded as continuous phases which are con-sidered in Eulerian coordinates, while solid phase as the dis-persed phase is described in Lagrangian coordinates, themotion of which is directly calculated from the forces acting onthem. The interaction forces for particle–fluid, particle–particlecollision interaction should also be considered into this model.Therefore, the simulation results of this model are also stronglydepends on the closure drag model. Moreover, the Lagrangianapproach is computational intensive and thus this method is fea-sible for low particle loading or small scale problems.

(d). CFD + FT/FC + DPM: In this model, the front tracking (FT) orfront capturing (FC) approach such as VOF, level-set methodcombined with discrete particle method (DPM) are applied for,respectively, dispersed gas bubbles and solid particles in the con-tinuous liquid phase that is modeled by Eulerian approach[91–99]. Because the complicated and different treatments forevery phase, the closure laws for interaction forces become par-ticularly important in achieving good results. For example, Zhanget al. [91] employed the Eulerian volume-averaged method, theLagrangian DPM, and the VOF method to describe the motionof liquid, solid particles, and gas bubbles, respectively. They suc-cessfully described single bubble rising behavior as well as parti-cle entrainment in a small scale three-phase FBR. These kinds ofmodels are not practicable for industrial scale reactor at presentdue to the limitation of computational resources.

Apart from themulti-scalemodelsmentioned above for three-phasesystem, researchers proposed some other models to simulate three-phaseflow in reactors. For instance, Sivaguru et al. [100] combinedmix-ture model for liquid and solid phases and DPM for gas phase to predictthe hydrodynamics of three-phase fluidization system. Zhang et al.[101] simulated liquid–gas-solid flows in three-phase slurry reactors,where liquid phase is treated as continuumphase using the Eulerian ap-proachwhile gas and solid phases are treated as dispersed phases usingthe Lagrangian approach, which dramatically improve the numericalcost in CFD simulations. Moreover, the bubble–bubble and particle–bubble interaction forces should be considered into the model, whichalso increases the modeling difficulties. Therefore, few open

Fig. 2. Modes of gas–liquid–

publications are reported by using this model to simulate gas–liquid–solid flow phenomena in three-phase reactors. Some researchers appliedfront capturing approach (like VOF, level-set method) to track bubbleseparately and the immersed boundary method (IBM) to introduce theparticle–fluid interactions into the model for three-phase simulations[102–104]. The advantage of these models is that they can generate in-sight to interphase interaction closures for other models. In three-phasefluidized bed reactors, there are two operation modes for gas–liquid–solidfluidization based on the differences inflowdirections of gas and liq-uid and in contacting patterns between the particles and the surroundinggas and liquid, namely, cocurrent and countercurrent three-phase fluidi-zation [15,53]. The details are shown in Fig. 2. As described in Fig. 2, thecocurrent three-phase fluidization with liquid as the continuous phaseis expressed as Mode-1-a, while the cocurrent three-phase fluidizationwith gas as the continuous phase is expressed as Mode-1-b. For counter-current three-phase fluidization, Mode-2-a that regarded as inversethree-phase fluidization defines the fluidizationwith liquid as the contin-uous phase andMode-2-b defines the fluidizationwith gas as the contin-uous phase. The choice of CFD modeling approach for individual phaseflow in three-phase fluidized bed reactors is also generally influencedby the flow directions and contacting patterns among phases. Table 1summarizes multi-scale models applied for gas–liquid–solid fluidizationsystem in various length scales of simulated FBRs under various operationmodes. The fundamentals of three-phase flow are rather complicated be-cause of the simultaneous interactions between the individual phases,such as particle-gas interaction, particle-liquid interaction, and particle–particle collision, which also affect the macroscopic flow behaviors inthree-phase FBRs. Asmentioned above, the suitable closure laws for inter-action forces between phases also strongly affect results accuracy of vari-ous multi-scale models in simulating three-phase fluidization system.Hence, understanding themicroscopic fundamentals in terms of these in-teractions is of paramount importance to the formulation of strategies forprocess development and control. In the following, the fundamentals ofthe coupling interactions among three phases for gas–liquid–solid fluidi-zation in FBRs will be discussed.

3. Fundamentals of coupling interactions for three-phase system inFBRs

3.1. Coupling between gas and liquid phases

In the modeling of three-phase system in FBRs, various treatmentsare proposed for the coupling of gas–liquid phases based on different

solid fluidization [15].

Table 2Summary of various drag coefficient models.

Models Equations

Schiller and Naumann [109]CD;gl ¼ 24ð1þ 0:15Re

0:687Þ=Re Re≤10000:44 Re≻1000

�(2)

Grace model [110] CD;gl ¼ Maxf 24Reg ð1þ 0:15Reg0:687Þ;Minð83 ; 4gdgΔρ3U2Tρl Þgε

2l

(3)

UT ¼ μ lρl db M−0:149ð J−0:857Þ (4)

M ¼ μ l4gΔρρl2δ

3(5)

J ¼ 0:9H0:7521 2bH≤59:3

3:42H0:441 HN59:3

�(6)

H ¼ 43 EoM−0:149ð μ lμref Þ−0:14

; μref ¼ 0:0009 kgm−1s−1 (7)Tomiyama model [111] CD;gl ¼ Maxf ;minð 24Reg ð1þ 0:15Reg

0:687Þ; 72RegÞ; 83 EoEoþ3gε−0:5l (8)

Reg ¼ ρl jvg−vl jdgμ l (9)

Eo ¼ gðρl−ρg Þdg2

δ(10)

Ihme et al. [114] CD;gl ¼ 24Reg þ 5:48Reg 0:587 þ 0:36 (11)

Reg ¼ ρl jvg−vl jdgμ l (12)Morsi and Alexander model[115]

CD;gl ¼ a1 þ a2Reg þa3

Reg 2(13)

a1; a2; a3 ¼

0;24;0 0bReb0:13:690;22;73;0:0903 0:1bReb11:222;29:1667; ‐3:8889 1bReb100:6167;46:50; ‐116:67 10bReb1000:3644;98:33; ‐2778 100bReb10000:357;148:62; ‐47500 1000bReb50000:46; ‐490:546;578700 5000bReb100000:5191; ‐1662:5;5416700 Re≥10000

8>>>>>>>>>><>>>>>>>>>>:

(14)

Ishii and Zuber model [116] CD,gl = min(CDvis,CDdis) (15)

CvisD ¼ 24Re ð1þ 0:15Re0:75Þ (16)CdisD ¼ 23 dgffiffiffiffiffiffiffiffiffiffiffiδ

gjρl−ρg j

q (17)Dalla Ville [117] CD;gl ¼ ð0:63þ 4:8ffiffiffiffiffiffiRegp Þ2 (18)Ma and Ahmadi [118] CD;gl ¼ 24Reg ð1þ 0:1Reg

0:75Þ½ 1ð1−ð 1−αs0:64356Þ

1:6089 Þ� (19)

Mei and Klausner [119] CD;gl ¼ 16Reg f1þ ½ 8Reg þ 0:5ð1þ 3:315ffiffiffiffiffiffiRegp Þ�g (20)Grevskott et al. [120] CD;gl ¼ 5:645Eogþ2:835 (21)Kurose et al. [121]

CD;gl ¼16Reg

ð1þ 0:15Reg0:5Þ Reg ≥116Reg

Regb1

((22)

Lain et al. [122]

CD;gl ¼

16Reg

Reg ≤1:514:9

Reg 0:781:5≤Reg ≤80

48Reg

ð1− 2:21ffiffiffiffiffiffiReg

p Þ þ ð1:86� 10−15ÞReg4:756 80≤Reg ≤15002:61 Reg ≥1500

8>>>><>>>>:

(23)

Snyder et al. [123]

CD;gl ¼

24Reg

Reg ≤1

24Reg

ð1þ 3:6Reg 0:313

ðReg−119 Þ2Þ 1≤Reg ≤20

24Reg

ð1þ 0:15Reg0:687Þ RegN20

8>><>>:

(24)

Clift et al. [124] CD;gl ¼ 24Reg þ 316 Regb0:01CD;gl ¼ 24Reg ½1þ 0:1315Reg

ð0:82−0:05wÞ� 0:01≤Reg ≤20CD;gl ¼ 24Reg ½1þ 0:195Reg

ð0:6305Þ� 20≤Reg ≤260; logðCD;glÞ ¼ 1:6435−1:1242wþ 0:1558w2 260≤Reg ≤1500; logðCD;glÞ ¼ −2:4571þ 2:558w−0:929w2 þ 0:1049w3 1500≤Reg ≤1:2� 104; logðCD;glÞ ¼ −1:9181þ 0:6370w−0:0636w2 1:2� 104 ≤Reg ≤4:4� 104; logðCD;glÞ ¼ −4:3390þ 1:5809w−0:1546w2 4:4� 104 ≤Reg ≤3:8� 105w ¼ ; logðRegÞ

(25)(26)(27)(28)(29)(30)

Gidaspow model [128] See page 17Arastoopour [131] ð17:3Re þ 0:336Þ

ρgdp

��ug−upjð1−εgÞεg−2:8 (31)Syamlal and O’Brien [132] 3

4ð1−εg Þεgρg jug−up j

V2r dpð0:63þ 4:8

ffiffiffiffiVrRe

qÞ2

Vr ¼ 12 ða−0:06Reþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið0:06ReÞ2 þ 0:12Reð2b−aÞ þ a2

qÞ

a ¼ εg4:14

b ¼ 0:8εg1:28 ðεg ≤0:85Þ

εg2:65 ðεgN0:85Þ�

(32)(33)(34)

O’Brien and Syamlal [133]34ð1−εg Þεgρg jug−up j

V2r;cor dpð0:63þ 4:8

ffiffiffiffiffiffiffiffiVr;corRe

qÞ2

Vr ¼ 12 ða−0:06Reþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið0:06ReÞ2 þ 0:12Reð2b−aÞ þ a2

qÞ

Vr;cor ¼ ð1þmReð1−εgÞenÞVrn ¼ −0:005ðRe−5Þ2−90ðεg−0:92Þ2

m ¼ 250 ðGs ¼ 98kg �m−2 � s−1Þ

1500 ðGs ¼ 147kg �m−2 � s−1Þ�

(35)(36)(37)(38)(39)


Table 2 (continued)

Models Equations

Xu and Yu [134]34ð1−εg Þεgρg jug−up j

dpð0:63þ 4:8

ffiffiffiffi1Re

qÞ2εg−α

α ¼ 3:7−0:65 ; expð− ð1:5− ;logReÞ22 Þ

(40)

(41)

Nieuwland [135] 34ð1−εg Þεgρg jug−up j

dpCD0 f ðεpÞ

f ðεpÞ ¼ 10:997þ443:35εp−1733:42ε2p εpb0:1276f ðεpÞ ¼ 129:22 εp ≥0:1276

(42)(43)(44)

TGS [137] Fdðϕ;ReÞ ¼ Fdð0;ReÞð1−ϕÞ2 þ5:81ϕð1−ϕÞ2 þ 0:48

ϕ1=3

ð1−ϕÞ3

þ ð1−ϕÞϕ3Re½0:95þ 0:61ϕ3ð1−ϕÞ2� 0:01≤Rem ≤300

Fdð0;ReÞ ¼ 1þ 0:15Re0:687 Reb1000 ϕ : 5 values in ½0:1;0:5�

0:4424 Re ReN1000

�(45)(46)

HKL [139] αðϕÞ ¼ 0:03365ð1−ϕÞ þ 0:106ϕð1−ϕÞþ 0:016ð1−ϕÞ4 þ

δFRe 40bReb120;ϕ : up to 0:641

(47)

van der Hoef et al. [140] Fdðϕ;0Þ ¼ 180ϕ18ð1−ϕÞ2 þ ð1−ϕÞ2ð1þ 1:5 ffiffiffiffiϕp Þ (48)

BVK [141] Fðϕ;ReÞ ¼ 10ϕð1−ϕÞ2 þ ð1−ϕÞ2ð1þ 1:5ϕ1=2Þ 20bReb1000

þ 0:413Re24ð1−ϕÞ2 ½

ð1−ϕÞ−1þ3ϕð1−ϕÞþ8:4Re−0:3431þ103ϕRe−ð1þ4ϕÞ=2 � ϕ : 6 values in ½0:1;0:6�

(49)

Tang et al. [143] Fðϕ;ReÞ ¼ 10ϕð1−ϕÞ2 þ ð1−ϕÞ2ð1þ 1:5ϕ1=2Þ Re≤1000

þ ½0:11ϕð1þ ϕÞ− 0:00456ð1−ϕÞ4 ϕ : 11 values in ½0:1;0:6�þ ð0:169ð1−ϕÞ þ 0:0644ð1−ϕÞ4ÞRe

−0:343�Re

(50)

Zhou and Fan [144]FD ¼ ð1−ϕÞð1−1:7601ϕ

1=3 þ ϕ−1:5593ϕ2 þ 3:9799ϕ8=3−3:0743ϕ10=3Þ−1ðϕb0:2Þ2:812 þ 2:621ϕþ 47:99ϕ2 þ 16:99ϕ3 ð0:2≤ϕ≤π=6Þ

(

ðfor simple cubic arrays spheresÞ

FD ¼ ð1−ϕÞð1−1:7918ϕ1=3 þ ϕ−0:3292ϕ2Þ−1 ðϕb0:1Þ

2:118 þ 6:512ϕþ 76:22ϕ2−16:3:0ϕ3 þ 281:5ϕ4 ð0:1≤ϕ≤ffiffiffi3

pπ=8Þ

(

ðfor body‐centred cubic arrays spheresÞ

FD ¼ ð1−ϕÞð1−1:7917ϕ1=3 þ ϕ−0:3020ϕ2Þ−1 ðϕb0:1Þ

; expð0:6767þ 5:796ϕ−3:959ϕ2 þ 4:664ϕ3 ð0:1≤ϕ≤ffiffiffi2

pπ=6Þ

(

ðfor face‐centred cubic arrays spheresÞFD ¼ 9:9ϕ=ð1−ϕÞ

2 þ ð1−ϕÞ3ð1þ 3ϕ0:6Þ ðϕb0:55Þ5:87 ; sinððϕ=0:637Þ1:75π=2Þ=ð1−ϕÞ2 ð0:55≤ϕ≤0:637Þ

�ðfor random arrays spheresÞ

(51)(52)(53)(54)


approaches employed for individual phases in three-phase fluidizationsystem. As the CFD simulations of gas and liquid phases are performedin Eulerian framework, the interphase momentum exchange terms arecomposed of different interactions like drag force, added mass force,Basset force, and lift forces including Magnus force and Saffman force,inwhich drag force is always as the predominant interaction to describeinterphase momentum exchange for gas and liquid phases. Because liftforce should not be included as long as no clear evidence of its directionand magnitude is available [105], and added mass force can be presentonly when high frequency fluctuations of the slip velocity occur [106].Furthermore, compared with drag force, other forces are much smallerand insignificant in most cases [106–108]. The gas–liquid interphasedrag force is always expressed as follows:

FD;gl ¼ CD;gl34ρl

εgεldg

vg−�� vlj vg−vl� �: ð1Þ

CD,gl, the drag coefficient between gas and liquid phases, has beendescribed through a multitude of models (See Table 2: Eqs. (2)–(30)),many of which such as Schiller and Naumann [109], Grace model[110], Tomiyama model [111], and Ihme et al. [114] have been exten-sively used to calculate the drag force between gas and liquid phasesin three-phase fluidization system [79–82,88–90]. For instance,Panneerselvam et al. [80] used Tomiyama model [111] and Gracemodel [110] to predict the gas hold up profilewith low superficial liquidand gas velocities, indicating that the gas hold up predicted byTomiyama model [111] is closer to the experimental data than that byGrace model [110]. In general, the drag force determines the gas phaseresidence time and velocity of the bubbles. The increased drag force ob-tained by Tomiyama model [111] reduced the bubble velocity and thusincreased the bubble residence time, which results in an increase in thegas holdup. Hamidipour et al. [82] studied the sensitivity of various

mean bubble sizes on model predictions without considering bubbledeformation effects by applying Schiller and Naumann [109] dragmodel. For spherical bubbles without accounting for bubble deforma-tion, the most popular model has been introduced by Schiller andNaumann [109]. While different bubble shapes like sphere, ellipse andcap shape can be considered through Ishii and Zuber model [116]. Thedragmodels listed in Table 2were proposed from theoretical derivationor experimental data, which greatly influences the reliability of themodeling results. The perspective that computing a pair or all the inter-facial forces insteadof the only drag forceswill lead themodeling resultsto be more reliable is increasingly recognized. Huang [88], Wen et al.,[89,90] and Cao [80] all considered the drag force that was illustratedthrough the relations developed by Mitra-Majumdar et al. [113] (Eq.(55)) andMagnus lift force that wasmainly caused by the local rotationof the continuous phase and proposed by Delnoij et al. [112] (Eq. (56))for the interactions between gas and liquid phases:

FD;gl ¼ 0:75CD:glεlεg2ρldb

vg−�� vlj vg−vl� � ð55Þ

Ll ¼ −Lg ¼ 0:75εlεgρl vlr−vgr� � ∂vlr

∂z: ð56Þ

On the other hand, as the VOF method is employed to simulate themotion and the topological change of gas or liquid phase, the continuumsurface force (CSF) model [125] is often introduced to calculate the vol-ume force that is the interface force between gas and liquid phases. Inthe gas–liquid free surfaces, the stress boundary condition follows theLaplace equation as:

ps ¼ p−pv ¼ δκ ð57Þ


where, the surface pressure ps is the surface tension induced pressurejumping across the interface. The CSF model converts the surface forceinto a volume force within free surfaces and is shown as follows:

Fgl x; tð Þ ¼ δκ x; tð Þ∇F x; tð Þ ð58Þ

where, δ is the surface tension of the bubble interface; κ is the curvatureof the gas-free surface; and F(x,t) is a scalar volume fraction function offluid, where F(x,t)=1 in the liquid or liquid–solidmixture, 0 b F(x,t) b 1at the gas bubble surface, and F(x,t) = 0 in the gas bubble. Fan's group[91–95] have applied front capturing method (VOF method or Level-Set method) to simulate the gas phase in the gas–liquid–solid flow influidized bed reactors, the authors all employed CSF model to couplethe interaction between the gas and liquid phases. In Van SintAnnalandet al.’s work, the front tracking (FT) method was applied for dispersedgas bubbles in the continuous liquid phase [96]. The surface tensionforces were computed from the tensile forces on the three edges l ofall interface markers m (Fig. 3) and the volumetric surface tensionforce appearing in the momentum equation is calculated as:

Fδ xð Þ ¼ 1ΔxΔyΔz�X

mX

kρm;kD x−xm;k� �

δ tm:k⊗nm:kð ÞXmX

kρm;kD x−xm;k� � ð59Þ

where, tm .k is the tangent vector (or edge) shared by element m andneighboring element k and nm .k its unit normal vector.

3.2. Coupling between solid and liquid phases

The particles will interact with the surrounding fluid, liquid phase,which also generates various solid–liquid interaction forces, such asdrag force, pressure gradient force, virtual mass force, Basset force,and lift forces [126,127]. In Eulerian framework, the coupling of the in-terphasemomentumexchange between particle and fluid phases is alsomainly considered via drag force. Table 2 (Eqs. (31)-(44)) shows thecommonly used solid–liquid drag models, of which Gidaspow model[128] that combinesWen and Yumodel [129] used for dilute phase cal-culation and Ergun Equation [130] used for dense phase calculationwasa popular one in multiphase flow. Panneerselvam et al. [81] andHamidipour et al. [82] both employed Gidaspow model [128] to de-scribe the drag model that solved solid–liquid interaction force in

Fig. 3. Surface tension force exerted by three neighboring surface elements on the centralsurface element [96].

three-phase FBRs, which is expressed as:

Fsl ¼ FD;sl ¼ Ksl vs−vlð Þ ð60Þ

when εl ≤ 0.8, the solid–liquid exchange coefficientKsl is of the followingform:

Ksl ¼ 150εs 1−εlð Þvl

εlds2 þ 1:75

ρlεsds

vs−vlj j: ð61Þ

When εl N 0.8

Ksl ¼34CD:sl

εsεlρlds

vs−vlj jεl−2:65 ð62Þ

where, CD,slis the drag coefficient obtained as:

CD;sl ¼24

εlRes1þ 0:15 εlResð Þ0:687h i

Res ≤1000

0:44 ResN1000

8<: ð63Þ

where, the Reynolds number is defined as:

Res ¼ ρlds vs−vlj jμ l: ð64Þ

A wide range of solid volume fractions and Reynolds numbers areencounteredwithin the computational domain in this dragmodel, lead-ing to convenient usage in CFD simulations.

As the motion of a particle in the three-phase fluidized bed wasgiven by Lagrangian coordinates systemwith its origin set at the centerof the moving particle, the solid–fluid interactions become much morecomplicated, since the presence of other particle reduces the space forliquid to form a sharp liquid velocity gradient, which results in an in-creased shear stress on particle surface. In order to calculate interactionforces acting on liquid phase from all particles in the cell, at each timestep, the interactions of solid and liquid phases on individual particlesin a computational cell are calculated first, and then the values summedto produce the particle-liquid interaction force at the cell scale, henceliterature suggested that solid–liquid interaction force Fsl in three-phase FBRs was calculated as follows [88–90]:

Fsl ¼ FD;sl ¼XNci¼1

Ksl vi−vlð Þ ð65Þ

where, Ksl is still defined via the Gidaspowmodel [128] in these studies.However, unlike the drag coefficient correlation (See Eq. (63))employing in Eulerian system, the influence of the presence of sur-rounding particles was considered in terms of local voidage through ex-ponent function for drag coefficient CD .sl' in Lagrangian system. CD .sl' isdescribed as follows (Wen and Yu [129]):

C0D:sl ¼ CD;slεl−4:7 ð66Þ

where, CD,sl is still calculated by Eq. (63).Fan's group [91–105] and Sivaguru et al., [100] however, considered

Newton's third law of motion for coupling of the solid–fluid interactionin three-phase fluidization system. Unlike the fundamental of couplingof solid and liquid phases in the above Lagrangian framework, theforce from liquid phase to each particle is calculated separately accord-ing to individual-particle velocity, and Fsl is described as:

Fsl ¼X

FklsΔVkij

; xkp∈Ωij ð67Þ


where, subscript ijdefines the location of a computational cell,Ω andΔVare the domain and volume of this cell respectively, and Fls representsthe forces acting on an individual particle from liquid, which includesthedrag force, the addedmass force andBasset force andhas the follow-ing form:

Fls ¼ FD;ls þ FAM þ FBA: ð68Þ

The drag force acting on a suspended particle, FD,ls, is proportional tothe relative velocity between the phases and given as:

FD;ls ¼12C

0D;lsρA vl−j vp

�� vl−vp� � ð69Þwhere, A is the cross-sectional area of the particle to the direction of theincomingflow. CD,ls' is also composed of the drag coefficient for an isolat-ed particle and the impact of other particles as described in Eq. (66). Theaddedmass force induced by the fluid mass resistance that is moving atthe same acceleration as the particle is described as:

FAM ¼ 12ρVpddt

vl−vp� �

: ð70Þ

The Basset force accounts for the particle acceleration or decelera-tion in liquid and is obtained by Mei and Adrian [136]:

FBA ¼ 3πμdPZ t

0K t−τð Þd vl−vp

� �dt

dτ ð71Þ

where,K(t−τ) in Eq. (71) is expressed as:

K t−τð Þ ¼ π t−τð Þνr2p

" #1=4þ 1

2π

U þ νp−ν� �3rpν f

3H Reð Þ

t−τð Þ2" #1=28<

:9=; ð72Þ

f H Reð Þ ¼ 0:75þ 0:105Re ð73Þ

Re ¼ Udp=ν: ð74Þ

As discussed above, the coupling of interphase momentum ex-change between particle and fluid phases require closure models(mean drag models) that are typically presented from a combinationof theoretical and experimental studies (see Table 2: Eqs. (31)–(44)).It should be noted that traditional closure models from theoretical andexperimental approaches are limited by assumptions, experiment con-ditions, or measurement techniques [137], which then restricts the reli-ability and predictive capability of these models. Recently, particle-resolved direct numerical simulation (PR-DNS) as a first-principles ap-proach has become a promising alternative tool to develop accuratedrag models for particle–fluid interphase momentum transfer, inwhich the fluid flow is solving by imposing boundary conditions atthe particle surfaces. Hence, the interfacial force can be “measured” di-rectly for arbitrary material and flow conditions in the simulations toprovide the insight and data to formulate improved drag models,which overcomes difficulties in theoretical and experimental ap-proaches [138]. Some new particle–fluid drag models have been pro-posed on the basis of PR-DNS approach with Lattice-BoltzmannMethod (LBM) [139–141,143] and/or IBM [137,142,145], covering awide range of Re and ϕ (see Table 2: Eqs. (45)–(54)). For details of thedrag model development via PR-DNS, see Tenneti and Subramaniam[138]. It isworthmentioning thatMagnus lift force that ismainly causedby the fast rotation of the dispersed phase (solids) in solid–liquid flowhas also been considered to formulate a new correlation in Zhou etal.'s work [144,145], indicating that theMagnus lift force can be anotherpossible factor that significantly affect the overall dynamics of particle–fluid flows other than the drag force.

3.3. Coupling between gas and solid phases

Due to the intricacy of interaction forces among three phases, someresearchers did not take the interaction force between gas and solid intoaccount in CFD simulation for three-phase flows [146]. However, ac-cording to published studies, the interaction force between gas andsolid has to be considered, since solid particles in the vicinity of bubblestend to follow the bubbles' trajectory [87,113,147–149]. Hamidipour etal. [81] coupled gas and solid phases by a drag force coefficient model(Schiller and Naumann) similar to that of solid and liquid phases, eventhough gas and solid were two dispersed phases. They believed it wasreasonable to model the drag force between solid particles and bubblesin the same way as that between the continuous and the dispersedphase because the two dispersed phases are assumed to be continuain their simulation. Similarly, in Panneerselvam et al.'s work, Wen andYumodel [129]was applied tomodel the interaction betweendispersedbubbles and dispersed solids in three-phase FBRs. [81] Eq. (65) that re-searchers employed to obtain the solid–liquid interaction force has alsobeen used for the coupling of the gas–solid interaction force in Huang[88] andWen et al.’s works [89,90].The difference was that the drag co-efficient using for interaction forces between gas and solid phases in Eq.(65)was the correlation developed by Jean and Fan [150]while the dragcoefficient applying for interaction forces between solid and liquidphases was the correlation presented byWen and Yu [129]. The formu-lation of Jean and Fan [150] is shown as:

CD;gs ¼ 11þ 1:147Re−2:876s ds2R

� �−4:0 : ð75Þ

In Fan and co-workers' CFD simulations of three-phase fluidizationsystem [91–95], they introduced a bubble induced force model (BIF)to calculate the interaction forces between gas and solid phases becausethe size of the computational cell is larger than the thickness of the filmof the gas–liquid interface:

Fgs ¼ Vpδκ x; tð Þ∇F x; tð Þ: ð76Þ

They demonstrated that when particlesmove close to the gas–liquidinterface, the surface tension force would act on the particles throughthe liquid film. If particles overcome this bubble-induced force, the par-ticles would penetrate into the bubble surface and the bubble surfacecan be broken at any time upon contact owing to particle penetration.The bubble may recover its primary shape upon the penetrating of par-ticle when the particle is small enough [151], otherwise, the bubblebreakage may take place.

3.4. Analysis for particle–particle collision

The contact force between two particles relating to many geometri-cal and physical factors such as the shape, material properties andmovement state of particles is very difficult to describe [152] in numer-ical simulation. Due to its complication, extensive effort has been putinto the development and further optimization for the particle–particlecollision simulationmodel. In general, the modeling approaches involv-ing the treatment of particle–particle collision through the Lagrangianmethod can be roughly divided into two types: hard-sphere approachand soft-sphere approach.

The hard-sphere approach was first proposed by Alder and Wain-wright [153] so as to investigate phase transitions in numerical simula-tion for molecular systems. Subsequently, Campbell and Brennen [154]applied the hard-sphere approach to simulate granular dynamics. Inthis approach, it is assumed that the spherical particles interact throughbinary, quasi-instantaneous collisions where contact takes place at apoint and are impulsive. These collisions are processed in sequence onthe basis of the order where the events occur. The interaction times

Fig. 4. The coordinate system used in the description of the discrete particle collision models [155].


are much smaller than the free flight times; therefore, the hard-sphereapproach is considered as event-driven. The coordinate system andequations [155] applied in the hard-sphere approach for two collidingspherical particles are illustrated in Fig. 4 and Table 3, respectively.Ever since the hard-sphere approach has been adopted by Campbelland Brennen [154] for granular system in numerical simulation, hard-sphere models have been employed extensively in various gas–solidtwo-phase systems [156–162].

The soft-sphere approach was originally presented by Cundall andStrack [163] and first adopted for gas-fluidized bed by Tsuij et al.[164]. In this approach, it is assumed that particles are experiencing de-formation when interacting with each other, where the contact forcesare calculated from the deformation history of the contact within a con-tact force scheme. Unlike the hard-sphere approach, the interactiontimes are more than free flight times in the soft-sphere approach,which makes the soft-sphere approach typified as time driven. More-over, a fixed time step is used in soft-sphere approach, and particlesare allowed to overlap slightly. The coordinate system and equations[155] for soft-sphere approach are shown in Fig. 5 and Table 3, respec-tively. The soft-sphere approach has been extended and further devel-oped in describing particle–particle collisions for gas–solid systems[165–167].The soft-sphere models found in publications were mainlydifferent from each other with regards to the contact force scheme[155] and Schäfer et al. [168] reviewed a variety of popular schemesfor repulsive inter-particle forces.

Table 3Comparison of particle–particle collision models.

Models Normal interaction

Hard-sphere approach [155] nab ¼ ra−rbjra−rb j (77)where vab = (va−vb) ‐ (Raωa + Rbωb) × nab

ma(va−va,0) = −mb(vb−vb,0) = JIaRaðωa−ωa;0Þ ¼ − IbRb ðωb−ωb;0Þ ¼ −nab �

Jn = −(1 + en)mab(vab,0 ⋅ nab)

Jt ¼− 27 ð1þ β0Þ �mabðvab;0 � tabÞ if μ 0 J‐μ Jn if μ 0 Jn≺

�J = Jnnab + Jttab

Soft-sphere approach [155] nab ¼ ra−rbjra−rb j (85)Fab,n = −knδ'nnab−ηn(vab ⋅ nab)nab (86)

δ'n = Ra + Rb− |rb−ra|(87)

ηn ¼−2 ; ln en

ffiffiffiffiffiffiffiffiffiffimabkn

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ2þ ; ln 2en

p if en≠02

ffiffiffiffiffiffiffiffiffiffiffiffiffimabkn

pif en ¼ 0

8<: (88)

As discussed above, the particle–particle collisions are independentof the local flow field and the shear force is always neglected in gas–solid flow system. Nevertheless, in gas–liquid–solid three-phase fluidi-zation system, as two particles move close to each other, especiallywhen the distance between two particles is less than 0.1dp, the liquidshear influence greatly affects particle–particle collisions in fluidizationbehaviors [169]. Li et al. [91] firstly incorporated close-distance interac-tion (CDI)model (Details see Reference [169]) presented by Zhang et al.into the hard-spheremodel for particle–particle collision to consider theliquid interstitial impact on colliding particles in three-phase FBRs. Sub-sequently, a similar model was reported by Fan's group [91–95] for de-scription of the particles in three-phase fluidization system. Van SintAnnaland et al. [96] also combined the CDI model with the hard-sphereapproach to describe particle collisions and the transport of particles inthe wake of a single bubble rising in a liquid.

3.5. Summary and discussion

The CFD simulations of three-phase fluidization systems can be clas-sified into three categories: the Eulerian approach, the Lagrangian ap-proach and the DNS approach. Several factors contribute to the choiceof these three approaches in three-phase FBRs, including reactor lengthscale, operation modes, computational cost, accuracy and required in-formation. For instance, gas phase is treated as continuous or dispersedphase in different operation modes (Fig. 2). If treated as continuous

Tangential interaction

tab ¼ vab;0−nab �vab:0jvab;0−nab �vab:0 j (78)(79)(80)

J (81)

(82)

n ≥ 27 ð1þ β0Þ �mabðvab;0 � tabÞ27 ð1þ β0Þ �mabðvab;0 � tabÞ

(83)

(84)

tab ¼ vab−ðnab �vabÞnabjvab−ðnab �vabÞnab j (89)

Fab;t ¼ ‐ktσ0t‐ηtvab:t if jFab;tj≤μ 0 jFab;nj

‐μ 0 jFab;n jtab if jFab;tj≻μ 0 jFab;nj�

(90)

vab .t = vab−vab,n (91)

ηt ¼−2 ; lnβ0

ffiffiffiffiffiffiffiffiffiffiffi27mabkt

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ2þ ; ln 2β0

p if β0≠0

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi27mabkt

qif β0 ¼ 0

8><>:

(92)

Fig. 5. The coordinate system used in the soft-sphere approach.


phase, gas phase is often simulated by the Eulerian approach while it issimulated by the Lagrangian or the DNS approach to get accurate andadequate information as dispersed phase. In general, none of model issuitable for all the problems encountered in multiphase systems.These three approaches have their own advantages and disadvantages,and researchers should select suitable one for specific situations andcombine various multi-scale models to simulate three-phase systemsof various length scales reactors on the compromise of computationalcost, required information and accuracy.

Coupling of the phases in three-phase fluidization system is rathercomplex due to its various interaction forces between phase, like dragforce, lift force, added mass force, Basset force, etc. The most universalmethod for coupling gas and liquid phases, solid and liquid phases,and gas and solid phases is solving drag force without consideringother insignificant forces, especiallywhen all the three phases are treat-ed in Eulerian framework. The only difference among these three cou-plings is the various drag coefficient models applied.

On the other hand, as the motion of gas phase is calculated throughthe DNS approach, like VOF and Level-set method, the coupling of gasand liquid phases is described by CSF model and the coupling of gasand solid is often through BIF model. When solid phase is treated inthe Lagrangian coordinate, the fundamentals of coupling of solid andliquid phases can be divided two schemes [170] in the previous CFDsimulation publications of three-phase FBRs. One scheme is that theforce from liquid phase to each particle is calculated separately accord-ing to individual-particle velocity; the other one is that at each timestep, the interactions of solid and liquid phases on individual particlesin a computational cell are calculated first, and then the values summedto produce the particle-liquid interaction force at the cell scale. The col-lisions between two separate particles are always described throughhard-sphere approach or soft-sphere approach. Unlike the collisionmodel employed in two-phase simulation, the liquid shear effect onthe collisions between two separate particles is added into the hard-sphere model to calculate particle–particle collisions in three-phase flu-idization system.

4. Applications for CFD simulations in three-phase FBRs

Understanding the hydrodynamics characteristics of three-phasesystems is crucial in optimization and control of three-phase FBRs oper-ations, particularly in the scale-up of small scale reactor behavior tolarger ones in industrial and commercial three-phase FBRs. Hence, hy-drodynamics characteristics of three phases have been a researchfocus in simulating three-phase FBRs phenomena in the past decades

[79–82,88–96,100], especially the gas phase hydrodynamics that great-ly affect the inherent transport and fluidization of three-phase FBRs.

4.1. Gas hydrodynamics

Bubble behaviors that obviously affect many unique properties inthree-phase FBRs have been studied by numerous researchers, such asbubble formulation, bubble rise velocity, size and shape, bubble wake,and gas holdup.

4.1.1. Bubble formulationTo have comprehensive understanding of bubble formulation in

gas–liquid–solid fluidization, Li et al. [94] proposed a numerical tech-nique to describe two-bubble formulation behavior in liquid and liq-uid–solid fluidized beds and the simulation results were verifiedthrough experimental and theoretical models. The typical bubble for-mulation process consists of expansion, detachment, separation and ris-ing (Fig. 6). They investigated the pressure impact on bubbleformulation process in liquid suspensions through two far enough ori-fices (35 mm) to avoid interaction between bubbles. It was found thatbubble formation process was not sensitive to system pressure withinconstant flow conditions. The inlet gas momentum increases and thebubble surface tension decreases with pressure increasing, whichhelps the detachment of the bubble. Yet the liquid viscosity increasesand buoyancy force decreases with pressure increasing, which hindersthe detachment of bubble. Consequently, the overall effect of systempressure on bubble formation is inconspicuous, which has also beenproved by Luo et al. [171] and Yang et al. [172]. The distance of two or-ifices plays an important role on two-bubble formation process becausebubble interaction occurs as the two orifices are located closely. Li et al.concluded that if the orifices were close enough, two forming bubblesinduced a high liquid velocity that leads to a lower pressure in the re-gion near the center of two orifices, which make bubbles draft towardthe center area rather than rise rectilinearly around original directionbased on the numerical simulation that two-bubble formation in a fluid-ized bed with an orifice separation distance of 20mmwasmodeling. Inorder to get quantitative sights on the influence of orifices distance onbubble formation process, they presented a simple analytical criterion(Fig. 7) to determine the exact distance between orifices to avoid the ig-norance of the bubble interaction during formation process. In this cri-terion, Ds represents the distance between the bubble edge andsymmetric plane, and Dcr is the distance between two orifices. WhenDs is smaller thanD, the bubble interaction can be negligible on the con-dition thatDcrwould be 3D. WhenDs is larger thanD, the interaction be-tween two orifices would not happen [94].

Fig. 6. Simulation results of multi-bubble formation in liquid–solid suspensions (εp =0.223, P = 6.6 Mpa) [94].

Fig. 7. Schematic of the analytical criterion for nozzle interaction effect [94].


Bubble formation is drastically affected by the absence and presenceof particles. Inertial force that impedes the bubble detachment acting onforming bubbles increases because of the presence of particles, hence,bubble formation process takes much longer at the existence of parti-cles, which further influences bubble diameter. According to Li et al.’sresearch [94], the initial bubble diameter in the three-phase flow waslarger than that in the gas–liquid flow. As the solids holdup increasedfrom 0 to 23%, the bubble diameter increased about 14%, which agreed

well with the experimental findings and empirical model predictions.Apart from the effects of solids concentration on bubble formulationprocess discussed above, the solids concentration also influences theconnection between forming bubbles. A similar study was performedby Chen et al. [95] who also disclosed the bubble formation processunder different solid concentrations in air-Paratherm-solid fluidizationsystem with solid particles of 0.08 cm in diameter and 0.896 g/cm3 indensity (Fig. 8). Their results suggested that at the initial period of0.3 s, there was almost no difference in bubble formation processesamong these cases, and the difference that first bubble was not wellconnected to the second bubble became obvious after 0.4 s, especiallyfor the high solids concentration cases.

4.1.2. Bubble rise velocity, size and shapeIn three-phase fluidization system, a variety of physical properties

and system parameters influence bubble rise velocity and shapes, suchas the surface tension and viscosity of the liquid, densities of liquidand particles, solid holdup, bubble size and system pressure and tem-perature. Zhang et al. [92] studied a single bubble rise velocity in thetwo-dimensional three-phase fluidized bed through a CFD approachcombining VOF and DPM models. The simulation was performed on3 × 8 cm2 domain within 20 × 50 grid points, in the center located1.5 cm above the bottom of which was positioned a 1.0 cm diameterspherical bubble initially. Fig. 9 revealed snapshots of the simulationsand experimental results of a single bubble rising in the bed at varioustimes. Quantitatively, the relative deviations between the simulatedand experimental bubble rise velocity have also been demonstrated inFig. 10, the magnitude of which is similar with that reported by GraceandHarrison. Subsequently, further study on the impact of system pres-sure on bubble rise velocity investigated by Zhang et al. [93] has beencarried out in three-phase FBRs. The results showed that bubble rise ve-locity decreasedwith the increasing system pressure, which significant-ly improved gasholdup of the three-phase FBRs at elevatedpressures. Inaddition, they also found the bubble rising trajectory was greatly affect-ed by the change of flow parameters. The trajectories of the rising bub-ble with various pressures and solid holdups were compared in Fig. 11.It has been observed that low pressure and high solid holdups contrib-uted more stability for the trajectory of the bubble rising.

Fig. 8. Simulation results of bubble formation and rising in ParathermNF heat-transfer fluid with andwithout particles (Nozzle size 0.4 cm I.D., liquid velocity 0 cm/s, gas velocity 10 cm/s,and particle density 0.896 g/cm3. (a) No particle; (b) 2000 particles; (c) 8000 particles; and (d) 8000 particles) [95].


During bubble rising in three-phase flow system, bubble shape al-ways changes due to the breakages or/and coalescence of bubbles,which further results in the change of bubble size. Additionally, bubblesize is of crucial importance to the overall gas holdup in three-phase flu-idization system. Panneerselvam et al. [81] developed a three-dimen-sional Eulerian model to simulate the local hydrodynamic of a gas–liquid–solid FBR. However, they did not consider the effect of bubblebreakup and coalescence on bubble size distribution and assumed amean bubble size in the simulation. Three different bubbles sizes (5,13 and 17 mm) were applied to investigate the gas holdup profilesalong the radial direction (Fig. 12). The results showed that the averagegas holdup ofmean bubble sizes of 13 and 17mmwaswellmatching Yuand Kim's experimental data [173]. Similarly, Hamidipour et al. [82] alsostudied the effect of mean bubble size on various flow parameters, in-cluding gas holdup, axial gas velocity and solid viscosity, without con-sidering bubble breakup and coalescence. Bubble shape is alwaysdefined in terms of the aspect ratio, h/b, represented as the ratio ofthe minor axis (vertical) over the major axis (horizontal) of the bubble.Zhang et al. [93] simulated a series of bubble shape changes for a bubblewith diameter of 10.0 mm rising in three-phase fluidization (Fig. 13).According to Luo et al., [174] a single bubble owns maximum stablesize (detail calculated model see reference [87]) in fluidization systemunder certain conditions, beyond which the bubble begins to break inthe rising. The figure shows that the shape of the single rising bubblechanges intensively and finally is broken into three parts, of which thelargest bubble that rises without further breakage almost has an equiv-alent diameter of 9.0mm that is consistentwith Luo et al.’smodel [174].Chen et al. [95] further investigated the shape and size changes of mul-tiple bubbles rising in ParathermNF heat-transferfluid on different inletgas velocities (Fig. 14). They confirmed that bubble coalescence and

breakage took place easily near the nozzles and free surfaces, respec-tively, which came to the result that bubble shapes are not as sphericalas in other areas; therefore, bubble size distributionwas nonuniform es-pecially after bubbles traveled for a distance. Moreover, high inlet gasvelocity enhances bubble coalescence and breakage.

4.1.3. Gas holdupGas holdup as one of the most vital flow properties has been exten-

sively studied by various researchers through means of theoretical cor-relations, experimental technique and numerical simulationmainly dueto its engineering meaning in industrial FBRs. Panneerselvam et al. [81]compared time averaged gas holdup profile along the dimensionless ra-dial direction between CFD simulation and the experimental results atthe axial position of 0.325 m (Fig. 15). The gas holdup profile agreeswell with experimental data at the center area while slightly varies atthewall area. In addition, the gas holdup profile decreaseswith increasein dimensionless radial position. However, a deeper research on gasholdup in three-phase fluidization system has been studied by Huang[88]who surveyed the detail influences of gas superficial velocity, liquidsuperficial velocity and bed diameter on local gas holdup in thefluidizedbed through experiments and numerical simulations. The radial distri-bution of local gas holdup, higher at the center and lower near thewall, was not uniform from both experimental and numerical results.It was found that the local radial nonuniformity of gas holdup decreasedwith the increase in liquid superficial velocity as well as bed diameterbecause the increment in liquid superficial velocity enhanced the turbu-lence of gas phase, which was beneficial to gas dispersion. However,with increasing of gas superficial velocity, local gas holdup was higherand the radial distribution of local gas holdup tended to be more non-uniform. Similar results that the effect of gas and liquid superficial

Fig. 9. Simulated and experimental results of a bubble rising in a liquid–solid fluidized bed [92].


velocities on radial distribution of local gas holdup in three-phase FBRswere reported byWen et al. [89,90] and Cao [80]. Besides, Cao [80] alsoinvestigated other vital parameters for the radial distribution of local gasholdup, liquid viscosity, which confirmed that higher liquid viscosity re-duced the radial nonuniformity of local gas holdup and improved localgas holdup.

4.2. Solid hydrodynamics

4.2.1. Solid velocityHuang [88] briefly predicted the axial solid velocities under various

bed diameters, gas and liquid superficial velocities in a three-phase flu-idized bed. The simulation results showed that both higher gas and

Fig. 10. Comparison of the simulation and experimental results of the bubble rise velocity[92].


liquid superficial velocities increased the radial solid velocity. And withthe improvement of bed diameter, the nonuniformity of axial solid ve-locity along radial directions was more obvious. Recently,Panneerselvam et al. [81] comprehensively described solid velocitieson axial and radial directions in a 3-D three-phase FBR CFD model. Forthe axial solid velocity profile, it was reported that the discrepancythat the axial solid velocity was higher in the central area (positive)and lower near thewall area (negative) led to a solid circulation pattern

Fig. 11. Bubble rising trajectory at different pressures and solids holdups. (a) p=0.1Mpa, εs=00.545, dB = 7.5 mm [93].

in gas–liquid–solid fluidization system, which was consistent with ex-perimental works even for the bed height of flow reversal occurring.Moreover, they discussed the impact of gas superficial velocity on thetime and spatially averaged the axial solid velocity. They found thatthe axial solid velocity increased with increasing the gas superficial ve-locity. This is because the increment in gas superficial velocity results incoalesced bubble flow regime that increases the axial solid velocity.Contrarily, dispersed flow regime under lower gas superficial velocitywill decrease the axial solid velocity. For the radial solid velocity, its dis-tribution was relatively uniform and themagnitude of axial solid veloc-ity was much higher than radial solid velocity.

4.2.2. Solid holdupTheCFD approach has been employed to study solid holdup in three-

phase fluidized bed reactors, whichmainly focuses on discussing the in-fluence ofmacroscopic factors on solid holdup, like gas superficial veloc-ity, liquid superficial velocity and bed diameter. As an example, Cao [80]experimentally and numerically analyzed the local solid holdup of athree-phase fluidization system under various gas superficial velocitiesand liquid superficial velocities. They concluded that the radial distribu-tion of local solid holdup is heterogeneous, lower at the central regionand higher near the wall region, which is contrary with that of localgas holdup as discussed before. As increase of the gas superficial veloc-ity, the local solid holdup increased slightly and the peak radial solidholdup disappeared, so better homogeneous radial distribution of thesolids holdupwas observed. However, with increase of the liquid super-ficial velocity, the local solid holdup became less and the radial nonuni-formity of local solid holdup was more apparent. Wen et al. [89,90] alsoanalyzed local solid holdup in a three-phase FBR through Two fluidmodel + DPM multi-scale model. The similar results of the effect ofgas superficial velocity and liquid superficial velocity on local solid

.384, dB=7.5mm; (b) p=17.3Mpa, εs=0.384, dB=7.5mm; and (c) p=17.3Mpa, εs=

Fig. 12. Effect of mean bubble size on the averaged gas holdup at gas superficial velocity of 0.04 m/s and liquid superficial velocity of 0.06 m/s [81].


holdup are found. Apart from these two factors, Huang [88] also simu-lated the local solid holdup under three reactor diameters, which indi-cated that larger reactor can enhance the radial uniformity of localsolid holdup due to the improvement of turbulent intensity that helpsparticles to disperse uniformly with increasing the reactor diameter.

4.2.3. Particle entrainmentAs discussed above, fluidized bed reactors have been extensively ap-

plied inmany industrial processes owing to their greatmerits: easy con-trol, high mass and heat transfer, uniform particle mixing andtemperature gradients, etc. However, there are still some drawbacksthat must be considered for fluidized bed reactors. One of the most per-sistent problems for fluidized bed reactors designer is particle entrain-ment that particles are always captured and then carried out ofreactors by the fluid, which will result in the loss of the yield powderor catalytic. In addition, the particles carrying with the fluid have to beseparated though equipment, which also increases the investment and

Fig. 13. The simulated sequence of bubble shape change and the bubble breakage

operation difficulties. Therefore, a comprehensive understandingof par-ticle entrainment in three-phase fluidized bed reactors is essential. Be-cause of this, in the CFD simulations of three-phase FBRs, particleentrainment has been analyzed qualitatively and quantitatively. Li etal. [91] and Zhang et al. [92] both simulated particle entrainment witha single rising bubble in three-phase fluidization system. They qualita-tively described the bubble emerging from the bed free surface andthe evolution of particle flow around a single bubble. In Zhang et al.’swork, the computational domain was 6 × 12 cm2 with 32 × 80 gridpoints and a spherical bubblewith a diameter of 0.8 cmwas initially im-posed at 1.5 cmabove thebed bottom. 1000 particles of glass beadswitha density of 2500 kg/m3 and a diameter of 0.7 mm is used as the solidphase and water is used as the liquid phase within 7.5 cm/s velocity.The simulation results showed a group of particles were dragged bythe bubble wake in the subsequent frames (Fig. 16). Further investiga-tionswere performed by Van Sint Annaland et al. [96]with the objectiveto characterize the differences of particle entrainment between a single

in a liquid–solid medium. (p = 17.3 Mpa, εs = 0.384, dB = 10.0 mm) [93].

Fig. 14. Simulation results of bubble formation from four nozzles and rising in Paratherm NF heat transfer fluid (nozzle size 0.4 cm I.D. and nozzle gas velocity (a) ~6 cm/s; (b) ~10 cm/s)[95].


rising bubble and multiple rising bubbles. They developed an equationto calculate the volume of drift zoneXd due to particle entrainment phe-nomena, which was shown as follows:

Xd ¼NdNp

VsVb

ð93Þ

where, Vs is the initial volume occupied by the suspension. Np is thenumber of suspended solid particles and Nd is the number of particlesin drift zone. Vd and Vp are bubble volume and particle volume, respec-tively. They compared the particle drift profiles at t=0.30s for the case

Fig. 15. Radial distribution of gas-phase holdup at liquid superficial liquid velocity Ul = 0.06

where five bubbles were present with the case where only the centralsingle bubble was present (Fig. 17), which revealed the isolated singlebubble carrying particles rose faster through the liquid than that offive bubbles because of the retarding downward flow of the displacedliquid owe to the presence of surrounding bubbles. Moreover, the com-parison of the volume of the drift zone as a function of time for the twocases above, calculated by Eq. (93), were also shown in Fig. 18 which il-lustrated that the volumeof drift zone of particle entrainment for an iso-lated single bubble was higher than that of five bubbles due to thecompetition with each other for the particles drifting behind themwhile they rose and carried particles through the suspension.

m/s and gas superficial liquid velocity Ug = 0.04 m/s at axial position of 0.325 m [81].

Fig. 16. Simulation of a bubble emerging in a liquid–solid fluidized bed [92].


Fig. 17. Comparison of the computed particle drift profiles at t = 0.30 s for a system with five bubbles (left) and a system with only the central bubble (right) [96].


4.3. Liquid hydrodynamics

The Applications of CFD approach in three-phase fluidized bed FBRsfor liquid hydrodynamic mainly focus on the study of liquid velocity. Asdiscussed above, Panneerselvam et al. [81] described solid hydrody-namics in a cylindrical pexiglas column with diameter of 0.1 m andheight of 1.5 m by CFD simulations, whose results were validated by

Fig. 18. Comparison of computed particle drift profiles for a bubble rising in isolation(closed circles) and a bubble rising together with four surrounding bubbles (opencircles). In both cases the bubbles rise through a layer of 60,000 particles [96].

the experimental data of Kiared et al., while they studied liquid hydro-dynamics in another cylindrical pexiglas column with diameter of0.254 m and height of 2.5 m, of which the simulation results were ver-ified by the experimental data of Yu and Kim [175].The comparison ofthe time averaged liquid velocity profile along the radial direction be-tween simulation and experimental results was illustrated in Fig. 19,which shows close matching although slight difference is found at thecentral area where the maximum liquid velocity exists. They alsofound a reversal flow near the wall region of the bed. Wen et al. [89,90] have carried out CFD research on axial liquid velocity and confirmedthe radial nonuniformity in a three-phase FBR. Additionally, the localaxial liquid velocity increased with the improvement of superficial gasvelocity and the radial distribution were more nonuniform. Acircumfluence flowwas observed near thewall where the local axial liq-uid velocity was negative. Huang [88] has extended this study and de-termined the effect of reactor diameter on the radial distribution ofthe local axial liquid velocity by means of CFD simulations and experi-ments, indicating that the nonuniformity is more prominent with theincrement of reactor diameter due to the weaker turbulence intensityof the flow.

4.4. Summary and discussion

Researchers have applied the CFD approach to study various aspectsin three-phase FBRs, of which hydrodynamics of individual phase areknown as important fundamental characteristics and surveyed a lot inthe publications. For gas hydrodynamics in three-phase fluidization sys-tem, CFD approach has successfully described bubble formulation pro-cess in a three-phase fluidized bed, which is strongly affected bysystem pressure, orifice distance, and solid concentration. Bubble rise

Fig. 19. Radial distribution of liquid velocity at liquid superficial liquid velocity Ul = 0.06 m/s and gas superficial liquid velocity Ug = 0.04 m/s [81].


velocity has been investigated qualitatively and quantitatively, whichshows that system pressure and solid concentration also have great in-fluence on bubble rise velocity. As bubble rises in three-phase flow, thebubble shape and sizewill change due to bubble coalescence and break-age,which further results in the nonuniformity of bubble size anddistri-bution. Gas holdup influenced bymany factors in three-phase FBRs, likegas and liquid superficial velocities, liquid viscosity and bed diameter, ishigher at the center and lower near the wall. The simulation resultsshow that with increase in liquid superficial velocity, liquid viscosityand bed diameter the radial direction of local gas holdup becomesmore uniform while the increment in gas superficial velocity decreasesthe radial uniformity of the local gas holdup distributionwithin increas-ing local holdup.

For solid hydrodynamics in three-phase fluidization system, solidsuperficial velocity and solid holdup have been studied by discussingthe impact of parameters as used in gas hydrodynamics. The radial dis-tributions of the axial solid velocity and solid holdup are heterogeneous.The nonuniformity velocity distribution shows a solid circulation pat-tern in the three-phase FBR, which is higher at the central area andlower near the wall area. The high gas superficial velocity has positiveinfluence on axial solid velocity and solid holdup as well as their radialdistributions. However, larger bed diameter increases the uniformity ra-dial distribution of solid holdupwhile decreases that of axial solid veloc-ity. The high liquid superficial velocity has negative impact on solidholdup and its radial distribution. Additionally, particle entrainment asthe most concerned engineering problem has been simulated by a sin-gle rising bubble and multiple rising bubbles, which illustrates that theisolated single bubble carrying with particles rises faster through theliquid than that of five bubbles because the presence of surroundingbubbles retards the downward flow of the displaced liquid. Moreover,a quantitative equation is proposed to calculate the particle drift zonedue to particle entrainment phenomena.

For liquid hydrodynamics in three-phase fluidization system, CFDsimulation has been mainly focused on investigating liquid velocity.The local axial liquid velocity is negative near the wall, indicating acircumfluence flow in three-phase FBRs, which is consistent with theexperimental result. Researchers have numerically and experimentallydiscussed the time averaged liquid velocity profile. With the gas super-ficial velocity increasing, the axial liquid velocity increases and its radialdistribution are more nonuniform. Similarly, the improvement of reac-tor diameter has negative effect on the radial distributions of axial liquidvelocity.

As discussed above, CFD has been used intensively to studymany as-pects in three-phase FBRs, which provide precious insights into thecomplicated flow behavior of three-phase fluidization system. Howev-er, these applications of CFD simulations for three-phase FBRs are all

performed based on cold models without taking chemical reactionsinto account. Moreover, the most of CFD simulation objects are basedon experimental scale plants (Table 1) which to some extent hinder ac-curate understanding of three-phase hydrodynamics due to the scale-up influence. The lack of knowledge of mass and heat transfer phenom-ena in three-phase fluidization system still puzzled researchers and re-actor designers. Consequently, future emphasis on CFD application inthree-phase FBRs should be placed more on the study of mass andheat transfer and large scale reactors.

5. Outlook

CFD simulations of gas–liquid–solid multiphase flow behaviors inthree-phase FBRs have been summarized and analyzed in the precedingsections, indicating that CFD is a very powerful tool to understand thefundamentals of three-phase fluidization system and useful to designthree-phase FBRs. However, gas–liquid–solid three-phase flow charac-terized by multi-scale phenomena and complicated hydrodynamicsposes unique modeling challenges and future effort remains to bedone to develop and further validate appropriate CFD models forthree-phase fluidization system in FBRs:

(i) As discussed above, the closure laws for the interphase momen-tum exchange (like drag force, added mass force, Basset force, orlift forces includingMagnus force and Saffman force) between in-dividual phases play a significant role in accurately modelingthree-phase fluidization system. For the drag force, until now,the drag models used in CFD modeling of three-phase flowhave been mostly proposed from theoretical, experimental, andcomputational studies in two-phase systems not three-phasesystems [109,110,114–119,121–124,128–135,137,139–145],which certainly limits the accuracy of closure laws in three-phase fluidization system. Furthermore, even though dragmodels, such as Grevskott et al. [120] and Tomiyama et al. [111]were conducted from three-phase experimental systems, theseexperiments were almost performed with air-water-solid sys-tems at ambient pressures and temperatures instead of the realsituations in industrial reactors that operating under high pres-sures and temperatures with organic liquids or gas. Hence, shar-ing same drag models in two-phase system will bring intoquestion the capabilities of extrapolating the schemes to three-phase system. For other forces, such as added mass force, Bassetforce, or lift forces, Fan's group [144,145] found that Magnus liftforce can be very significant, and even larger than the dragforce in particle–fluid two-phase system, as the rotational Reyn-olds number increases up to O (102), especially for low solid


volume fractions while there is no open literature to discuss theim

cfd simulations of gas-liquid-solid flow in fluidized bed ......review cfd simulations of...

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