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Simulation of Heterogeneous Structure in a Circulating

Fluidized-Bed Riser by Combining the Two-Fluid Model with the

EMMS Approach

Ning Yang,* Wei Wang, Wei Ge, Linna Wang, and J inghai Li

M ul ti phase Reaction L aborator y, In stitu te of Pr ocess Engin eeri ng, Chi nese Academy of Sciences,

P . O. B ox 353, B ei j i ng 100080, P . R. Chi na

To consider the critical effect of mesoscale structure on the drag coefficient, this paper presentsa drag model based on the energy-minimization multiscale (EMMS) approach. The proposedstructure par a meters ar e obta ined from t he EMMS m odel, a nd then t he avera ge drag coefficientcan b e calcu late d from th e stru ctu re p ara m e te rs a n d fu rth e r in corp orat e d in to th e tw o-f luidm o d e l to s im u late th e g as-solid flow in a circulating fluidized-bed riser. Simulation resultsin d icate t h at th e sim u late d f low stru ctu res a re d i f fe ren t for t h e EM M S -b ase d d ra g m o d el a n dthe hy brid model using the Wen a nd Yu correlat ion a nd t he Er gun equa tion. The former showsits improvement in predicting t he solids entra inment r a te, th e mesoscale heterogeneous st ructurein volvin g clu sters or s tra n d s, a n d th e r ad ia l a n d axial void ag e d istr ib u tion s. Th e sim u lationre sults su p port th e id e a th a t th e a verag e d ra g coe ff icien t is a n im p orta n t factor fo r th e t wo -fluid model and suggest tha t t he EMMS a pproach could be used a s a kind of multiscale closurelaw for dra g coefficient.

1. Overview of Modeling Approaches

In circula ting fluidized beds (CFB s), gas a nd pa rticlesexhibit a complex flow behavior that is quite differentfrom t ha t in t he regime of dilute tra nsport. For insta nce,a relatively dense suspension exists with a solids volumefraction ranging from 0.1 to 0.3 if a sufficiently effectivecyclone is provided to return the solids that escape fromthe top to the lower pa rt of the system, a nd th e effectiveslip velocities between ga s a nd pa rticles a re usua lly 20-30 times terminal velocity. To explain these complexp h en ome n a , Y eru s h a lmi e t a l .1 a n d L i a n d K w a u k2

supposed that most particles did not remain single, buttend to aggregate in the form of so-called clusters or

strands. Later, these ideas were confirmed by experi-menta l investigations such a s measurements employinghigh-speed photography, 3 v ide o c a me ra s a n d o p t ic a lfibers ,4 and capacitance probes.5 I t wa s f ou n d t h a t t h eclusters are generally irregular in shape with variablesize. This leads t o significa nt local heterogeneity a t t hescale of both t ime and space, posing great challengesfor th e ana lysis and m odeling of gas-solid flow beha viorin CF B systems. Conventional empirical models, how-ever, cannot be employed to investigate the flow struc-ture extensively. Recent decades have witnessed theemergence of models and simulation based on compu-tat ional f luid dynamics (CFD).

G e n e ra l ly s pe a k ing , t h e s e mo dels f a l l in t o t h re ecategories, namely, two-fluid models, discrete-particle

models, an d pseudo-part icle models. The tw o-fluidmodels (TFMs) are bas ed on a pure Euleria n tr eat mento f bo t h g a s a n d s o l id p h a s e s a s c o n t in u o u s a n d f u l lyinterpenetra t ing, tha t is , both th e gas a nd solid phasesa re de scribed by t h e loca l ly a v e ra g e d con s erv a t ionequations with closure laws including the respectiveconstitutive correlations for stress-strain relat ionshipof g a s a n d s ol ids , a n d t h e t ra n s f e r c orre la t io n s f ordescribing the interactions between the two phases. 6-10

The second category is the so-called discrete particlemo de l ( D P M ) in wh ic h t h e g a s p h a s e is t re a t e d a s ac o n t in u u m a n d t h e s o l id p h a s e is t re a t e d a s dis c re t epart icles. The gas phase is st ill described by locallya v e ra g e d c o n s e rv a t io n e q u a t io n s wit h t h e g a s-solidinteraction terms representing the coupling between thetwo phases; while the part icle motion is resolved intothe suspension process in which individual part icle istracked by the applicat ion of Newtons second law ofmo t io n , a n d t h e c o l l is io n p ro c e s s in wh ic h t h e h a rdsphere or soft sphere approach is used to describe theparticle-part icle int eract ions 11-15. The t hird categoryis a kind of pure Lagrangian model for both the gas andsolid phases, treat ing the fluid phase as composed ofpseudo-particles 16 a n d a t t r ibut in g a l l t h e in t e ra c t ion sto collision processes between pseudo-particles and realpart icles. For the last two categories, enormous com-putat ional resources are needed to handle the motionof individual part icles, thereby hindering, or perhapseven rendering impossible, their application in indus-t r ia l s y s t e ms , a t le a s t a t p re s e n t . O n t h e o t h e r h a n d,two-fluid models are employed in current commercialC F D p a ck a g es t o s im u la t e r ea l i nd u st r i a l s y st e m sbecause of their relat ively lower computat ional costs.

One of the key points for two-fluid models is how toestablish t he constitut ive correlat ions for t he relat ion-s h i p b e t w e e n t h e s o l i d s t r e s s a n d t h e s t r a i n . S o m eresearchers 17,18 h a v e u s ed e mpirica l corre la t io n s t o

calculate the so-called solids viscosity and solidspressure, whereas the kinetic theory of granula r f low(KTGF), originating from the kinetic theory for nonuni-form dense gases, has been employed by others. 19,20 I nt h e K T G F , t h e s o l ids p re s s u re a n d v is c o s i t y a re re -garded as functions of the so-called gra nular temper-a ture, a mea sure of the pa rticles velocity fluctua tions.An addit ional part ial differential equation for the bal-a n c e o f t h e f l u c t u a t i n g e n e r g y o f t h e p a r t i c l e s w a sre q u ire d. Alt h ou g h t h e K TG F s ee ms t o be a morereasonable approach to calculating t he solids stress tha ne mpirica l corre la t io n s, P o la s h en s k i a n d Ch e n21 a n d

* To whom correspondence should be addressed. E-mail :[email protected].

5548 In d. Eng. Chem. Res. 2004, 43 , 5548-5561

10.1021/ie049773c CC C: $27.50 2004 American C hemical SocietyP ubl ish ed on Web 07/27/2004

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B ussing a nd Reh 22 found that their experimental datafor solids stress were quite different from the resultspredicted w ith the K TG F. B ussing a nd R eh22 suggestedfurther that particle clusters should be considered as amomentum transferring structure for the KTGF.

Another key point for TFMs is how t o establish th ecorrelat ions for the interphase dr ag coefficient , w hichrepresents t he intensity of momentum tr an sfer betw eent h e g a s p h a s e a n d t h e s o l id p h a s e . I n t h e l i t e ra t u re ,

several synonyms for this parameter have been used,such as interphase frict ion coefficient , interphase ex-change coefficient, and interphase momentum transfercoefficient. All of th em, in essence, refer t o the a verag edrag coefficient for particles in a unit volume. In mostliterature reports, the drag coefficient was calculatedby modifying the sta nda rd dr ag coefficient for individualparticles 23,24 o r wa s de riv e d f ro m t h e p re s s u re dro pequations for f ixed beds such as t he Ergun equation.25

In other works, the drag coefficient was calculated eitherfrom some pressure drop correlat ions such as that ofGibilaro et al .26 or some correction factors such as thatproposed by Di Felice.27 Recently, Mostoufi and Cha-ouki28 and Ya ng and Renken29 proposed s ome impr oveddrag correlations based on their respective experimental

results. However, almost all of the above correlat ionswere derived from experimenta l results for liquid-solidfluidized-bed or f ixed-bed systems. The question iswh e t h e r t h e s e c orre la t io n s cou ld re pre se n t t h e re a lmechan isms of gas-solid interact ions in C FB systems,which are, in common cases, more heterogeneous thanliquid-solid fluidized-bed or fixed-bed systems.

B y using t he so-called energy-minimiza tion multiscale( E M M S) mo de l , L i e t a l . 30 compared the calculat ionresults of slip velocities a nd d ra g coefficients for d ensecluster phases, dilute broth phases, and the interphaseb et w e en cl us t er s a n d d i lu t e b r ot h s , s h ow i n g t h ei rdra ma t ic di ff ere n ce . L i a n d K wa u k31 invest igated theavera ge dra g coefficients for different connections of

dense and dilute phases, as w ell as the uniform distri-bution of par ticles in a u nit volume, show ing the str ongdependence of the dra g coefficient on simple stru ctura lchanges. Their furth er study 32 indicat ed tha t both localand global structural changes led to a decrease in thea v e ra g e dra g coe ff icien t . H o we ve r , t h e p res en t dra gcoefficient correlat ions used in TFMs relate the dragcoefficient only to the local average voidage and slipvelocity, losing sight of the structure of control volumes.

OBr ien and Sya mlal33 reported that the present dragcorrelat ions must be corrected to account for clustereffects for f ine part icles and further claimed that theunphysical a djustment of the solids stress w as not thecorre ct a p p roa c h . O n t h e ba s is of t h e e xp erimen t a lpressure drop data, they presented an empirical cor-relation for two operating conditions. Qi et al. 34 reportedtha t t he part icles fed into the riser would be elutriat edimmediately, leading to a ra ther dilute flow if the Er gunequation was used to calculate the drag force in simula-t ion. They claimed that the present drag correlat ionswere suitable only for lower gas velocit ies and coarseparticles, in which case the terminal velocity was equalor close to the superficial gas velocity. Agrawal et al. 35

reported tha t the so-called coarse-grid simulat ion,which completely ignored the subgrid microstructure,would overest imat e the dra g force. Zha ng a nd Vander-Heyden 36 proposed a tw o-a verag e a pproa ch for conser-vation equations to resolve the added-mass force anddrag force due to mesoscale structure from those at the

part icle length scale, showing their essential roles inthe macroscopic momentum equations. Helland et al. 37

studied the influence of porosity functions on correctionfactors for the drag coefficient , indicat ing that homo-geneous fluidizat ion is unsta ble to sma ll porosity per-turba tions because of n onlinear dra g-inducing clusterf o rma t io n a n d t h a t t h e c o l l is io n p a ra me t e rs h a v e as ig n if ica n t i nf lu en ce on t h e s h a pe a n d v oi da g e ofclusters. The work of Li and Kuipers 38 indicated alsot h a t n on li n ea r d r a g p la y s a n e ss en t i a l r o le i n t h epatt ern format ion of heterogeneous flow structures indense fluidized beds. Zha ng a nd Reese39 reported thatthe a verage dra g force needs not only to incorporat e theinfluence of solids volume fraction but also to addressthe effect of the random fluctuational motion of par-ticles.

O n t h e o t h e r h a n d, e x pe rime n t a l re su lt s h a v e s u g -gested th at the effective dra g coefficient decreases uponthe format ion of clusters a nd, therefore, tha t the Wenan d Yu correlation and E rgun equat ion were inadequa tefor chara cterizing CFB systems. For example, Gunn a ndMalik40 f o u n d t h a t , i f p a rt ic le s we re g ro u p e d in a g -gregates for a given rate of fluid flow and overall voidageof the system, the measured drag coefficient decreased

because of the d ecrease of the ga s f low penetrat ing thea g g re g a t e s a n d t h e in c re a s e o f t h e g a s f lo w p a s s in garound the aggregates. Muller and Reh 41 found that theformation of part icle strands resulted in considerabledrag reduction, leading to a longer acceleration regionin risers. In view of the deficiency of the correlation ofWen and Yu and t he Ergun equat ion for CFB systems,Li42 a n d B a i et a l .43 proposed their respective empiricalcorrelations for the drag coefficient based on the averagev a lu es of e xp erimen t a l p res s u re dro p da t a ov er t h ewh o le cros s -s ect ion a l a re a . R e ce n t ly , M a k k a wi a n dWright 44 also pointed out that it was far from accuratefor most num erica l models to assum e tha t t he exponentof the correction factor for the drag coefficient was aconstant, and they proposed several correlations of theexponent of the correction factor for the wall and forthe center region of a fluidized bed on th e basis of theirexperimental measurements.

2. Motivation of the Present Study

J udging the survey of literature, it can be concludedthat the present drag correlations are too crude to reflectt h e t r u e m e ch a n i sm s of g a s-solid interact ions forheterogeneous systems, calling for a further investiga-tion of the relat ionship between t he dra g coefficient an dthe formation of mesoscale structure. In our opinion,schemes to solve this problem can be classified into fourcategories, as described below.

2.1. Tracking the I nterface between Gas andParticles. This category includes so-called direct nu-merical simulat ion (DNS ), pseudo-part icle modeling(PPM), and the lat t ice-Boltzmann method (LBM). Byusing th ese models, the interfa ce betw een each part iclea n d t h e s u r rou n di ng g a s ca n b e t r a ck ed a n d t h einteract ions between them need not t o be describedexplicit ly. In DNS (Hu 45), the moving body-fitted un-s t ru c t u re d g r ids we re f re q u e n t ly g e n e ra t e d wit h t h emotion of particles. The influence of particles on fluidwa s reflected by the nonslip boundar y condition of fluidf low f ie ld wh ich wa s t h e n c omp u t ed by s olv in g t h eNavier-St okes equa tion, wherea s the influence of fluidon particles was reflected by the integration of the stresson ea ch element a round the surfa ce of par ticles. In P P M

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(G e a n d L i16) and LB M, the ga s phase is r esolved on alength scale sma ller tha n the part icle size and th e dragforce is calculated by the collision process betweenpseudo-particles and real particles.

2.2. Refining theComputational Grids. The simu-lat ion work of some researchers 35,46 s h o w e d t h a t t h etwo-fluid model with high grid resolution, using theconventiona l correla tions for the a verag e dra g coefficientor even using correlat ions for the standard drag coef-ficient for a single par ticle, is capable of simulatin g themesoscale structure. This might be due to the fact thatheterogeneity is weakened when the grid size is reduced.

2.3. Extracting the Closure Correlations fromMicroscopic Simulations. With a ll of these modeling

a p p ro a c h e s , i t is g e n e ra l ly a g re e d t h a t a c a s c a de de -scription, i.e., extra cting t he closure correlations for t hedra g coefficient fr om microscopic simulat ions, ca n su g-gest a pract ical way to explore the mult iscale hetero-geneity. For exam ple, Agra wa l et al.35 took the gr id sizeused in the coarse-grid simulation of a large unit as thedomain size, imposed periodic boundary conditions inall directions, and carried out a two-fluid simulation toobtain the average slip velocity, which could then beused to extract an approximate drag law. Recently, anat tempt wa s mad e to compute the drag force with LB Mby K a n dh a i e t a l . 47

2.4. Establishing a Structure Model for ControlVolumes. Although the above-mentioned schemes arelogical a nd funda menta l, they ar e genera lly difficult toimplement at present because of the complexity of themodels and the enormous computa tiona l cost. A simplebut rea sona ble approach is to a ssume a structure foreach control volume. A structure model is establishedto describe the mult iscale heterogeneity through theresolution of structure a nd the definit ion of structurepara meters. The a verage dra g coefficient can then becalculated from structure models.

The motivat ion of this study arises from the fourthscheme because of its simplicity. The structure modelis established according to the so-called energy-mini-miza tion mult iscale model (EMMS ) proposed by L i a ndK w a u k .48 The relat ionship between the average dragcoe ff icien t f or a con t ro l v o lu me a n d t h e s t ru ct u re

parameters is deduced from the structure model. Also,a simplified drag model is incorporated into the two-fluid model to simulat e the ga s-solid flow behavior inthe riser sect ion of a circulat ing fluidized bed. As anextension of our previous work,49,50 this paper combinesthe analysis of f low structure and the implementat ionof combinations of the two models. Furthermore, thisapproach is employed to investigate the effect of thesolids inventory on the axial voidage profile.

3. Dependence of the Drag Coefficient on theFlow Structure

3.1. Reiteration of theStructure Model. Figure 1shows the concept of structure resolution according to

the definition of the E MMS model. The het erogeneousstructure is resolved into a particle-rich dense clusterphase and a gas-rich dilute phase. Eight parameters areassigned to describe the phases, nam ely, c, f, dcl, Ugc,a nd Upc for t he dense phase a nd f, Ugf , a n d Upf for t hedilute phase. Correspondingly, the gas-solid interactioni s r es ol ve d i n t o t h e i nt e ra c t ion s b et w e en g a s a n dpart icles w ithin each of the phases and the intera ct ionbetween the dense cluster phase and the dilute brothphase. In a ddition to the resolution of the structure a ndof t h e g a s-solid intera ct ions, t he E MMS model st ipu-lat es further tha t th e total energy consumption per unitma ss of part icles, NT, can be resolved in to componentsNst t h a t is n e e de d f o r s u s p en din g a n d t ra n s p ort in gpart icles and Nd tha t is dissipated in other dissipat iveprocesses. The particle-fluid flow is subject not only tom a s s a n d m om en t u m con s er v a t i on b ut a l s o t o t h esystem st ability condit ion, i .e., to the minimizat ion ofthe energy consumption, N st , for suspending an d tra ns-port ing part icles. For an extensive discussion of thederivat ion and development of the EMMS model, thein t e re s t e d re a de r is re f e rre d t o t h e wo rk s o f L i a n dK w a u k ,48 X u a n d L i ,51 Li et al . ,52 a n d G e a n d L i .53

Originally, this model was proposed to describe mul-tiscale heterogeneity for global fluidized-bed systems.I n t h is s t u dy , we e x t e n d t h is mo de l in t o t h e c o n t ro lv o lu me o f t h e t wo - f lu id mo de l a n d t a k e t h e a v e ra g eacceleration, a, of a control volume into consideration.The ma thema tical m odel is formulat ed by considering

Figure 1. Mult iscale resolut ion of s tructure and gas-solid interact ion in the EMMS model.

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the ma ss a nd momentum ba lances, the correlat ions forstructure para meters, and the st ability condit ion. Themass balances for gas and solids require

For the dense pha se, the resulting dr a g force comprises

t h a t i n d u c e d b y g a s-part icle interact ions inside thede ns e p h a s e a n d t h a t in du ce d by g a s-cluster interac-t ions between the cluster phase and the surroundingdilute broth. For the dilute phase, the drag force refersonly to that induced by gas-particle interactions insidethe dilute phase. Then, the momentum equations forpart icles in the dense an d dilute phases per unit volumecan be expressed respectively as

a n d

where adenotes the a verage a ccelera tion term. Accord-ing to the EMMS model, the gas pressure drop in thede ns e p h a s e s h ou ld be ba la n c ed by t h a t in t h e di lu t ep h a s e p lu s t h a t in du c e d by t h e in t e ra c t io n be t we e ndense clusters an d t he dilute broth

Rearra ngement of eqs 3-5 leads t o the following threeequivalent equat ions

wh e re

The left-hand sides of eqs 6-8 represent the threekinds of gas-solid int era ctions proposed by th e EMMSmodel, i .e. , the int eract ions betw een ga s a nd pa rt iclesin the dense phase (eq 6) and in the dilute phase (eq 8)an d the intera ction between the dense cluster phase a ndthe dilute broth phase (eq 7). The r ight-han d sides,correspondingly, represent the effect ive pa rt of th epart icle weight plus the inert ial force balanced by thet h re e re s p ect iv e dra g f orce s. I n ot h e r wo rds , t h e s eequations show that only part of the gravitat ional andinertia l force for th e dense phase (kc) is balanced by thed r a g f o r c e i n d u c e d b y t h e g a s-part icle interact ionsin s ide t h e de n s e p h a s e, a n d t h e re ma in in g p a rt (ki) isba la n c e d by t h e g a s-cluster interact ion between thedense cluster and the dilute broth, whereas all of thegravitat ional and inert ial force for the dilute phase isba la n c e d by t h e g a s-part icle interact ions inside thisphase. Furthermore, the resultant dra g force for a ll ofthe particles per unit volume FD should be balanced bythe w eight of part icles plus the inert ia l force

wh e re represents the average drag coefficient for acontrol volume (type A, see Gidaspow20) a n d Us i s t h esuperficial slip velocity.

I n a ddit io n t o t h e ma s s a n d mo me n t u m e q u a t io n s ,an equation should be specified to relate the overallvoidage t o those of the dense pha se an d t he dilute phase,i.e.

The hydrodyna mic equivalent cluster dia meter can becorrelated to the energy consumption for suspendinga n d t ra n s p ort in g Nst , a s p rop os ed by L i a n d K wa u k48

a n d X u a n d K a t o54

wh e re

Here, the para meter ma xdenotes the ma ximum voidagefor the aggregates of particles (0.9997, see Matsen 55).

Finally, the stability condition is proposed to close theabove equat ions. Tha t is , the energy consumption forsuspending and transport ing, Nst , w i t h re s p ect t o t h et o t a l e n erg y , NT, t e n d s t o b e m i n i m a l a t t h e s t a b l econdition so that

wh e re

Ug ) f Ugc + (1 - f)Ugf (1)

Up ) f Upc + (1 - f)Upf (2)

3

4CD c

f(1 - c)

dpFgUsc

2+

3

4CD i

f

dclFgUsi

2)

f(1 - c)(Fp - Fg)(g+ a) (3)

3

4CD f

(1 - f)(1 - f)

dpFgUsf

2)

(1 - f)(1 - f)(Fp - Fg)(g+ a) (4)

CD c

1 - c

dpFgUsc

2) CD f

1 - f

dpFgUsf

2+

f

1 - fCD i

1

dclFgUsi

2

(5)

f(1 - c)

dp3

/6CD c

1

2FgUsc

2

4dp

2) f(Fp - Fg)(g+ a)(1 - c)kc

(6)

f

dcl3/6

CD i1

2FgUsi

2

4dcl

2) f(Fp - Fg)(g+ a)(1 - c)ki

(7)

(1 - f)(1 - f)dp

3/6

CD f12FgUsf

24

dp2)

(Fp - Fg)(g+ a)(1 - f)(1 - f)kf (8)

kc ) 1 -

1 - c(9)

ki ) - c

1 - c(10)

kf ) 1 (11)

FD )

Us

) (Fp - Fg)(1 - )(g+ a) (12)

) cf+ f(1 - f) (13)

dcl )

dp[ Up

(1 - ma x)- (

Umf + Upmf

(1 - mf ) )]g

NstFp

(Fp - Fg)- (

(Umf + Upmf )

(1 - mf ) )g

(14)

Nst ) [Ug - f - 1 - f(1 - f)Uf](g+ a)Fp - Fg

Fp(15)

Nst

NT)

[Ug(1 - ) - f Uf(f - )(1 - f)]

Ug(1 - ) ) m in (16)

NT )Fp - Fg

FpUg(g+ a) (17)


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Th e re lev a n t p a ra me t e rs in e q s 1-17 a re l is t e d inTable 1. With the value of Ug, Gs, a n d specified, th estructure parameters can be calculated by solving theabove nonlinear equa tions, which constitute a nonlinearoptimization problem.

3.2. More Insight into the Structural Heteroge-neity. Computat ion wa s carried out w ith t he specifiedsuperficial gas and solids velocities for the fluid catalyticcra cking (FC C)/a ir sys tem (Fp ) 930 kg /m 3, dp ) 54 m,Ug ) 1.52 m/s, Gs ) 14. 3 kg/m 2s). Our previous work 49

s h o we d t h a t , wi t h de c re a s in g a v e ra g e v o ida g e , t h es o l ids v o lu me f ra c t io n in t h e de n s e p h a s e f(1 - c)in cre a s ed a n d t h a t in t h e di lu t e p h a s e (1 - f)(1 - f)

decreased. Meanwhile, the superficial gas velocity in thedilute phase,Ugf , increased, and t ha t in the dense pha se,Ugc, decreased. The calcula tion results indicate th a t th eparticles tend to enter into clusters instead of suspend-i n g i n d i l u t e b r o t h , w h e r e a s t h e g a s t e n d s t o p a s sa r ou n d , i ns t ea d of p en et r a t i n g t h r ou g h, t h e d en s ecluster pha se. These conclusions a re conceptua lly con-sistent with the observed phenomena in particle-fluidsystems, and, in t urn, they show the reasonableness ofthe EMMS approach.

A s de f in e d in e q s 9 a n d 10, kc a nd ki in dic a t e t h eeffect ive part of the weight plus inert ial force in thedense pha se bala nced by different gas-solid dra g forces,and their variat ion is closely related to a nd c. Ourprevious work49 s h o we d t h a t

c f

mf wh e n f

mf,

wh e re a s c f wh e n f1. Therefore, we can deducefrom eqs 9 an d 10 that kc f1 a n d ki f0 when fmf ,wh e re a s kc f0 a n d ki f1 when f1. Figure 2 showst h a t kcincreases w hereas ki decreases w ith d ecreasingaverage voidage . Because of the difficulties inherentin solving the nonlinear equations when f ma x, t h eright sides of the curves a re plotted w ith da shed lines.The left ( f mf ) a n d r ig h t ( f1) sides of the curvescan be considered to represent the two extreme casesof fluidiza tion respectively, i.e., par ticulat e fluidizat ionand dilute tran sport . Thus, we found tha t t he absolutev a lu e of t h e di ff ere n ce , k, bet we e n kc a nd ki, a srepresented by vert ical l ines in Figure 2, can be usedto est imate the extent of heterogeneity of the system.

Lar ger values ofk represent weaker heterogeneity ofthe system. The intersection of the two curves wherek is equal to zero indicates that the gravitat ional andinert ial forces of the dense cluster phase is one-halfbala nced by t he dra g force result ing from ga s-particleinteract ions inside the dense pha se, whereas the otherha lf is balanced by the dra g force due to the gas-clusterinteract ions. At the right side of the intersection, kincreases with increasing voidage, indicat ing that thesystem h eterogeneity decreases quickly and approachesthe sta te of dilute tra nsport in the extreme case. At t heleft side, k f irst increases gradually with decreasingvoidage, indicat ing essentially the decrease in hetero-geneity; its sudden increase a t a turn ing point indicatesthe tendency that the system becomes homogeneousmore quickly and approaches another extreme case,na mely, part iculat e fluidiza tion. The turning point herecan be considered to reflect a kind of regime transitiona n d a ls o a p p ea re d in t h e v a ria t ion of o t h e r s t ru ct u rep a r a m et e rs s u ch a s f, dcl, a n d c w i t h v oi da g e , a spresented in our previous work.49

3.3. Structure-Dependent Drag Coefficient. Th eov era l l mome n t u m e q u a t ion (e q 12) s h ows t h a t t h eavera ge a ccelera t ion term a is determined by only thea verage dr a g coefficient if t he superficial s lip velocityand the average voidage are specified, indicat ing that ,

Table 1. Summary of Related Expressions for the EMMS Model

den se ph a se dilut e ph a se in t er ph a se

effective dra g coefficient CDc ) CD0cc-4.7 CDf ) CD0ff-4.7 CDi ) CD0i (1 - f)-4.7

stan dard drag coefficient CD0 c ) 24

Rec+

3.6

Rec0.313

CD0 f ) 24

Ref+

3. 6

R ef0.313

CD0 i ) 24

R ei+

3.6

Rei0.313

chara cterist ic Reynolds number Rec )FgdpUsc

gRef )

FgdpUsf

gRei )

FgdclUsi

g

superficial slip v elocity Usc ) Ugc -cUpc

1 - cUsf ) Ugf -

fUpf

1 - fUsi ) (Ugf -

fUpc

1 - c)(1 - f)drag force act ing on asingle particle or cluster Fdense ) CD c

dp2

4

Fg

2Usc

2Fdilute ) CD f

dp2

4

Fg

2Usf

2Fcluster ) CDi

dcl2

4

Fg

2Usi

2

number of particles orclusters per unit volume Mc )

f(1 - c)

dp3/6

Mf )(1 - f)(1 - f)

dp3

/6

Mi ) f

dcl3

/6

result ing drag force FD ) McFdense + MiFcluster + MfFdilute

avera ge a ccelerat ion a)FD

(Fp - Fg)(1 - )- g

avera ge dra g coefficient : ) FD

Us/

Figure 2. Variat ion of s tructure parameters with average void-age.


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if cannot be accurately calculated for the two-fluidmodel, a and its related flow field cannot be obtainedcorrectly. In fact , i t is the a verage a ccelerat ion term atha t determines t he simulation a ccura cy of the flow fieldand the corresponding concentrat ion field when two-fluid models a re used.

Summing the left a nd right sides of eqs 6-8 leads to

Combining eq 12 and 18, one obta ins

wh ere the relevan t pa ra meters, as given in Table 1, canbe calculated from the st ructure para meters.

For comparison, we now give the derivat ion of thedrag correlation used in the common two-fluid models.Assuming that the effective drag force for each particlein a contr ol volume is th e sa me, i.e., ignoring th e effectof the heterogeneous stru cture, the resultan t dra g forcefor all of the part icles per unit volume, FD , should beequal to the effect ive drag force for a single part iclemultiplied by the number of particles in the unit volume

The resulting dra g force per unit volume is a lso definedin the two-fluid model as

Combining eqs 20 and 21, one obtains the drag correla-

tion employed in the present two-fluid models

Comparing eqs 19 and 22, one can see tha t t he structureinformation is considered in the former, whereas it isneglected in the la t ter.

Figure 3a compares the results of correction factorsfor effective dra g coefficients computed using t he EMM Sa pproa ch an d t he Wen a nd Yu/Er gun correlat ions. Thecorrection factors ar e calculat ed from

wh e re

It can be seen that the correction factor computed withthe EMMS model is much sma ller th an tha t computedwith the Wen a nd Yu/Ergun correlat ions, which is inreasonable agreement with the conclusions from experi-mental observations that the drag coefficient decreasesas a result of cluster formation. At the left side of thecurve calculated with the EMMS model, the correctionfactor increases abruptly with decreasing average void-a g e a t t h e p re vi ou s ly d is cu s se d t u r n in g p oi n t a n dapproaches the curve calculated w ith th e Ergun equa -t io n . T h e e x t e n de d da s h e d l in e s a t t h e r ig h t s ide o fcurves indicate t ha t t he correction fa ctors from both th eWen an d Yu correlat ion a nd t he EMMS model approachunity. This extreme value corresponds to the sta nda rddra g coefficient.

Figure 3b compares the average drag coefficient computed with the E MMS model and t he Wen and Yu/Ergun correlat ions. I t also shows that the value com-p u t e d wit h t h e E M M S mo de l is mu c h le s s t h a n t h a tcomputed with the Wen a nd Yu/Ergun correlat ions.

Figure 3. Varia t ion of drag coefficients with avera ge voidage.

)3

4

(1 - )

dpFg|ug - up|CD 0

-2.7 (22)

) /0 (23)

0 )3

4

(1 - )

dpFg|ug -up|CD 0 (24)

FD ) McFdense + MiFcluster + MfFdilute (18)

)

2

Us(McFdense + MiFcluster + MfFdilute) (19)

FD )(1 - )

6dp

3

1

2Fg(ug - up)|ug - up|

2

4dp

2CD 0

-4.7

(20)

FD )

(ug - up) (21)


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Furt hermore, the tw o curves tend t o coincide with eachother at the left and right sides.

Figure 3c shows the average accelerat ion a a t t h ec ro s s s e c t io n o f a CF B a s c o mp u t e d wit h t h e E M M Smodel and the Wen a nd Yu/Er gun correlat ions. The da tafor the local superficial gas velocity, solids flow rate, andavera ge voidage a re ta ken from t he experimenta l resultsof Bader et al .56 I t can be seen tha t using the Wen andY u c orre la t io n le a ds t o mu c h h ig h er v a lu es of t h ea v e ra g e a c ce le ra t ion t h a n u s in g t h e E M M S mode l,e sp ecia l ly in t h e wa l l re g ion . Th e h ig h er v a lu e o faccelerat ion seems to be unreasonable unless a largep a rt o f i t c o u ld be ba la n c e d by t h a t du e t o p a rt ic le-part icle interact ions. Muller and Reh 41 reported that ,f or a p a r t icl e s t r a n d of 1-m m d ia m e t er f a l li ng i nstagnant air, the part icle velocity is much higher thanthe terminal velocity for a single suspended particle, andthe a ccelera t ion of part icles inside the st ran d is equa lto the a ccelerat ion of free fall. The la tt er result im pliest h a t n o dra g f orce a c t s o n t h e p a rt icles in t h e s t ra n d.

In some litera ture reports, th e effect of the dra g forcei s e xp r es s ed a s a f un ct i on of t h e a v er a g e p a r t icl erelaxat ion t ime, wh ich indicat es the speed of responsefor particles to entrainment by the surrounding gas and

can be generally formulated as

Thus, the average relaxat ion t ime can also be givenby

Figure 3d shows th e computed part icle average relax-a t i on t i m e . Th e r es ul t s i nd ica t e t h a t t h e v a lu ecomputed with the Wen and Yu correlation is much lessthan that computed with the EMMS model. This impliesthat , for the Wen and Yu correlat ion, the response ofpart icles to the entra inment of surrounding ga s is veryfast , and the simulated solids entrainment rate for gaswould be higher than that from the EMMS model.

4. Combination with the Two-Fluid Model

4.1. Model Description. Ta ble 2 s u mma rizes t h emass and momentum conservation equations and con-stitutive equations of the two-fluid model. For simplic-ity, empirical models were employed to calculate thesolids pressure a nd viscosity. The gra dient t erm of solidspressure wa s employed to prevent th e solid pha se frombeing too densely packed. The correlat ion for solidsv is c o s i t y wa s t a k e n f ro m t h e wo rk o f M il le r a n d G i-daspow.57 To compare the effect of the average dragcoefficients, we adopted two kinds of drag correlations.Drag model A refers to the hybrid of the Wen and Yucorrelat ion and the Er gun equat ion. For drag model B,we employ a simplified version of the EMMS model tode riv e t h e corre ct ion f a c t ors f or di ff ere n t ca s e s ofoperating conditions, as shown in Table 2. Because thecomputation procedure for solving the nonlinear equa-t ions involves t wo la yers of iterat ion a nd t he optimiza-tion process for the EMMS model, it would be difficultto directly apply t his model to each contr ol volume wit hthe local superficial gas velocity and solids flow rate asthe input parameters. To circumvent this problem, wefirst simplified the model by assuming that the voidage

of the dense phase is a constant. Then, with the overallsuperficial ga s velocity an d solids flow r a te a s the inputp a ra me t e rs , we c a n c a lc u la t e t h e a v e ra g e dra g c o e f -f icien t f or a s pe ci fied v oida g e a n d t h e reby obt a in acorrelat ion for the relat ionship () between the cor-rection factor for t he dra g coefficient a nd the voidage.Extending this correlation into each control volume a ndemploying the local slip velocity and avera ge voidage,the avera ge dra g coefficient for each control volumecan thus be calculated as

The simulat ion wa s car ried out for th e riser sect ionof a circulating fluidized bed, as illustrated in Figure 4.Th e g a s en t er ed t h e r i s er f r om t h e b ot t om w i t h aspecified superficial velocity and left the bed at the topfrom t wo side outlets. S olids w ere first sta cked up to acertain height in the bottom of the riser. When fluidized,the solids were allowed to leave from the t op outlets a ndin tur n be fed back into the riser from the bottom inletsby s p ecif yin g t h a t t h e s ol ids f low ra t e a t t h e t w o s idein le t s wa s e q u a l t o t h a t a t t h e t o p ou t let s . Th e in i t ia l

bed height was evaluated from the experimental dataof t h e a x i a l v oi d a g e p r of il es i n t h e r i se r. F or t h eop era t in g con dit ion s wit h t h e s a me s u pe rficia l g a svelocities and solids circulation rates but different solidsin ve n t ories , t h e e va lu a t e d in it ia l bed h e ig h t s we redifferent becau se of the difference in experimenta l a xialvoida ge profiles. In th is w a y, w e could consider the effectof s ol ids in ve n t ory on t h e s imula t e d a x ia l v oida g eprofiles. The initia l bed voidage w a s set t o be incipientfluidizat ion voidage. F or simplicity, t he grids a doptedin the simulat ion w ere uniformly distributed in both thea x i a l a n d r a d i a l d i r e c t i o n s . F o r t h e g a s p h a s e , t h enonslip bounda ry condition wa s used at t he wa ll, so that

For the solid phase, th e free-slip bounda ry condition wa schosen (Ding a nd Gida spow58)

Th e s im u la t i on w a s ca r r i ed ou t b y e mp loy i ng t h ecommercial CFD package CFX4.4 (AEA Technology) inwh ich t h e a bo ve -me n t ion e d t wo dra g mode ls we reincorporat ed t hrough user-defined routines. The highlycoupled part ial differential equations of the two-fluidmo de l we re s o lv e d by t h e in t e rp h a s e s l ip a lg o ri t h m(IPSA) of Spalding.59 Th e s imu la t ion p a ra me t e rs a re

listed in Table 3.

4.2. Results and Discussion

4.2.1. Solids Entrainment Rate. The solids massflux at the top outlets can be obtained by dynamicallychecking t he outlet solids velocity a nd concentra tion inthis location during the simulation. Figure 5 illustratest h e v a ria t ion of t h e o u t le t s o lids ma s s f lux w it h t imefor two solids inventories. At initial times (t< 5 s), th eoutlet solids mass flux is almost equal to zero. Becausethe bed expands with the incipient fluidization voidagein t h is p e rio d, t h e E rg u n e q u a t io n is a lwa y s u s e d byboth dra g models. After t his period, the curves of outletsolids f lux oscillate in different ways. For drag model

1

2FgUs

2

4dp

2CD ) Fp

6dp

3Us

(25)

)Us

(g+ a) (26)

)3

4

(1 - )

dpFg|ug - up|CD 0() (27)

ug, w) vg, w) 0 (28)

up,w) 0, vp,w) -dp

(1 - )1/3

vp,w

x (29)


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A, the curve rises abruptly and falls back because ofthe resistance of the outlets, and finally, i t oscillatesstea dily a t a n a verag e of about 39.03 kg/m 2s for I) 15kg. Considering drag model B, the outlet solids massf lu x a lwa y s o s c i l la t e s s t e a di ly a ro u n d a n a v e ra g e o fa bout 12.17 kg/m 2s a t t h e s a m e s ol id s i nv en t or y .According to the experiments of Li and Kwauk, 48 t h eoutlet solids flux (14.3 kg/m 2s ) w a s ca l cu la t e d b yme a s u rin g t h e p res s u re drop a n d a c cu mu la t e d bedheight a bove the closed butterfly va lve in the sta ndpipewithin a certa in t ime. I t can be seen tha t dra g model Aoverpredicted whereas drag model B well predicted thesolids mass f lux. The higher solids entrainment ratepredicted by dra g model A obviously results from theoverpredict ion of t he avera ge dra g coefficient a ndtherefore the average accelerat ion a for each controlvolume. The sa me phenomenon can also be found for I) 20 k g i n F ig ur e 5b t h a t d r a g m od el A g r ea t l yoverpredicted the average value of the solids outlet massflu x (77.8 kg /m 2s), whereas the simulation result fromdra g model B (12.1 kg/m 2s ) w a s c o m p a r a b l e t o t h eexperimental result .

4.2.2. Dynamic Flow Structure. F ig u res 6 a n d 7illustra te the gas and solids velocity vectors from dra gmodels A and B, respectively, at t) 20 s. Figure 8 showssnapshots of t he contours for t he voidage distributiona t t ) 2 0 s . I t c a n b e s e e n t h a t t h e s i m u l a t e d f l o w s t ru ct u re s f or t h e t wo dra g mode ls a re di ff ere n t . F o rd r a g m o d e l A , t h e d i r e c t i o n s o f t h e g a s a n d s o l i d svelocity vectors a re alw a ys upright, an d the distributionof solids concentration is rather homogeneous across thewhole bed except for the denser region near the two sideinlets at the bottom. The phenomena predicted by dragmodel A a re much more like the flow st ructure of dilutetra nsport tha n tha t of fast f luidizat ion. For dra g modelB, the solids velocity vectors are upright at the top andchaotic at the bottom, showing that the lateral motiono f t h e s o l ids is s t ro n g a t t h e bo t t o m, a n d t h e s o l idsthemselves are entrained upward in a snakelike wrig-gling way. The corresponding snapshot of the solidsconcentra tion distribution, as show n in Figure 8, showstha t the t op section of the bed is dilute a nd t he bottoms ect i on i s d e n se a n d i n a t u r bu le nt s t a t e . C l ea r l yobserved is t he format ion of clusters protruding from

Table 2. Mathematical Equations in the Two-Fluid Model

Conservat ion Equations(1) Mass C onservat ion Equa tions

(a) gas pha se

t(Fg) + (Fgug) ) 0

(b) solid pha se

t[(1 - )Fp] + [(1 - )Fpup] ) 0

(2) Momentum C onservat ion Eq uations

(a) gas pha se

t(Fgug) + (Fgugug) ) -pg + Tg - (ug - up) + gFgg

(b) solid pha se

t[(1 - )Fpup] + [(1 - )Fpupup] )

-(1 - )pg + Tp - (up - ug) + (1 - )Fpg

Constitut ive Equations(1) St ress-Strain Correlat ions

(a) gas pha se Tg ) g{[ug + (ug)T] -2

3(ug)I}

(b) solid pha se Tp ) -ppI + (1 - )p{[up + (up)T] -2

3(up)I}

(2) Model for Solids Viscosity an d P ressure(a) solids viscosity p ) 0.5(1 - )

(b) solids pressure pp ) 10-8.686+6.385(1 - )(3) Model for G as-Solid Intera ct ion

(a) drag model A ) {

3

4

(1 - )

dpFg|ug - up|CD 0

-2.7( > 0.8)

150(1 - )

2g

dp2 + 1.75

(1 - )Fg|ug - up|

dp( e 0.8)

(b) dra g model B ) {

3

4

(1 - )

dpFg|ug - up|CD 0() ( > *)

150(1 - )

2g

dp2 + 1.75

(1 - )Fg|ug - up|

dp( e *)

* ) 0.74 for Ug ) 1.52 m/s, Gs ) 14. 3 kg /m 2s

() ) {-0.5760+

0.0214

4( - 0.7463)2+ 0.0044

(* < e 0.82)

-0.0101+ 0.0038

4( - 0.7789)2+ 0.0040

(0.82 < e 0.97)

-31.8295 + 32.8295 ( > 0.97)


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the w all a nd of stra nds a bove the bottom dense region.Part icles fall along the wall and then stack together toform ra dial protrusions. Some particles are dyna micallyextracted from the protrusions by the up-flowing gasa n d re a g g reg a t e in t o s t ra n ds a bo ve t h e bot t o m de n s eregion. The cluster protr usions resist the motion of up-f lowin g g a s a n d s ol ids , le a din g c on s eq u e n t ly t o t h ewriggling movement. These phenomena predicted byd r a g m od el B a r e i n r e a s on a b le a g r ee me nt w i t h t h eexperimenta l observa tions of Rhodes et a l. ,60 wh o foundt h a t t h e p redomin a n t mo t ion is ra dia l ly in wa rd a t t h einterfa ce between th e upper dilute region a nd th e lowerdense region. Davidson 61 insisted that clusters of par-t ic le s de t a c h e d f ro m t h e f i lm f a l l in g a g a in s t t h e wa l land further decomposed in the dilute phase to contrib-ute par t icles to the d ilute phase. Because coar se gridsa re u s ed in t h is s t u dy , t h e s imu la t e d s t ru c t u re migh t

represent a kind of group behavior of microstructures.4.2.3. Time-Average Flow Structure. The time-a v er a g e v oi da g e ca n b e o bt a i n ed b y a v e ra g i n g t h ecalculated transient values over 20-30 s, and t he time-avera ge velocity for a control volume can be obtainedas follows

wh e re i ,j a nd ui ,j den ot e t h e t ra n s ie nt v oida g e a n dvelocity for a contr ol volume, respectively.

F ig u re 9 s h o ws t h e ra dia l p ro f i le s o f t h e re s u lt in gtime-avera ge voidage ba sed on dra g models A a nd B forI) 15 kg. Figure 10 displa ys t he ra dia l voida ge profilesfor I ) 20 kg. With experimentally measured voidagesobtained using optical fibers, Tung et al. 62 a n d Z h a n ge t a l .63 corre la t e d t h e ra dia l v oida g e p ro fi le wit h t h ecros s -s ect ion a l a v e ra g e v oida g e, t h u s a l lowin g e a s ycompar isons of our simulat ed profiles with calculationsf r om t h ei r cor r el a t i on . I t ca n b e s e en t h a t , f or t h e

Figure 4. Schematic drawing of the simulated riser.

Table 3. Parameter Settings for the Simulation

pa r a m et er va lue

pa r t icle dia m et er 54 mpa r t icle den sit y 930 kg/m 3

t im e in t er va l 5.0 10-4 sgrid size (x) 2.25 10-3 mgrid size (y) 3.5 10-2 mm a x im u m v oid a g e f or p a r t icle a g g r e g a t ion 0 .9 99 7voida ge of t h e den se ph a se 0.69super ficia l ga s velocit y 1.52 m /ssolids inventory I) 15 kg H) 1.225 msolids inventory I) 20 kg H) 1.855 m

ui,j)

t

i,jui,j

t

i,j

(30)

Figure 5. Comparison of outlet solids ma ss f lux.

Figure 6. S imulat ed velocity vector w ith dra g model A at t) 20s : (a ) g a s , I ) 15 kg; (b) solids, I ) 15 kg; (c) gas, I ) 20 kg; (d)solids, I ) 20 kg.


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E M M S-ba s e d mode l ( dra g mode l B ), t h e s imu la t e dresults ar e in good agr eement w ith tha t of experimenta lcorrelat ion in t he dense a nnulus, but un derpredict thevoidage at the dilute core. Because most part icles area g g re g a t e d in t h e a n n u lus f or f a s t f lu idiza t io n, t h eunderprediction of th e voida ge in th e dilute core w ouldh a v e l it t le e ff ect on t h e a c cu ra c y of h y drody n a mics im u la t i on s . I t s h ou ld b e n ot e d t h a t t h e s im u la t e dvoidage near the w all is generally larger tha n tha t from

Tung et al. s correlation. According to the recent work

of Xu et al.,64 this might indicate tha t the simulat ion ismore representa tive of experiment becau se the Tung etal. correlat ion a ctually underest imat es the voidages onthe wall. For the hybrid model based on the Wen andYu correlation and the Ergun equation (drag model A),the simulat ed radia l voidage profile is rat her flat, w hichactua lly shows th e homogeneous str ucture discussed inthe previous sections. It can be seen tha t t he simulat ionresults obta ined with dra g model A deviate significa ntlyfrom the experimenta l correlat ion, overpredict ing thevoidage in the a nnulus a nd underpredicting th e voidagein the core. Figure 10b shows that the simulated voidageobtained with drag model A is much lower than thosemeasured in experiments a nd obtained from the simu-lat ion with dra g model B. B ecause the solids entra inedto the outlet a re all fed back into the inlet , the avera gesolids concentra t ion for t he w hole riser is a lwa ys heldas a constant. For drag model A, the overprediction ofthe voidage in t he bottom a nnular region w ould lead t oan underpredicted voidage for the top region. If a solids-feeding loop were not a llowed a nd a consta nt solids flowra t e we re s p e cif ie d a t t h e bot t o m in le t s , t h e s ol idscon ce n t ra t ion p redict e d by dra g mode l A wo u ld bera t h e r s ma ll . I n t h is c a s e , u s in g dra g mo del A wo u ld

overpredict the voidages throughout the bed, and thesystem then appears to transport part icles quickly, asreported by Qi et a l .34

F i gu r e 11 s h ow s t h e t i m e-a v er a g e a x i a l v oi d a g eprofiles from calculat ions ba sed on the tw o drag models.The plott ed experiment a l a xial voida ge profiles ar e fromL i a n d K w a u k.48 For dra g model A, the result ing a xialvoidage profile is almost vertical except near the inletan d outlet. Using dr ag model B improves the prediction,leading t o the S -sha ped voidage profile, alth ough devia-t ion s s t i ll e xis t p roba bly beca u s e of t h e in a ccu ra t ee v a lu a t io n o f t h e s o l ids in v e n t o ry in t h e r is e r a t t h ein it ia l t ime a n d t h e u n re a l ist ic s et u p o f t h e in let a n doutlet boundary condit ions in the 2D simulat ion.

To illustrate the effect of the solids inventory on the

Figure 7. S imulated velocity vector with dra g model B a t t) 20s : (a ) g a s , I ) 15 kg; (b) solids, I ) 15 kg; (c) gas, I ) 20 kg; (d)solids, I ) 20 kg.

Figure 8. S imulated voidage distribution at t) 20 s : (a) I) 15 kg, dra g m odel A; (b)I) 15 kg, drag model B. (c) I) 20 kg, drag modelA; (d) I ) 20 kg, drag model B.


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axial voidage profile, results for the axial voidage profiles imu la t e d wit h dra g mo de l B a re re p lo t t e d in F ig u re12. As discussed previously, the set t ing of the solidsin ve n t ory in t h e s imula t ion is a ch iev ed by s et t in gdifferent initial bed heights evaluated from experimen-tal results of axial voidage profiles corresponding todifferent solids inventories. It can be seen tha t t he onlydifference betw een the tw o curves is th e inflection point .The cross-sectional voidages in both the dilute top regionand the dense bottom region ar e invaria ble. Moreover,the simulat ed values of the outlet solids m ass f lux for

t h e t w o s ol ids in v e nt o rie s, a s s h own in F ig u re 5, a rea l m os t t h e s a m e, i n di ca t i n g t h a t t h e e nt r a i n m en treaches the sta te of satur at ion. This reasonably a greeswith the conclusions from experimenta l f indings t hat ,wh en the solids inventory I is increased within a certainra nge at a given superficial ga s velocity Ug, the solidsflow rate Gs remains essentially consta nt , a nd th e onlyvariation for the S-shaped voidage profile is the move-ment of the inflection point. Therefore, it is insufficientto determine t he a xial voidage profile by specifying onlyUg a nd Gs, as reported by Li et al . 31 a n d X u a n d G a o.65

F ig ure 13 s h o ws t h e t ime-a v e ra g e a x ia l v eloci t yprofiles for gas and solids in radial direction for I) 15k g o bt a in e d w it h t h e t wo dra g mo dels . I t c a n be s ee nfrom Figure 13a a nd c tha t, for drag m odel A, the sha peof the solids velocity curve looks too similar to that ofthe ga s velocity, indicat ing tha t t he solids tra ck the ga sflow very quickly and the particle relaxation time shouldbe very short . This phenomenon can also be found inFigure 6, where the gas and solids velocity fields in thewhole bed simulated using drag model A are almost thesam e, showing the q uick response of the part icles to gasentrainment as well. The results from the simulat ionba sed on drag model B a re obviously different , especiallyin the bottom region, as shown in Figure 7. Moreover,the velocity f ields of the gas and solids simulated w ithdra g model A ar e higher tha n those obta ined with dra g

Figure 9. Radia l voidage profile for I ) 15 kg.

Figure 10. Radia l voidage profile for I ) 20 kg.

Figure 11. Axial voidage profile for (a) I) 15 kg, (b) I) 20 kg:- - -, dra g model A; s, d r a g m od e l B ; 9, experimental.

Figure 12. Effect of solids inventory on axial voidage profile:- - -, I ) 15 kg; s, I ) 20 kg.


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model B. Thus, the choice of a suitable drag model isnecessarily needed for CFD simulat ion.

5. Conclusions and Prospects

Complexity in gas-solid flow in circula ting fluidizedbeds lies in the formation of heterogeneous structuresexisting a t different time a nd spa ce scales. Although theemergence of CF D-based models provides a powerfult o ol f o r s imu la t in g f low s t ru ct u re in CF B s y s t ems ,models that can reflect the heterogeneous nature arising

from the mutual compromise among different kinds ofintera ctions a re la cking for such complex systems. Theobjective of this study w as to understa nd t he relat ion-ship betw een the mesoscale heterogeneity an d the ga s-s o l id dra g c o e f f ic ie n t . Co mp a re d t o t h e h y brid dra gmode l of t h e W en a n d Y u c orre la t io n a n d t h e E rg u ne q u a t ion , t h e dra g mode l p res en t e d in t h is s t u dy wa sestablished on the basis of a structure model tha t t akesthe effect of mesoscale st ructure int o considerat ion.Consequently, a combina tion of the dra g model wit h thetwo-fluid model shows the reasonableness and improve-ment in simulat ions of the dyna mic forma tion of localh e t erog e ne ou s s t ru ct u re , s u ch a s t h e f orma t io n ofclu st e rs o r s t ra n ds , a n d in p redict ion s of t h e t ime-avera ge ma croscale heterogeneity. Although this E MMS-ba s e d mo de l n e e ds t o be f u rt h e r v a l ida t e d f o r o t h e rsystems, it would be expected to be used as a closurelaw for drag coefficients.

R ece nt l y , G e a n d L i53 proposed tha t the tw o-fluidmodel could be extend ed to th e so-called t w o-pha se tw o-fluid model. This means tha t t he part icles and fluid int h e de n s e a n d di lu t e p h a s e s c ou ld a l l be re g a rde d a scontinua, with similar but double average conservationequations of the present TFM. The interactions betweene a ch p a ir of t h e m c ou ld be de t e rmin e d in a f u rt h ergeneralized version of the EMMS-based drag model.Therefore, future w ork might be needed to improve theproposed model and incorporate it into a two-phase two-fluid model. I t is also worthw hile to esta blish a kind of

multiscale approach to evaluate the relative dominanceand mutual compromise among different interact ionsoccurring betw een gas a nd solids, solids a nd solids, a ndmixture and wa ll. This might allow for a proper evalu-at ion of the ma gnitude of the a ccelera tion terms inducedby these interactions in the average conservation equa-tions of the two-fluid models.

Acknowledgment

This w ork wa s f inancially supported by t he Nat iona l

Natural Science Foundation of China (Nos. 20336040,20221603).

Notation

a) avera ge a cceleration of part icles for a contr ol volume,m/s 2

CD ) effective dra g coefficient for a part icleCDc , CD f ) effective drag coefficients for a particle in the

dense phase and the dilute phase, respectivelyCDi ) effective dra g coefficient for a cluster

CD0 ) s t a nda rd dra g coefficient for a pa rt icledcl ) cluster diameter, m

dp ) pa rt icle dia met er , mf) volume fraction of dense phaseg) gra vita tional accelera tion, m/s 2

Gs ) solids flow ra te, k g/m 2sH ) init ia l bed height , mI ) solids invent ory, kgNst ) ma ss-specific energy consumption for suspending an d

tr a nsportin g pa rt icles, W/kgNT ) mass-specific total energy consumption for particles,

W/kgUg ) superficial fluid velocity , m /s

Ugc, Ugf ) superficial fluid velocities in the dense phasean d in the dilute pha se, respectively, m/s

Umf ) superficial fluid velocity at minimum fluidization,m/s

Up ) superficial par ticle velocity, m/s

Figure 13. S imulat ed velocity profile for ga s a nd solids for I) 15 kg.


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Upc, Upf ) superficial pa rticle velocities in the dense pha sean d in the dilute pha se, respectively, m/s

Us ) superficia l s lip velocity, m/sUsc, Usf , Usi ) superficial slip velocities in the dense phase,

the dilute pha se, and the int erphase, respectively, m/s

Greek Symbols

) dra g coefficient in a cont rol volume, kg/m 3s ) voidagec, f ) voida ges of t he dens e pha s e a nd t he dilut e pha s e,

respectivelyma x ) ma ximum voidag e for part icle a ggrega tion (0.9997)mf ) voidage at minimum fluidization (0.5)g, p ) ga s or solids viscosities, respectively, P a sFg, Fp ) fluid a nd solids densit ies, respectively, kg /m 3

) a vera ge pa rt icle rela xa t ion t ime, s ) correction factor

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Receiv ed for r evi ew March 22, 2004Revised man uscript received May 31, 2004

Accepted J une 9, 2004

IE049773C


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