Significanceof core diameter variation in the lateral solid transfer model for the upper dilute regionof circulatingfluidized beds

MASAYUKI HORIO and HONGWEI LEI Department of ChemicalEngineering, Tokyo University of Agriculture andTechnology,Koganei, Tokyo 184,JapanReceived 20 August1997;accepted13 November 1997 Abstract-The formulation of an axial solid concentration profilein the upperdiluteregionofcirculatingfluidized beds is analyzedbased on the lateral solid transfer model which includes mass transfer of particlesbetween the core and the annulus,analogousto the countercurrent absorptionprocess.Theoperating region ismappedon a phase plane of solid concentration in the coreregionversus that in theannulusregion.It is demonstrated that if the core diameter isassumedconstant,the model faces a serious paradoxofhavingeither a negative decay factor/positive solid fraction εpC,∞or a positivedecay factor/negative solid fraction εpC,∞.Thisparadoxwas found to be solved bythe introduction of a variable core diameter assumption.The calculated results of solid concentration alongthe riser with Werther correlation for core diameter distribution showed a fair agreementwithprevious experimental results.

NOMENCLATUREa decayfactor(m-1)A sectional area (m2)dp diameter of particles(m)D column diameter (m)Fr Froud number Uol(gD)0.5(-)9 gravitation acceleration, g =9.81 m/s2 Gs solid flux (kg/m2s)k lateral solid transfer or solid depositioncoefficient(m/s)K equilibriumconstant, K =kAC/kcA(-)L riserheight(m)Re Reynoldsnumber,Re= DUop//L(-)s Stokesnumber,S=dgUoPp/18/LD(-)u' turbulentvelocityfluctuationinequation(11) (m/s)

108Va superficial gas velocity (m/s)t) solid velocity(m/s)v() superficialsolidvelocity,vo= G> /pp (m/s)w solid flow rate (kg/s)z height(m)Greeka' dimensionless core diameter,a = Dc/ Dt (-)s volumevoidage(-)JL viscosity (Pa s) p density (kg/m3)

Subscripts1 riserexitA annulus C cored denseregion,depositioneq equilibriump particlet tubevex) above TDH (transport disengaging height)

1.INTRODUCTIONCore-annulusflow,i.e. solid upflowin the center (core)and downflow in the wall boundary layer (annulus),is the basic featureofthemacroscopicflow structure in circulatingfluidized beds (CFBs).The existence of core-annulus flow at least in the upperdiluteregionof CFBs has been confirmed for both laboratory-scaleandlarge-scale units [1 ]. It was found that the formation of the core-annulus structure is due to the radial distribution of turbulent intensityin the dilute regionasinvestigated by Horio et al. [5]and Pemberton and Davidson [6]for the freeboardregionofbubblingfluidized beds. It can be easily postulated that the turbulent intensitynear the wall should be smaller than that in the region away from the wall boundary layer; onceparticlesor clusters are captured by theboundary layer, they cannot diffuse back outof this layer. Accordingly, lateral solid transfer from the core to the annulus takes place against the solid fraction gradient.Fromexperimentalevidence,sucha radial movement of clusters to the wall boundary layer in CFBs has been visualized bythelaser sheet technique[7].

Sofar,hydrodynamicmodelsproposedto characterize the solid hold-updistributionin CFBs as a function of gas velocity, solid mass flux,risergeometryandparticle

109characteristics can be roughlyclassified into three categories:(1)those concerned onlywith the axial voidage profile [8-11]; (2) thoseapproximatingtheradialprofile by thedivision of the flow into two or more regions, e.g. core-annulus models [3, 12-22]; and(3)thosepredicting two-phase flow based on the fundamental conservation equa-tionsand with the aid of computationalfluiddynamic techniques [23-25].

Amongthem,categories(1)and(3)are either too simpleor too complicatedforpractical design purposes. Incategory(2),core-annulus flow structure has found widespreadacceptance,but further examination is requirednotonlyfor the feasibilitybut also for the reliabilityof the model parameters.Inpreviousmodels of this kind,e.g.in the Senior and Brereton model [17],asmanyas 10 parameters(atleast three tentativeones)need to be determined. Inthe model of Berruti et al. [19,21,22],theriser was supposedto be divided into an acceleration regionand a fully developed region.However,thismodel,astheyalsodeclared,can not account for 'the situa- tions where a dense bed exists at the base of the riser' [22].Moreover,thismodelemphasizedthe effect of the acceleration term on the pressure drop; however,theacceleration contribution onlyremains less than 1% comparedwith the gravitytermundertheirhighsolid flux conditions 210kg/m2 s), let alone their low solid flux conditions. Therefore,it is difficult to explainthe solid fraction decrease alongthe riser heightwith the model of Berruti et al. The three-phasemodel of Kunii and Levenspiel[20]maybe more realistic than the two-phasemodel but the necessityofsuch a complicationis doubtful. In addition,amongthesepreviousmodels,a constant core diameter was assumed for the upperdiluteregion[3,12-15,18-21]which is in contrast with the experimentalfindingsreviewedbyWerther[1]and Bi et al. [26].

Accordingly,we will focus on the effect of core diameter indevelopingthe CFB model,and then proposeasimplelateralsolid transfer model between the core and the annulus so that the solid concentration profilein the dilute regionof CFBs can be morereliably predicted.

2. FORMULATION OF THE LATERAL SOLID TRANSFER MODEL FORTHEDILUTE REGION OF CFBS

In this lateral solid transfer model,the downflow of solid inthe annulus and upflowin the core are considered as illustrated in Fig.1. The followingthreelumpingassumptionsare made for the presentformulation:(1)Solid fractions and velocities are radiallyuniform in the core and the annulus. (2)The solid transfer flux from regionx (x= C or A)toy (y = A or C)is

proportionalto the solid fraction in regionx,so the net radial solid flux from the coreto the annulus is written as

wherekxyis the lateral solid transfer coefficient from region x toregion y. (3)Solidvelocities in the core and intheannulus, uc andVA,respectively,are

assumed constant over the height.

110

Figure1. Lateral solid transfer model.Defining thesolid fraction in the core in equilibriumwith that in the annulus,

as

we can rewrite equation(1)as

since the net particletransfer is from the core to the annulus,K must be less than unity.The overall solid mass balance at each elevation can be written as

wherewcandwAdenotethe solid upflowrate in the core and the downflow rate intheannulus,respectively.Thearea-averagedsolidvelocity vx (x=C or A)can be definedin terms of wx,thearea-averagedsolid fraction Epxand the dimensionless core diameter a as

Thenequation(4)canbe rewritten as

111wherevois the superficialsolidvelocity.Therefore,mass balances for the solid upflowand downflow are givenas

As an additionalrelationship,the radial averagesolid fraction 8pcan be expressedas:

After the introducingofequation(5)intoequation(7)andrearrangementofequa-tion(6),respectively,thegoverningequationsofthepresentsystemare summarized as

and

Equation(10)expressestherelationshipbetween-,cand-pA-In the later analysis,forconvenience,equation(10)is termed the operatingline of EpcandspAforCFBs,where(1 - (2) ( - v A) I œ2 Vc is termed the slopeoftheoperatingline.

3. PREDICTION OF MODEL PARAMETERS In the above formulation,we have K,UC,VAanda as modelparameters,whicharedeterminedin the followingmanners.

3.1. Lateral (coretoannulus)solidtransfercoefficientkcAIn the dilute regionof a bubblingfluidizedbed,the lateral solid diffusion appearstobe induced bytheturbulence caused bythe bubble eruptionor,in other words,bytheeddies of 'ghostbubbles'[5, 6]. AssumingSherwood number : 4(forturbulentflow)andeddywave number = 4/Dt,Pemberton and Davidson [6]relatedthedepositioncoefficient to the amplitudeofvelocityfluctuations and proposed

where Stokes number S was defined as

112

superficial gas velocity Uo[m/s]Figure2.Comparisonofexperimentalresults for depositioncoefficient(reproducedfrom Bolton and Davidson [27]).

Forcirculatingfluidizedbeds,Bolton and Davidson [27]modified the amplitudeofturbulentvelocityfluctuation u' by

wheretheReynoldsnumberRewasdefined as

Thentheyused the depositioncoefficientkdto estimate the mass transfer coefficient kCAin the upperdiluteregionof CFBs. Based on the correlations in the litera- ture[28-32],theexperimentalconditions of Bolton and Davidson were confirmed to fall in the fast fluidization regime.However,theirexperimentaldata for kdscatteredwidelyas shown in Fig.2(relativedeviation: -35 to +170%)and the effect of the solid flux on kd was not included. Accordingly,in the presentwork,a different kCAisapplied by alsotakinginto account of the effect of solid flux (GSfrom 11.7 to 251kg/m2s in the literature [2-4]),i.e.kcA * kd = 0.02 m/s is assumed for both cases of Weinstein et al. [2]and Horio et al. [3],which lies in the tolerance rangeofthis correlation (+50%),and the estimatedkcA=kd=0.30m/s is used forthecaseof Arena et al. [4].3.2.Equilibriumconstant K There has beanlittleresearch on theequilibriumbetween the solid fraction in thecore and thatinthe annulus. However,theinvestigationof Wirth and Seiter[33]and Gohla [34]showed that there was a linearrelationshipbetweenspc.eq/lpand

113

Figure3.Equilibriumsolid fractions in the core and the annulus (datafrom Wirth and Seiter [33]and[34]).Table 1. CalculatedequilibriumconstantK

EpA,eq/Spas shown in Fig.3 for various operatingconditions,from which equilibriumconstants were estimated and summarized in Table 1. If we justtake account of them in the upperdiluteregion,i.e.z>4.18 m for Gohla's case,it can be noted that the equilibriumconstant K liesalmost in the rangeof 0.35-0.75 forUo=2-7 m/s and K is almost keptconstantalongthe riser heightunder the same operatingconditionsexceptfor the last pairof data. In the followingcalculation,K= 0.35 is chosen for the analysisof Weinstein et al.'s[2]experimentalresult,K = 0.5 for that of Horio et ul.'s [3]experimentalresultand K = 0.10 for that of Arena et ul.'s [4]experimentalresult under the consideration of the solid flux effect.

114Table 2. Calculatedvc,VAandVCIVAbased on experimentaldata

3.3. The choice of area-averaged solidvelocityratiovc/vAInsome of the previousmodels,the solid velocityinthe annulus was assumed con- stant,e.g. -1.1 m/s s in the Senior and Brereton model [17],oras the particleterminalvelocity,e.g.inthe Berruti and Kalogerakismodel[13],which was not consistent withtheexperimentallyobserved results. Fromequation(10)andthe later stated boundaryequation(27),it can be found that we justneedto know the value of uc/uAotherthan each specificvalue of ucorvAsinceucisdetermined from the operatingsolidflux and the exit solid fraction. Table 2 lists the experimentresults in the literatureforvc, VA and the ratio uc/L'A-Itcan be found that v?/vAlies in the regionof -1.75 to -2.61 for Uo= 1.17-4.0 m/s and Gs= 11.7-98kg/m2 s. Forsimplicityof the modelcalculation,the ratio of the solid velocityinthe core to that in the annulus is chosen as -2, i.e.VC/VA= -2.3.4.Dimensionless core diameter a alongtheheightWithrespectto the core diameter Dc,several correlations have been proposed[1, 26, 39, 40], which are summarized in the form of dimensionless core diameter a as follows:

115As for the PatienceandChaoukicorrelation,the effects of gas velocity, solid cir-

culation flux and particle density were taken intoaccount,but it cannot predictthecore diameter variation with height,whichimpliedthat a did not varyalongthe riser height.In the latter two correlations of Bai et al. andBietal.,the solid fraction in the riser should be known first. In the Werther correlation,the effects of solid circulation flux and exit restriction are not included,but this correlation has been confirmedalsobythe data from large-scaleboilers,i.e.Duisburg plant (Dt= 8m) [1] and EDF E. Huchet plant(Dt= 9.65m,125MW) [41], so it seems to be a goodcandidate.Accordingly,inthepresentcalculation,Werther's correlation is adopted.

4. SOLUTION FOR A CONSTANT CORE DIAMETER AND ITS ASSOCIATED DIFFICULTY

EliminatingspAbetweenequations(2)and(10),we can rewriteEpC,eqas

AfterintroducingEpC.eqintoequation(9)andintegratingfor-pcwith a constant core diameter,we obtain

wherezdis the lower denseregion height and

116

Figure4. Paradox of operatinglinesin the phase plane ofFpAversusEpcfor constant a.

spoois obtained by eliminating EpAbetweenequations(8)and(10)andintroducingequation(22)intoequation(8)forz-+ 00 as

Although equation (20)has the same form as the empirical expression proposed by Zenzand Weil [42],there is a serious difficultyinequations(21)and(22) (or (23)). That is ifuo, uc and(-vA)are all positive,wemayhave either negative decay factora or negativesolidfraction Both a and are all positive only whenthe solid flow in the annulus is upwardor the net solid flux is downward,which is analyzedas follows:

Inequations(21)and(22),the same partis

If a and are all positive,

and

117should have the same signforpositiveornegative.Because the solid flow is upwardin the core of a CFB,i.e.vc>0,we have the positive .

Therefore,

should be positive.Thiswilllead to two situations: (1)whenvoispositive,VAshouldbepositive(i.e.the solid flow VAin the annulus is upward),whichimpliesthere is no solid downflow in the annulus and (2)when solid flowinthe annulus is downward(i.e.VA<0), uo should be negative,whichimpliesthecirculatingsolid flux intheriser is downward (i.e.there is no upward circulating solid flux intheriser).Figure4demonstrates this paradoxwhenvo,vcand(-vA)are all positive.ForoperatinglineI, <0 while a >0;foroperatinglineII,eP?,?> 0 while a <0.

5. SOLUTION TOVARIABLECORE DIAMETER IN THE LATERAL SOLID TRANSFER MODEL

Fromequations(9)and(10),theoperatingline for a variable a case satisfies thefollowing equation (referto the derivationinAppendixI):

where

Accordingly,theslopeof the operatinglinechangeswiththeheightin contrast to thefollowingconstantslopeforthe case of constant a

Sinceequation(24)cannotbe solved analytically,it is numerically integrated withWerther's correlation (i.e.equation(16)).Inthiscase,theoperatingline,i.e.EpAversusEpcforms a curve instead of a straightone.

118Theboundaryconditionsatz= L,which are derived from equations(2)and(16)

and limit analysisonequations(6), (8), (10) and(24) (refer to the derivation inAppendixII),are as follows:

By solving equations (8)and(10),theinitialvalues of solid fractions epa.dandEpC.dat the denseregion height Zdforintegrationare obtained from

Forthe time being,8pdatZdis determined from experimentaldata. The computationprocedureis shown in Fig.5.

Figure5. Calculation procedureof the lateral solid transfer model for variable a.

1196.DEMONSTRATION OF THESIGNIFICANCEOF VARIABLE CORE DIAMETER

INTHE LATERAL SOLID TRANSFER MODEL To validate the model derived above,acomparisonwas made for the calculated and experimentalaxial solid fraction profile.Theoperatingconditions and experimentalresults in the literature [2-4]are listed in Table 3. The numerical integrationfor the solid fraction was performedwith the Runge-Kuttamethod from the dense regionheightZdto the riser exit;boundaryconditions and initial values are determined from equations(26)-(31).

The calculated results are givenin Table 4. Figure6a shows the phaseplaneandcomparisonof the calculated with the observed axial profileof8pfor Horio et al.,Fig.6b for Weinstein et al. and Fig.6cfor Arena et al. For the variable corediameter,theoperatingline bends whenitproceedsto the equilibriumline in theupper region as shown in the insets in Fig.6. The axial solid fraction profilesshow fair agreementwith the experimentalresults. This indicates that the decay phenomenon inthe dilute regionof CFBs can be interpreted by thepresent two-region lateralsolid transfer model.

In the case of a variable core diameter,the net solid flux is alwaysequalto the experimentalGs.However,if the core diameter is assumedconstant,it turns out thatthe net solid flux becomes negativewith the realistic value of dspc/dspAasanalyzedabove.Accordingto the numerical results of Horio et al. [3]in Table 4,theslopeofspAversusEpcatz = Zdis calculated from equation(24)as:

Adoptingthe above 'realistic' slopeof the operatingline at z=zdandsubstitutingitintoequation(25)for the constant core diameter model,weobtain

Table 3. Operatingconditions and experimentalresults

120Table 4. Calculatedresults for the variable core diameter

Figure6.Significanceof variable a in the lateral solid transfer model and comparisonwithexperimentaresults:(a)forHorioetal.[3]; (b) for Weinstein et al. [2]; (c) for Arena et al. [4].

Then the net solid flux is calculated as

This is an exampleof the problemsforthe constant core diameter model.

121

Figure6.(Continued).7. EFFECTS OF MODEL PARAMETERS kCAANDuc /uA ON THE OPERATING

PHASE PLANE AND AXIAL SOLID FRACTION PROFILES InFig.7,theeffect of the lateral solid transfer coefficientkcAon the operating phase planeand solid fraction profilesis shown. If kCAis too big,the contribution of variable core diameter is suppressedand even negative8p,ooappear;if it is too small,the effect of the variable core diameter alongtheheightis so significantthat the equilibriumwillbe reached in the middle of the riser height.Accuratepredictionof

122

Figure7.Effect of keAonoperatinglines and axial solid voidage profiles.

Figure8. Effect of UC/VAonoperatinglines and axial solid voidage profiles.

kcAis thus necessaryin the lateral solid transfer model to predictthe solid fraction in the dilute region.

The effect of v?/vAon the operating phase plane and solid fraction profileisshown in Fig.8. It can be known from equations(30)and(31)that the variation ofVCIVAleads to different solid fractions inthe core and the annulus at the dense region height Zdwhichcorrespondto the initial drivingforce of the lateral solid transfer.Ifuc/(-uA)is too small,thedrivingforce at Zdis too big,so that the

123solid transfer equilibriumcannot be reached at the riserexit;on the other hand,ifuc/(-uA)is too large,theinitialdrivingforce at Zdis too small so the solid transferreaches the equilibriummuch below the riser exit. Since ucin this model is justdeterminedbytheoperatingconditions(seeequation(27)),the variation of VC/(-VA)reflectsthe variation of -Up.Such a variation of -vA willstronglyinfluenceparticleagglomerationandclusteringintheannulus,and then affects the lateral solid transfer from the core to the annulus.

Therightdirection of CFB research should be to focus on the clusteringbehavior of suspensionasalreadystressedbyIshii et al. [14]and such a meso-scale suspensionstructure has been recently investigated ina more precise way by the laser sheet technique[7, 42]; somesimplifiedmodels are still preferredtocompletethepresentmodelcapableofpredictingthe axial distribution onlyfromoperatingconditions.

8. CONCLUSIONS A lateral solid transfer model based on the mass transfer between the core and the annulus in the dilute regionof CFBs is solved for both constant and variable core diameter cases. The variable core diameter model with anempiricalcore diameter correlation can predictthe solid volume fractionsfairlywell. The constant core diam- etermodel, however, fails to satisfyall the requirementsofthepositive equilibrium solidfraction,thepositive decay factor and the positivesolid mass flux. It was made clear that in the core-annulus solid transfer model the core diameter variation is a keyfactor to obtain realistic predictions,whichindicates that the variable core diameter has a significantrole in developingtheCFB model. Further investigationis needed for the determination of kCAandVCIVA.

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APPENDIX I: DERIVATION OF EQUATION (24) Rearranging equation (9),we obtain

Fromequations(5)and(7),we can write

Accordingly,we have

126Introducing equation (1.1)intoequation(1.3)yields

Substitutingspi= '0 intoequation(10)andrearrangingwe obtain

Accordingly, equation (24)is obtained by introducing equation (1.5)intoequation(1.4):

APPENDIXII: DERIVATION OFEQUATION (29) Substituting equations (27)and(28)intoequation(10)andperformingthe limit anal- ysisforz L,wecan write

On the other hand,dividing equation (1.1)by(1/œ2)(dœ2/dz),we obtain

Whenz - L,we have (KspA - 8,C)z=L = 0. Accordingly, equation (11.2)isrewrittenas

127

Eliminating6p,betweenequations(IL 1) and(11.3)andrearranging,we can write

Introducing equation (11.4)intoequation(25)forz-j L,weobtainequation(29)as follows: