EVALUATION OF A SUB-GRID MODEL IN A TURBULENT CIRCULATING FLUIDIZED BED

Transcription

1 Proceedin of COBEM 007 Copyriht 007 by ABCM 19th International Conre of Mechanical Enineerin November 5-9, 007, Braília, DF EVALUATION OF A SUB-GRID MODEL IN A TURBULENT CIRCULATING FLUIDIZED BED Ivan Carlo Geor SINMEC Laboratory of Numerical Simulation in Fluid Dynamic and Computational Heat Tranfer Dept. of Mechanical Enineerin Federal Univerity of Santa Catarina Potal Office Box 476 Zip Code: Florianopoli, SC, Brazil ivan@inmec.ufc.br Clovi R. Malika SINMEC Laboratory of Numerical Simulation in Fluid Dynamic and Computational Heat Tranfer Dept. of Mechanical Enineerin Federal Univerity of Santa Catarina Potal Office Box 476 Zip Code: Florianopoli, SC, Brazil malika@inmec.ufc.br Luimar Marque Porto Dept. of Chemical Enineerin Federal Univerity of Santa Catarina Florianopoli, SC, Brazil luimar@enq.ufc.br Abtract. The 3D tranient hydrodynamic of a-olid flow in a turbulent circulatin fluidized bed i numerically tudied. The rid refinement i carefuly handled to be adequate for capturin the tructure of alomerated particle (cluter and trand) which are found in the o called meo-cale. A ub-rid model (LES) i applied for the a phae and extended for the olid phae. The numerical reult are compared with the experimental one with ood approximation for the averae axial velocitie for the olid phae. The ub-rid model employed ive ood behavior at near wall and intermediate reion, with deviation in the central reion (dilute) by over-etimatin the dra. Thi deviation eem to be caued by the way dra i calculated by the two-fluid model. Spectral analyi i alo performed for the tudied cae. The inertial reion wa oberved by the patial and temporal reolution, to have a -5/3 lope, indicatin that the refinement in pace and time wa adequate. Keyword: two-phae a-olid flow, turbulence, fluidized bed, cluter, ub-rid model. 1. INTRODUCTION Circulatin a-olid fluidized bed (CFB) are common device found in many operation in the chemical, petroleum, pharmaceutical, aricultural, biochemical, food and power eneration indutrie. The hydrodynamic behavior of fluidized ytem i non-linear and therefore very complex. Scale-up from lab-cale to indutrial cale, a tron requirement when deinin new equipment, i a major problem. The nature of hydrodynamic interaction at variou cale in uch device preent reat challene when attemptin to model them mathematically. The mot common practice i to ue averaed tranport equation for two-phae flow, van Wachem (000), Zhan and VanderHeyden (001), Arawal et al. (001), Huilin et al. (003), amon everal other. While thee effort preented ome ucce, the etate-of-the-art for two-phae flow imulation i till far from ideal. Meo-cale tructure in the form of cluter and trand are commonly oberved in dilute a-particle flow. Thee tructure affect the overall flow behavior, with reatly influence in momentum and enery exchane, and hould therefore be accounted for in computational fluid dynamic (CFD) imulation of CFB. Unfortunately, the meo-cale tructure cannot be adequately reolved in CFD imulation of typically ized CFB. The oal of the preent tudy i to evaluate a ub-rid model to account for the effect of the unreolved meocale tructure in a circulatin a-olid fluidized bed and compare to a well characterized fluidized bed experiment performed by van den Moortel et al. (1998). In what follow, we dicu ome backround on the averaed equation for two-phae a-olid flow and preent the ub-rid model ued and the reult compared with experiment.. CONTEXT OF THE GAS-SOLID FLOW PROBLEM Computational fluid dynamic (CFD) i an emerin technique for predictin the flow behavior of many ytem required cale-up, dein or optimization. Althouh inle-phae CFD tool are widely and uccefully applied

2 (Anderon, 1995), multiphae flow remain unreolved due to the complex mathematic and the difficulty in the phyical decription (van Wachem, 000). CFD model for a-olid flow can be divided into two roup, Laranian and Eulerian model. Laranian model or dicrete particle model, calculate the path of each individual particle. The interaction between the particle can be decribed by a potential force and by colliion dynamic. The drawback of the Laranian technique are the lare memory requirement and the lare calculation time. Moreover, the decription of the dra force from the a phae actin on each particle i difficult to model accurately, and thi i in practice baed on experimental correlation. In other word, it i impoible to track all a/olid interface which exit in a real flow. Eulerian model treat the particle phae a a continuum performin an averae on the cale of individual particle. Computation by thi method can predict the behavior of dene-phae flow on a realitic eometry, van Wachem (000). The drawback of thi method, however, i that one require complex tatitic to tranlate the behavior of many particle into one continuum. For thi purpoe, often ranular kinetic theory i enerally employed. At a baic level, interaction between the fluid and particle are imply modeled with a dra force. A we know from inle-phae flow, the nonlinearity of the convective term in the momentum equation ive rie to the Reynold turbulence tenor. In two-phae flow, there are additional nonlinearitie that rie from the mamomentum couplin in both the ma and momentum conervation equation, Zhan and van der Heyden (001). Both thee nonlinearitie ive rie to meo-cale phenomena called particle cluterin in which particle areate into dynamic tructure with hiher than averae volume fraction of olid. Thi phenomenon i oberved both experimentally and numerically. Cluter formation i oberved in fine-rid (5-10 particle diameter (Arawal et al. 001 and particle diameter (Zhan and van der Heyden, 001, Geor, 005). When the fine rid calculation reult are averaed and compared with coare-rid calculation reult, different flow field are obtained. Coare-rid calculation filter out the meo-cale effect. Becaue of the meo-cale effect influence the macro cale behavior, they hould be properly taken into account in the coare-rid calculation by ue of appropriate ub-rid model for the apparent vicoity, for example. Horio and Kuroki (1994) meaured the typical cluter diameter which wa time the particle diameter. The downward motion of a cluter caue a major variation in the local velocity profile and hould therefore lead to a inificant increae in the production of a-phae turbulence ariin form tiff radient in mean velocitie, Geraldine et al. (004) and Geor (005). Thi idea i upported by the computational data of Arawal et al. (001), in which a tron increae in a-phae turbulence production reultin from cluter formation wa found. Cluterin can increae the a-phae turbulence throuh other mechanim a well. Becaue the effective cluter diameter i lare, the cluter will interact with the a imilarly to a lare particle, even if only temporarily. It i known that a lare diameter particle can inificantly enhance the a-phae turbulence throuh the vortex-heddin mechanim and throuh the formation of temporary wake behind the (larer) particle or the cluter, Geraldine et al. (004). Another mechanim i by the vortex tretchin aociated with the mall area available for the a flow caued by cluter formation near wall, Geor (005). The purpoe of the preent tudy wa to better undertand the baic behavior of the imple a-olid flow equation before introducin any particle-particle interaction model. By comparin numerical reult with experimental data we are able to make ome quantitative aement of the adequacy of the imple a-olid flow equation and the ub-rid model propoed. We alo ue tranient hih reolution imulation to attempt to fully reolve the meo-cale tructure in the flow. 3. AVERAGED EQUATIONS FOR TWO-PHASE FLOWS The majority of the author, to obtain the overnin equation for multiphae flow, refer to the pioneerin work of Anderon and Jackon (1967) or Ihii (1975). Anderon and Jackon (1967) and Jackon (1997) ue a formal mathematical definition of local mean variable to tranlate the Navier-Stoke equation for the fluid and the Newton equation of motion for a inle particle directly into continuum equation repreentin momentum balance for the fluid and olid phae. The local variable are averaed over reion lare with repect to the particle diameter but mall with repect to the characteritic dimenion of the complete ytem, to account for variation in the domain. The averain procedure, (ee Jackon, 000), can be applied to the equation of motion. Startin with the equation of ma conervation, for a incompreible fluid, it take the form ( φ ) + ( ρ φ U ) = 0 ρ, (1) t For the olid phae, one ha t ( φ ) + ( ρ φ U ) = 0 ρ, ()

3 Proceedin of COBEM 007 Copyriht 007 by ABCM 19th International Conre of Mechanical Enineerin November 5-9, 007, Braília, DF where φ and φ are the volume fraction for the a and particle phae and U and U are the averaed velocitie of the a and particle phae. The reultin momentum balance for the fluid and olid phae are a follow ( φ U ) + ( ρ φ U U ) = P + [ t ] + φ ρ + β ( U U ) ρ (3) t ρ. (4) t ( φ U ) + ( ρ φ U U ) = P + [ t ] + φ ρ + β ( U U ) The firt term on the riht hand of the a phae of motion repreent the a preure, the econd term i the averaed hear tenor of the a phae, the third term repreent the ravity force and the forth term i the traction exerted on the a phae by the particle, the dra force. The firt term on the riht hand ide of Eq. (4) i the olid preure, the econd term i the averaed hear tenor of the olid phae the third term repreent the ravity force on the particle and the forth term i the dra force contribution to the olid phae. The a tre wa calculated uin conventional Newtonian form and a modified one uin Smaorinky-like ubrid model a follow [ t ] = φ [ U + U ] µ φ [ U ]I T µ, e, e 3 1/ 3 ( 0,1 ) ( t ) 1/ = ( x y z), e = µ + ρ t,, (5) µ (6) where µ, e i the effective vicoity of the a phae modeled by a Smaorinky ub-rid model and µ i the a phae vicoity. In the ame way for the particle olid phae, one ha [ t ] = φ [ U + U ] µ φ [ U ] I 3 T µ, e, e (7) 1/ 3 ( 0,1 ) ( t ) 1/ = ( x y z) µ (8), e = µ + ρ t, where µ, e i the effective vicoity of the olid phae and µ i the olid phae vicoity and i the filter width. The olid preure are iven by Gidapow (1994) relation P P = G( φ ) φ, G( r ) =, (9) φ G ( φ ) G exp( c( φ )), (10) = 0 φmax where φ max = 0,637, G 0 = 1 Pa, c = 600. c i the fictitiou peed of ound for the particle material when it i cloely packed, and φ max i the cloe packin volume fraction for particle. The effect of particle preure radient term i to keep the particle apart o that the calculated particle concentration doe not exceed the maximum concentration obtainable for a iven ize ditribution of the particle. The a-olid inter-phae dra coefficient, Gidapow (1994), are 3 ρ φ U U 1, 65 β, = CD φ, φ 0,8 4 d p p 3 150φ µ ρ φ U U β, = CD + 1,75, φ < 0,8 4 d φ φ d p (11) (1)

4 0, ( 1+ 0,15Re ) Re < 1000, C = 0,44 Re C D = D (13) Re ρ φ U U d p Re =. (14) µ Where d p i the particle diameter. Equation (1-14) are known a the Two Fluid Model and it model an iothermal a-olid two-phae flow. In the olid and a phae a Smaorinky like ub-rid model i incorporated in one of the imulated cae (Cae ). 4. DESCRIPTION OF NUMERICAL SIMULATION Simulation performed in thi tudy ued the eometry decribed by van den Moortel et al. (1998). The croection of the fluidized bed i 0x0 cm and the total heiht i 00 cm. The eometry and rid ditribution are illutrated in Fi. 1. The dimenion of each volume of the rid wa 0.65x0.65x1.33 cm 3, with volume. Fiure 1. Computational domain and rid. van den Moortel et al. (1998) reported they ued particle with mean diameter of 10 µm with tandard deviation of the particle diameter of 0 µm. In thi paper we inored the particle ize ditribution and ued the mean diameter. Uin thi particle diameter we have about 5x5x110 particle encloed in a x,y,z volume for the rid refinement ued. Vicoity of the particle phae wa et to be k/m., Gidapow (1994), and the particle material denity ρ wa et to be 400 k/m 3 a in the experiment. We ued a a denity of 1. k/m 3 and vicoity of 1.7 x 10-5 k/m.. The numerical imulation were carried out uin the CFD code ANSYS CFX 5.6, which ue an element-baed finite volume cheme. We implemented the ub-rid model in the multiphae code verion of CFX 5.6 uin hiher reolution cheme for pace dicretization and a econd order backward difference cheme for the time dicretization. The time tep ued for all imulation wa 1x10-3 econd. In the imulation, the boundary condition on the olid wall wa treated a no-lip for both phae. Generally the free lip boundary condition are ued for the olid phae. Jackon (000) uet a boundary condition for de olid phae an enery balance. Andrew et al. (005) conclude that different wall boundary condition only contribute to a mall chane near the wall reion. A we do not account for enery balance, and baed on the recommendation of Andrew (005), one chooe no lip boundary condition for the olid phae. The boundary condition for the inflow velocitie at the bottom of the device i aumed to be uniform. The flux of the a i determined by the uperficial a velocity reported by van den Moortel et al. (1998). The volume fraction of particle inflow i aumed to be 0.4, which i the value for the minimal fluidization condition. The uperficial velocity of the a phae wa 1 m/ for accordin to van den Moortel et al. (1998) data. Simulation tart with the particle uniformly ditributed in the fluidized bed and the velocitie for both phae et to zero. The imulation were continued until about 15 econd of real time. The tatitical teady tate wa reached in

5 Proceedin of COBEM 007 Copyriht 007 by ABCM 19th International Conre of Mechanical Enineerin November 5-9, 007, Braília, DF our imulation after about econd of real time, knowin that the characteritic time of the flow i 0. econd and the relaxation time for the particle phae i 0.11 econd, providin a Stoke number of 0.55 and a particle Reynold number, Re p ~6. For the a phae the Kolmoorov lenth cale i η ~0.015 cm and the time cale i about of 0.00 econd. Note that with thi cale we do not reolve all the lenth cale for the a flow. For the particle flow the rid ize and time reolution i adequate to account for the preence of cluter and trand upported by the experimental reult of Horio and Kuroki (1994). 5. NUMERICAL VERSUS EXPERIMENTAL RESULTS Experimental reult of van den Moortel et al. (1998) for the cae of 1 m/ uperficial a velocity were ued. We compare our averaed numerical reult, after tart-up in the imulation. Firt we how the fine meo-cale tructure, particle cluter and trand. a) b) c) d) e) f) Fiure. Particle volume fraction contour on a mid-plane (z=10 cm). Fiure a), c) and e) denote Cae 1, without ubrid model, at 3.8, 4.0 and 4. econd; Fiure b), d) and f) denote Cae, with ub-rid correction for both phae, at 3.8, 4.0 and 4. econd. Gray cale repreent particle alomeration and white color repreent a phae. Fiure how imulation for two cae, Cae 1, without uin ub-rid model and Cae uin ub-rid model for both phae. What we ee in thi fiure i qualitatively imilar to reult obtained by Arawal el al. (001). The trand are thin and oriented in the axial direction and cluter are preent near the wall. The bottom reion of the bed i dener than the top reion, characterizin a turbulent fluidized bed reime. We can ee that trand do not tay only near the wall, appearin at the center of the bed, oriented ome time in the lateral direction. Cluter and trand are continuouly formed and detructed, ivin rie to lateral fluctuation of the olid volume fraction. A the a phae pae throuhout the cluter it carrie olid, formin new tructure, but now more dilute. Cae 1 i dener near the bottom of the bed than Cae, but for both cae the meo-cale are preent and were adequately captured. For the Cae the tructure appear thinner and the top of the bed i more dilute than Cae 1. A mentioned above, Arawal et al. (001) obtained computed meo-cale tructure in their a-olid flow imulation. There are, however, ome key difference between their reult and the one preented in thi work. Arawal et al. (001) computed on a mall two-dimenional 1cm x 4 cm periodic domain, which i much maller than ued in thi work. While our mallet pace increment wa 0.65 cm, Arawal et al. (001) computed pace increment

6 a fine a cm. For larer meh ize, the numerical reult inificantly over-predict the particle ma flux, becaue the meo-cale tructure are not reolved adequately by coare rid, Geraldine et al. (004). Inide a particle cluter the particle phae feel le dra ince it fall in the wake enerated by the leadin part of the cluter. In the cae of a coare rid, thi effect i not reolved, o particle dra i overetimated, Arawal et al. (001). Thi effect i more pronounced at dilute reion of the bed a we will ee in our averaed reult. However, the reult preented in Fi, reveal qualitatively all meo-cale tructure pointed out by Arawal et al. (001) and the characteritic of the turbulent fluidized bed reime are adequately repreented. The averaed reult are calculated uin Favre averae of volume fraction and velocity of each phae, the ame method ued by van den Moortel et al. (1998) for their experiment, a follow φk t φ k =, (15) t u i, k = φkui, k t. φ t k The ubcript i=,p for the a and the particle phae, repectively. Averaed velocity of each phae, defined above, i weihted by the volume fraction of the particle phae. Fiure 3 how the averaed particle vertical velocity for Cae 1 and. Calculated particle velocitie aree with the trend of the experimental reult overetimatin the downward velocitie near the wall, while in the center reion (dilute) the areement i ood, Fi. 3 (a). (16) a) Averaed Particle Vertical Velocity [m/] van den Moortel et al. (1998) Cae 1 - Without Sub-Grid Model dilute reion Lateral Ditance [m] wall reion b) Averaed Particle Vertical Velocity [m/] dilute reion van den Moortel et al. (1998) Cae - Sub-Grid Model for both phae wall reion Lateral Ditance [m] Fiure 3. a) Averaed particle vertical velocity profile for Cae 1 at z= 1m from the bottom of the bed; b) averaed particle vertical velocity profile for Cae at z= 1m form the bottom of the bed. Experimental reult from van den Moortel et al. (1998). When the reult for Cae are compared with experiment, Fi. 3 (b), the particle velocitie near the wall aree very well, however, at the dilute reion the velocitie are even more overetimated more than for Cae 1. Comparin the reult for Cae 1 and we ee that the ub-rid model obtained a very ood approximation at the dene reion of the bed what doe not occur when no ub-rid correction i applied. Thee reult how that the ue of the ub-rid model can be appropriated accordin to the reion of the flow. Near the wall we would ue the ub-rid model, and at ome ditance from the dene reion it could be ued a modification of the preent ub-rid model. We tre that we do not make any tet to modify the ub-rid contant (Eq. 6 and 8). 5.1 Power pectrum denity In thi ub-ection we how the power pectrum denity (PSD) for the two cae tudied. Fiure 4 how the total enery pectrum for Cae 1, Fi. 4 (a), and the particle velocitie fluctuation power pectrum denity, Fi. 4 (b). A een in Fi 4 (a), the total enery i reater at the central reion than near wall, for the lare lenth cale. The inertial zone bein around 0.4 Hertz for the center velocitie and 0. Hertz for the wall. Similar frequency ha been reported by Neri and Gidapow (000), uin kinetic theory approach for particulate phae. Thi frequency repreent the formation and detruction of cluter tructure. Thee frequency ocillation of a-olid flow could be ued to know the minimum time required to conduct proper time averain, which for our cae i 5. Note that we take averae over 13.

7 Proceedin of COBEM 007 Copyriht 007 by ABCM 19th International Conre of Mechanical Enineerin November 5-9, 007, Braília, DF (a) 0.1 (b) 0.01 PSD of particle velocitie at wall reion PSD of particle velocitie at central reion PSD of particle velocitie fluctuation at wall reion PSD of particle velocitie fluctuation at central reion 1E-3 1E-3 1E-4 1E-4 Power Spectrum Denity 1E-5 1E-6 1E-7 1E-8 Slope = E-5 1E-6 1E-7 1E-8 Slope = E-9 1E-9 1E-10 1E-10 1E E Frequency [Hz] Frequency [Hz] Fiure 4. a) Power pectrum denity of the imulated particle velocitie for Cae 1; b) Power pectrum denity of the imulated particle velocitie fluctuation for Cae 1. (a) E-3 PSD of particle velocitie at wall reion PSD of particle velocitie at central reion (b) E-3 PSD of particle velocitie fluctuation at wall reion PSD of particle velocitie fluctuation at central reion 1E-4 1E-4 Power Spectrum Denity 1E-5 1E-6 1E-7 1E-8 1E-9 1E-10 Slope = E-5 1E-6 1E-7 1E-8 1E-9 1E-10 Slope = E-11 1E-11 1E-1 1E-1 1E-13 1E-13 1E E Frequency [Hz] Frequency [Hz] Fiure 5. a) Power pectrum denity of the imulated particle velocitie for Cae ; b) Power pectrum denity of the imulated particle velocitie fluctuation for Cae. The power pectrum denity for the fluctuation of the particle velocitie for the central reion, Fi. 4 (b), ha more enery than for the wall reion. The inertial reion decay with -5/3 bein about 0.4 Hertz for the center and about 0.7 Hertz for the near wall reion. The turbulent kinetic enery at central reion i reater than wall reion. From 10 Hertz the lope for the particle velocitie PSD at the wall i aumented inificantly, indicatin major diipation at thi reion, where cluter are encountered with major frequency which contribute with the reater diipation. The Kolmoorov -5/3 law i obeyed in the inertial rane, like the experimental reult for a-liquid-olid flow obtained by Cui and Fan (004) with the ame particle diameter ued here and there obtained by Moran and Glickman (003) for the ame particle diameter for a-olid flow. Fiure 5 (a) how the total pectrum of enery for the particle velocitie at central and near wall reion for Cae. The central reion ha more enery aociated with lare cale than wall reion. Inertial ub-rane bein about 0.45 Hertz for central reion and 0.3 Hertz for wall with -5/3 decay. The power pectrum denity of the velocitie fluctuation, Fi 5 (b), ha the ame behavior a Cae 1, Fi 4 (b). From 10 Hertz the diipation of kinetic enery i major than Cae 1 for near wall reion. In both cae 1 and, at low frequencie, we ee coherent tructure repreented by the reat peak. A the lare cale tructure are broken thee peak are moothed to form the inertial reion. The firt peak appear about 0.3 Hertz repreentin the natural frequency of a-olid flow in a fluidized bed. The reult preented in thi ection how that the ub-rid model ued wa appropriate to repreent the characteritic behavior of a-olid flow in a circulatin fluidized bed, a mentioned above, epecially near the wall reion of the bed. The very ood approximation of the averaed velocity to the experimental data near the wall for the

8 ub-rid model reult, Cae, how the applicability of thi methodoloy. The power pectrum denity reult are coherent with experimental data and the reult obtained by kinetic theory. 6. CONCLUSIONS Numerical imulation uin the Two Fluid Model for multiphae flow were performed with the appropriate rid refinement to account meo-cale tructure like cluter and trand. Comparion were made with experimental reult howin ood areement for certain reion of the flow uin a ub-rid model. The outcome of the work uet that Smaorinky-like ub-rid model could be ued in a-olid flow with proper modification for the dilute reion of the circulatin fluidized bed. Very ood areement for the near wall averaed particle velocitie wa found. The characteritic behavior of a turbulent fluidized bed wa well repreented with the bottom of the bed more dene than the top with cluter and trand preented in all location of the bed. The reult for the Cae 1 howed that the bottom reion of the bed wa dener than for the Cae. Spectral profile how that the diipation i reater near the wall and the decay of the power pectrum obey the Kolmoorov -5/3 law, indicatin that the refinement in pace and time wa adequate. The natural frequency calculated by kinetic theory i captured. Generally in a-olid flow the particle vicoity i modeled a an empirically function of the particle volume fraction (e.. Huilin et al. 003). Reult preented in Fi. 3 (a) for Cae 1 (where the vicoity wa maintained contant), uet that thi function i not needed for the fluidized bed tudied here. For the Cae the vicoity wa modeled with ub-rid correction which are a function of the train tenor. Reult for Cae, Fiure 3 (b) how a modified profile of the Favre averaed particle velocitie in the central reion when compared with the Favre averaed particle velocitie for Cae 1. Becaue the vicoitie of Cae 1 i contant and for Cae are not, we may conclude that the deviation of reult for Cae i oriinated from thi difference. A the tre for a and olid phae, Eq. 5 and 7, are dependent on the volume fraction, we could conclude that the volume fraction already contribute to modify the tre and may not necearily be included in the vicoity formulation. It i important to mention that the averaed reult for Cae 1, Fi. 3 (a), uet that a refinement of the rid will produce better reult, dipenin the ue of a ub-rid correction, but aumentin the computation cot. 7. ACKNOWLEDGEMENTS The author ratefully acknowlede the financial upport for thi work by the PRH09 - ANP/MCT (Aência Nacional de Petróleo, Gá e Biocombutívei), throuh a cholarhip for the firt author. 8. REFERENCES Andrew IV, A.T., Loeezo, P.N. and Sundarean, S., 005, Coare-rid Simulation of Ga-Particle Flow in Vertical Rier, Ind. En. Chem. Re. Vol. 44, No. 16, pp Arawal K., Loezo, P.N, Syanmlal, M., Sundarean, S., 001, The role of meo-cale tructure in rapid a-olid flow, Journal of Fluid Mechanic, Vol. 445, pp Anderon, J.D., 1995, Computational Fluid Dynamic: The baic with application. Mechanical Enineerin Serie. McGraw-Hill, New York. Anderon, T.B, and Jackon, R., 1967, A fluid mechanical decription of fluidized bed. Ind. Enn. Chem. Fundam., Vol. 6,, No. 4, pp Cui, Z. and Fan, L.S., 004, Turbulent enery ditribution in bubblin a-liquid and a-liquid-olid flow ytem, Chemical Enineerin Science, Vol. 59, pp Gidapow, D., 1994, Multiphae Flow and Fluidization. Academic Pre, San Dieo, firt edition. Geor, I. C., 005, Modelaem e imulação numérica tridimenional traniente do ecoamento á-ólido em um reator de craqueamento catalítico em leito fluidizado, PhD Diertation, Departamento de Enenharia Mecanica da Univeridade Federal de Santa Catarina. Geraldine, J.H., Da, A. K., de Wilde, J., Marin, G.B., 004, Effect of Cluterin on Ga-Solid Dra in Dilute Two- Phae Flow, Ind. En. Chem. Re., Vol. 43, pp Horio, M., Kuroki, H, 1994, Three-Dimenional Flow Viualization of Dilutely Dipered Solid in Bubblin and Circulatin Fluidized Bed, Chemical Enineerin Science, Vol 49, pp Huilin, L., Gidapow, D., Bouillard, J., Wentie, L., 003, Hydrodynamic imulation of a-olid flow in a rier uin kinetic theory of ranular flow, Chemical Enineerin Journal, Vol. 95, pp Ihi, M., 1975, Thermo-Fluid dynamic theory of Two-Phae Flow. Direction de etude et Recherche d Electricité de France. Pari, France. Jackon, R., 1997, Locally averaed equation of motion for mixture of identical pherical particle and a newtonian fluid, Chem. En. Sci., Vol. 5, pp

9 Proceedin of COBEM 007 Copyriht 007 by ABCM 19th International Conre of Mechanical Enineerin November 5-9, 007, Braília, DF Jackon, R., 000, The Dynamic of Fluidized Particle, Cambride Univerity Pre, Cambride Monoraph on Mechanic. Moran, J.C. and Glickman, L.R, 003, Mean and fluctuatin a phae velocitie inide a circulatin fluidized bed, Chemical Enineerin Science, Vol. 58, pp Neri, A. Gidapow, D., 000, Rier hydrodynamic: imulation uin kinetic theory, AIChe Journal, Vol. 46, pp van den Moortel, T. Azario, E., Santini, R., Tadrit, L., 1998, Experimental analyi of the a-particle flow in a circulatin fluidized bed uin a phae Doppler particle analyzer, Chemical Enineerin Science, Vol. 53,, No. 10, pp van Wachem, B., 000, Derivation, Implementation, and Validation fo Computer Simulation Model for Ga-Solid Fluidized Bed, PhD Diertation, Delft Univerity of Technoloy. Zhan, D.Z., van der Heyden, W.B., 001, Hih-reolution three-dimenional numerical imulation of a circulatin fluidized bed, Powder Technoloy, Vol. 116, pp RESPONSIBILITY NOTICE The author() i (are) the only reponible for the printed material included in thi paper.