A REVIEW OF SOME EXSISTING DRAG MODELS DESCRIBING THE INTERACTION BETWEEN PHASES IN A BUBBLING FLUIDIZED BED

Transcription

1 A REVIEW OF SOME EXSISTING DRAG MODELS DESCRIBING THE INTERACTION BETWEEN PHASES IN A BUBBLING FLUIDIZED BED Joachim Lundber 1, Britt M. Halvoren 1,2 1. Telemark Univerity Collee Telemark, Norway 2. Telemark Technoloical R&D Centre (Tel-Tek) Telemark, Norway joachim.lundber@hit.no (Joachim Lundber) Abtract Thi work repreent a computational tudy of ow behaviour in a bubblin uidized bed. The imulation are performed by uin the commercial computational uid dynamic (CFD) code, Fluent 6.3. The advantae of uin a commercial CFD code i that correpondin cae for indutrial application can be imulated by uin the ame model without havin very deep knowlede about the ource code and the olvin alorithm. In CFD imulation of uidized bed, it i important to decribe the interaction between the particle and the momentum tranfer between the phae. Di erent model are developed for thi purpoe. The kinetic theory of ranular ow decribe the interaction between particle and i baed on the kinetic a theory. In a bubblin uidized bed there are reion with rather low fraction of particle and reion with hih particle concentration. The bed can be decribed by two ow reime, the vicou reime and the frictional reime. In the vicou reime the kinetic and the colliional tree are dominatin. The frictional reime occur at hih particle concentration and in thi reime the ow behaviour i decribed by friction and rubbin between the particle. The interaction between the particle and the continuou a phae are decribed by a dra model, and everal dra model are developed for thi purpoe. The model decribe the momentum exchane between the phae. The aim of thi work i to tudy how the di erent dra model in uence on the ow behaviour in a bubblin uidized bed. Five di erent dra model have been tudied. The dra model are the Gidapow dra model, a dra model developed by Syamlal O Brien, a cutomized iterative verion of the dra model by Syamlal O Brien, the modi ed Hill Koch Ladd dra model and the newly developed RUC dra model. Two of the dra model are included in Fluent 6.3, the other model are implemented by the author. The reult from the imulation with the di erent dra model are compared, and the dicrepancie are dicued. 1 Introduction Fluidized bed are widely ued in indutrial operation, and everal application can be found in chemical, petroleum, pharmaceutical, biochemical and power eneration indutrie. In a uidized bed a i pain upward throuh a bed of particle upported on a ditributor. Fluidized bed are applied in indutry due to their lare contact area between phae, which enhance chemical reaction, heat tranfer and ma tranfer. The e ciency of uidized bed i hihly dependent of ow behavior and knowlede about ow behavior i eentially for calin, dein and optimization. Gravity and dra are the mot dominatin term in the momentum equation of the ranular phae. The application of di erent dra model ini - cantly impacted the ow of the ranular phae by in uencin the predicted bed expanion and the particle concentration in the dene phae reion of the bed. Thi paper will focu on the prediction of dra force in a uidized bed. By other reearcher everal dra model are developed. In thi review ome of thi are further invetiated and compared. To decribe the propertie of the particle or ranular phae, model with oriin in the kinetic theory for ranular ow by Lun et al. [1] i choen. Thi paper i baed on the mater thei by the author [2].

2 2 Theory of Modelin Granular ow Modelin of ranular ow i a fairly lare ubject and the focu ha been on the model included in the commercial oftware Fluent 6.3. Fluent ue the Navier Stoke equation to olve a Finite volume model. The Navier Stoke equation are conervation equation of ma, momentum and enery. The enery equation ha not been tudied or ued becaue compreible ow or heat tranfer i not included. The mot relevant multiphae model for imulation of bubblin uidized bed in Fluent 6.3 i the Eulerian multiphae model. Thi model will calculate one tranport equation for momentum and one for continuity per phae. The theory for thi model i taken from the reference [3]. The volume fraction for each phae i calculated with an continuity equation. Equation (1) i a example of the a phae volume fraction equation. + r u = ( _m _m ) =1 (1) Index i the ranular phae and i the a or uid phae. i the volume fraction, i the denity, u i the phae velocity and _m i the ma tranfer from one phae to another. Equation (1) i valid for both the a phae and the ranular phae. Total continuity will be all the volume fraction equation added. The therm r i the reference denity, or the volume averaed denity. The riht hand ide of equation (1) i ued where ma tranfer between phae occur. The momentum equation for the a i like u + r u u (2) = rp + r + nx + (K ( u u ) + _m u _m u ) =1 F + + F lift; + F vm; p i the hydrotatic preure hared by all the phae, i the ravitational force, F i an external body force, F lift and F vm i a virtual ma force. K i the interfacial momentum exchane or dra. Equation (2) can be impli ed to a impler expreion when aumin no ma tranfer between the phae, no lift force and no virtual ma force. The impli ed expreion will be like equation (3). u + r u u = rp + r + +K ( u u ) (3) The i the a phae tre-train tenor. = r u + r u T r u I (4) i the vicoity i the bulk vicoity I i the unit tenor. The aumption for the ranular phae equation (5) i the ame a for the ( u ) + r ( u u ) (5) = rp + r + rp + +K ( u u ) The momentum equation for a and ranular phae i quite imilar except for the ranular preure p in the ranular phae. The tre-train tenor for the ranular phae i like equation (6). = r u + r u T r u I (6) In the modelin of ranular ow a new concept of enery i introduced, ranular temperature. The normal or thermal temperature which i mot common way of thinkin temperature, i a meaurement of the random uctuation of the molecule in any ubtance. Random uctuation will be at a micro level in the molecule. Thi theory i extended to the macro cale where the molecule are ubtituted with particle. Thi i called the Kinetic Theory for Granular Flow (KTGF) and i decribed by Lun et al. [1]. The ranular temperature i a conerved calar olved by a partial di erential equation. The conervation equation for ranular temperature for ranular phae, can be written a equation (7) [4]. ( ) + r ( u ) = : r u r q 3K (7) In word thi equation (7) can be explained a equation (). Tranient term + Convective term () = ranular phae tre -Flux of uctuatin enery - Colliional enery diipation + Exchane term with phae

3 Generation of ranular temperature i due to tree within the ranular phae. i the ranular phae tre and can be written a equation (9). = [ p + r u ] I 2 S (9) S i the deformation rate and i written a equation (10). S = 1 2 h r u + (r u ) T i 1 3 r u I (10) The therm r q decribe the di uive ux of uctuatin or ranular enery [3]. q can be written a equation (11). q = k r (11) k i the conductivity of ranular temperature. i the diipation of ranular temperature. Due to colliion between particle in the ranular phae, the enery in the particle will diipate. The alebraic equation for colliional enery diipation i derived by Lun et al. [1] and howed in equation (12). = 12 1 e2 0; p p d 2 3 (12) When the retitution factor e oe to 1, the diipation of the ranular temperature oe to zero. Thi mean that the particle are perfectly elatic [1]. The exchane coe cient K i the dra factor of the particle. The retitution coe cient e pecify the the coef- cient of retitution for colliion between particle. The coe cient compenate for colliion between particle to be inelatic. In Fluent 6.3 other propertie model are included [3], but it i choen the model mot related to the kinetic theory for ranular ow by [1]. The model ued are Granular vicoity Syamlal et al. [5] = ;col + ;kin (13) ;col = 4 5 0; (1 + e ) r (14) ;kin = p d (15) 6 (3 e ) (1 + e ) (3e 1) 0; Granular bulk vicoity Lun et al. [1] = 4 3 d (1 + e ) r Granular conductivity Syamlal et al. [5] k = 15d p 4 (41 33) (4 3) 0; (41 33) 0; = 1 2 (1 + e ) Radial ditribution function Lun et al. [1] 0; = " 1 ;max (16) (17) 1 # 1 3 (1) Fluent 6.3 ha an option for includin model for frictional reime at hih particle concentration [3]. Thi will e ect the vicoity and preure in the dene part of the uidized bed. It i aumed that thi will not e ect the imulation and i not included [2]. 3 Decription of the dra model The dra model decribed here are derived in different manner. In the Mater Thei by the author [2] mot of the dra model are derived in detail. 3.1 Syamlal O Brien The Syamlal O Brien dra model, hown in equation (19), i derived for a inle pherical particle in a uid, and modi ed with a relative velocity correlation v r. The relative velocity correlation v r i the terminal ettlin velocity of a particle in a ytem divided by the terminal ettlin velocity of a inle phere [6]. K = 3 4d vr 2 C D j u u j (19) In thi model i the volume fraction of uid or in thi cae a, i the particle volume fraction, i the uid or a denity, d i the particle diameter and j u u j i the abolute relative interracial velocity of the particle compared to the uid. Since thi model i derived for a inle pherical particle

4 the dra factor C D i alo a inle particle model from Dalla Valle [7]. Thi C D i modi ed with the relative velocity correlation and hown in equation (20). 2 3 C D = 40:63 + 4: q Re v r 5 2 (20) The relative velocity correlation v r ued in thi model i baed on a analytical model of experimental data by Richardon and Zaki []. Thi model i iven by Garide and Al-Dibouni [9] and hown in equation (21). v r = 1 [A 0:06 Re] (21) 2 q(0:06 Re) 2 + 0:12 Re (2B A) + A 2 Re i the particle Reynold number decribed in equation (22). The olution alorithm for the relative velocity correlation v r i 1. Calculate particle Reynold number. 2. Gue a initial value for the relative velocity correlation v r e Calculate the modi ed Reynold number with equation (25). 4. Ue the calculated Re m to calculate the parameter n in equation (24). 5. Calculate riht hand ide of equation (23) A = 4:14 0: 1:2 B = 2:65 0:5 > 0:5 The Reynold number ued in thi model i the particle Reynold number Re = d j u u j (22) The main idea abut thi model i the aumption that the Archimede number i the ame in a inle particle and a multiparticle ytem. The Archimede number relate the ravitational force to the vicou force [6]. 3.2 Richardon Zaki Thi model i imilar to the Syamlal O Brien model and the aumption are the ame. The di erence i the formulation of the relative velocity correlation v r [2]. In thi model the experimental reult from Richardon and Zaki are ued directly. Experimental data from Richardon and Zaki provide a formula to nd the v r []. The formula ive v r implicit and v r ha to be found with a alorithm [6]. The relative velocity correlation v r i v r = n 1 (23) n i the Richardon Zaki parameter hown in equation (24). 4:65 Re m < 0:2 >< 4:4 Rem 0:03 0:2 > Re n = m < 1 4:4 Rem >: 0:1 (24) 1 > Re m < 500 2:4 Re m > 500 The Reynold number ued i a modi ed Reynold number and i hown in equation (25). Re m = Re v r (25) 6. Check if the ueed v r and the calculated v r in tep 5. match. If not ue the new v r in tep 3 and calculate v r one more time until converence. The error accepted in thi work i 10 5 [2]. 3.3 Gidapow The Gidapow dra model i a combination of the Wen and Yu dra model and the Erun equation [10]. The Wen and Yu dra model ue a correlation from the experimental data of Richardon and Zaki. Thi correlation i valid when the internal force i neliible which mean that the vicou force dominate the ow behavior. The Erun equation i derived for a dene bed and relate the dra to the preure drop throuh porou media. The Wen and Yu dra model can be written a equation (26). K = 3 (1 ) 4d p C D j u u j 2:65 (26) The dra factor C D i the dra factor for a pherical particle iven by [11]. C D = 24 Re h 1 + 0:15 ( Re ) 0:67i (27) The Erun equation i hown in equation (2). K = 150 (1 ) 2 (d ) 2 +1:75 (j u u j) (1 ) d (2) The Erun equation i a combination of the Kozeny Carman equation and the Burke Plummer equation [12]. The Kozeny Carman i the rt part of the Erun equation and decribe the vicou, low Reynold number ow. The Burke Plummer equation i the econd part of the Erun equation and decribe the kinetic, hih Reynold number ow [13]. The contant i a hape factor for the particle. In thi work it ha been et to one, meanin completely pherical particle.

5 The combination of the two dra model (26) and (2) in the Gidapow dra model i hown in equation (29) [10]. K (Wen Yu) K = > 0: (29) K (Erun) 0: 3.4 RUC The RUC or Repreentative Unit Cell model i a dra model derived from preure drop throuh porou media. It wa rt propoed by Du Plei and Maliyah [14] and further developed later. The baic principle of thi model i eometrical averain of a porou media. It ha the ame form a the Erun equation (2), but the two emi empirical contant (150 and 1:75) i chaned with A and B. Thi contant are mathematically baed. A = B = 26: 3 (1 ) 2=3 (1 (1 ) 1=3 )(1 (1 ) 2=3 ) 2 (30) 2 (1 (1 ) 2=3 ) 2 Thi model i valid for all volume fraction of particle. 3.5 Hill Koch Ladd The Hill Koch Ladd dra model di er omewhat from the other dra model becaue thi i baed on reult from computer imulation. Thi i reult from Lattice Boltzmann imulation. Thi technique i rather new becaue repreentative reult from thi imulation demand hih computational e ort [15]. The model ued in thi work i a modi ed verion of the Hill Koch Ladd dra model made by the reference [15]. The modi ed Hill Koch Ladd model i K = 3 C D j u u j (31) 4 d p The dra factor C D i modeled a C D = 12 2 Re r F (32) F i a dimenionle dra factor which correlate the dra to the Reynold number and particle concentration. In thi model the Reynold number Re r i baed on the radiu of the particle rather than the diameter [15]. Re r = d p j u u j (33) 2 The model ha ome factor w, F 0, F 1, F 2 and F 3 hown in equation (34, 35, 36, 37, 3). w = e ( 10(0:4 )=) (34) >< F 0 = >: F 1 = >< F 2 = >: (1 w) h +w p 1+3 =2+(135=64) ln( )+17:14 1+0:61 :4 2 +:163 i 10 ; 0:01 < (1 ) 3 < 0:4 10 ; (1 ) 3 0:4 (35) ( q :01 < 0:1 0:11 + 0:00051e (11:6a) > 0:1 (36) F 3 = (1 w) p 1+3 =2+(135=64) ln( )+17:9 1+0:61 11: :413 h i +w 10 ; (1 ) 3 < 0:4 10 ; (1 ) 3 0:4 (37) 0: :03667 < 0:0953 0: : :0232= (1 ) 5 0:0953 (3) Thi factor are ued in the dra model to model the dimenionle dra factor F which i a piecewie function of Reynold number and particle concentration. The piecewie function for F i hown in equation (39). F = 1 + 3= Re r F = F 0 + F 1 Re 2 r F = F 2 + F 3 Re r < : < : 0:01 and Re r (F 2 1) (3= F 3 ) > 0:01 and Re r F 3 + p F3 2 4F 1 (F 0 F 2 ) 2F 1 fotherwie (39) 4 Comparion of dra model In the previou ection the dra model wa decribed. In the work thi paper i baed on a et of initial condition ued, i the ame for all the dra model. Dra or interfacial momentum exchane for two phae, one a and one particle phae, i a function of the volume fraction of a, a denity and vicoity, lip velocity and particle diameter. For comparion, all of thi are aumed to be contant except the lip velocity. The a will move fater if the volume fraction of particle are increaed. It i aumed that the lip velocity i the uidization a velocity divided by the a volume fraction. The value for the propertie of the ow i iven in Tab. (1) [2]. The dra factor K from the di erent model are iven in Fi. (1) a a function of particle volume fraction.

6 Diameter 154 m Ga denity k/m 3 Ga vicoity 1.794x10 5 k/m Slip velocity 0.133/ m/ Table 1: Parametrer ued a the input parameter in the dra model Diameter 154 m Ga denity k/m 3 Particle denity 245 k/m 3 Ga vicoity 1.794x10-5 k/m Inlet velocity m/ Initial bed heiht 75 cm Initial particle fraction 0.6 Operatin preure pa Gravitational acceleration 9.1 m/ 2 Table 2: Parametrer and boundary condiion for imulation parameter [2]. Both of thi model i baed on experimental data form the reference [] from 1954, meanin the meaurement equipment i from before Thi equipment miht not be a accurate a today equipment. Fiure 1: The di erent dra predicted by the different dra model [2]. The Hill Koch Ladd ha the mot characteritic form. At thi initial value it ha a local maximum at the volume fraction of particle around 0:4. Thi i due to the tranition between low and hih Reynold ow at moderate volume fraction. It will alo have a tranition from low to hih Reynold ow at a lower volume fraction, but thi i di - cult to ee at the raph. Thi behavior i a well known phenomena when the ow ha a tranition from laminar to turbulent reime. The Gidapow model will have a "witch" at a volume fraction of particle at 0:2 [10]. Thi i due to the tranition between Wen Yu dra model and Erun equation. Thi i not noticeable at the raph. The Richardon Zaki dra model ha four di erent reion, but it will chane with the modi ed Reynold number Re v r. To et thi clearly the initial condition of the cae ha to be chaned. The Syamlal O Brien model i baed on a analytical expreion baed on data from Richardon Zaki and ha a rather at curve for the dra [6]. The RUC dra model i baed on aymptote matchin and will have a mooth curve. Thi model i continuou for all volume fraction [14]. The RUC and Hill Koch Ladd dra model will predict the hihet dra at the volume fraction of a which i mot likely to occur in a dene uidized bed. The problem with the Hill Koch Ladd model i that it i jut valid for one particle phae [5]. In literature it i claimed that the particle ize ditribution will e ect the reult. The Syamlal O Brien and the Richardon Zaki model will predict the lowet dra for thi initial 4.1 Simulation reult Bubblin frequency a a function of radial poition wa ued to compare the di erent model. The bubblin frequency wa collected at 0:39 cm from the bottom of the bed. The computational etup i like explained in previou chapter. Experimental data from [16] i made on a bubblin uidized bed with the dimenion [ ] cm with uniform air upply in the bottom cro-ection. Due to computational e ort the imulation domain i impli ed to 2-D. The rid ued i [25200] cm, where every cell i one by one cm. The coordinate ued i Carteian. The baic propertie and boundary condition i decribed in Tab. (2). The reult and experimental data from [16] i hown in Fi. (2). Fiure 2: Reult from imulation comparred to experimental reult [2]. Reult from the imulation the RUC, Hill Koch Ladd and Gidapow ive reult cloe to the experimental data from [16]. In the work thi paper i baed on [2] ome other parameter wa inveti-

7 very well with the experimental reult. Reference [1] C. K. K. Lun, S. B. Savae, D. J. Je rey, and N. Chepurniy. Kinetic Theorie for Granular Flow: Inelatic Particle in Couette Flow and Slihtly Inelatic Particle in a General Flow Field. J. Fluid Mech., 140: , 194. Fiure 3: Modi ed reult for the RUC model [2]. ated. Thi modi cation wa to include more particle phae reardin the ize ditribution particle ued in the experiment have. The other thin invetiated wa the wall function. By combinin thi reult a modi ed reult and the RUC dra model i achieved and howed in Fi.(3). The modi ed reult how imulation reult very cloe to the experimental reult. The reaon it i not a mooth a the experimental data miht be that the experimental data i the mean bubblin frequency of 30 minute and the imulation i for 30 econd [2]. 5 Concluion Five di erent dra model have been decribed, dicued and compared to each other. The dra model are Syamlal and O Brien dra model, Gidapow dra model, Richardon Zaki dra model, RUC dra model and Hill Koch Ladd dra model. The RUC and the Hill Koch Ladd model predicted the hihet dra. Syamlal and O Brien dra model and Gidapow dra model are default dra model in Fluent 6.3, wherea the three other model have been implemented by the author. Simulation with Fluent 6.3 are performed on a 2D uidized bed with a uniform a ditribution. The reult from the imulation with the di erent dra model are compared with repect to bubble frequencie. The computational reult are alo compared to experimental reult. The RUC model, the Hill Koch Ladd model and the Gidapow model ive the bet areement with experimental data. Modi cation of the reult achieved with the RUC model are performed. The modi cation are baed on the in uence on bubble frequencie due to includin multiple particle phae. The reult are alo modi ed to account for the e ect of wall function on bubble frequencie. The modi cation are baed on reult from imulation with Gidapow dra model. The modi ed imulation reult aree [2] J. Lundber. CFD tudy of a bubblin uidized bed. Mater thei Proce Technoloy Høkolen i Telemark, 200. [3] Fluent. Fluent 6.3 uer uide. Fluent Inc., Lebanon, N.H, USA, [4] J. Din and D. Gidapow. A Bubblin Fluidization Model Uin Kinetic Theory of Granular Flow. AIChE J., 36(4):523-53, [5] M. Syamlal, W. Roer, and T. J. O Brien. MFIX Documentation: Volume 1, Theory Guide. National Technical Information Service, Sprin eld, VA, [6] M. Syamlal, and T. J. O Brien. The Derivation of a Dra Coe cient Formula from Velocity-Voidae Correlation. Unpublihed report, April 197. [7] J. M. Dalla Valle. Micromeritic. Pitman, London, 194. [] J. F. Richardon, and W. N. Zaki. Sedimentation and Fluidization: Part I. Tran. Int., Chem. En., 32:35-53, [9] J. Garide, and M. R. Al-Dibouni. Velocity- Voidae Relationhip for Fluidization and Sedimentation. I & EC Proce De. Dev., 16: , [10] D. Gidapow. Multiphae Flow and Fluidization-Continuum and Kinetic Theory Decription. Academic Pre, San Dieo, [11] L. Schiller, and Z. Naumann. Ver. Deutch. In., 77:31, [12] S. Erun. Fluid Flow throuh Packed Column. Chem. En. Pro., 4(2):9-94, [13] R. K. Niven. Phyical iniht into the Erun and Wen & Yu equation for uid ow in packed and uidized bed. Chemical En. Science, 57: , [14] J. P. Du Plei, and J. H. Maliyah. Mathematical Modelin of Flow Throuh Conolidated Iotropic Porou Media. Tranport in Porou Media, 3: , 19.

8 [15] S. Benyahia, M. Syamlal, and T. J. O Brien. Extenion of Hill Koch Ladd dra correlation over all rane of Reynold number and olid volume fraction. Powder Tec., 162: , [16] B. Halvoren. An Experimental and Computational Study of Flow Behavior in Bubblin Fluidized Bed. Doctoral thei at NTNU, 2005.