Catalytic Cracking (FCC) particles and air were used as the solid and gas phases,

Transcription

1 Simulation of Particle and Ga Flow Behavior in the Rier Section of a Circulatin Fluidized Bed Uin the Kinetic Theory Approach for the Particulate Phae S. Benyahia a, H. Aratoopour a,*, T.M. Knowlton b, and H. Maah c a Illinoi Intitute of Technoloy, Chicao, IL b Particulate Solid Reearch Inc., Chicao, IL c Fluent, Inc., Lebanon, NH Abtract Ga/particle flow behavior in the rier ection of a circulatin fluidized bed (CFB) wa imulated uin a computational fluid dynamic (CFD) packae by Fluent Inc. Fluid Catalytic Crackin (FCC) particle and air were ued a the olid and a phae, repectively. A two-dimenional, tranient and iothermal flow wa imulated for the continuou phae (air) and the dipered phae (olid particle). Conervation equation of ma and momentum for each phae were olved uin the finite volume numerical technique. Thi approach treat each phae eparately, and the link between the a and particle phae i throuh dra, turbulence, or enery diipation due to particle fluctuation. Ga and particle flow profile were obtained for velocity, volume fraction, preure, and turbulence parameter for each phae. The computational value areed reaonably well with the available experimental reult. Our computational reult howed that the inlet and outlet dein have inificant effect on the overall a and olid flow pattern and cluter formation in the rier. However, the effect of the initial condition tended to diappear after ome time. The main frequencie of ocillation of the * Correpondin author. Tel.: ; Fax: ; hamid.aratoopour@iit.edu 1

2 ytem were obtained in different reion of the rier. Thee frequencie are important in comparin the computational reult with the available time-averaed experimental data. Keyword: Kinetic theory; Numerical imulation; Circulatin bed; Fluidization; FCC particle Introduction Ga/olid flow ytem are an eential part of many chemical procee, and contribution to the undertandin of the behavior of uch flow ytem can inificantly enhance the dein and, in turn, the productivity of uch procee. The firt tep in the fundamental undertandin of fluidization i uually attributed to Davidon [1] for hi analyi of a inle bubble motion in an infinite fluid bed. The development of Davidon' model wa carried by Jackon [], Murray [3], Piford and Baron [4], and Soo [5,6]. Solid vicoity and preure are cloure that can either be determined experimentally or theoretically by uin more fundamental approache. In any cae, olid vicoity i an important parameter to include in imulation of the rier ection of a circulatin fluidized bed (CFB) ince wall effect can be inificant. In recent year, the dynamic of inter-particle colliion baed on the concept of the work of Chapman and Cowlin [7] have been conidered by Soo [5], Jenkin and Savae [8], Lun et al. [9], Johnon and Jackon [10], Sinclair and Jackon [11], and Din and Gidapow [1]. The kinetic theory approach ue a one equation model to determine the turbulent kinetic enery of the particle (ranular temperature) and aume either a Maxwellian ditribution for the particle, or a non-maxwellian ditribution [13], which conider both dilute and dene cae. Moreover, Kim and Aratoopour [14] conidered

3 the coheivene of the particle and extended the kinetic theory model for coheive particle. The kinetic theory approach for ranular flow allow the determination of the preure and vicoity of the olid in place of empirical relation. The interaction between a and particle ha been conidered in the form of dra force in mot of the publication in the literature. Loue et al. [15] were probably the firt to include both a turbulence and particle-particle interaction. Recently, mot of the invetiation have focued on very dilute ytem with a turbulence and, in ome cae, lare-cale particulate phae fluctuation [16, 17, 18]. Thi could be due to the fact that mot a phae turbulence model [19] have been developed for inle-phae flow, and their application could mot probably be limited to a very dilute flow of particle. For the cae of dene particle flow, the interaction between a turbulence and particle fluctuation i inificant and, in ome cae, the flow tend to be imilar to laminar flow. The kinetic theory formulation include an interaction term between a turbulence and particle fluctuation in the ranular temperature equation [13]. However, the validity of thi expreion to decribe the interaction between a turbulence and particle fluctuation i yet to be invetiated. FCC unit uually operate at hih olid throuhput of k/m. The rier ection of a CFB can be characterized by a core-annular flow reime with particle formin tructure at the wall in the form of cluter and heet. Lateral ereation in hih denity CFB' wa probably firt found experimentally by Bader et al. [0] who imultaneouly meaured the olid denity and ma flux acro the diameter of the rier. Simulation of lateral ereation in the rier i poible only by uin at leat a twodimenional model. Tuo and Gidapow [1] were amon the firt to predict the core- 3

4 annular flow reime uin multiphae flow equation. They uccefully ued an empirical relation for the olid vicoity that wa later ued by Bin [] and Benyahia et al. [3]. However, theoretically derived olid vicoity and preure are the preferred relation to fully predict the a/olid in the rier. Recently, Knowlton et al. [4] preented a challene problem at the fluidization VIII conference to compare different CFB hydrodynamic model. The experimental data were produced mainly at the Particulate Solid Reearch Inc. (PSRI). Thee experimental data included the olid concentration and flux at a certain location of the rier, a well a the axial a preure drop. Benyahia et al. [3] conducted a imulation uin an empirical relation for olid vicoity and compared their reult with the pilot-cale experimental data. The analyi in thi paper will include the ue of the kinetic theory for olid particle; the main oal i to invetiate the effect of initial condition, inlet and outlet dein, and increaed rier diameter on the overall a/olid mixin in the rier. Furthermore, frequency analye of the ocillatory a/olid flow were performed. Thi enable an accurate comparion of the time-averaed computational reult with the experimental data. Ga/Solid Multiphae Flow Model Decription The complexity of the hydrodynamic equation make obtainin an analytical olution very unlikely. Therefore, numerical olution hould be conidered. In the Fluent computer proram that wa ued in thi tudy, the overnin equation were dicretized uin the finite volume technique [5]. The dicretized equation, alon with the initial and boundary condition, were olved to obtain a numerical olution. 4

5 Conervation equation of ma and momentum were developed uin the Eulerian approach, and were olved imultaneouly. Thi approach treat each phae eparately, and the only link between the two phae i throuh the dra force in the momentum equation. The dicretization cheme for the convection term in the momentum equation ue the power law interpolation cheme which provide a formal accuracy between firt and econd order. Thi cheme i more robut and le computationally intenive than hiher order cheme. The SIMPLE iterative alorithm [5] i ued by Fluent to relate the velocity and preure correction to recat the continuity equation in term of a preure correction calculation. The Inter-Phae Slip Alorithm (IPSA) method that ue the Partial Elimination Alorithm (PEA) developed by Spaldin [6] i ued by Fluent The Full Elimination Alorithm (FEA), which i a fully coupled implicit approach ued in the new verion of Fluent-4.5, inificantly enhance converence of the numerical cheme. Governin Equation The followin are the equation of conervation of ma and momentum for the a and particulate phae a well a the particulate phae fluctuatin enery [13, 7]: The conervation equation of ma of phae i (i = a, olid) t ( ρ ε ) ( ρ ε U ) = 0 i i +. i i i (1) with the contraint ε i =1 () 5

6 6 The conervation of momentum of phae I (i = a, olid, k i) ( ) ( ) ( ) U U T U U U i i k i i i i i i i i i. P. t ε ρ β ε ε ρ ε ρ + + = + i (3) The fluctuation that occur in the olid phae were modeled from the kinetic theory of ae modified to account for inelatic colliion between particle. The equation for the turbulent fluctuatin enery of olid, called ranular temperature, Θ ( i U 3 1 = Θ ), may be written a: ( ) ( ) ( ) k t C C U : T U + + = + β β Θ γ Θ Θ ε ρ Θ ε ρ (4) The lat two term in Equation (4) repreent the interaction between a turbulence and particle fluctuation, and their derivation may be found in Gidapow [13]. Numerically, thee term can be important. For example, the term before the lat term may be about three time larer than the diipation term. However, thee term were not included in the preent model ince no turbulence model in the a phae wa ued. Contitutive Equation Contitutive relation are needed to cloe any overnin relation. The followin repreent the contitutive relation ued in the current model. Aratoopour et al. [8] ued a a/olid dra force that ive continuou value over all rane of olid volume fraction: ( ) p e p e d R d R µ ρ ε ε ρ β U U U U = + = (5)

7 Solid phae tre: ( + ε µ b U ) I + ε T = P µ S (6) Deformation rate: S = 1 T 1 [ U + ( U ) ] ( U )I 3 (7) Ga phae tre: T = ε µ S (8) The olid preure ( P ), vicoity ( µ ), and conductivity ( λ ) are compoed of two part: a kinetic term that dominate in the dilute flow reion and a colliion contribution that i inificant in the dene flow reion. P + ρ + ( 1 e ) ε o Θ = ε ρ Θ (9) Where e i the retitution coefficient, and o i the radial ditribution function which become very lare a the olid volume fraction of olid approache the maximum packin ( ε, max Gidapow [1]: = 0.6 in thi paper). The followin expreion wa ued by Din and o ε 3 1 = 5 ε,max (10) Solid phae hear vicoity: µ = µ, dil ( 1 + e ) o o ε 4 5 ( 1 + e ) + ε ρ d ( 1 + e ) p o Θ π 1/ (11) Solid phae dilute vicoity: µ, dil 5 ρ d p Θ π = (1) 96 7

8 The olid bulk vicoity account for the reitance of the ranular particle to compreion and expanion, and i iven by: µ b 4 = ε 3 ρ 1/ Θ d p o ( 1 + e ) (13) π The diffuion coefficient for the olid phae enery fluctuation i: 150 ρ d p Θ π 6 Θ k = 1 ( 1 ) + ( 1 + ) 384 (1 ) + o ε + e + 5 ε ρ d p o e (14) e π o The colliional diipation of enery fluctuation i: ( 1 e ) 1/ 4 Θ γ = 3ε ρ o Θ. U (15) d p π The diipation of enery fluctuation due to interaction of particle with the a phae wa not conidered. The term (.U ) in Equation (15) can be poitive or neative dependin on the direction of the flow in the rier. Since the term (.U ) wa about two order of manitude le than the poitive term on it left, Equation (15) will alway be a diipative term in the ranular temperature Equation (4). It i important to note that the tron dependency of the diipation of the ranular temperature on the retitution coefficient (e) appear in term ( 1 e ). An increae in the retitution coefficient from 0.9 to 0.99 would lead to an order of manitude decreae in the diipation ( γ ). Thi decreae in the colliional diipation yield to a inificant increae in the ranular temperature. Pita and Sundarean [9,30] pointed out the hih enitivity of their computed olid denity and ma flux to the retitution coefficient. They howed that only a value of e = 1 wa appropriate to accurately compute the experimental data of Bader et al. [0]. However, thi value of retitution coefficient i not realitic ince the 1/ 8

9 particle-particle colliion in the FCC rier are not elatic. For particle like FCC catalyt, the retitution coefficient may be deduced from indirect meaurement. Gidapow et al. [31] etimated the retitution coefficient to be cloe to unity from meaurement of the Reynold tre and ranular temperature. In thi tudy, a particle-particle retitution coefficient (e) of 0.95, and a particle-wall retitution coefficient (e w ) of 0.9 were ued. In the literature, the reported meaured retitution coefficient were for larer particle than FCC particle [3]. Boundary Condition At the inlet, all velocitie and volume fraction of both phae were pecified. The preure wa not pecified at the inlet becaue of the incompreible a phae aumption (relatively low preure drop ytem). At the outlet, only the preure wa pecified (atmopheric). The other variable were ubject to the Newmann boundary condition. At the wall, the a tanential and normal velocitie were et at zero (non-lip condition). The olid normal velocity wa alo et at zero. The followin boundary equation apply for the olid tanential velocity and ranular temperature at the wall: v,w = 6 µ 3 π φ ρ ε ε,max 0 Θ v,w n (16) 3 / k Θ Θ 3 π φ ρ,w ε v,lip 0 Θ Θ,w = + (17) γ,w n 6 ε,max γ,w Thee equation were developed by Hui et al. [33], and Johnon and Jackon [10]. Accordin to their work, the lip velocity between particle and the wall can be obtained by equatin the tanential force exerted on the boundary and the particle hear tre cloe to the wall. Similarly, the ranular temperature at the wall wa obtained by equatin 9

10 the ranular temperature flux at the wall to the inelatic diipation of enery, and to eneration of ranular enery due to lip at the wall reion. Later, thee boundary condition were uccefully applied to dene rier flow [34]. Simulation and Reult of Ga/Solid Flow in a -D Rier Sytem Decription Fiure 1 how the rier ection of a CFB ued in the preent numerical imulation of a/olid flow. The eometry of the 0-cm rier i imilar to the experimental et-up ued in the challene problem by Knowlton et al. [4] for the cae of 489 k/m olid ma flux inide the rier. Solid particle were fed from both ide of the rier near minimum fluidization condition. At thee condition, an expected low value of ranular temperature wa ained to the particulate phae. It i important to note that the real eometry of the rier i cylindrical with the olid enterin from one ide only. To obtain mixin at the entrance zone imilar to the 3- D experiment, a two-inlet eometry wa elected. A one-inlet dein for the -D rier could lead the inlet a to flow to the oppoite ide of the olid inlet, therefore, creatin low mixin throuhout the heiht of the rier. The effect of inlet dein will be demontrated in thi paper. Initially, the velocity of both the a and olid wa et at zero. Ga and olid exited the ytem throuh two ymmetric ide outlet et 0.3 m below the cloed top of the rier with a width of 0.1 m (imilar to the ide-inlet). The total heiht of the rier wa 14. m. The olid particle in thi imulation conited of FCC material of 76 µ m in 10

11 diameter and 171 k/m 3 in denity. The imulated ytem wa iothermal at 300 K, and the initial preure wa et at 1 bar. In thi imulation, the computational domain conited of 18 rid radially and 10 non-uniform rid alon the axi of the rier. A total of 3780 fluid cell reulted from thi rid ditribution. A contant time tep of ec wa ued, and the imulation wa conducted on an SGI Oriin 00 computer for 40 ec of real fluidization time correpondin to 3-4 week of computational time. Reult and Dicuion The flow pattern in the 0-cm rier ection of a CFB conited of a core-annular flow reime typical of a hih denity CFB a hown in Fiure 11. The olid denity i hiher near the wall reion and more dilute at the center of the rier. The olid velocity i hih at the center of the rier and olid down-flow at the wall reion i hown in Fiure 11 by the arrow plot of the olid velocity vector. Moreover, the olid down-flow wa motly in the form of cluter decendin at low velocitie at the wall reion. Fiure how the comparion between the calculated time-averaed (from 15 to 40 ec) olid denity ditribution ( ρ ε ) at 3.9 m heiht inide the rier with the experimental data taken at the ame heiht. Thi fiure how a dilute reion in the center of the pipe and a hih concentration of particle at the wall. Thi ditribution reflect the etablihment of a core-annular reime hown by the experiment and the computational reult. However, contrary to the experiment, the computational reult howed a maller core width in the core-annular ytem. By mean of ma conervation, the olid concentration at the wall wa predicted to be le than the experimental value. Moreover, it wa experimentally proven that bier particle in a mixture (a in an FCC 11

12 particle ize ditribution) have a reater tendency than the maller particle to accumulate at the wall reion [35, 36]. Thi could reult in a hiher olid concentration at the wall than that predicted in thi tudy uin mono-ized particle. In addition, a more accurate boundary condition that account for the poible electrotatic force at the wall could increae the olid denity to a value cloer to the experimental data. Fiure 3 how the comparion between the calculated time-averaed olid ma flux ditribution in the rier at a heiht of 3.9 m with the experimental data taken at the ame heiht. The computational reult areed reaonably well with the experimental data. The olid flux wa at it maximum value at the center of the rier, althouh the olid denity wa at it lowet value. Thi i due to the very hih olid (and a) axial velocity at the center of the rier. Near the wall reion of the rier, there wa down-flow of olid mainly in the form of cluter (ee Fiure 11). The olid down-flow wa due to the weiht of the olid in the annular reion that exceeded the axial preure drop. The velocity boundary condition ued in thi cae uccefully captured the velocity manitude of olid down-flow at the wall. Fiure 4 how the time-averaed axial preure drop in the rier compared with experimental data. The hih preure drop at the bottom of the rier wa due to the effect of olid feedin in that reion. The preure drop then decreaed alon the heiht of the rier due to the decreae in the olid concentration. The preure drop wa in reaonable areement with the experimental data, althouh it wa under-predicted. Thi could be mainly due to the prediction of a bier dilute core in the -D rier (ee Fiure 11), which yielded lower flow reitance and, in turn, a lower calculated preure drop. 1

13 Fiure 5 how the intantaneou variation of the ranular temperature with the olid volume fraction after 10 ec of imulation. Gidapow and Huilin [37] howed that, in the dilute reion, the ranular temperature i proportional to the olid volume fraction raied to the power of /3. Thi i imilar to the increae of an ideal a temperature upon compreion. In the dene reion, the decreae in the ranular temperature i due to the decreae of the mean free path of the particle. The trend of the computed reult hown in Fiure 5 areed well with the experimental reult of Gidapow and Huilin [37]. However, the ranular temperature wa an order of manitude lower than the experimental data. Thi could be due to the nelect of inlet a and, in turn, lare-cale fluctuation and the relatively low retitution coefficient ued in the imulation. An increae in the retitution coefficient will definitely increae the ranular temperature ince the diipation of fluctuatin enery will decreae coniderably. Fiure 5 alo how that the ranular temperature wa very low in the dene flow reion. The mot probable reaon could be that the lare-cale ocillation that were meaured experimentally were not included in the computational reult [38]. The time-averaed experimental meaurement alway include mall- and lare-cale ocillation [39], and the computational reult, in thi cae, include only the mall-cale ocillation. Therefore, a direct comparion between the computation and experimental data would be accurate only if the lare-cale ocillation are ubtracted from the experimental data. Unfortunately, Knowlton et al. (1995) did not obtain a time erie analyi of their experimental data. Therefore, calculation of the lare-cale ocillation for thi experiment were not poible. On the other hand, lare-cale ocillation may be 13

14 calculated by computin the time-averaed quantity: U U uin the reult of our imulation. Fiure 6 how the variation of the vicoity of the olid particle with the olid volume fraction in the rier after 10 ec. The olid vicoity meaured by Gidapow and Huilin [37] wa alo found to be an order of manitude hiher than the computed reult in Fiure 6. Thi wa mainly due to the under-prediction of the ranular temperature (ee Fiure 5). In fact, a correct ranular temperature will lead to a more realitic prediction of vicoity uin Equation 11, and a linear increae of the olid vicoity with the olid volume fraction would be obtained [37]. Fiure 7 how the power pectrum denity at different frequencie of ocillation in the rier. The Fourier tranform method ha been ued to calculate the power pectrum of the olid volume fraction. Thi analyi wa done in the dilute core reion of the rier at a heiht of about 1 m with a amplin time of 0.1 ec. The calculated averae olid volume fraction wa about 5%. The main frequency of the olid particle wa Hz, which correpond to 5-7 ec time-period. A imilar frequency ha been reported experimentally in a different rier by Gidapow et al. [40] and computed uccefully by Neri [34] uin the kinetic theory approach for the olid phae. Fiure 8 how the power pectrum of different frequencie of ocillation in the dene reion at about 3 m above the bottom of the rier. The main frequency of ocillation of particle in the dene reion wa about 0.14 Hz, which correponded to 7-8 ec time-period. The dene reion had an averae olid volume fraction of about 13%. The main reaon for calculatin the frequency of ocillation of the a/olid flow wa to know the minimum time required to conduct proper time averain of the computational 14

15 data. In the pecific imulation that wa run, there wa not a inificant difference between the frequencie of ocillation in the dilute and dene reion. Thi could be due to the mall difference between what we called the dene and dilute olid denitie. In the cae of larer difference in olid denity, Gidapow et al. [40] howed that lare difference could exit between the main frequencie of the dene and dilute reion. It i important to mention that the computational value for the frequency analyi were taken at leat 3 m away from the olid inlet zone. In the inlet reion, a hih concentration of olid alon with olid inlet confiuration led to a very mall ocillation. Therefore, it i advied to perform time erie analyi away from the inlet reion in order to capture the riht ocillatory flow pattern inide the rier. Experimentally, the meaurement of Knowlton et al. [4] did not include a frequency analyi of their ytem, and comparion with the experimental data wa, therefore, not poible. Effect of Initial Condition A cae where initially the olid particle filled the bottom of the rier to a heiht of 3 m at minimum fluidization condition ( ε = 0.4) wa conidered. All the other initial, mf and boundary condition are the ame a the previou cae tudy. Fiure 9 how the effect of initial condition on the olid denity ditribution in the rier at 3.9 m heiht compared with experimental data taken at the ame heiht. At the beinnin of the imulation, the olid initially preent in the rier moved toward the wall reion. Thi enhanced the olid denity at the wall in the beinnin of the imulation. However, after a loner period of time, the olid initially preent at the bottom moved toward the top and mixed with the particle that were initially in the bed. The loner averain time (10-40 ec) howed no inificant difference between thi imulation and 15

16 the previou one tartin with an empty rier (ee Fiure ). Therefore, the effect of the initial condition ued in thi imulation tend to diappear after everal econd of operation. Effect of Inlet and Outlet Dein To invetiate the inlet and outlet effect, only one inlet and outlet that are diametrically ymmetric are conidered. Geometrically, thi imulation eem to be in many feature cloer to the experiment ince the 3-D experiment conit of only one inlet and outlet. The inlet olid velocitie were U = V = m/ with a feed inclination of 45 o, and a minimum fluidization olid volume fraction of 0.4. Thi aured a 489 k/m olid ma flux in the rier imilar to the previou cae. Fiure 10 how the time-averaed olid denity at 3.9 m heiht in the rier compared with the experimental data at the ame heiht. The olid denity wa not ymmetric due to inlet and outlet effect. Mot of the olid particle are concentrated at the inlet ide of the rier, and the oppoite ide of the rier wa maintained at a relatively dilute concentration. The typical core-annular flow reime wa not clearly oberved in thi imulation. It i important to mention that the experimental data were taken from one ide of the rier to it center only. Therefore, it i not certain that the experimental data were ymmetric. In the previou fiure, radial ymmetric behavior of the data wa aumed for the ake of comparion with the computational value. The exact experimental inlet and outlet confiuration and condition cannot be implemented unle a 3-D imulation i conducted in a eometry that i more complex. Thi require very hih computational time with today' worktation capabilitie. 16

17 Effect of Doublin the Rier Diameter To invetiate the effect of increain the rier diameter on the flow profile and preure drop alon the rier, a 40-cm rier operatin at the ame olid flux of 489 k/m a the 0-cm rier wa conidered. The diameter of the ide inlet and outlet wa 0 cm. All other initial, inlet, and outlet condition are decribed in Fiure 1. Fiure 11 how a comparion of the flow behavior in a 0- and 40-cm rier after 40 ec. The core-annular flow reime can be een clearly in thi fiure ince the core of both rier i more dilute than the near wall reion. The velocity of the particulate phae i hiher at the center, and down-flow of olid occurred at the wall of the two rier mainly in the form of cluter. It i noticeable that the lenth of the core of the 40-cm rier wa relatively maller than that of the 0-cm rier. Thi wa due primarily to olid ide inlet dein. A chane in the inlet dein could inificantly chane the olid mixin and flow pattern in the lower reion of the rier. Fiure 1 how a comparion between the time-averaed axial preure drop inide the 0- and 40-cm rier. From inle-phae flow calculation, it could be hown [41] that a hiher diameter would lead to a lower preure drop inide the rier. However, the effect of olid inlet condition inificantly affected the mixin in the inlet reion of the rier and, in turn, reulted in hiher preure drop in the rier with a larer diameter. The hiher preure drop wa alo due to the hiher olid hold up in the rier a hown in Fiure 11. Therefore, pecial care hould be taken when deinin the olid inlet and outlet. 17

18 Concluion A two-dimenional tranient model incorporatin the kinetic theory for the olid particle ued in the Fluent code wa capable of predictin reaonably well the complex a/olid flow behavior in the rier ection of a CFB. The core-annulu flow oberved in the dene rier flow wa predicted by thi model. The flow reime wa inificantly affected by the olid down-flow in the form of cluter near the wall of the rier. The calculated olid flux and the axial preure drop inide the rier compared reaonably well with the available lare-cale experimental data. However, the calculated olid denity deviated from the experimental data at the wall reion. We believe that experimentally verified boundary condition that account for particle tructure due to electrotatic effect, particle coheivene, and multi-ized particle are needed to accurately predict the olid denity at the wall reion. The kinetic theory hould be extended for multi-ized particle with the proper interaction term between particle of different ize. The kinetic theory for olid particle predict well the trend and behavior that are experimentally oberved for the ranular temperature and olid vicoity. The difference are due to the retitution coefficient, lare-cale fluctuation, and ue of proper a phae turbulence interaction with the particulate phae. The fluctuation in a/olid flow predicted by thi model howed that the main frequency of the ocillation wa about 0.15 Hz in both the dilute and dene reion (e.., 5% and 13% of olid volume fraction). Therefore, an adequate averain time i neceary for comparion with the time-averaed experimental reult. 18

19 The initial preence of olid inide the rier did not have a inificant effect on the olid denity ditribution. The initial condition ued in thi imulation tended to diappear after everal econd of imulation. The effect of inlet and outlet dein are inificant in the overall olid flow pattern and preure drop alon the rier. Real inlet and outlet condition can be implemented only when uin a 3-D imulation with complex eometry. However, computational time i till the major limitin factor to imulate a/olid flow in complex 3-D eometry. Nomenclature -D, 3-D - Two-dimenional, three-dimenional Ci - Intantaneou velocity vector of phae i d p - Particle diameter e - Particle-particle retitution coefficient e w - Wall-particle retitution coefficient o - Radial ditribution function k - Diffuion coefficient of ranular temperature P - Preure P - Solid preure R e - Particle Reynold number T i - Stre tenor of phae i U i - Velocity vector of phae i β - Inter-Phae dra coefficient - Normal radient to the wall n ε i - Volume fraction of phae i φ - Specularity coefficient (φ = 0.01) γ - Colliional diipation of ranular temperature µ i - Vicoity of phae i ρ i - Denity of phae I Θ - Granular temperature 19

20 0. m Ga + Solid Outlet Experimental Computation 14. m At Time = 0 V = V = 0 ε = 1. Θ 1e 05 m / = Solid Denity (k/m 3 ) Ga + Solid Inlet 0.1 m 0.3 m U = U = m/ ε = 0.4, Uniform Ga Inlet Ditribution, V = 5. m/ Θ 1e 05 m / = Fiure 1 Schematic Drawin of a -D Rier with Inlet and Initial Condition Radial Poition (m) Fiure Solid Denity Ditribution in the Rier at 3.9 m Compared with Experimental Data. Solid Ma Flux (k/m ) Experiment Compuation Radial Poition (m) Preure Drop (Pa/m) Experimental Computation Heiht (m) Fiure 3 Solid Ma Flux Ditribution in the Rier at 3.9 m Heiht Compared with Experimental Data. Fiure 4 Time-Averaed Axial Preure Drop in the Rier. 0

21 Granular Temperature (m / ) Solid Vicoity (Pa.) Solid Volume Fraction Solid Volume Fraction Fiure 5 Granular Temperature Variation with Solid Volume Fraction in the Rier After 10 ec. Fiure 6 Solid Vicoity Variation with Solid Volume Fraction in the Rier After 10 ec. 3 5 Power Spectrum Denity 1 Power Spectrum Denity Frequency (Hz) Frequency (Hz) Fiure 7 Power Spectrum Denity Analyi of the Solid Denity Fluctuation in the Dilute Reion of the Rier (1 m Above the Inlet). Fiure 8 Power Spectrum Denity Analyi of Solid Denity Fluctuation in the Dene Reion of the Rier (3 m Above the Inlet). 1

22 Solid Denity (k/m 3 ) Experimental Computation (10-15 ec) Computation (10-40 ec) Radial Poition (m) Fiure 9 Effect of Initial Condition on the Solid Denity Ditribution in the Rier at 3.9 m Heiht. Solid Denity (k/m 3 ) Experimental Computation Radial Poition (m) Fiure 10 Solid Denity Ditribution at 3.9 m Heiht in the Rier with One Inlet Compared with Experimental Data cm Rier 40-cm Rier Preure Drop (Pa/m) Heiht (m) Fiure 1 Comparion of the Time Averaed Axial Preure in a 0 and 40-cm Rier.

23 0-cm Rier 40-cm Rier Fiure 11 Comparion of the Solid Volume Fraction and Velocity Profile in a 0 and 40-cm Diameter Rier After 40 ec. 3

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