Predictions of gasrparticle flow with an Eulerian model including a realistic particle size distribution

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1 Powder Technology wwweleviercomrlocaterpowtec Prediction of garparticle flow with an Eulerian model including a realitic particle ize ditribution Vidar Mathieen a,b, Tron Solberg a,c, Bjørn H Hjertager a,c,) a Telemark Technological R& D Centre ( Tel-Tek ), Porgrunn, orway b Telemark UniÕerity College ( HiT-TF ), Porgrunn, orway c Aalborg UniÕerity Ebjerg, DK-6700 Ebjerg, Denmark Abtract Thi paper preent a computational tudy of the flow behavior in a cold-flow pilot-cale circulating fluidized bed A multi-fluid Computational Fluid Dynamic Ž CFD model ha been developed and verified againt experimental data reported in the literature The flow model i baed on an Eulerian decription of the phae where the kinetic theory of granular flow form the bai for the turbulence modelling in the olid phae The model i generalized for one ga phae and number of olid phae to enable a realitic decription of the particle ize ditribution in garolid flow ytem Each olid phae i characterized by a diameter, denity and retitution coefficient The imulation are performed with different uperficial ga velocitie, initial olid concentration and tandard deviation of the particle ize ditribution Mot emphai i given to tudy the effect of different particle ize ditribution and to tudy the fluctuating behavior of the dilute garolid flow ytem Altogether, the imulation reult are in very good agreement with experimental data Both mean diameter, axial and radial mean and turbulent velocitie, and ma fluxe are calculated uccefully q 000 Elevier Science SA All right reerved Keyword: Eulerian CFD model; Kinetic theory of granular flow; Ga particle flow; Circulating fluidized bed 1 Introduction Many indutrial fluid flow procee include multiphae garparticle phenomena umerical imulation of thee multiphae flow procee provide a new tool for deign and optimization of eg chemical reactor, uch a fluidized bed ytem Computational Fluid Dynamic Ž CFD technique are derived from the equation governing the fluid flow, in the form of partial differential equation repreenting the conervation of ma, momentum and energy The partial differential equation are reduced to an approximate and equivalent et of algebraic equation, which are olved numerically to give the flow field at dicrete point in the calculation domain Although CFD model are fairly well etablihed for ingle phae flow, ) Correponding author Chemical Engineering Laboratory, Aalborg Univerity Ebjerg Ž AAUE, iel Bohr Vej 8, 6700 Ebjerg, Denmark Tel: q ; fax: q addre: bhh@aueaucdk Ž BH Hjertager multiphae flow and complex geometrie make the olution even more challenging However, CFD in multiphae flow ha during the recent year become more accepted in modelling of garolid flow ytem, and much progre ha been made toward developing computer code which calculate the flow behavior of fluidized bed Mot model reported in the literature are baed on a two-phae decription, one ga and one olid phae, where all the particle are aumed to have identical diameter, denity and retitution coefficient ormally a Eulerian decription of the phae i given, where the contitutive equation decribing the olid phae are baed on kinetic theory for granuwx 1 Ref wx did imulation of a lar flow, eg Ref circulating fluidized bed uing both Lagrangian and Eulerian approache to decribe the granular material They demontrated that, at the preent time, the Eulerian approach i preferable when conidering garolid flow ytem uch a fluidized bed Ref wx 3 have given a general review of different Eulerian two-phae flow model applied to fluidization In garolid flow ytem, particle egregation due to different ize andror denity will affect the flow behavior r00r$ - ee front matter q 000 Elevier Science SA All right reerved PII: S

2 ( ) V Mathieen et alrpowder Technology ignificantly To decribe uch phenomena, an extenion to multiple particle phae i eential Ref wx 4 extended the kinetic theory for granular flow to binary mixture Equipartition of energy wa aumed with mall correction for the individual phae temperature Baed on thi turbuwx 5 developed a multiphae garolid lence model, Ref flow model and performed imulation with one ga and three olid phae The model predicted egregation effect fairly well and a good agreement with experimental data wa demontrated Ref wx 6 and wx 7 extended the kinetic theory to binary mixture of olid with unequal granular temperature between the phae, where each olid phae i uniquely defined by particle diameter, denity and retitution coeffiwx 8 propoed a modi- cient Baed on their reearch, Ref fied multiphae garparticle flow model The model wa generalized and made conitent for one ga phae and number of olid phae to enable decription of realitic particle ize ditribution In thi work, the flow model i preented and applied to imulation of a cold-flow pilot-cale circulating fluidized bed Three olid phae Ž with different particle diameter are ued to enable a realitic decription of the particle ize ditribution The numerical reult are dicued and comwx 9 pared againt the experimental data of Ref Experimental etup A detailed decription of the circulating fluidized bed ytem and meaurement technique i given by Ref wx 9 Fig 1 give a ketch of the experimental etup The rier ha a cro-ectional area of 0=0 m and i 0 m high The primary air inlet i located at the bottom of the rier where the ga i paing through an air ditributor to provide a uniform flow of air The air ha ambient temperature and preure At the top of the rier the upended gla particle enter a cyclone and are recycled via a return loop Ref wx 9 ued laer Doppler anemometry Ž LDA and phae Doppler anemometry Ž PDA meaurement technique to meaure flow parameter, uch a mean diameter, particle flux and mean and fluctuating velocitie LDArPDA i a non-invaive optical technique that doe not diturb the flow, ha a high patial reolution with a fat dynamic repone and range The particle velocity meaurement i baed upon that when two coherent Gauian laer beam are interecting each other; the interection will caue interference fringe The particle velocity of a paing particle will be the Doppler frequency time the fringe pacing The Doppler frequency i obtained by a photo detector The particle ize may be meaured uing the PDA ytem The technique i baed upon that when two adjacent photo detector are ued to collect cattered laer light; they will how a phae difference that i linearly proportional to the diameter of a mooth and pherical particle a paing through the meauring volume The experiment were performed with two different particle ize ditribution with a contant mean particle diameter at 10 mm The meaurement were conducted with two different uperficial ga velocitie, 07 and 10 mr, repectively The initial olid concentration in the garolid ytem wa dilute with a olid concentration of 1% and 3% of the rier volume otice that all the wx experimental data wa originally reported by Ref 9 3 CFD model wx Fig 1 The experimental etup 9 Baed on an Eulerian decription of the phae, a multiphae CFD model for turbulent garolid flow i preented The Eulerian approach conider the ga a well a each olid phae a continuum In our model we ued M phae, one ga and olid phae Each olid phae i uniquely defined by a diameter, denity and retitution coefficient The preence of each phae i decribed by a volume fraction, varying from zero to one The law of conervation of ma, momentum and granular temperature are atified for each phae individually The dependent variable, ie the volume fraction and the momentum, are olved for each phae All the phae hare a fluid preure The ga phae turbulence i modelled by a ubgrid cale Ž SGS model The larget cale are imulated directly, and the mall cale are modelled with the SGS turbulence model A conervation equation for granular temperature Ždefined a one-third time the fluctuating velocity quared i olved for each olid phae

3 36 ( ) V Mathieen et alrpowder Technology The tranport equation for the continuou ga phae wa derived from a control volume approach For the olid phae, the Reynold tranport theorem wa ued to develop the well-known Boltzmann equation A complete derivation of the equation for two-phae garparticle flow i given by Ref wx 1 The particle are aumed inelatic, mooth, pherically and non-rotating Moreover, baed on the ditribution of the particle, the contitutive relation, which cloe the tranport equation are provided by thi method Thee contitutive law pecify how the phyical parameter of the phae interact with each other Ref wx 7 give an extenion to binary mixture of particulate material and derive tranport equation for each olid phae Hi approach involved kinetic theory for binary granular mixture with unequal granular temperature between the phae The contitutive equation come from the interaction of the fluctuating and the mean motion of the particle The hear tree are the um of a colliional and a kinetic part The preure of the olid phae, which include both kinetic and colliional preure, were determined from an equation of tate In thee expreion the retitution coefficient were introduced along with the radial ditribution function The hear vicoitie for dilute and dene flow are a product of mean free path time an ocillation velocity time particle denity The coupling between the variou particle phae i through particle preure, binary radial ditribution function, vicoitie, particle colliion diipation and conductivitie In addition to ga particle drag, particle particle drag i introduced in the model The mathematical model i incorporated in the CFD model, FLOTRACS-MP-3D, which i three-dimenional Ž 3D in Carteian coordinate with the following governing conervation equation A detailed decription, including a dicuion of the conitency of the multiphae garparwx 8 ticle model i given by Ref 31 GoÕerning equation wx The mathematical model developed by Ref 8 i ummarized in thi ection 311 Continuity equation The continuity equation for phae m i given by: E E Ž b v m rm q Ž b i m rmui,m 0 Ž 1 Et Ex i where m, rm and Ui,m are the phae volume fraction, denitie and the ith direction velocity component, repectively bv i the volume poroity and bi i the area poroity in i-th direction The volume and area poroitie have value between zero and unity, where zero i a totally blocked and unity i totally open o ma tranfer i allowed between the phae 31 Momentum equation The momentum equation in the j-direction for phae m may be expreed a: E Et E b r U q b r U U Ex i v m m j,m i m m i,m j,m EP E y b q bp Ex Ex v m i ij,m j i M Ý qb r g qb F U yu v m m j v mk j,k j,m k1,k/m P and g j are fluid preure and j-direction component of gravity, repectively Fm k i drag coefficient between the phae m and k Hence, the term on the right ide repreent preure force, vicou force, ma force and drag force, repectively Both ga particle and particle particle drag are included in the total drag We conider each phae a an incompreible fluid But due to the extenion to multiphae flow, the divergence term i included Hence the tre tenor Pij,g for the ga phae g i given by: ž / EUj EUi EUk Pij,gmeff,g q y dij Ž 3 Ex Ex 3 Ex i j k g where dij i the Kroenecker delta The ga phae turbu- lence i modelled by the SGS model propoed by Ref w10 x The technique i a large eddy imulation Ž LES, where the larget cale are olved directly, wherea the mall cale are modelled with the SGS turbulence model Thu, the effective vicoity m may be etimated a: eff,g eff,g g lam,g g gž t ij,g ij,g m m q r c D S :S EUj EUi D' D xd yd z and Sij,g q Ex Ex ( i j g The turbulence model contant ct i etimated to 0079 by uing the renormalization group Ž RG theory w 11 x D i the characteritic length cale for the reolved eddie The total tre tenor Pij, for each olid phae i the um of a colliional and a kinetic part: / EU ž k / ijž / EUj EUi Pij,yP dijqm q ž Ex Ex i j q j y m d Ž 5 3 Ex k Wherea the bulk vicoity i approximated to be zero for mot gae, a done here the bulk vicoity j hould be included when conidering granular material The olid phae preure P, which include a colliional and a kinetic term, i determined from an equation of tate imilar to

4 ( ) V Mathieen et alrpowder Technology van der Waal equation of tate for gae, and i the equation of tate for a granular material: Ý P P q r u Ž 6 C,n n1 The preure denoted P i caued by colliion C,n between the olid phae and n, and ha the expreion: p 3 P Ž 1qe d g n n 3 C,n n n n n = m0uu n ½ Ž Ž m rm u q Ž m rm u 5 n n n ½ Ž m rm 5 0 uu n = u q Ž m rm u Ž u qu n n n 3r Ž 7 Here, the binary retitution coefficient, diameter and ma are defined repectively by: 1 en Ž e qen Ž 8 1 dn Ž d qdn m0mqmn The retitution coefficient i unity for fully elatic and zero for platic colliion We aume the particle to be pherical Hence the number of particle and the particle ma are given, repectively: 6 pd 3 r n and m Ž 9 3 p d 6 The radial ditribution function gn i an expreion for the probability of particle colliion, and i near one when the flow i dilute and become infinite when the flow i o dene that motion i impoible Baed on the ingle olid phae model given implicitly by Ref w1 x, Ref w8x conider the binary radial ditribution function a: 1 y1 1y g 3 g 1y Ž 10 0 ž /,max ½ 5 g 0 gn Ž q n 1y g where,max i the maximum total volume fraction of olid, in thi work et to 063 The olid phae bulk vicoity or volume vicoity may be written a: d n Ý C,n Ž n n n1 3 j P u q m rm u = )pu u u q Ž m rm u n n n Ž 11 The olid phae hear vicoity, m, conit of a colliional term, m Ž col, which i proportional to the bulk vicoity : d n col, Ý C,n Ž n n n1 5 m P u q m rm u = )pu u u q Ž m rm u and a kinetic term: m kin, where = n n n m dil, 1 Ý Ž 1qen gn n1 4 ½ Ý 5 n n n n1 5 Ž 1 1q g Ž 1qe Ž m u,av m l Ž d p dil, ( The mean free path l i given by: 1 d l Ž 15 6' To enure that the dilute vicoity, m dil,, i finite a the volume fraction of olid approache zero, the mean free path i limited by a characteritic dimenion, in thi work et to the minimum control volume length The average granular temperature denoted u,av i ob- tained from Ref wx 7 : u m u ) Ý,av n d m rm u n n 0 n 3r ž /ž / ~ S n1 n d Ž u q Ž m rm u ß n n Ž m0rm uu n S u q Ž m rm u Ž u qu n n n Ž 16

5 38 ( ) V Mathieen et alrpowder Technology For gf 08 the ga particle drag coefficient i baed on the well-known Ergun equation w13 x: 1y m r < u yu < g lam,g g g Fg 150 q175 Ž 17 d d g For g )08, the drag coefficient i baed on the ingle phere expreion derived by w14 x: 3 r< u yu < g g g Fg Cd f Ž g Ž 18 4 d where f Ž y65 Ž 19 g g The relation given in Eq Ž 19 act a a correction of the Stoke law for free fall of a ingle particle and i introduced due to the preence of other particle in the fluid w15 x The drag coefficient CD i related to the Reynold number by w16 x: Cd Ž 1q015Re ReF1000 Ž 0 Re Cd044 Re )1000 d r < u yu < g g g Re m lam,g The particle particle drag coefficient, f n, i proportional to the particle colliional preure wx 7 : ½ ) 3 Ž muqm nun Fn P C,n dn p m0uu n 1 uu n =un q = ln q < u yu < n uqun un n =u lnž mun y q3= Ž 1 u lnž m u Turbulent kinetic energy equation Conervation equation for granular temperature i olved for each olid phae: 3 E E Ž b v ru q Ž b i ru i,u Et Ex i ž / ž / i i i EUj, E Eu bv Pij, q bk i Ex Ex Ex ybvgy3b vfgu The term on the right ide of the equation repreent production due to hear, diffuive tranport of granular temperature, diipation due to inelatic colliion and diipation due to fluid friction A correponding production term due to fluctuation in drag ha been aumed negligible Thi i a reaonable aumption for the relatively large and heavy gla particle conidered in thi work Hence, the particle repone time i aumed to be much longer than the characteritic time cale for the turbulent fluid motion The conductivity of granular temperature k, and the diipation due to inelatic colliion g are determined from the kinetic theory for granular flow The granular conductivity i given by a dilute and a dene part a: kdil, 6 ½ Ý n nž n 1 5 n1 5 Ý Ž 1qen gn n1 u ( Ý n n n p n1 k 1q g 1qe where q r d g Ž 1qe Ž 3 5 m u,av k l Ž 4 3 p dil, ( The firt term in Eq 3 dominate in dilute flow and the econd term in dene regime A for the hear vicoity, the mean free path i limited by the minimum control volume length to enure that the dilute granular conductivity i finite a the olid volume fraction approache zero The diipation of the turbulent kinetic energy due to particle colliion i given by: g 3 Ž 1yen PC,n 4 d n Ý n1 = u u n 4 )p Ž mrm0 uq Ž mrm0 un Ž mrm0 uq Ž mnrm0 un EUk, ydn Ž 5 ž Ž m rm u q Ž m rm u / Ex 0 n 0 n k The divergence term i often neglected, but we have retained it in our calculation However, the term may caue production intead of diipation, and i handled with pecial care 3 umerical olution procedure The governing equation are olved by a finite volume method, where the calculation domain i divided into a finite number of non-overlapping control volume At main grid point placed in the center of the control volume, volume fraction, denity, and turbulent kinetic energy are tored A taggered grid arrangement i adopted for the velocity component that are tored at the main control volume urface The conervation equation are integrated in pace and time Thi integration i performed uing

6 upwind differencing in pace and fully implicit in time The et of algebraic equation i olved by the tridiagonalmatrix algorithm Ž TDMA, except for the volume fraction where a point iteration method i ued Due to the trong coupling between the phae through the drag force, the two-phae partial elimination algorithm Ž PEA w17 x, i generalized to multiple phae and ued to decouple the drag force The IPSA Ž interphae-lip algorithm i ued to take care of the coupling between the continuity and the velocity equation w18 x 33 umerical flow parameter ( ) V Mathieen et alrpowder Technology The imulation geometry with internal obtruction and grid node i hown in Fig The D calculation domain i divided into 38=10 grid node, in the radial and axial direction, repectively The grid i choen to be uniform in the axial direction, wherea a non-uniform grid i ued in radial direction in order to have maller control volume where the gradient are expected to be large Although a grid dependence tudy i clearly deirable, the long computational time involved make uch a tudy not feaible Baed on previou experience, eg Ref w19x and w0 x, the grid reolution appear neverthele to be adequate To enable a realitic particle ize ditribution, three olid phae are ued in the imulation The Sauter mean particle diameter i held contant at 10 mm and the tandard deviation,, are 18 and 40 mm, repectively For the particle ize ditribution with a tandard deviation of 18 mm, the three olid phae I, II and III have diameter of 84, 10 and 156 mm, repectively and a volume fraction of particle of 15%, 750% and 15%, Fig 3 Initial particle ize ditribution repectively Fig 3 how the numerical and real particle ize ditribution with thi tandard deviation A comparion how that with three olid phae, the CFD-model i able to decribe the real particle ize ditribution fairly well For the wide-ranging particle ize ditribution with a tandard deviation of 40 mm, the three olid phae I, II and III have diameter of 63, 10 and 177 mm, repectively and a volume fraction of particle of 50%, 500% and 50%, repectively The particle denity i 400 kgrm 3 and the retitution coefficient for all the olid phae are et to 10 in all imulation In order to get reaonable computational reult to compare againt the experimental data, time averaged reult are obtained between 16 and 0 of real imulation time The olid phae are initially perfectly mixed in the bed and the initial bed height depend on the concentration of olid The air inlet are modelled a 1D plug flow The outlet i located at the top of the cyclone where a continuity condition i ued for the ga phae o particle are allowed to leave the circulating fluidized bed ytem At the wall, no-lip condition are ued for the olid phae a well a the ga phae A zero-flux boundary condition i applied to the energy fluxe 4 umerical reult 41 Temporal eõolution Fig Simulation geometry with internal obtruction and grid point In order to tudy temporal variation of the garolid flow in the circulating fluidized bed, a 60- real time imulation with a uperficial ga velocity of 07 mr, initial olid concentration of 10% and a particle ize ditribution with a tandard deviation of 18 mm i performed The fluctuation are tudied at the center axi, 10

7 40 ( ) V Mathieen et alrpowder Technology Fig 4 Computed large olid cale velocity fluctuation, h10 m, V 07 mr SUP m above the main ga inlet that i aumed to be located in the freeboard zone In Fig 4, the computed velocity fluctuation of the three olid phae are preented, wherea the meaured velocity fluctuation are reproduced in Fig 5 A both figure how, the circulating fluidized bed never reach a normal teady tate condition, but exhibit a trong fluctuating behavior However, the imulation how that the ytem need almot 10 to reach thi condition After the firt 10 the fluctuation have a contant mean value and frequency Each of the olid phae ha the ame frequency and the relative velocity between the olid phae eem to be almot contant A expected, the particle velocity increae with decreaing particle diameter A omewhat wider range of velocity fluctuation i oberved in the imulation compared with the meaurement The ocillating velocity include negative and poitive value, but at thi poition in the center of the rier, a much higher fraction of the particle are flowing upward than downward The frequency of the fluctuation i in the ame order of magnitude a the experimental, but they are a little bit higher in the meaurement than in the imulation Thi may be related to the obervation that the meaured velocity fluctuation include mall cale a well a large cale Fig 6 Computed diameter fluctuation, h10 m, V 07 mr fluctuation, wherea the computed fluctuation are only produced by the large cale effect A cloer dicuion of thee large-cale fluctuation a well a the mall cale will be treated later Computed and meaured particle diameter fluctuation v time are plotted in Fig 6 and Fig 7, repectively The mean diameter are in a fairly good agreement and the frequency of the fluctuation eem to be in the ame order of magnitude The meaured ocillation are coniderably larger than the computed Thi i probably becaue the real particle ize ditribution i more wide-ranging than the numerical one By increaing of number of olid phae, the ocillating diameter would have been more correctly predicted It hould be oberved that particle diameter a large a 00 mm and a mall a 10 mm are experimentally detected wherea the particle ize ditribution in Fig 3 how no particle above 160 mm and below 50 mm The PDA technique i baed upon the obervation that the particle are mooth and pherical Probably, ome of the particle were not completely pherical and hence a mall or large particle ize wa detected for thee particular particle If the puriou data were removed, comparion with the computation hown in Fig 6 would be even better SUP wx Fig 5 Meaured velocity olid fluctuation 9 wx Fig 7 Meaured diameter fluctuation 9

8 ( ) V Mathieen et alrpowder Technology Mean diameter profile A factor that ignificantly affect the magnitude of the egregation i the tandard deviation of the particle ize ditribution A narrow particle ize ditribution ha obviouly a more uniform mean diameter in a garolid flow ytem than a wider ize ditribution Ref wx 9 did a comparion of the mean diameter for two different particle ize ditribution with a contant mean diameter and howed a ignificant increae in axial egregation by ize when the tandard deviation increaed Simulation are performed with the two different particle ize ditribution and compared againt the experimental data in Fig 8, which how the number averaged mean diameter along the median axi of the rier The uperficial ga velocity and concentration of olid are held contant at 10 mr and 3%, repectively The imulation reult are in a good agreement with the experimental data and have the ame form For the narrow ize ditribution Ž 18 mm, the particle ize i almot contant along the median axi and only inignificant deviation from the experimental data are oberved in the upper part of the rier Thee deviation are probably caued by the difference in imulated and actual outlet geometry and 3D effect For the wide-ranging particle ize ditribution Ž 40 mm the egregation by ize i coniderable The multiphae model calculate almot a correct level of egregation, although the predicted mean diameter are omewhat maller than the meaured The computed mean diameter a well a the experimental data confirm that by uing a large particle ize ditribution for the olid in the rier, an increae of the ma flow rate of fine particle will occur Hence a lower mean particle diameter along the median axi i obtained 43 Velocity profile wx Ref 9 meaured axial velocity profile in the freeboard zone of the rier The meaurement were performed Fig 8 Axial particle diameter profile for different particle ize ditribution Fig 9 Velocity profile for different olid phae, h150 m with a uperficial ga velocity of 10 mr, 3% initial volume fraction of olid and a tandard deviation of the particle ize ditribution of 18 mm Fig 9 how the particle velocity profile for different olid phae at height 150 m above the primary ga inlet Solid phae I, II and III have particle diameter of 84, 10 and 156, repectively The computed velocity profile are compared againt a number averaged experimental velocity profile that doe not eparate between different particle ize A the figure how, the agreement between the computed and experimental reult are very well A core annulu flow, with an almot contant velocity in the core and a down-flow in the wall region, i obtained The different olid phae have a correponding flow behavior The imulation how that the mallet particle have a larger velocity than the experimental, wherea the intermediate and larget particle have a lower velocity The relative velocity between the larget and the middle particle i omewhat larger than between the intermediate and the mallet diameter The relative velocity i approximately 0 mr in the central part of the rier and decreae toward the wall An overall mean particle velocity from the individual olid phae velocitie i obtained in order to compare thi to the experimental one The number averaged velocity i calculated from the following relation, where the ubcript indicate the olid phae number: 1 3 V1q Vq V d1 d d3 V Ž q q d d d 1 3 In Fig 10, the predicted mean velocity profile are compared againt experimental data The mean particle velocity profile are obtained 10, 150 and 190 m above the primary ga inlet The imulation i in good agreement with the meaurement At all three height, the core annulu flow i correctly predicted and only mall deviation

9 4 ( ) V Mathieen et alrpowder Technology Fig 11 Radial particle velocity, h10 m about the median axi and a correct time averaging i performed, the radial velocity in the center hould be zero A mall deviation from zero i found in the imulation Different particle ize ditribution will caue different mean velocitie In Fig 1, the computed axial mean velocity along the median axi i compared againt experimental data for the two particle ize ditribution with a tandard deviation of 18 and 40 mm, repectively The imulation how the ame trend a the experiment both in form and magnitude although ome dicrepancie are oberved The widet particle ize ditribution ha the larget mean velocity at the centerline 44 Particle RMS Õelocity profile The Eulerian garparticle flow model i baed on the kinetic theory for granular flow The equation come from interaction of the fluctuating and mean motion of the particle Thee interaction generate tree and give rie to effective vicoitie for the olid phae that relate the random fluctuating motion to the mean motion of the olid phae In the derivation of the kinetic theory and in the definition of the granular temperature, an iotropic Fig 10 Axial particle velocity profile at different height from the experimental data are oberved A in the experiment, the velocity profile become flatter a the height above the inlet increae The radial mean velocity i plotted and compared againt experimental data in Fig 11 The radial velocity i obtained at height 10 m above the primary ga inlet and i plotted from the wall to the centerline In thi figure, the radial velocity i defined a poitive toward the wall from the centerline The numerical reult are in good agreement with the experiment, both in form and magnitude The radial velocity i very low in the center of the rier, but increae toward the hear layer If the rier i ymmetrical Fig 1 Mean particle diameter along the median axi for different particle ize ditribution

10 ( ) V Mathieen et alrpowder Technology Fig 13 Radial RMS velocitie, h10 m behavior of the particle fluctuation i aumed Thu the axial and radial fluctuation hould be the ame Ref wx 9 howed experimentally that the magnitude of the axial and radial velocity fluctuation are quite different In the multiphae garolid flow model, a turbulent velocity i obtained for each olid phae and hence i not directly comparable to the experimental data However, a number averaged granular temperature i etimated from the individual granular temperature The turbulent velocity obtained from thi granular temperature i conidered a the mall-cale turbulence The large cale mean velocity fluctuation are hown in Fig 4 The mall- and large-cale fluctuation are aumed tatitical independent and hence a total particle RMS velocity ) may be obtained: q q d1 d d3 VRMS 3u q Ž q q d d d 1 3 where the tandard deviation of each olid phae i given by: ) 1 i Ý Ž ÕiyV Ž 8 Ž y1 i i1 Experimental tudie confirm a non-contitutive iotropic behavior of the fluctuation in a garolid flow ytem In modelling of uch ytem, an aumption of iotropic granular temperature i not deirable if thee fluctuation are the only ocillation of the ytem However, it i reaonable to believe that the granular temperature only capture the mall-cale fluctuation which will occur in a garolid ytem The exitence of large-cale fluctuation i clearly and evidently hown by Fig 4 7 and i alo eay to oberve viually in a garolid flow ytem That the imulation never reache a normal teady tate condition indicate alo that there exit fluctuating effect a the granular temperature i not captured There i no doubt that large-cale a well a mall-cale fluctuation are included in the meaurement Computed turbulent velocity from mall- and large-cale fluctuation are compared againt the experimental data in Fig 13 and 14 for radial and axial direction, repectively Fig 13 how a ignificant difference in the computed radial RMS velocity whether the large cale fluctuation are added or not The mall-cale fluctuation have an almot contant value in the central part of the rier and increae ignificantly in the wall region A comparion between the experimental data and the numerical mallcale turbulence how that it i correctly predicted in the wall region, wherea quite ignificant dicrepancie are obtained in the core region In uch a dilute garolid ytem conidered in thi imulation, the granular temperature i mainly produced by the hear force, and hence larget in the annulu zone When adding the large-cale turbulence, the curve change both in form and magnitude A more contant and correct turbulence behavior i obtained and the agreement with the meaured RMS velocitie are fairly well In Fig 14 the experimental data how that the axial velocity fluctuation are approximately three time the radial RMS velocity The computed mall-cale turbulence i ignificantly too mall according to the experimental data By adding the large-cale turbulence, a correct order of magnitude i obtained although ome deviation are till oberved Comparion of Fig 13 and 14 how that the large-cale fluctuation are ignificantly higher in the axial than in the radial direction 45 Ma flux profile wx Ref 9 alo meaured and reported net ma flux profile at three different height Particle flux profile are computed and compared againt thee experimental data in Fig 15 The computed fluxe are the um of the individual olid phae fluxe The flux profile are obtained with a uperficial ga velocity of 10 mr, a tandard deviation of Fig 14 Axial RMS velocitie, h10 m

11 44 ( ) V Mathieen et alrpowder Technology Thi reult indicate that a too large axial mixing i computed, although the total net flux eem to be reaonable The particle velocity profile preented in Fig 10 howed a very good agreement with the experimental data Thee profile together with the net flux profile indicate that the preence of olid at height 1 and 15 m i omewhat too high The volume fraction of olid i a little too high in the core a well a in the annulu region and probably the bed height i omewhat overpredicted 5 Concluion Fig 15 et olid flux profile at different height the particle ize ditribution of 18 mm, 3% initial olid volume fraction, and 1, 15 and 19 m above the main ga inlet The imulation how a fairly good agreement with the experimental data Bet agreement are obtained at a height 190 m above the primary ga inlet and thi indicate that the total olid flux circulating in the loop i reaonably predicted At height 10 and 150 m above the inlet, larger dicrepancie are oberved The CFD model compute a too high flux upward in the core region, wherea a too high negative net flux i computed in the annulu region Two-dimenional imulation of a pilot cale circulating fluidized bed are performed The computational reult are compared againt experimental reult reported in the literature The imulation are performed with different uperficial ga velocitie, initial olid concentration and tandard deviation of the particle ize ditribution Mot emphai i given to tudy the effect of different particle ize ditribution and to tudy the fluctuating behavior of the dilute garolid flow ytem Altogether the imulation reult are in very good agreement with experimental data The CFD model i hown to be able to capture the large cale fluctuation which will evidently be preent in a circulating fluidized bed ytem Axial and radial largecale velocity fluctuation are calculated and added to the mall-cale turbulence from the iotropic granular temperature The computed mall-cale turbulence i ignificantly too mall according to the experimental data By adding the large-cale fluctuation, a correct order of magnitude i obtained, although ome dicrepancie are till oberved The calculation confirm that the axial fluctuation are much larger than the radial fluctuation and hence the fluctuating behavior of the circulating fluidized bed i evidently not iotropic Simulation are performed with two different particle ize ditribution and compared againt experimental data along the median axi of the rier The computed mean diameter are in good agreement with the experimental data The axial velocity along the median axe are calculated for the two different particle ize ditribution a well The imulation how the ame tendency a the experiment, although ome dicrepancie are oberved A higher velocity i correctly obtained for the large particle ize ditribution Mean particle velocity profile are obtained at three different height with only inignificant deviation from the meaurement A typical core annulu flow i calculated with a nearly contant velocity in the central part of the rier and a down flow of particle in the wall region A in the experimental data, the velocity profile became flatter when the height above the inlet increae

12 ( ) V Mathieen et alrpowder Technology A mean radial velocity i calculated a well The agreement with experimental velocity i very well and the reult how that the radial velocity i approximately zero at the centerline, then increae toward the hear layer and further decreae to the wall et ma flux profile are computed and preented The calculated flux profile how a fairly good agreement with the experimental data Bet agreement i obtained in the upper part of the rier and the total flux circulating in the loop eem to be reaonably predicted At lower height, larger deviation are obtained 6 Lit of ymbol Cd drag coefficient ct contant in Sub Grid Scale model d particle diameter d mean particle diameter 05 Ž d q d n n e retitution coefficient e mean retitution coefficient 05 Ž e q e n n gi i-direction component of gravity g0 radial ditribution function for a ingle olid phae gn binary radial ditribution function l mean free path M number of phae m ma of a particle m0 binary mamqmn number of olid phae n number of particle P fluid preure PC colliional preure P olid phae preure Re particle Reynold number U i, Uj i- and j-component of velocity u velocity vector V particle velocity Greek ymbol volume fraction,max maximum total volume fraction of olid bi area poroity in i-direction bõ volume poroity g colliional energy diipation dij Kroenecker delta u granular temperature1r3² C : k tranport coefficient of granular temperature m hear vicoity j bulk vicoity Pij tre tenor, olid phae r denity tandard deviation tij,g tre tenor, ga phae F drag coefficient Subcript av average col colliional cor correction dil dilute eff effective g ga phae kin kinetic lam laminar m ga phae or olid phae m n olid phae n RMS Root Mean Square olid phae SUP uperficial turb turbulent Acknowledgement The author would like to expre their thank to ork Hydro, Statoil and The Reearch Council of orway Ž FR for their financial upport of thi work Reference wx 1 D Gidapow, Multiphae Flow and Fluidization, Academic Pre, Boton, 1994 wx E Helland, T Van den Moortel, R Occelli, L Tadrit, BH Hjertager, T Solberg, V Mathieen, in: J Werther Ž Ed, Circulating Fluidized Bed Technology VI, DECHEMA, Frankfurt, Germany, 1999, p 61 wx 3 H Enwald, E Peirano, A-E Almtedt, Int J Multiphae Flow Ž wx 4 JT Jenkin, F Mancini, J Appl Mech 54 Ž wx 5 V Mathieen, T Solberg, E Manger, BH Hjertager, in: J Kwauk, L Li Ž Ed, Circulating Fluidized Bed Technology V, Science Pre, Beijing, China, 1997, p 46 wx 6 D Gidapow, L Huilin, E Manger, XIXth Int Cong of Theoretical and Appl Mech, Japan, 1996 wx 7 E Manger, PhD-thei, Telemark College, orway, 1996 wx 8 V Mathieen, T Solberg, BH Hjertager, Int J Multiphae Flow 6 Ž wx 9 E Azario, L Tadrit, R Santinini, J Pantaloni, Proc of the nd Int Conference on Multiphae Flow, Japan Vol w10x JW Deardorff, J Comp Phy 7 Ž w11x Y Yakhot, SA Orzag, J Scientific Comput 1 Ž w1x RA Bagnold, Proc R Soc A 55 Ž w13x S Ergun, Chem Eng Prog 48 Ž w14x CY Wen, YH Yu, Chem Eng Prog Symp Ser 6 Ž w15x D Gidapow, B Ettehadieh, I&EC Fundam Ž w16x P Rowe, Tran Int Chem 39 Ž w17x DB Spalding, in: RW Lewi, K Morgan, JA Johnon, WR Smith Ž Ed, Computational Technique in Heat Tranfer, Pineridge Pre, 1985, pp 1 44 w18x DB Spalding, umerical Propertie and Methodologie in Heat Tranfer, Hemiphere Publihing, 1983, pp w19x A Samuelberg, BH Hjertager, Int J Multiphae Flow Ž w0x A Samuelberg, BH Hjertager, AIChE J 4 Ž