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Adjustment of drag coefficient correlations in three dimensional CFD simulation of gas–solid bubbling fluidized bed Ehsan Esmaili, Nader Mahinpey ⇑ Dept. of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, AB, Canada T2N 1N4 a r t i c l e i n f o Article history: Received 13 November 2009 Received in revised form 8 November 2010 Accepted 10 March 2011 Available online 9 April 2011 Keywords: Multiphase flow Fluidized bed Computational Fluid Dyn
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  Adjustment of drag coefficient correlations in three dimensionalCFD simulation of gas–solid bubbling fluidized bed Ehsan Esmaili, Nader Mahinpey ⇑ Dept. of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, AB, Canada T2N 1N4 a r t i c l e i n f o  Article history: Received 13 November 2009Received in revised form 8 November 2010Accepted 10 March 2011Available online 9 April 2011 Keywords: Multiphase flowFluidized bedComputational Fluid DynamicsInter phase drag modelCoefficient of restitutionEulerian–Eulerian model a b s t r a c t Fluidized beds have been widely used in power generation and in chemical, biochemical, and petroleumindustries. 3D simulation of commercial scale fluidized beds has been computationally impractical due tothe required memory and processor speeds. In this study, 3D Computational Fluid Dynamics simulationof a gas–solid bubbling fluidized bed is performed to investigate the effect of using different inter-phasedrag models. The drag correlations of Richardon and Zaki, Wen–Yu, Gibilaro, Gidaspow, Syamlal–O’Brien,Arastoopour, RUC, Di Felice, Hill Koch Ladd, Zhang and Reese, and adjusted Syamlal are reviewed using amultiphase Eulerian–Eulerian model to simulate the momentum transfer between phases. Furthermore,a method has been proposed to adjust the Di Felice drag model in a three dimensional domain based onthe experimental value of minimum fluidization velocity as a calibration point. Comparisons are madewith both a 2D Cartesian simulation and experimental data. The experiments are performed on a Plexi-glas rectangular fluidized bed consisting of spherical glass beads and ambient air as the gas phase. Com-parisons were made based on solid volume fractions, expansion height, and pressure drop inside thefluidized bed at different superficial gas velocities. The results of the proposed drag model were foundto agree well with experimental data. The effect of restitution coefficient on three dimensional predictionof bed height is also investigated and an optimum value of restitution coefficient for modeling fluidizedbeds in a bubbling regime has been proposed. Finally sensitivity analysis is performed on the grid intervalsize to obtain an optimum mesh size with the objective of accuracy and time efficiency.   2011 Elsevier Ltd. All rights reserved. 1. Introduction Gas–solid fluidized bed reactors are used in many industrialoperations, such as energy production and petrochemical pro-cesses. Some of the distinct advantages of gas–solid fluidized bedreactors over other methods of gas–solid reactors are controlledhandling of solids, isothermal conditions due to good solids mixingand the large thermal inertia of solids, and high heat flow and reac-tion rates between gas and solids due to large gas-particle contactarea. Hence, the fluidized bed reactors are widely used in gasifica-tion, combustion, catalytic cracking and various other chemicaland metallurgical processes. Two approaches are typically usedfor CFD modeling of gas–solid fluidized beds. The first one isLagrangian–Eulerian modeling [1–6], which solves the equationsof motion individually for each particle and uses a continuousinterpenetrating model (Eulerian framework) for modeling thegas phase. In large systems of particles, the Lagrangian–Eulerianmodel requires powerful computational resources because of thenumbers of equations that are being solved. Bokkers et al. [5] havestudied the effect of implementing different drag models on simu-lation of gas–solid fluidized bed using Discrete Particle Model(DPM) which assume a Lagrangian–Eulerian model for the multi-phase fluid flow. van Sint Annaland et al. [6] have also studiedthe particle mixing and segregation rates in a bi-disperse freelybubbling fluidized bed with a new multi-fluid model (MFM) basedon the kinetic theory of granular flow for multi-component sys-tems. The second approach is Eulerian–Eulerian modeling [7–13],which assumes that both phases can be considered as fluid andalso take the interpenetrating effect of each phase into consider-ation by using drag models. Therefore, applying a proper dragmodel in Eulerian–Eulerian modeling is of a great importance.Many researchers have applied 2D Cartesian simulations tomodel pseudo-2D beds [1,7,11,13]. Behjat et al. [11] applied a two-dimensional CFD (Computational Fluid Dynamics) techniqueto the fluidized bed in order to investigate the hydrodynamic andthe heat transfer phenomena. They concluded that the Eulerian–Eulerian model is suitable for modeling industrial fluidized bedreactors. Their results indicate that considering two solid phases,particles with smaller diameters have lower volume fraction atthe bottom of the bed and higher volume fraction at the top of the bed. They also showed that the gas temperature increases asit moves upward in the reactor due to the heat of polymerization 0965-9978/$ - see front matter    2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.advengsoft.2011.03.005 ⇑ Corresponding author. Fax: +1 403 284 4852. E-mail address:  nader.mahinpey@ucalgary.ca (N. Mahinpey).Advances in Engineering Software 42 (2011) 375–386 Contents lists available at ScienceDirect Advances in Engineering Software journal homepage: www.elsevier.com/locate/advengsoft  reaction leading to the higher temperatures at the top of the bed[11]. Peiranoa et al. [14] investigated the importance of three dimensionality in the Eulerian approach simulations of stationarybubbling fluidized beds. The results of their simulations show thattwo-dimensionalsimulationsshouldbeusedwithcautionandonlyfor sensitivity analysis, whereas three-dimensional simulations areable to reproduce both the statics (bed height and spatial distribu-tion of particles) and the dynamics (power spectrum of pressurefluctuations)ofthebed.Inaddition,theyassumedthattheaccuratepredictionof the drag force (the force exerted by the gas on a singleparticle in a suspension) is of little importance when dealing withbubbling beds. However, in the present study, it is found that usinga proper drag model can significantly increase the accuracy of results in the 3D simulation of bubbling fluidized beds.Cammarata et al. [8] compared the bubbling behavior predictedby 2D and 3D simulations of a rectangular fluidized bed using thecommercial software ANSYS-CFX (a CFD software). The bed expan-sion, bubble hold-up, and bubble size calculated from the 2D and3D simulations were compared with the predictions obtained fromthe Dartonequation [15]. A more realistic model of physical behav-ior for fluidization was obtained using 3D simulations. They alsoindicated that 2D simulations can be used for sensitivity analyses.Xie et al. [10] compared the results of 2D and 3D simulations of slugging, bubbling, and turbulent gas–solid fluidized beds. Theyalso investigated the effect of using different coordinate systems.Their results show that there is a significant difference between2D and 3D simulations, and only 3D simulations can predict thecorrect bed height and pressure spectra. Li et al. [12] conducted athree-dimensional numerical simulation of a single horizontalgas jet into a laboratory-scale cylindrical gas–solid fluidized bed.They proposed a scaled drag model and implemented it into thesimulation of a fluidized bed of FCC (Fluid Catalytic Cracking) par-ticles. They also obtained the jet penetration lengths for different jet velocities and compared them with published experimentaldata, as well as with predictions of empirical correlations. Zhanget al. [16] suggested a mathematical model based on the two-fluidtheory to simulate both homogeneous fluidization of Geldart Aparticles and bubbling fluidization of Geldart B particles in athree-dimensional gas–solid fluidized bed. The usage of their mod-el is easy since it does not include adjustable parameters. It is capa-ble of predicting the fluidization behavior leading to similar resultsas the more complex Eulerian–Eulerian models.Li and Kuipers [17] studied the formation and evolution of flowstructures in dense gas-fluidized beds with ideal collisional parti-cles (elastic and frictionless) by employing the discrete particlemethod, with special focus on the effect of gas–particle interaction.They have concluded that gas drag, or gas–solid interaction, plays avery important role in the formation of heterogeneous flow struc-turesin dense gas-fluidized beds with ideal and non-ideal particle–particle collisionsystems. They discoveredthat the non-linearityof gasdrag hasa ‘‘phase separation’’ functionby acceleratingparticlesin the dense phase and decelerating particles in the dilute phase totrigger the formation of non-homogeneous flow structures.Goldschmidt et al. [13] investigated a two-dimensional multi-fluidEulerian CFD model to study the influence of the coefficient of restitution on the hydrodynamics of a dense gas–solid fluidized Nomenclature  A  constant in RUC-drag model (–)  A  constant in Syamlal–O 0 Brien drag model (–) B  constant in RUC-drag model (–) B  constant in Syamlal–O 0 Brien drag model (–) C  n  drag factor on multi-particle system (–) d s  diameter of solid particles (m) e  restitution coefficient of solid phase (–) F   drag factor in HKL drag model (–) Fr   friction factor from Johnson et al. frictional viscosity (–) F  0 ,  F  1 ,  F  2 ,  F  3  drag constants in the HKL drag function (–)  g   the gravitational acceleration (=9.81) (m s  2 )  g  0  the general radial distribution function (–) I   the unit tensor (–) I  2 D  the second invariant of the deviatoric stress tensor (–) K  sg   drag factor of phase s in phase g (kg m  3 s  1 ) k H s  conductivity of granular temperature (kg m  1 s  1 ) n  coefficient in the Richardson and Zaki drag correlation(–) P   pressure (Pa) P  s  solids pressure (Pa) P  s ,  fric   frictional pressure (Pa) D P   pressure drop (Pa) r    q s  diffusive flux of fluctuating energy (kg m  1 s  3 ) Re  the Reynolds number (–) Re m  the modified Reynolds number in the Richardson Zakicorrelation (–) Re s  the particle Reynolds number (–) t   time (S) D t   time interval (S) U  mf   minimum fluidization velocity (m s  1 ) u s , i , u s ,  j  solid phase velocity in the i and j direction (m s  1 ) ~ V   velocity (m s  1 ) v  r   the relative velocity correlation (–) w  factor in the HKL drag correlation (–) Greek letters b  angel of internal friction(  ) e  g   gas phase volume fraction (–) e s  solid phase volume fraction (–) c H s  dissipation of granular temperature (kg m  1 s  3 ) D  change in variable, final–initial (–) r  the Dell operator (m (  1) ) H s  granular temperature(m 2 s  2 ) k s  bulk viscosity (kg m  1 s  1 ) l  g   gas viscosity (kg m  1 s  1 ) l s  granular viscosity (kg m  1 s  1 ) l s , col  collisional viscosity (kg m  1 s  1 ) l s , kin  kinetic viscosity (kg m  1 s  1 ) l s ,  fric   frictional viscosity (kg m  1 s  1 ) l dil  dilute viscosity in Gidaspow kinetic viscosity model(kg m  1 s  1 ) p  the irrational number  p  (–) q  g   gas density (kg m  3 ) q s  solid density (kg m  3 ) s  the stress–strain tensor (Pa) Subscriptscol  collisional dil  dilute  fr   frictional  g   gas or fluid phase kin  kinetic max  maximum mf   minimum fluidization condition min  minimum q  general phase  qs  solid phase 376  E. Esmaili, N. Mahinpey/Advances in Engineering Software 42 (2011) 375–386   beds. They demonstrated that, in order to obtain reasonable beddynamics from fundamental hydrodynamic models, it is signifi-cantly important to take the effect of energy dissipation due tonon-ideal particle–particle encounters into account.A few works in the literature have investigated the effect of using different drag models in 3D simulation of fluidized beds toobtain an optimum drag model for simulation of bubbling gas–so-lid fluidized beds. Therefore, the underlying objective of this studyis to present an optimum drag model to simulate the momentumtransfer between phases and to compare the results of 3D and2D simulations of gas–solid bubbling fluidized beds. Furthermore,a method has been proposed to adjust the Di Felice Drag Model[18] based on the experimental value of minimum fluidizationvelocity as the calibration point. The effect of restitution coefficienton the three dimensional prediction of bed height is also investi-gated and an optimum value of restitution coefficient for modelingfluidized beds in bubbling regime has been proposed. 2. Experimental setup Experiments were carried out in the Department of Chemicaland Biological Engineering at the University of British Columbia.The fluid bed is a Plexiglas rectangular shape column consistingof spherical glass beads with ambient air as the gas phase. Thecolumn dimensions are 0.280 (m) in width, 1.2 (m) in length,and 0.0254 (m) in depth. Ambient air is uniformly injected intothe column via a gas distributor which is a perforated plate witha hole to plate cross sectional area ratio of approximately 1.2%.Pressure drops were measured using three differential pressuretransducers located at the elevations of 0.03, 0.3 and 0.6 (m) abovethe gas distributor. Fig. 1 illustrates the shape of the column usedin this research, along with its dimensions and pressure transducerlocations. Spherical, non-porous glass beads, Geldart group B parti-cles, with a particle size distribution of 250–300 ( l m) and densityof 2500 (kg/m 3 ) were used as the granular parts. The static bedheight is 0.4 (m) with a solid volume fraction of approximately60%. Several experiments were conducted at steady-state bedoperations in order to calculate the void fraction and minimumfluidization velocity. In order to estimate the minimumfluidizationvelocity, measurements were carried out at increasing velocityincrements from fixed bed to high inlet velocity (0.6 (m/s)). Fromthe data obtained, minimum fluidization velocity is estimated as U  mf   = 0.065 (m/s). 3. Hydrodynamic model In this study the general model of multiphase flow based onEulerian–Eulerian approach has been derived. The model solvessets of transport equation for momentum and continuity of eachphase and granular temperature for the solid phase. These sets of equations are linked together through pressure and interphasemomentum transfer correlations (drag models). The solid phaseproperties have been obtained using the kinetic theory of granularflow.  3.1. Continuity equation The continuity equation in absence of mass transfer betweenphases is given for each phase by: Fig. 1.  Geometry of 3D Plexiglas fluidized bed. E. Esmaili, N. Mahinpey/Advances in Engineering Software 42 (2011) 375–386   377  @ @  t  ð e  g  q  g  Þþ r ð e  g  q  g  ~ V   g  Þ¼ 0 ;  ð 1 Þ @ @  t  ð e s q s Þþ r ð e s q s ~ V  s Þ¼ 0 :  ð 2 Þ And the volume fraction constraint requires  e  g   +  e s  = 1. where  e ,  q , and  ~ V   are the volume fraction, the density and theinstantaneous velocity, respectively. By considering the masstransfer between the phases, the term  ð  _ m  gs    _ m sg  Þ  would then beadded to the right hand side of the above equations, where,  _ m  isthe rate of mass transfer between phases.  3.2. Gas phase momentum equation Assuming no mass transfer between phases and no lift and vir-tual mass forces, the conservation of momentum for the gas phasecan be expressed as: @ @  t  ð e  g  q  g  ~ V   g  Þþ r ð e  g  q  g  ~ V   g  ~ V   g  Þ¼ r   s  g    e  g  r P  þ e  g  q  g   g  þ K  sg  ð ~ V  s  ~ V   g  Þ ; ð 3 Þ where  P   is the pressure,  g   is the gravity and  K  sg   is the drag coeffi-cient between the gas and the solid phase which will be explainedin detail in Section 3.5. The gas stress tensor   s  g   is given by:  s  g   ¼ e  g  l  g   r ~ V   g  þð r ~ V   g  Þ T    þ e  g   k  g  þ 23 l  g    r  ~ V   g   I  :  ð 4 Þ  3.3. Solid phase momentum equation Assuming no mass transfer between phases and no lift and vir-tual mass forces, the conservation of momentum for the solidphase can be expressed as: @ @  t  ð e s q s ~ V  s Þþ r ð e s q s ~ V  s ~ V  s Þ¼ r   s s  r P  s þ e s r P  þ e s q s  g  þ K  sg  ð ~ V  s  ~ V   g  Þ ; ð 5 Þ @ @  t  ð e s q s ~ V  s Þþ r ð e s q s ~ V  s ~ V  s Þ¼ r   s s  r P  s þ e s r P  þ e s q s  g  þ K  sg  ð ~ V  s  ~ V   g  Þ ; ð 6 Þ where  P  s  is the granular pressure, derived from the kinetic theory of granular flow, and is composed of a kinetic term and a term due toparticle collisions. In the regions where the particle volume fraction e s  is lower than the maximum allowed fraction  e s , max , the solid pres-sure is calculated independently and is used in the pressure gradi-ent term  r P  s  It can be expressed as (Lun et al. [19]): P  s  ¼ e s q s H s þ 2 q s ð 1 þ e Þ e 2 s  g  0 H s ;  ð 7 Þ where  H s  is the granular temperature;  e  is the restitution coeffi-cient of granular particles and  g  0  is the radial distribution function.Different values for the coefficient of restitution, from 0.73 to 1,have been proposed in literature. In this study the effect of restitu-tion coefficient onthe simulation of bubblingfluidized bed has beeninvestigated in order to obtain an optimum value for the entirerange of study. The results are presented in Section 5.3. For the ra-dial distribution function,  g  0 , the following correlation has beenproposed by Ibdir and Arastoopour [20] and it is well related tothe data from the molecular simulator by Alder and Wainwright[21].  g  0  ¼ 35 1   e s e s ; max   13 #  1 :  ð 8 Þ In momentum equation,   s s  is the solid stress tensor and can bewritten as:  s s  ¼ e s l s  r ~ V  s þð r ~ V  s Þ T    þ e s  k s þ 23 l s   r  ~ V  s  I  ;  ð 9 Þ where  k s  is the granular bulk viscosity that is the resistance of gran-ular particles to compression or expansion. The following model isdeveloped from the kinetic theory of granular flow by Lun et al. [19]for  k s : k s  ¼ 45 e s q s d s ð 1 þ e Þ  ffiffiffiffiffiffi H s p r   ;  ð 10 Þ where  d s  is the particle diameter. In the solid stress tensor equation  l s  is the granular shear vis-cosity that consists of a collision term, a kinetic term, and a frictionterm: l s  ¼ l s ; col þ l s ; kin þ l s ;  fric  :  ð 11 Þ The collisional viscosity is a viscosity contribution due to collisionsbetween particles and has the highest contribution in the viscousregime. The corresponding correlation is taken from the kinetic the-ory of granular flow by Lun et al. [19]. l s ; col  ¼ 45 e s q s d s ð 1 þ e Þ  ffiffiffiffiffiffi H s p r   :  ð 12 Þ The kinetic viscosity is expressed by Gidaspow model [22,23] as: l s ; kin  ¼  2 l dil  g  0 ð 1 þ e Þ  1 þ 45 ð 1 þ e Þ e s  g  0   2 ;  ð 13 Þ l dil  ¼ð constant Þð bulk density Þð mean free path Þð osccillation velocity Þ l dil  ¼ 5  ffiffiffiffi p p  96  ð e s q s Þ  d s e s   ffiffiffiffiffiffi H s p   :  ð 14 Þ The Schaeffer expression [24] for the frictional viscosity can bewritten as l s ;  fric   ¼ P  s ;  fric   sin b 2  ffiffiffiffiffiffi I  2 D p   ;  ð 15 Þ where  P  s ,  fric   is the frictional pressure, the constant  b = 28.5   [25] isthe angel of internal friction and  I  2 D  is the second invariant of thedeviatoric stress tensor which can be written as I  2 D  ¼ 16 ½ð D s 11  D s 22 Þ 2 þð D s 22  D s 33 Þ 2 þð D s 33  D s 11 Þ 2 þ D 2 s 12 þ D 2 s 23 þ D 2 s 31 ;  ð 16 Þ D sij  ¼ 12 @  u s ; i @   x  j þ @  u s ;  j @   x i   :  ð 17 Þ  Johnson et al. [26] made a simple algebraic expression for the solidpressure in the frictional region: P  s ;  fr   ¼ Fr  ð e s  e s ; min Þ n ð e s ; max  e s Þ  p  ;  ð 18 Þ Fr  ¼ 0 : 1 e s :  ð 19 Þ In which  e s , min  = 0.5,  n  = 2, and  p  = 3 are all experimental basedparameters.  3.4. Kinetic theory of granular flow (KTGF) The transport equation for granular temperature of solid phase H s  can be written as: 378  E. Esmaili, N. Mahinpey/Advances in Engineering Software 42 (2011) 375–386 

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