Numerical study of segregation using multiscale models Jin Sun a *, Heng Xiao b and Donghong Gao c a 155 Ewing Street, Princeton, 08540, USA; b Department of Civil and Environmental Engineering, Princeton University, Princeton, USA; c Metso Minerals, Optimization Service, 621 S. Sierra Madre, Colorado Springs 80903, USA (Received 24 October 2008; final version received 7 January 2009) Segregation occurs ubiquitously and is an important process in mineral processing. Numerical simulation of the segregation phenomena can improve the understanding of the mechanisms and hence helps the optimisation of equipment and process designs. In this article, a multi-fluid model and a hybrid model are presented. Their capabilities of addressing various aspects of segregation problems in gas–solid flows are demonstrated. Specifically, the multi-fluid model treats both particles and fluid at the macroscopic continuum level. It can be employed to simulate segregation, given different particle types are treated as different solid phases. The hybrid model is derived using a concurrent multiscale modelling approach, where the discrete element method for solid particles is coupled with continuum fluid mechanics for the fluid phase. It is demonstrated to be able to capture segregation, which may be difficult to be predicted by the current multi-fluid model in some cases. Keywords: segregation; multi-fluid model; discrete element method; hybrid method; multiscale modelling 1. Introduction It is common to encounter mixtures of particles with different physical properties, such as size and density, in mining processes. In such processes, segregation is either to be suppressed as in mixers and bubble distributors or to be promoted as in classifiers and separators. Knowledge of the degree and rate of this kind of segregation phenomena is very important in those processes. In this article, the segregation phenomena in gas–solid fluidised beds will be studied using numerical simulations. The dynamics of fluidised beds can be described at different levels of details (Jackson 2000). At the most fundamental level, the translation and rotation of each particle are detemined by Newtonian equations of motion, and fluid dynamics is determined by the Navier–Stokes and continuity equations. The fluid motion and particle motion are linked by the no-slip condition at each particle boundary. At the second level, the fluid velocity at each point is replaced by an average taken over a spatial volume that is large enough to contain many particles but still small enough compared with the whole region occupied by the flowing mixture. The Newtonian equations of motion are solved for each particle in a Lagrangian framework. The coupling force between the fluid and a particle is a function of the particle’s velocity relative to the local fluid velocity and of the local concentration of the particle assembly. A model at this level of detail is referred to as the hybrid model in this article. At the third level, both the fluid velocity and the particle velocity at a point are averaged over a local spatial volume. A description of particulate flows at this level of details is often referred to as the Eulerian–Eulerian model, or two-fluid model, or multi-fluid model for the extension to two or more solid phases. Accord- ingly, the computational cost of models being able to capture more details of motion is higher than that of coarse models. For applications in mining industry with millions of particles, the most accurate models are so expensive that they are often infeasible. In this work, we will be focussed on hybrid and multi-fluid models. Segregation phenomena in gas-fluidised beds have been extensively studied. Nienow and Naimer (1980) performed experiments on systems consisting of particles of equal size but different density. Hoffmann et al. (1993) experimentally studied systems consisting of particles of equal density but of different sizes. They showed segregation patterns where the larger particles migrated to the bottom, whereas the smaller particles concentrated at the top of the bed. These studies indicate that mixing and segregation of bubbling fluidised beds are largely determined by the bubble dynamics. Bubbles act as vehicles for both mixing and segregation. Goldschmidt et al. (2003) measured the bed expansion and segregation dynamics in a dense gas-fluidised bed using digital image analysis. *Corresponding author. Email: [email protected]International Journal of Computational Fluid Dynamics Vol. 23, No. 2, February 2009, 81–92 ISSN 1061-8562 print/ISSN 1029-0257 online Ó 2009 Taylor & Francis DOI: 10.1080/10618560902736491 http://www.informaworld.com Downloaded By: [Sun, Jin] At: 14:37 9 March 2009
Transcript
Numerical study of segregation using multiscale models
Jin Suna*, Heng Xiaob and Donghong Gaoc
a155 Ewing Street, Princeton, 08540, USA; bDepartment of Civil and Environmental Engineering, Princeton University, Princeton,USA; cMetso Minerals, Optimization Service, 621 S. Sierra Madre, Colorado Springs 80903, USA
(Received 24 October 2008; final version received 7 January 2009)
Segregation occurs ubiquitously and is an important process in mineral processing. Numerical simulation of thesegregation phenomena can improve the understanding of the mechanisms and hence helps the optimisation ofequipment and process designs. In this article, a multi-fluid model and a hybrid model are presented. Theircapabilities of addressing various aspects of segregation problems in gas–solid flows are demonstrated. Specifically,the multi-fluid model treats both particles and fluid at the macroscopic continuum level. It can be employed tosimulate segregation, given di!erent particle types are treated as di!erent solid phases. The hybrid model is derivedusing a concurrent multiscale modelling approach, where the discrete element method for solid particles is coupledwith continuum fluid mechanics for the fluid phase. It is demonstrated to be able to capture segregation, which maybe di"cult to be predicted by the current multi-fluid model in some cases.
It is common to encounter mixtures of particles withdi!erent physical properties, such as size and density,in mining processes. In such processes, segregation iseither to be suppressed as in mixers and bubbledistributors or to be promoted as in classifiers andseparators. Knowledge of the degree and rate of thiskind of segregation phenomena is very important inthose processes. In this article, the segregationphenomena in gas–solid fluidised beds will be studiedusing numerical simulations.
The dynamics of fluidised beds can be described atdi!erent levels of details (Jackson 2000). At the mostfundamental level, the translation and rotation of eachparticle are detemined by Newtonian equations ofmotion, and fluid dynamics is determined by theNavier–Stokes and continuity equations. The fluidmotion and particle motion are linked by the no-slipcondition at each particle boundary. At the secondlevel, the fluid velocity at each point is replaced by anaverage taken over a spatial volume that is largeenough to contain many particles but still smallenough compared with the whole region occupied bythe flowing mixture. The Newtonian equations ofmotion are solved for each particle in a Lagrangianframework. The coupling force between the fluid and aparticle is a function of the particle’s velocity relativeto the local fluid velocity and of the local concentrationof the particle assembly. A model at this level of detail
is referred to as the hybrid model in this article. Atthe third level, both the fluid velocity and the particlevelocity at a point are averaged over a local spatialvolume. A description of particulate flows at this levelof details is often referred to as the Eulerian–Eulerianmodel, or two-fluid model, or multi-fluid model forthe extension to two or more solid phases. Accord-ingly, the computational cost of models being able tocapture more details of motion is higher than that ofcoarse models. For applications in mining industrywith millions of particles, the most accurate modelsare so expensive that they are often infeasible. In thiswork, we will be focussed on hybrid and multi-fluidmodels.
Segregation phenomena in gas-fluidised beds havebeen extensively studied. Nienow and Naimer (1980)performed experiments on systems consisting ofparticles of equal size but di!erent density. Ho!mannet al. (1993) experimentally studied systems consistingof particles of equal density but of di!erent sizes. Theyshowed segregation patterns where the larger particlesmigrated to the bottom, whereas the smaller particlesconcentrated at the top of the bed. These studiesindicate that mixing and segregation of bubblingfluidised beds are largely determined by the bubbledynamics. Bubbles act as vehicles for both mixing andsegregation. Goldschmidt et al. (2003) measured thebed expansion and segregation dynamics in a densegas-fluidised bed using digital image analysis.
International Journal of Computational Fluid DynamicsVol. 23, No. 2, February 2009, 81–92
ISSN 1061-8562 print/ISSN 1029-0257 online! 2009 Taylor & FrancisDOI: 10.1080/10618560902736491http://www.informaworld.com
Downloaded By: [Sun, Jin] At: 14:37 9 March 2009
Both multi-fluid models (Goldschmidt et al. 2001,Sun and Battaglia 2006) and hybrid models (Hoomanset al. 2000, Feng et al. 2004, Feng and Yu 2004, 2007)have been employed as useful tools to study segrega-tion phenomena. However, the relative advantagesbetween them have not been extensively discussed.Some pitfalls in the developments and applications ofthese two kinds of models for segregation simulationsremain unclear.
In this article, a hybrid model is presented to solvefor the dynamics in particulate flows. The hybridmodel couples the control volume CFD for gas phasedynamics with the DEM for solid particle motion. Itcan directly account for motions of individual particlesand the influence of each particle property. It is auseful tool to study dense particulate flows with wideparticle size and/or other property distributions. Incontrast, it is di"cult to model such flows in a multi-fluid model. The results associated with segregationobtained from the hybrid model will be compared withthose from a multi-fluid model.
The following sections present the theoretical andnumerical methodology for the multi-fluid and hybridmodels. They are followed by the simulation detailsand results.
2. Multi-fluid model
In the multi-fluid model, the fluid phase and thedispersed (solid) phase are described as interpenetrat-ing continua. The particle mixture is divided into adiscrete number of phases, each of which can havedi!erent physical properties, e.g. particle diameter. Theformulation of this model is essentially based on thecontinuum fluid dynamics.
The governing equations for the multi-fluid modelare (Syamlal et al. 1993):
For gas phase:
@
@t!egrg" #r $ !egrgvg" %
XNg
n%1Rgn; !1"
@
@t!egrgvg" #r $ !egrgvgvg" % r $ Sg # egrgg&
XM
m%1Igm;
!2"For mth solid phase:
@
@t!emrm" #r $ !emrmvm" %
XNm
n%1Rmn; !3"
@
@t!emrmvm" #r $ !emrmvmvm"
% r $ Sm # emrmg# Igm &XM
l%1l6%m
Iml: !4"
Translational granular temperature equation (Agrawalet al. 2001):
where the translational granular temperature is definedas:
ym;t %1
3hC2
pii: !6"
For details of this model, please refer to previouscomplete descriptions (Syamlal et al. 1993). A briefsummary of the constitutive relations used in themodel is given here.
The constitutive equations for the solid phases werederived for granular flows (Syamlal 1987a). There aretwo distinct flow regimes in granular flows: a viscousor rapidly shearing regime in which stresses arise dueto collisional or translational momentum transfer, anda plastic or slowly shearing regime in which stressesarise due to Coulomb friction between grains in closecontact. The granular stress equation based on kinetictheory for granular flows (Syamlal 1987b) is applied tothe viscous regime. The solid stress tensor in theviscous regime only takes into account the contribu-tions of particle translational momentum flux andbinary collisions. In the plastic flow regime, the solidstress tensor was derived based on the plastic flowtheory (Jenike 1987) and critical state theory (Schae!er1987).
Particle rotation can also be accounted for in themulti-fluid model (Sun and Battaglia 2006) using asimple model from kinetic theory for rapid flow ofidentical, slightly frictional, nearly elastic spheres(Jenkins and Zhang 2002).
3. Hybrid model
From the above review of the multi-fluid model, wecan see that n sets of governing equations have to besolved for a multiphase flow with n phases. Moreover,a number of constitutive equations are required forclosure of the model, and the number increases greatlywith the number of phases. Therefore, for particulateflows with a wide property distribution, modellingerrors are easily accumulated and computational costsare amplified.
To overcome the problems with the multi-fluidmodel, we present a hybrid model which couplesparticle-based DEM with grid-based control volumeCFD. The models for particle motion and fluiddynamics are first described, and then the details oncoupling them together are given.
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3.1. Discrete element method
Individual particle motion in a fluidised bed can bedescribed by Newtonian equations of motion. Thediscrete element method employs numerical integra-tion of the equations in a Lagrangian approach toresolve particle trajectories (Cundall and Strack 1979).The translational and rotational motions of a particleare governed by:
midvpidt% fci # fgpi #mig; !7"
Iidoi
dt% Ti; !8"
where fci is the particle–particle contact force, fgpi is thefluid–particle interaction force, mig is the gravitationalforce, Ti is the torque arising from the tangentialcomponents of the contact force, and Ii, vpi, oi are themoment of inertia, linear velocity and angular velocity,respectively. The net contact force, fci, and torque, Ti,acting on each particle result from a vector summationof the force and torque at each particle–particlecontact. A linear spring-dashpot model is employedfor the contact force model due to its simplicity andreasonable accuracy (Cundall and Strack 1979). Thebasic principles of the contact model are brieflydescribed next.
Two contacting spherical particles {i, j} are shownin Figure 1 with radii {ai, aj} at positions {ri, rj}, withvelocities {vi, vj} and angular velocities { oi, oj}. Thenormal compression dij, relative normal velocity vnij
and relative tangential velocity vtij are (Silbert et al.2001):
dij % d& rij; !9"
vnij % !vij $ nij"nij; !10"
vtij % vij & vnij & !aioi # ajoj" ' nij; !11"
where d % ai # aj, rij % ri 7 rj, nij % rij/rij, with rij %jrijj and vij % vi7vj. The rate of change of the elastic
tangential displacement utij, set to zero at the initiationof a contact is:
dutijdt% vtij &
!utij $ vij"rijr2ij
: !12"
The last term in Equation (12) arises from the rigidbody rotation around the contact point and ensuresthat utij always lies in the local tangent plane of contact.Normal and tangential forces acting on particle i are:
Fnij % f!dij=d"!kndijnij & gnmeffvnij"; !13"
Ftij % f!dij=d"!&ktutij & gtmeffvtij"; !14"
where kn,t and gn,t are the spring sti!ness andviscoelastic constants, respectively, and me! % mimj/(mi # mj) is the e!ective mass of the spheres withmasses mi and mj. The corresponding contact force onparticle j is simply given by Newton’s third law,Fji % 7Fij. The function f(dij/d) % 1 is for the linearspring-dashpot model, and f!dij=d" %
##########dij=d
pis for
Hertzian contacts with viscoelastic damping betweenspheres.
Static friction is implemented by keeping track ofthe elastic shear displacement throughout the lifetimeof a contact. The static yield criterion, characterisedby a local particle friction coe"cient m, is modelledby truncating the magnitude of utij if necessary tosatisfy jFtijj5 jmFnij
j. Thus, the contact surfaces aretreated as ‘sticking’ when jFtijj 5 jmFnij
j, and as‘slipping’ when the yield criterion is satisfied. Thetotal contact force and torque acting on particle i arethen given by:
fci %X
j
!Fni j # Fti j"; !15"
Ti % &1
2
X
j
rij ' Ftij: !16"
The amount of energy lost in collisions is char-acterised by the inelasticity through the coe"cient ofrestitution e. For the linear spring-dashpot model, thecoe"cient of normal restitution en and contact time tccan be analytically obtained:
en % exp!&gntc=2"; !17"
tc % p!kn=meff & g2n=4"&1=2: !18"
The value of the spring constant should be largeenough to avoid particle interpenetration, but not too
Figure 1. Schematic of two particles i and j in contact andposition vectors ri, rj, respectively, with overlap dij.
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large to require an unreasonably small time step dt,because an accurate simulation typically requires dt–tc/50. After the contact force is calculated, the equationsof motion, which are ordinary di!erential equations,can be numerically integrated to get the particletrajectories.
3.2. Model for fluid phase
The locally averaged governing equations for in-compressible flow are used for the fluid phase (Rusche2002):
@
@t!eg" #r $ !egug" % 0; !19"
@
@t!egug" #r $ !egugug" % r $ !egRg" &
egrprg# egg&
Igrg;
!20"
where Rg represents the viscous stress.The gas volume fraction eg is not solved but
obtained from the relation eg % 1 7 es, where thesolid volume fraction es is from averaging particle datafrom DEM simulations.
3.3. Coupling between fluid and particles
The fluid phase momentum Equation (20) is coupledto the DEM particle equations of motion throughinterphase momentum transfer terms. The two majorforces fluid–particle drag and buoyancy are consideredhere. The buoyancy force acting on a single particle bythe gas is computed as:
b % &rpVp; !21"where Vp is the volume of the particle. The drag forceis calculated using the following formulation
fd % b!us & ug"; !22"where b is the coe"cient of momentum transferthrough drag, and (us7ug) is the slip velocity betweenthe two phases. The specific form of the coe"cient bdepends on the drag correlations used. In this study,the drag correlation by Syamlal et al. (1993) isadopted:
b % 3
4
Cd
V2r
rgjus & ugjdp
eseg; !23"
Cd % 0:63# 4:8##############Vr=Re
p$ %2; !24"
where the particle Reynolds number Re is defined as
Re %dpjus & ugjrg
mg!25"
and Vr is the ratio of the terminal velocity for a multi-particle system to that for a single particle. Thefollowing form for Vr is used (Garside and Al-Dibouni1977):
The drag force between the two phases arecomputed for each cell according to Equation (22).The volume fraction and velocity of the solid phase areneeded during the calculation of drag. The volumefraction of each cell is obtained from the particlespatial distribution and the volume of each particlethrough the averaging processes:
es %1
Vc
XNp
i%1K!jxi & xcj"Vi; !29"
where the subscript c denotes a cell and i denotes aparticle. Np is the total number of particles in thesystem. K is a kernel function with compact support,such as the Gaussian function K(x) % exp[7(x/w)2],where x % (jxi7xcj)/w and w is the bandwidth. In thisstudy, the following kernel function is used:
K!x" %(
1& x2) *4
if jxj < 10 if jxj ) 1
!30"
This function is very close to a Gaussian function(with scaled bandwidth w % 0.45), as shown inFigure 2. This kernel function is used because it ismore e"cient to compute than the Gaussian function.Although the expression in Equation (29) theoreticallyrequires looping over each particle when computingvolume fraction of a cell, resulting in a computationalcomplexity of O(Np Nc), where Nc is the total numberof cells, in practice only the particles in the
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neighbouring cells are examined because the kernelfunction in Equation (30) has compact supports.
Similarly, the solid phase velocity is obtained viathe following coarse-graining procedure:
ucs %PNp
i%1 K!jxi & xcj"ViuiPNp
i%1 K!jxi & xcj"Vi
!31"
The drag of Equation (22) is the average drag forceacting on the particles in a cell. For the Lagrangianequations of the particles, the drag applied on eachparticle, fdi, is needed. This can be obtained from thedrag of the cell whose centre is nearest to the particle.Each particle shares the drag force proportional to itssurfaces area. To investigate the sensitivity of the resultsto the drag distribution scheme, the scheme ofdistributing drag forces proportional to particle volumewas also examined and the results are presented anddiscussed in Section 5.2.3. As a summary, the forceexerted on a particle by the fluid is thus:
fgpi % fdi &rpVp !32"where drag and buoyancy terms are included.
4. Numerical schemes
The multi-fluid model equations are solved using afinite volume approach in a staggered grid for thediscretisation of the governing equations to reducenumerical instabilities (Syamlal 1998). Scalars such aspressure and volume fraction are stored at the cellcentres and the velocity components are stored at cellsurfaces. A second-order discretisation is used forspatial derivatives and first-order discretisation for
temporal derivatives. A modified SIMPLE algorithmwith partial elimination of interphase coupling isemployed to solve the discretised equations (Syamlal1998).
For the hybrid model, the fluid equations aresolved using a solver provided by a library in theOpenFOAM (Open Field Operation and Manipula-tion) toolbox. This solver is used in the current studywith modifications to accommodate the fact that onlythe fluid phase is solved and the dispersed phase istracked in the Lagrangian framework. The finitevolume method is used to discretise the equations onan unstructured mesh. For the time integration, theEuler implicit scheme is used, which has only first-order accuracy but is unconditionally stable. Theconvection and di!usion terms are discretised with ablend of central di!erencing (second-order accurate)and upwind di!erencing (first-order accurate). Detailsof the numerical schemes are given in Rusche (2002).
The velocity–pressure coupling is dealt with usingthe modified PISO algorithm (Rhie and Chow 1983).In this algorithm, the momentum equation is firstsolved to get a predicted velocity field, and then thepressure equation (obtained by combining the momen-tum equation with the continuity equation) is solvedfor a corrected velocity field. This process is repeateduntil the velocity field satisfies the continuity equation.The PISO algorithm prevents pressure–velocity decou-pling and oscillation in the solution, eliminating thenecessity of a stagger grid. Therefore, a colocated gridis used in hybrid models, where all the variables arestored at the cell centres, thus it is a significantsimplification over a stagger grid.
The particle motion equations are handled using aclassical molecular dynamics simulation solver,LAMMPS (Large-scale Atomic/Molecular MassivelyParallel Simulator), developed at the Sandia NationalLaboratory (Plimpton 1995). It has been extensivelyused in previous studies of granular flows (Silbert et al.2001, Sun et al. 2006). The theories and details ofLAMMPS were discussed in Section 3.1.
5. Results and discussions
To study particle segregation due to size and densitydi!erence, several computational cases comparablewith the experiments by Goldschmidt et al. (Gold-schmidt 2001, Goldschmidt et al. 2003) have beensimulated. Specifically, the following four cases weresimulated using the hybrid model:
(1) Segregation of two types of particles withdi!erent densities. Inlet velocity is 1.0 m/s.
(2) Segregation of two types of particles withdi!erent sizes. Inlet velocity is 1.0 m/s.
Figure 2. Comparison of scaled Gaussian functionK(x) % exp[7(x/0.45)2] and the kernel function used in thisstudy.
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(3) Segregation of two types of particles withdi!erent sizes. Inlet velocity is 1.5 m/s.
(4) Segregation of two types of particles withdi!erent sizes. Inlet velocity is 1.0 m/s. Dragdistributed from cells to particles proportionalto particle volumes (as opposed to particlesurface areas).
To reduce computational cost while still keepingthe essential physics of the process, a domain smallerthan that in the experiments of Goldschmidt et al. wasused. Consequently, a smaller number of particles(totalling 2160) is used in the simulation. The dimen-sions of the bed are shown in Figure 3. The aspect ratioof the bed in the experiment and that in thecomputation are the same. The wall boundary forparticle–wall interactions conditions are specified onall the boundaries. The particle properties and contactparameters with the wall are shown in Table 1. Thedomain geometry, grid size, boundary conditions andinitial conditions are specified in Table 2.
A pseudo-3D simulation was conducted, i.e. theparticle simulations were conducted in 3D while thefluid simulations were conducted in 2D. This isjustified because the variation of fluid motion in thethickness direction is small due to the very smallthickness of the bed. The height of the computationaldomain is chosen to allow particle expansion in thisheight direction without elutriation.
In addition to the size-segregation studies describedabove, a case with density segregation in the samecomputational domain was also investigated in orderto examine the hybrid model capability of predictingdensity segregation.
For all the cases, the dynamic viscosity of the gas(air) mg % 1.8 6 1075 [Pa s] was used. The gravityconstant is 9.8 m/s2. When estimating the minimumfluidisation velocities, the minimum fluidisation voidageof all the particle types was assumed to be 0.417. Theminimum fluidisation velocities estimated were close tothat reported in the experiments (Goldschmidt 2001).
To characterise the bed expansion, an averageparticle height is defined as
hhpi %PNp
i mihiPNp
i mi
!33"
To investigate the segregation extent and rate, thesegregation percentage is defined (Goldschmidt et al.2003):
sp %Sp & 1
Sp;max & 1; !34"
and sp equals 0 when the particles are perfectly mixedand 1 when the mixture has completely segregated. The
parameter Sp is the ratio of the average heights of thesmall and large particles according to Equation (33):
Sp %hhsmallihhlargei
: !35"
The parameter Sp,max is the maximum degree ofsegregation, which can be calculated in terms of
Table 1. Particle properties of the computational cases.
Figure 3. Geometry of the pseudo 3D computationaldomain and boundary conditions.
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mixture composition (assuming the maximum packingdensity in the fluidised state for small particles equalsthat for large particles) from:
Sp;max %2& xsmall
1& xsmall: !36"
5.1. Segregation predictions using the multi-fluidmodel
The segregation due to size di!erence of the two typesof particles has been simulated using the multi-fluidmodel and the results have been reported in a previouspaper (Sun and Battaglia 2006). It has been demon-strated that the multi-fluid model can predict thesegregation percentage to a satisfying degree whenboth large and small particles were fluidised andparticle rotation were taken into account. The agree-ment between model predicted segregation percentagesand the experimental results was shown in Figure 4, inwhich the experimental data were averaged over 1 stime intervals. However, when the large particles werenot fluidised, the multi-fluid model predicted a muchfaster segregation rate and larger segregation percen-tages than the experimental data (see Figure 5). Themajor discrepancy between numerical results andexperimental results is due to the ‘hindrance e!ect’(Gera et al. 2004) arising between the small particlesand the reluctant large particles. It is di"cult to modelthis e!ect within the multi-fluid framework.
5.2. Segregation predictions using the hybrid model
In each of the four cases set up in the previous section,the average particle height of each particle type, as well
as the segregation percentage, were studied. In addi-tion, for cases (1)–(3), the configuration of the bed att % 0 s and that at t % 6 s (when stable segregationwere achieved) are presented. Note that cases (2)–(4)share the same initial bed configuration. These fourcases are discussed in the following sections.
5.2.1. Density segregation
In case (1), two types of particles with the same size butdi!erent densities were perfectly mixed initially. The
Table 2. Computational domain and general initial andboundary conditions.
GeometryHeight of domain 12 cmWidth of domain 4 cmDepth of domain 0.75 cmHorizontal grid size, Dx 0.4 cmVertical grid size, Dy 0.4 cmGrid size in depth, Dz 0.75 cm
Initial conditionsInitial bed heightdensity-segregation cases *2.5 cmsize-segregation cases *4 cm
Boundary conditionsUniform gas inflow Specified velocityPressure Zero gradient at outletWall boundary for gas phase No slipWall boundary for solid phase Partial slip
Figure 4. Segregation percentages from simulations andexperiments (Goldschmidt 2001) at the gas inlet velocity of1.30 m/s.
Figure 5. Segregation percentages from simulations andexperiments (Goldschmidt 2001) for the gas inlet velocity of1.00 m/s.
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applied gas inlet velocity (1.0 m/s) was larger than theminimum fluidisation velocity of the light particles butsmaller than that of the heavy ones.
The simulation was conducted for 20 s. The averageparticle heights and the segregation percentage resultswere presented in Figure 6. It can be seen that thesegregation occurred rapidly during the first 2 s, withthe heavy particles settling to the bottom and thus someof the light particles being squeezed to the top. After theinitial stage of rapid separation, the segregationcontinued with a slower but steady rate, achieving asegregation rate of approximately 40% at t % 20 s.During this stage, the light particles initially at thebottom were gradually transported upward, causingslight settling of the heavy particles. In addition, thecompacting of the heavy particles probably alsocontributed to the decrease of their average height,raising the segregation percentage. The segregationprocesses were confirmed by the two snapshots inFigure 7. The left panel showed the initial configuration,with the two types of particles perfectly mixed andrandomly packed. After 6 s, most of the heavy particleswere settled to the bottom. At this time there were stilllight particles trapped at the bottom region by the heavyones, which were to move upward subsequently.
The hybrid model is capable of capturing thedensity segregation as demonstrated in this case.
5.2.2. Particle size segregation
For cases (2) and (3), two types of particles with thesame density but di!erent sizes were perfectly mixedinitially. Two scenarios were examined. In case (2), gasinlet velocity is Vg % 1.0 m/s, which was larger thanthe minimum fluidisation velocity of the small particles
but smaller than that of the large ones; in case (3), gasinlet velocity was Vg % 1.5 m/s, which was larger thanthe minimum fluidisation velocity of the large particles.
The simulation for case (2) was conducted for 20 s.The results of average heights and segregation percen-tages are presented in Figure 8. It can be seen that thesegregation percentages gradually increased to about30% after 20 s. Compared with the fast segregation ratepredicted by the multi-fluid model (the segregationpercentage reached 50% in 5 s) (Sun and Battaglia 2006)(see Figure 5), the hybrid model was able to predict amuch slower segregation rate, which is closer to theexperimental results (see Figure 5). This result demon-strates that the hybrid model can capture the defluidisa-tion feature of the large particles and the hindrancee!ect to the small particles. The discrepancy betweenthe actual segregation percentages predicted by thehybrid model and the experimental results (about 15%after 20 s) could be due to the inter-phase momentumtransfer treatment used for the hybrid model. Thefollowing approaches can be tested to further improvethe accuracy of the hybrid model. First, the distributionof drag force from the continuum level to the particlelevel can be altered. Second, the drag forces can becomputed for each particle using the local fluid velocityat the particle position. These approaches will be furtherdiscussed in the next section.
For case (3) with gas inlet velocity Vg % 1.5 m/s,the simulation was also conducted for 20 s. It can beseen from Figure 9 that the large particles initiallysegregated to the top with a segregation percentageof 710% and then the large and small particlesremained mixed at that level. This mixing behaviouris consistent with the experimental results (comparablewith the similar case in Figure 4) although the
Figure 6. Segregation time evolution for case (1): segregation due to density di!erences. Left: average particle heights for smalland large particles as a function of time. Right: segregation percentage as a function of time.
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Figure 7. Case (1): Two-dimensional snapshots of the density segregation with inlet velocity 1.0 m/s. Dark dots are largeparticles and light ones are small particles. Left: initial configuration. Right: configuration after 6 s. The figure is zoomed in toshow the lower half of the domain.
Figure 8. Segregation time evolution for case (2): segregation due to particle size di!erences, with inlet velocity Vg % 1.0 m/s.Left: average bed heights for small and large particles as a function of time. Right: segregation percentage as a function of time.
Figure 9. Segregation time evolution case (3): segregation due to size di!erence with inlet velocity Vg % 1.5 m/s. Left: averagebed heights for small and large particles as a function of time. Right: segregation percentage as a function of time.
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segregation percentages are di!erent. The hybridmodel is thus capable of predicting the mixingbehaviour when both types of particles are fluidised.
The initial bed configuration as well as those att % 20 s for both case (2) and case (3) are presented inFigure 10. The left panel shows that initially theparticles were perfectly mixed and randomly packed.The central panel shows that in case (2) after 20 s, thesegregation was noticeable with higher concentrationof large particles towards the bottom. The right panelshows that for case (3) the two types of particles werestill mixed at a certain level.
5.2.3. Roles of drag distribution scheme
The roles of the scheme distributing drag force fromthe continuum level to the particle level were investi-gated. Case (4) was conducted with the same domainsetup and particle properties as in case (2), except thatthe drag forces were distributed to the particlesproportionally to their volumes. The average bedheight and the segregation percentage are presentedin Figure 11. No segregation was observed. Instead,after about 6 s, the equilibrium segregation percentage(about 78%) was reached with some fluctuations, i.e.large particles rose above the small particles. This isbecause the drag force, the buoyancy and the
gravitational force are all proportional to the particlevolume when the drag force is distributed proportion-ally to the volume. Therefore, the drag force does notdynamically di!erentiate small and large particles. Asa result, kinetic sieving dominated in this process,during which the smaller particles percolated to gapsbetween particles and the large particles were movedupward.
It can be seen that in an application with particlesof multiple sizes, the drag distribution scheme couldinfluence the qualitative nature of the segregationresults. Therefore, for the hybrid model, the capabilityof predicting size-segregation processes depends sig-nificantly on the drag distribution schemes.
As an alternative to this direct drag forcedistribution, other approaches can also be used toaddress particle property di!erences. An approachsimilar to the multi-fluid model can be used to dividethe particle into di!erent classes according to theirproperty. The drag force can be calculated for eachclass and then distributed to the particles within thisclass. As a further step, the drag force can also becalculated for each particle using the local fluidvelocity at the particle position, which would increasecomputational costs compared with the drag forcedistribution approach. It should be pointed out thatthe performance of each approach depends heavily on
Figure 10. Cases (2) and (3): two-dimensional snapshots of size segregation with inlet velocity Vg % 1.0 m/s and Vg % 1.5 m/s.Dark dots denote large particles and light ones denote small particles. Left: initial configuration. Centre: configuration after 20 sfor case (2) with inlet velocity Vg % 1.0 m/s. Right: configuration after 20 s for case (3) with inlet velocity Vg % 1.5 m/s.
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the drag force model. Accurate prediction of segrega-tion calls for drag force models that explicitly accountfor the particle size or density distribution, such asthe newly developed drag law by Yin and Sundaresan(2008).
6. Conclusion
Segregation in gas–solid fluidised beds has been studiedusing a multi-fluid model and a hybrid model. It hasbeen demonstrated that the hybrid model is capable ofcapturing particle mixing and segregation behavioursdue to density and size di!erences. The segregation ratespredicted qualitatively agree with corresponding experi-mental measurements although there are quantitativediscrepancies in segregation percentages. The multi-fluid model does not work well in the defluidised regimedue to its intrinsic di"culty to model multi-particlecontacts. The hybrid model has advantages in simulat-ing the particle interactions and thus is able to capturesegregation hindrance due to defluidisation.
Issues in the hybrid model development have alsobeen discussed. In particular, the drag distributionscheme was investigated. It has been shown that thedrag distribution scheme could qualitatively influencethe segregation results. Alternative approaches fordrag force calculation have also been suggested forfuture study.
Nomenclature
C Translational particle velocity fluctuationd Particle diameter
Figure 11. Segregation due to particle size di!erences, with drag distributed proportionally to particle volumes. Inlet velocity1.0 m/s. Left: average particle height for small and large particles. Right: segregation percentage.
e Coe"cient of normal restitutionf Forceg Gravityb Buoyancyw Bandwidth of kernel functionI Interphase momentum transfermi Mass of particle iNp Number of particles in the systemRe Reynolds numbersp Segregation percentageR Stress tensorU Fluidisation velocityu Velocity for gas and solids phasesx Mass fractionx Position of cell or particle
Greek lettersm Coe"cient of frictione Volume fractionr Densityx Generic independent variable for kernel
function
Superscripts/Subscriptsc Cell quantitiesi Particle quantitiesg Gas phasei Particle indexmf Minimum fluidisationmax Maximum valuep Particles Solid phasew Wall boundary
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