prediction of solids accumulation in slurry bubble columns with polydispersed solid loadings

Prediction of Solids Accumulation in Slurry Bubble Columns withPolydispersed Solid LoadingsIon Iliuta and Faïcal Larachi*

Department of Chemical Engineering, Laval University, Quebec, G1 V 0A6, Canada

ABSTRACT: A core-annulus multicompartment pseudo-2D two-bubble class model accounting for gas and slurry recircula-tion and coupled with catalyst sedimentation/advection/dispersion/lateral exchange balance equations for monodispersedand polydispersed solid particle systems was proposed to study the hydrodynamics of slurry bubble column facing solidsaccumulation in the reactor. The distribution of bubbles was represented by a bimodal two-class distribution: large andsmall bubbles. The model was coupled with a Prandtl−Nikuradse mixing length shear turbulence model or Sato bubble-induced and shear turbulence model to obtain the radial slurry velocity profile. A criterion for setting the onset of particlesaccumulation in the slurry bubble column was established. In solids polydispersed feeds, the influence of operating conditions(column diameter and height, solids feed mass flux and concentration, particle size distribution, gas velocity, and liquidviscosity) on the threshold particle size corresponding to the onset of particles accumulation in the reactor was analyzedsystematically.

■ INTRODUCTION

Although different reactor types are used for gas-to-liquidprocesses, the majority of attention during the past 10 years,from both academic and industrial interests, was paid to theslurry reactor. Slurry bubble columns have the advantages ofsimple construction, excellent heat transfer performanceenhanced by the agitation of the slurry phase, online catalystaddition and withdrawal, and a reasonable interphase masstransfer rate with low energy input. Also, slurry bubble columnsoffer operational flexibility, nearly isothermal operation, largecatalyst loadings and high productivity. These strengths arecounterbalanced by the detrimental back-mixing, a difficultliquid−solid separation downstream of the reactor, catalystattrition, and a persisting uncertain scale-up. The multiphaseinteractions in slurry bubble columns are very complex whichexplains why, despite being of a simple construction, theirdesign and scale-up are not yet fully understood. Hence, forefficient design of slurry bubble columns, it is desirable todevelop sound modeling with the capability to capture physicaland chemical features and to sort them out according to thedominant and the idle phenomena.Numerous investigators in the past have attempted to

develop mathematical models for slurry bubble columns. Thesuspension of solids in liquid was often modeled as a singlepseudohomogeneous slurry phase given the fairly small size ofthe solid particles. Within the slurry phase, the solids velocitywas assumed to be equal to the liquid velocity; hence slurrybubble columns were most often modeled as two-phase gas-slurry systems. The majority of the literature models assumegradientless concentration of the solids throughout the reactor.Closer to physical reality and following the work of Kato el al.,1

some models estimate the solids axial distribution by means ofsedimentation−dispersion models2,3 or mechanistic modelswhich takes into account the entrainment/de-entrainment inthe wake of rising bubbles.4 These models revealed that, despitesmall particle sizes, the axial profile of the solids concentration

in the slurry may not be flat and strongly depends on the slurrysystem properties and the operating mode and conditions.Slurry bubble columns operate either in the homogeneous or

heterogeneous flow regime. The hydrodynamic behavior, heatand mass transfer, and mixing behavior are quite different in thehomogeneous and heterogeneous regimes. In homogeneous(bubbly) flow regime, the gas phase is adequately modeledassuming uniformly sized bubbles.4,5 In the heterogeneous(churn-turbulent) flow regime, bubbles differing in size andshape exist. On the basis of experimental observations andmeasurements in the churn-turbulent regime,6,7 bubble sizedistributions have been approximated as two lumping classes:large and small bubbles size populations. Large bubbles risestraight up through the column and disengage without recir-culation, while small bubbles, due to their lower rise velocity,may experience recirculation within the vessel before they even-tually disengage. The bimodal bubble size distribution approachhas been used in modeling slurry bubble columns by de Swart,8

Maretto and Krishna,9 van der Laan et al.10 and Rados et al.11

Cross-flow interactions between the small and the large bubblesis either neglected12 or modeled as infinitely fast to homogenizethe species compositions in both bubble classes.13 As suggestedby Rados et al.,11 both of these approaches are limiting caseswhile the extent of physical interactions between small andlarge bubble phases would lie somewhere halfway and wouldlead to some disparities in the species concentrations.Intraphase back-mixing has been modeled using ideal flow

patterns (i.e., perfectly mixed slurry phase and gas plug flow)and axial dispersion models. These models represent the sim-plest descriptions of the flow patterns in slurry bubble columnsand seem inadequate in properly describing the actual fluid

Received: May 24, 2012Revised: September 10, 2012Accepted: September 14, 2012Published: September 14, 2012

Article

pubs.acs.org/IECR

© 2012 American Chemical Society 13100 dx.doi.org/10.1021/ie301361p | Ind. Eng. Chem. Res. 2012, 51, 13100−13112

pubs.acs.org/IECR

dynamics prevailing in the reactor. If simplistic approaches rely-ing on the use of ideal flow patterns can lead to costly and un-acceptable unit oversizing, uncertainties related to the estimationof axial dispersion coefficients renders the design of slurrybubble columns based on axial dispersion models, for the least,questionable. For recirculation dominated convective flows,such as those in slurry bubble columns, the application of axialdispersion modeling to describe the state of fluids mixing lacksphysical basis and has had limited success, except at fittingexperimental data.14,15

The confident scale-up and optimal design of commercial-scale slurry bubble columns still necessitate improved under-standing and quantification of fluid dynamics and trans-port phenomena. Availability of engineering-type models forimproved design and scale-up, based on the observed physicalphenomena, is particularly attractive16 because the currentrepresentation of the complex flow pattern in slurry bubblecolumns by the axial dispersion model attempts to lump thedescription of too many physical phenomena into a single dis-persion coefficient which cannot be done in a precise mannerand because the computational fluid dynamics (CFD) codes,based on the first principles, are not yet sufficiently developedto be effective tools for the design of technical slurry bubblecolumns.17,18

In some industrial processes, slurry bubble columns operatein conditions for which not all the solid particles may be keptin suspension. In the long run, such reactors may not operatestably and efficiently and it is necessary for the sake ofsafe design to gain knowledge about the critical (or cutoff)particle diameter associated with solids accumulation in thereactor. This work proposes a core-annulus multicompart-ment pseudo-2D two-bubble class model accounting for gasand slurry recirculation and coupled with the catalystsedimentation/advection/dispersion/lateral exchange balanceequations for monodispersed and polydispersed solid particlesystems to study the hydrodynamics of slurry bubble columnsassociated with solids accumulation in the reactor. A strategywas developed to explore the axial solids concentrationdistribution of polydispersed particle systems and appropriateboundary conditions were formulated. A criterion for settingparticles accumulation in the slurry bubble column wasestablished. In solids polydispersed feeds, the influence of theoperating conditions (column diameter and height, feed massflux and concentration of the solids, particle size distribution,gas velocity and liquid viscosity) on threshold particle sizecorresponding to the onset of particles accumulation in thereactor was analyzed.

■ MATHEMATICAL MODELThe hydrodynamic platform consists in a pseudo-2-Daxisymmetric hydrodynamic model which describes thecatalyst sedimentation/advection/dispersion/(core-annulus)-lateral exchange coupled to gas and slurry (re)circulation. Theproposed approach views the hydrodynamics in slurry bubblecolumns to obey a core-annulus type of flow structure.14,15

The distribution of bubbles sizes and shapes is represented bya bimodal bubble population consisting of two bubble classes:large and small bubbles. The pseudo-2-D feature comes fromthe fact that the model was split in two blocs to handle theaxial dependence of the reactor hydrodynamics due to gasphase contraction/expansion, besides a radial dependenceimposed by the core-annulus flow structure. In the first bloc,a two-fluid turbulent flow model was used for computing the

radial profiles of slurry and gas velocities, of gas holdup andbubble size. In the second, a coupled axial multicompartmentmodel accounting for sedimentation, dispersion, advection,and lateral exchanges (liquid−liquid, gas−gas and solid−solid) was solved for obtaining the longitudinal distributionsof the hydrodynamic variables.The two-fluid turbulent flow model has been presented

elsewhere19,20 and will only be described briefly. The overallmomentum balance for the gas/slurry system assuming a fullydeveloped flow was coupled with the gas holdup universalprofile model21 (structurally similar to that holding for bubblecolumns) and with Prandtl−Nikuradse mixing length shearturbulence model or Sato bubble-induced and shearturbulence model to obtain the radial slurry velocity profile:

ε ξ ρ ε ξ ρ

ξ ξξ ε ξ τ ξ

= − + − −

− −

gPz

R

0 ( ( ) (1 ( )) )dd

1 dd

[ (1 ( )) ( )]

g g g ls

cg ls

(1)

ε ξ ε ξ= ++ −

−mm c

c( )2

2 2(1 )m

g g,tot (2)

The Prandtl-Nikuradse mixing length shear turbulence modelincludes only the turbulence induced by the single-phase flow.The turbulence is independent of bubbles agitation:

τ ξ ρ υ υξ

= − +v

R( ) ( )

ddls

cls ls

turb ls

(3)

υ ξξ

ξ λ

ξξ λ

=

− ≤

≥

⎧

⎨⎪⎪

⎩⎪⎪

l

vR

vR

( )

dd

dd

turb 2

ls

c

ls

c (4)

ξ ξ ξ= − −l R( ) (0.14 0.08 0.06 )2 4c (5)

In the Sato bubble-induced and shear turbulence model,22 theeddy diffusivity to express the turbulent structure of the liquidphase is subdivided into the two components, one for theinherent turbulence independent of bubbles agitation and theother for additional turbulence caused by bubbles:

ξ ≤ 0.9

τ ρτρ

ξ ξξ

ε

ρ εξ

ε

= − − + −

− −

k v

kd

vv

R

6[1 ][1 2 ]

dd

(1 )

2d

d(1 )

ls

ls

c

lsturb

lsw

ls

2 2g

1 ls g b g(6)

ξ > 0.9

τ ρτρ

ξ ξξ

ε

ρ εξ

ε

ξυ

τρ

= − − + −

− −

× − −−+

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎧⎨⎩

⎫⎬⎭

⎧⎨⎩

⎡⎣⎢⎢

⎤⎦⎥⎥⎫⎬⎭

k v

kd

vv

R

RA

6[1 ][1 2 ]

dd

(1 )

2d

d(1 )

1 exp(1 )

g

ls

c

lsturb

lsw

ls

2 2 ls

1 ls g b g

c

ls

w

ls

2

(7)

Industrial & Engineering Chemistry Research Article

dx.doi.org/10.1021/ie301361p | Ind. Eng. Chem. Res. 2012, 51, 13100−1311213101

The gas velocity profile results from solving the gas momentumbalance equation after neglecting viscous and turbulent shearstresses in gas phase coupled with a bubble diameter radialdistribution model:

ε ξ ρ ξ

ξ

ρ

ρ

ρ

−− | − | *

= −⎜ ⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝

⎞⎠

C

dv v v v

Pz

g

34

(1 ( )) ( )

( )( )

dd

g ls Dls g ls g

g

g

0.25

g (8)

Here, we describe the compartment model for recirculation inslurry bubble columns highlighting the sedimentation/advection/dispersion/lateral exchange model to describe thesolids axial concentration profiles in the core and annulusregions for monodispersed and polydispersed particulate sys-tems. Figure 1 depicts the flow structure, and the mechanisms

and flow variables intervening in the segmentation of thereactor. The radial location of slurry velocity inversion waschosen to separate the core ascending region from theannulus descending region. Volume fraction of small bubblesin core and annulus and associated interstitial velocities,volume fraction of large bubbles in core and associatedinterstitial velocity, core and annulus liquid and slurry sus-pension volume fractions and associated interstitial velo-cities, core and annulus solid mass concentrations andassociated interstitial velocities are obtained by aggregat-ing the radial profiles for the holdups and velocities, andcombining them with slurry compositional relationships,Richardson and Zaki23 liquid−solid slip velocity correla-tions, sedimentation/advection/dispersion/lateral exchangeequations for transport of catalyst and their correspondingboundary conditions:

Gas holdup in annulus:

∫ε ε ξ ξ ξ=ξ

2 ( ) dSBII

1

g(9)

Small bubbles velocity in annulus:

∫εε ξ ξ ξ ξ=

ξ v v

2( ) ( ) dSBII

SBII

1

g g(10)

Gas holdup in core:

ε ε ε ε+ = −SBI LB g,tot SBII (11)

Homogeneous saturation of dense phase above transition:ε ε=SBI SBII (12)

Core and small bubbles volumetric fluxes:

ε ε ε

ε ε

+ = −

+ =

v v U v

v v U

SBI SBI LB LB g SBII SBII

SBI SBI SBII SBII SB (13)

Volume conservation equations in reactor core and annulus:

ε ε ε ξ ε ε ξ+ + = + = − 1SBI LB LSI2

SBII LSII2

(14)

Core and annulus slurry volume fractions:

ε ερ

ε ε ερ

ε= + = +C C

LSI LIcI

pLSI LSII LII

cII

pLSII

(15)

Core and annulus slurry velocities:

∫

∫ε

ε ξ ξ ξ ξ

εε ξ ξ ξ ξ

= −

= −

ξ

ξ

v v

v v

2(1 ( )) ( ) d

2(1 ( )) ( ) d

LSILSI 0

g ls

LSIILSII

1

g ls(16)

Core and annulus slurry volumetric fluxes:

ε ερ

ε

ε ερ

ε

= +

= +

v vC

v

v vC

v

LSI LSI LI LIcI

pLSI SI

LSII LSII LII LIIcII

pLSII SII

(17)

Core and annulus solid slip velocities23for monodispersedsolid particles:

ρ

ρ

− = −

− = −

∞

−

∞

−

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

v v vC

v v vC

1

1

n

n

LI SIcI

p

1

LII SIIcII

p

1

(18)

Core and annulus solid slip velocitiesfor polydispersed solidparticles:

ρ

ρ

− = −

− = −

∞

−

∞

−

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

v v vC

v v vC

1

1

i i

n

i i

n

LI SIcI

p

1

LII SIIcII

p

1

i

i

(19)

The value of n (ni) in eqs 18 and 19 for the three-phase systemswas correlated according to Matsumoto et al.:24

−−

= −nn

Ga2

56i

ii

1/2

(20)

Nonsteady-state and steady-state alternatives of the sedi-mentation/advection/dispersion/lateral exchange model wereproposed to describe the solids axial concentration profiles

Figure 1. Conceptual flow structure and reactor segmentation.



in the core and annulus regions for monodispersed andpolydispersed particulate systems. It is important to definethe net particle flux relative to the column for a solidparticle species preceding the development of the solidflow model. According to Ramirez and Galvin25 the netparticle flux relative to vessel is given by the followingequation:

ϕ=N vi i ip (21)

where i represents a single particle species having a specificsize and density, vp

i is the particle velocity relative to thevessel, and ϕi is the local volume fraction of the speciesconcerned (ϕi = εLSCc

i ). Introducing the usual representa-tions for dispersion and segregation, the following governingequation is obtained:25

ϕ ϕ ϕε

ε= = − ∂∂

+ = −∂

∂+N v D

zv D

Cz

C v( )i i i

ii i

ii i

p S s SLS c

LS c s

(22)

The dispersion of species i is given by the product of thedispersion coefficient and the species concentration gra-dient. The segregation flux, which is equivalent to theparticle flux in the absence of dispersion, is the product ofthe segregation velocity vs

i , and the local volume fraction ofsolids. The segregation velocity is obtained using eqs 18and 19.For the multicomponent mixtures of solid particles, the

sedimentation/advection/dispersion/lateral exchange modelconsiders that the interaction of large and small particles hasa negligible effect on the hindered settling velocities andsolids mixing of the respective particle sizes.26 Separating thedistribution of particles size into narrow fractions allows theformulation of the mass balance equations on each individualnarrow fraction of particles size. So, the mass balance equa-tions of the solids transport are applied to each fraction of apolydispersed mixture according to the weight fraction ofeach size.27 The nonsteady state alternative of the sedimentation/advection/dispersion/lateral exchange model for polydispersedsolid systems is

Core sedimentation/advection/dispersion transport of solidparticlesmass balance of class i:

ε ε ε

ξε εε ε

∂∂

+ ∂∂

= ∂∂

∂∂

−

+ −

⎛⎝⎜

⎞⎠⎟t

Cz

C v Dz

Cz

Rk C C

( ) ( )

2( )

( )

i i iS

i

i i

LSI cI LSI cI SI LSIcI

LSI LSII

c LSI LSIIcI cII

(23)

Table 1. Experimental Conditions Used for Hydrodynamic Model Validation

parameters

no. ref Authors Ug m/s ρp kg/m3 ρg kg/m

3 Rc cm H m Cc0 kg/m3 dp micrometers

1 Kato et al. (1972) 0.082 2520 1.29 3.3 2.01 100 88

2 Rados (2003) 0.140 2490 1.29 8.08 1.78 226.6 131

3 O’Dowd et al. (1987) 0.081 2420 1.29 5.4 2.44 244 904 Smith et al. (1986) 0.067 2420 1.29 7.62 1.54 0.27 48.5

81.6192

Figure 2. Comparison with the experimental data for monodispersedsolid particles systems of Kato et al. (1972) (a), Rados (2003) (b), andO’Dowd et al. (1987) (c).



Annulus sedimentation/advection/dispersion transport of solidparticlesmass balance of class i:

ε ε

εξε εε ε

∂∂

+ ∂∂

= ∂∂

∂∂

−

+ −

⎛⎝⎜

⎞⎠⎟

tC

zC v

Dz

Cz R

k C C

( ) ( )

2( )

( )

i i i

ii i

LSII cII LSII cII SII

S LSIIcII LSI LSII

c LSI LSIIcII cI

(24)

Boundary conditions (constant inventory of the solid in thereactor; uniform solid loading in the whole column section):

Core:

ε ε

ε ε

− | −∂∂

+

= | −∂∂

==

==

−

−

+

+

⎛⎝⎜⎜

⎞⎠⎟⎟v C D

Cz

D g

S

v C DCz

i iz

i

z

i

i iz

i

z

LSII SII cII 0 S LSIIcII

0

m

LSI SI cI 0 S LSIcI

0 (25)

∂∂

==

Cz

0i

z H

cI

(26)

Annulus:

∂∂

==

Cz

0i

z

cII

0 (27)

Figure 3. Comparison with the experimental data for polydispersedsolid particles systems of Smith et al. (1986).

Figure 4. Particle size distribution of different solid particles in slurry:(1) dp = 30 μm, (2) dp = 35 μm, (3) dp = 40 μm, (4) dp = 45 μm, (5)dp = 50 μm.

Figure 5. The transient axial solid concentration profiles in the coreregion for (a) 35-μm class (mass fraction = 0.24) and (b) whole solidparticles in a slurry bubble column with D = 0.46 m and H = 4.6 m,assuming constant gas velocity in the axial direction: standardparameters used in simulation.

Table 2. Standard Parameters Used in Simulations

D, m 0.46 H, m 4.6Ug, m/s 0.15 Cc0, kg/m

3 240μl, Pa s 4.0 × 10−4 ρl, kg/m

3 650ρp, kg/m

3 1370 ρg, kg/m3 6.20

σ, N/m 0.003 dp, μm 30−35−40−45−50P, MPa 19.0 Dm, kg/s 0.1



ε ε

ε ε

− | −∂∂

−

= | −∂∂

==

==

+

+

−

−

⎛⎝⎜⎜

⎞⎠⎟⎟v C D

Cz

D g

S

v C DC

z

i iz H

i

z H

i

i iz H

i

z H

LSI SI cI S LSIcI m

LSII SII cII S LSIIcII

(28)

Initial conditions: uniform concentration distribution in coreand annulus.For the steady-state alternative of the sedimentation/

advection/dispersion/lateral exchange model, the boundaryconditions (constant inventory of the solid in the reactor) are

Core:

ε ε ε

ε

− | −∂∂

+ =

| −∂∂

==

==

−

−

+

+

⎛⎝⎜⎜

⎞⎠⎟⎟v C D

Cz

D g

Sv

C DCz

i iz

i

z

i i

iz

i

z

LSII SII cII 0 S LSIIcII

0

mLSI SI

cI 0 S LSIcI

0 (29)

∂∂

==

Cz

0i

z H

cI

(30)

Annulus:

∫ ε ε ε ε+ = +C C z H C g( ) d ( )H

i ii0

LSI cI LSII cII LSI LSII c0 (31)

ε ε

ε ε

− | −∂∂

−

= | −∂∂

==

==

+

+

−

−

⎛⎝⎜⎜

⎞⎠⎟⎟v C D

Cz

D g

S

v C DC

z

i iz H

i

z H

i

i iz H

i

z H

LSI SI cI S LSIcI m

LSII SII cII S LSIIcII

(32)

Equations 23 and 24 were obtained by the substitution of eq 22in the dynamic material balance equation of single particlespecies i:

ϕ∂∂

= − ∂∂

− Φt

Nz

i i

tri

(33)

Note that a common dispersion coefficient has been used forconvenience, given that the majority of solid particles are

Figure 6. Transient axial solid concentration profiles in the coreregion for (a) 35-μm class (mass fraction = 0.24) and (b) wholesolid particles in a slurry bubble column with D = 0.46 m andH = 4.6 m assuming variable gas velocity in the axial direction:standard parameters used in simulation (Ug

in = 0.15 m/s; Uge =

0.1 m/s).

Figure 7. The transient axial solid concentration profiles in the coreregion for (a) 45 μm class (mass fraction = 0.24) and (b) 300 μm class(mass fraction = 0.24) in a slurry bubble column with D = 0.46 m andH = 4.6 m: standard parameters used in simulation.



similar in size. The core−annulus mass transfer exchangecoefficient and dispersion coefficient were evaluated using anisotropic theory to relate the turbulent kinetic energy, thedissipation of turbulent energy, the eddy size distribution andthe bubble size diameters in the slurry phase:19

= −k gU d1.4

5 120

5/3

g LB3(34)

= −D d gU d12.6620

(5 1)s5/3

LB g LB3(35)

The terminal velocity in the Stokes and intermediate regime ofsedimentation was evaluated using the general Wiener−Churchill correlation:

μρ

= − + +∞v d

b b a Gaa

( 4 )4

l

l p

2

2(36)

where a = 0.525, b = 24.The mean size of the large and small bubbles in the core and

annulus are obtained by aggregating the radial profiles for theholdups, effective bubble diameter and gas−liquid interfacial

area and assuming that core and annulus small bubbles have thesame diameter.19

It is worth mentioning that we have made the assumptionthat the time scale for the gas-slurry hydrodynamics to evolve intime because of the sedimentation of particles is much shorterthan the time scale for particles sedimentation. Hence changesin time of the sedimentation profiles do not influence the gasand liquid velocities and holdups. This is easily recognizable inthe formalism of eqs 1−20 which are in steady state, whereasthe equations for the sedimentation process, eqs 23 and 24are cast as transient forms for the core and the annulus,respectively.

■ NUMERICAL IMPLEMENTATIONAspen Custom Modeler from Aspen Tech was used to generatethe numerical platform to solve the mixed ODE/algebraicsystem which models the hydrodynamics of the slurry bubblecolumn reactor. A first-order backward finite difference methodwas used for the discretization in the radial direction and asecond -order backward finite difference method in the axialdirection. A nonlinear solver based on the Newton methodwas used to solve the set of simultaneous model equations.

Figure 8. The transient axial solid concentration profiles in the coreregion for (a) 45 μm class (mass fraction = 0.24) and (b) 800 μm class(mass fraction = 0.24) in a slurry bubble column with D = 0.92 m andH = 9.2 m: standard parameters used in simulation.

Figure 9. The inlet/outlet solid concentration in the core (a)and annulus (b) regions as a function of particle diameter in aslurry bubble column with D = 0.46 m and H = 4.6 m: stan-dard parameters used in simulation (35 μm class, mass fraction =0.24).



The residual convergence determined by the difference betweenthe left and right-hand sides of the equations was adopted.

■ RESULTS AND DISCUSSION

Model Validation. Validation of the cold hydrodynamicmodel is based on the experimental works of Kato et al.,1

O’Dowd et al.,28 and Rados29 for monodispersed solid particlesand of Smith et al.26 for polydispersed solid particles. Ambienthydrodynamic tests (room temperature and atmosphericpressure) were run using water as the liquid phase and air asthe gas phase. The experimental conditions are listed in Table 1.Figures 2 and 3 show comparisons between experimentalresults, expressed as the solid concentration in the core regionversus z/H (or z) and those simulated using the steady-statesedimentation/advection/dispersion/lateral exchange modelfor the solid longitudinal distribution. The two turbulencemodels were tested. It can be observed from these figures thatthe model predicts the concentration distribution of mono-dispersed and polydispersed systems reasonably well. The solidconcentration distribution as described by the sedimentation/advection/dispersion/lateral exchange model is valid for com-pletely suspended solids in slurry. This model may be useful fordetermining the onset of particles accumulation in the reactor.A criterion for setting the particles accumulation in the slurrybubble column was thus established.Simulations. The standard parameters used in the simula-

tions are listed in Table 2. The model describes satisfyingly the

slurry flow developed by the advection of the bubbles: theadvection velocity of the slurry in the core region is less thanthe gas velocity. Also, the model reproduces the core−annulusstructure observed in the experiments. The radius demarcatingthe slurry velocity inversion is relatively close to the onecorresponding to the gas velocity inversion (5% difference) andthis validates the hypothesis regarding distinction of one uniqueinversion radius for the gas−slurry system. The dimensionlessradius of core−annulus separation is also in agreement with theexperimental observations and corresponds to 0.7 ≤ ξ ≤ 0.85.The model indicates that a non-negligible fraction of the gasphase is recirculated in the annulus region and this providespost facto justification of the necessity to consider a gascompartment in the wall region. The model is incapable toquantify the influence of the gas density on the slurry velocityradial profile and this is not supported by the experimentalobservations of Rados29 and Chen et al.30 This nonagreement isdue to the fact that the correlations of Riquarts31 and Zehner32

used in the present model disregarded the effect of gas densityin the estimation of the centerline slurry velocity.In the following analysis, the axial solids concentration dis-

tribution of polydispersed particle systems was explored and acriterion for setting the particles accumulation in the slurrybubble column was established. The influence of the operatingconditions (column diameter and height, inlet mass flow rateand concentration of the solids, particle size distribution, super-ficial gas velocity and liquid viscosity) on the threshold

Figure 10. The inlet/outlet solid concentration in the core region as a function of particle diameter in slurry bubble columns with D = 0.3 m, H = 3m (a), D = 0.6 m, H = 6 m (b) and D = 0.92 m, H = 9.2 m (c): standard parameters used in simulation (35 μm class, mass fraction = 0.24), Cc0 = 120kg/m3, ρp = 2000 kg/m3.



(or cutoff) particle size, which corresponds to the onset of theparticles accumulation in the slurry bubble column forpolydispersed solid systems was explored.Figure 5 shows theoretical transient axial solid concentration

profiles in the core region for polydispersed solid systems(standard parameters used in simulation, Table 2). The particlesize distribution is illustrated in Figure 4 (solid particles weredivided in five fractions with diameters ranging between 30 and50 μm). A uniform concentration distribution in the core andannulus regions was assumed at t = 0. Steady-state is attainedafter 100 s. The concentration profiles are the consequence ofthe superposition of the convective (segregation-promoting)and dispersive (mixing-promoting) mechanisms. In Figure 5,the gas superficial velocity was considered constant in the axialdirection and equal with the inlet value. This choice may notbe representative for industrial slurry bubble columns wherechemical reactions and thermal effects may drastically influencethe exit gas superficial velocity. Therefore, such changes of thesuperficial gas velocity in the axial direction were emulated inthe following simulation by considering an ad hoc change(linear axial profile) corresponding to an exit velocity equalto 66.6% of the inlet superficial velocity. Figure 6 shows thetransient axial solid concentration profiles in the core region for

the 35-μm class and for the whole solid particles (standardparameters used in simulation as in Table 2) assuming Ugvariable in the axial direction. The same figure presentsthe steady-state axial solid concentration profiles underconstant superficial gas conditions. The difference between

Figure 11. The inlet/outlet solid concentration in the annulus regionas a function of particle diameter in a slurry bubble column with D =0.46 m and H = 3 m (a) and H = 7 m (b): standard parameters used insimulation (35 μm class, mass fraction = 0.24).

Figure 12. The inlet/outlet solid concentration in the annulus regionas a function of the particle diameter in a slurry bubble column withD = 0.46 m and H = 4.6 m for different values of liquid viscosity: (a) μl =0.0004 kg/ms; (b) μl = 0.0015 kg/ms; (c) μl = 0.003 kg/ms. Standardparameters used in simulation (35 μm class, mass fraction = 0.24).



the steady-state solid concentration simulated trends assumingconstant or variable Ug in the axial direction is not veryimportant, therefore, in the following analysis we assumedconstant Ug in the axial direction.Figure 7 shows the transient axial solid concentration profiles

in the core region for two different sizes of solid particles: 45and 300 μm (standard parameters used in simulation as inTable 2; particle size distribution of different solid particles inslurry is the same as in Figure 4). The figure indicates a partialsegregation of the two solid particle classes with no singlecomponent zone and close to perfect mixing. For polydispersedsolid systems, the concentration of larger particles exceeds thatof the smaller particles at the bottom, while the concentrationof smaller particles exceeds that of the larger ones at theapproach of the disengagement zone. A uniform concentrationdistribution in the core and annulus was assumed at t = 0 andsteady-state was attained after ca. 100 s.Figure 8 shows the transient axial solid concentration profiles

in the core region for the 45 and 800 μm solid particles classesin a slurry bubble column with a higher diameter (geometryscale-up factor = 2, aspect ratio H/D = invariant). For largecolumn diameters, the slurry interstitial velocity increases and

therefore a part of the small solid particles with lower settlingvelocity are transported easily toward the exit of the column. Asthe sedimentation is less important and the advective velocityof the slurry is higher the solid concentration becomes uniformin the reactor. The simulations indicate a partial segregation ofthe two classes of solid particles with no single componentzone.From Figure 9, it can be observed that for polydispersed solid

systems, the larger particles concentrate near the bottom of thecolumn where the solids concentration is the greatest. Incontrast near the column top, the solid concentration decreaseswith increased particle diameters. Particles accumulation isconsidered to start at a particle diameter whereby the solidconcentration in the core and annulus regions drops to zero atz = H. The particle size threshold shown in Figure 9 wasobtained by increasing the particle diameter of the solid fraction0.24 from 35 to 980 μm. Particle size distribution of differentsolid particles in slurry is the same as in Figure 4.Figure 10 shows the effect of the column diameter on the onset

of particles accumulation in the slurry bubble column. With theincrease of the column diameter (aspect ratio H/D = invariant),

Figure 13. The inlet/outlet solid concentration in the core region as afunction of the particle diameter in a slurry bubble column with D =0.46 m and H = 4.6 m for different values of the solid feed mass flowrate: (a) Dm = 0 kg/s; (b) Dm = 0.1 kg/s). Standard parameters used insimulation (35 μm class, mass fraction = 0.24), μl = 0.0015 kg/ms.

Figure 14. The inlet/outlet solid concentration in the annulus regionas a function of particle diameter in a slurry bubble column with D =0.46 m and H = 4.6 m for different values of the average concentrationof the solids charged in the column: (a) Cc0 = 100 kg/m3; (b) Cc0 =240 kg/m3. Standard parameters used in simulation (35 μm class, massfraction = 0.24, Dm = 0 kg/s; μl = 0.0015 kg/ms.



particles accumulation is shifted toward larger particle diametersbecause of the intensified recirculation rate of the slurry.At a higher slurry recirculation rate, the small solid particleswith lower settling velocity are transported easily to the exit ofthe column and cannot accumulate in the reactor. So, inindustrial-scale reactors, solids accumulation begins at largerparticle sizes.Figure 11 shows the impact of the column height on the

onset of the particles accumulation in the slurry bubble column.With the increase of the column height (aspect ratio H/Dchanged from 6.5 and 15.2) the solid holdup in the columnupper part becomes lower. In these conditions, theaccumulation in the inlet region determines the depletion inexit region at lower particle diameter. It appears that in slurrybubble columns with lower aspect ratio, solids accumulation isinitiated at larger particle sizes.Figure 12 shows the effect of the liquid viscosity on the onset

of the particles accumulation in the slurry bubble column.Increasing the liquid viscosity contributes to an increase in theslurry viscosity. This gives a higher particle-fluid drag and alower particle settling velocity. As the result, the particles

accumulation is shifted to higher threshold particle diameters inspite of weaker slurry recirculation rate in the reactor by thehigher viscosities.Figure 13 reveals that the cutoff diameter demarcating the

start of particles accumulation is sensitive to the incoming feedsolid mass flow rate. This figure compares the case of solidsbatch operation (Dm = 0 kg/s) to a continuous operation withsolids throughflow across the bubble column (Dm = 0.1 kg/s).Because of the higher solids concentration and flux near thebottom of the column, under solid feed conditions, particlesaccumulation was displaced to a lower threshold size. Thissimulation reveals that critical diameter must be set inthroughflow operation of the solids and by taking into accountthe most realistic possible recirculation pattern of the slurryphase in the vessel.Figure 14 shows the effect of the average concentration of

the solids charged in the column on the onset of particlesaccumulation. In a manner reminiscent of the increase in liquidviscosity itself as discussed in Figure 12, increasing the solidsaverage concentration contributes to increase the slurry vis-cosity. This contributes, as expected, to increase the particle−fluid drag and to decrease the particles’ settling velocity. Asthe result, the particle accumulation process begins at higherthreshold particle diameters.Figure 15 shows the influence of superficial gas velocity on

the onset of particles accumulation. Increasing gas velocityintensifies slurry recirculation and hence contributes to displaceparticles settling toward increased cutoff particle diameters.Particle size threshold as discussed in Figures 9−15 was

obtained by initializing different particles diameters which werekept constant during the dynamic simulations and maintain-ing the same particle size distribution of different solid particlesin slurry as in Figure 4. The diameters were progressivelyincreased until the slurry concentration at the exit (at z = H)attained zero. In solids throughflow operation this cutoff pointcorresponds to particles entering the feed but not exiting thebubble column reactor thus causing accumulation of particlesin the vessel. The criterion for setting particles accumulation inthe slurry bubble column can be established, also, by assuming

Figure 15. The inlet/outlet solid concentration in the annulus regionas a function of particle diameter in a slurry bubble column with D =0.46 m and H = 4.6 m for different values of the gas velocity: (a) Ug =0.15 m/s; (b) Ug = 0.3 m/s. Standard parameters used in simulation(35 μm class, mass fraction = 0.24), Dm = 0 kg/s; μl = 0.0015 kg/ms,Cc0 = 100 kg/m3.

Figure 16. The transient axial solid concentration profiles in the coreregion for the 40 μm class (mass fraction = 0.40) in a slurry bubblecolumn with D = 0.46 m and H = 4.6 m, assuming constant Ug in theaxial direction and time variable particle diameter: standard parametersused in simulation.



variable particle diameters in time. Figure 16 shows thetransient axial solid concentration profiles in the core regionfor the 40-μm class (standard parameters used in simulation,Table 2) assuming variable particle diameter in time (ddp/dt =5 × 10−6). In the time, near the bottom of the column the solidconcentration increases with the increase of the particlediameter. On the other side, near the top of the column thesolid concentration decreases with the increase of the particlediameter. Particles accumulation starts at the particle diameterwhere the solid concentration in the core and annulus regions(at z = H) drops to 0.

■ CONCLUSIONA core−annulus multicompartment pseudo-2D two-bubbleclass model which takes into account the recirculation of theslurry and gas phases coupled with the catalyst sedimentation/advection/dispersion/lateral exchange balance equations formonodispersed and polydispersed solid particle systems wasdeveloped to study the hydrodynamics of slurry bubblecolumns with solids accumulation in the reactor. The proposedapproach views the hydrodynamics of slurry bubble columns toobey a core−annulus type of flow structure. The distribution ofbubbles sizes and shapes was represented by a bimodal bubblepopulation consisting of two bubble classes: large and smallbubbles. The model was coupled with a Prandtl−Nikuradsemixing length shear turbulence model or Sato bubble-inducedand shear turbulence model to obtain the radial slurry velocityprofile. A criterion for setting the particles accumulation in theslurry bubble column was established, and the influence ofreactor operating conditions on the threshold particle size,which corresponds to the onset of particles accumulation in thereactor, in polydispersed feeds was analyzed. The followingconclusions can be drawn from the simulation results:

− Steady-state solid concentration distribution obtained usingthe model is in good agreement with the experimental data.

− Axial solid concentration distribution for polydispersedsolid systems indicates a significant segregation when theparticles size ratios increase.

− In large-scale reactors, solid accumulation seems to beginat larger particle sizes due to intensified slurry recircula-tion rate.

− Lower liquid viscosity initiates solid accumulation atlower threshold particle diameters due to the largerterminal particle velocities opposing slurry recirculation.

− Higher gas velocity generates solid accumulation at largerparticle diameters due to intensified slurry recirculationrate and lower settling velocity.

− Higher solid concentration causes solid accumulation tooccur at higher particle diameter as a result of largerslurry viscosity and lower settling velocity.

− In slurry bubble column reactors exhibiting lower aspectratio solids accumulation starts at larger particle sizes.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected] authors declare no competing financial interest.

■ NOTATIONA+ = empirical parameter in eq 7c = empirical parameter in eq 2

Cc0 = average mass solid concentration, kg/m3slurry

Cci = mass solid concentration in i compartment, kg/m3slurry

CD = drag coefficientdp = catalyst particle diameter, mdLB = diameter of large bubbles, md = local bubbles diameter, mD = column diameter, mDm = feed solid mass flow rate, kg/sDs = axial dispersion coefficient of the solid phase, m2/sg = gravitational acceleration, m/s2

gi = mass fraction of i-class solid particlesGa = modified Galileo number, Ga = (4)/(3) ((ρp − ρl)dp

3ρlg)/(μl2)

H = height of the reactor, mk = mixing length constantk = mass transfer coefficient at the core-annulus interface, m/sk1 = empirical constantl = mixing length (Nikuradse), mm = curvature parameterP = reactor pressure, Par = core-annulus separation rayon, mr = radial coordinate, mRc = reactor radius, mUg = superficial gas velocity, m/sUSB = volumetric flux of the small bubble, m/svb = relative velocity of bubbles, m/svg = interstitial gas velocity, m/svi = interstitial velocity of compartment i, m/svl = interstitial liquid velocity, m/svls = interstitial slurry velocity, m/sv∞ = terminal velocity, m/sz = axial coordinate, m

Greek Lettersεg = local gas holdup, m3/m3

reactorεg,tot = total gas holdup, m3/m3

reactorεi = volumetric fraction of compartment i, m3/m3

reactorΦtr = lateral exchange mass transfer term, kg/(m3

reactor s)λ = dimensionless radius which correspond to maximumnegative slurry velocityυturb = turbulent cinematic viscosity, m2/sυls = kinematic viscosity of slurry, m2/sμl = dynamic viscosity of liquid phase, Pa sμls = dynamic viscosity of slurry, Pa sρg = gas density, kg/m3

ρg* = gas density at atmospheric pressure, kg/m3

ρi = density of compartment i, kg/m3

ρl = liquid density, kg/m3

ρls = suspension density, kg/m3

ρp = catalyst density, kg/m3solid

τls = shear stress, kg/(m s2)τlsturb = turbulent shear stress, kg/(m s2)τw = wall shear stress, kg/(m s2)ξ = dimensionless radial coordinateξ = dimensionless radius of core-annulus separation

Superscriptsi = i-classin = inlete = exit

AbbreviationsLI = liquid in core regionLII = liquid in annulus region



mailto:[email protected]

LB = large bubblesLSI = slurry in core regionLSII = slurry in annulus regionSI = solid in core regionSII = solid in annulus regionSBI = small bubbles in core regionSBII = small bubbles in annulus

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prediction of solids accumulation in slurry bubble columns with polydispersed solid loadings

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