Chemical Engineering Journal 236 (2014) 149–157

Contents lists available at ScienceDirect

Chemical Engineering Journal

journal homepage: www.elsevier .com/locate /cej

Simulation of gas–solid flow in 2D/3D bubbling fluidized bedsby combining the two-fluid model with structure-based drag model

1385-8947/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cej.2013.09.067

⇑ Corresponding authors. Tel.: +86 01062556951 (H. Li).E-mail address: hzli@home.ipe.ac.cn (H. Li).

Xiaolin Lv a,b, Hongzhong Li a,⇑, Qingshan Zhu a,⇑a State Key Laboratory of Multi-Phase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, P.O. Box 353, Beijing 100190, PR Chinab University of Chinese Academy of Sciences, Beijing 100049, PR China

h i g h l i g h t s

� Seven local structural parameters (ee, Uge, fb, Ub, Ugb, db, ab) have been solved.� The structure-based drag correlation has been built.� The solid concentration profiles in 2D/3D bubbling fluidized beds are predicted.

a r t i c l e i n f o

Article history:Received 2 July 2013Received in revised form 29 August 2013Accepted 16 September 2013Available online 26 September 2013

Keywords:FluidizationSimulationHydrodynamicsDrag force modificationBubbling fluidized beds

a b s t r a c t

In this work, a new method was developed to simulate bubbling fluidized beds with Geldart A particlesby modifying the conventional drag correlations through considering the effect of the non-uniform flowstructure. In this method, seven local structural parameters (ee, Uge, fb, Ub, Ugb, db, ab), which were used todescribe the internal flow structure of bubbling fluidized beds, were obtained by solving seven indepen-dent equations made up of momentum conservation equation, mass conservation equation, and empir-ical correlations. The structure-based drag correlation was incorporated into the two-fluid model tosimulate the hydrodynamics of Geldart A particles in bubbling fluidized beds, where the simulatedresults showed good agreements with those experimental data for the axial and radial solid concentra-tion profiles. The solid circulation pattern in the 2D bubbling fluidized bed was also captured.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

As an important chemical reactor, bubbling fluidized beds havebeen widely used for their high heat and mass transfer rates be-tween gas and solids [1]. Predicting the hydrodynamics of gas–so-lid flows has great significance in scale-up of bubbling fluidizedbeds. Nowadays, substantial progress has been made in simulatingbubbling fluidized beds by using Eulerian approach [2]. However,the standard Eulerian model [3,4], which adopts the conventionaldrag models that assume homogeneous conditions inside the con-trol volume to calculate the drag coefficient, cannot describe thehydrodynamics of bubbling fluidized beds accurately. Li andKwauk [5,6], Beeststra et al. [7], Ma et al. [8] have proved thatthe standard Eulerian model would make the drag coefficientbigger since the standard model did not consider the effects ofthe meso-scale structure on the momentum transfer. In order tosolve this problem, it is necessary to develop a drag model that

takes the meso-scale structure effect into account. Gao et al. [9]demonstrated that the prediction accuracy could be much im-proved through replacing the particle size with a cluster size inthe standard Eulerian model even if the cluster size was assumedto be constant. This method is further extended to the simulationof Geldart C particles by Zou et al. [10]. However, if the local struc-tural parameters could be solved correctly, the hydrodynamics offluidized beds would be predicted well with incorporating the dragcorrelation based on the local structure into the Eulerian model.For solving the local structural parameters and combining thetwo-fluid model with the drag model based on the local structure,the energy minimization was adopted as the stability conditions inYang’s [11] and Shi’s [12] works. With a scaling factor predicted byenergy minimization multi-scale (EMMS) model, Yang et al. [11]incorporated his structure-dependent drag model into the Eulerianmodel. Shi et al. [12] extended this method to the bubbling fluid-ized beds and proposed a bubble-based EMMS model to considerthe effect of heterogeneous structures on the inter-phase dragforce. Without considering the gas through the bubbles, the expan-sion characteristics of bubbling fluidized bed still could be reason-ably predicted.

Nomenclature

ab bubble accelerationae the acceleration of emulsion phaseCD averaged drag coefficientCDbo drag coefficient of single bubble (dimensionless)CDe drag coefficient of multi-particle (dimensionless)CDb drag coefficient of multi-bubble (dimensionless)D fluidized bed diameter (m)db bubble diameter (m)dbm maximum bubble diameter (m)dbo initial bubble size (m)dp diameter of particles (m)e elastic coefficientfb ratio of the bubble phaseFDe the drag force of flowing gas on single particle in the

emulsion phaseFDen the drag force of flowing gas on the particles in the

emulsion phaseFDb the drag force of single bubble on the particles in the

emulsion phaseFDbn the drag force of bubbles on the particles in the emul-

sion phaseFD the total drag force on the particles in a unit volume of

bedg gravitational acceleration (kg m s�2)h height of the reactor (m)Ho initial packing height (m)Hd heterogeneous indexRe Reynolds numberDt time step (s)U superficial velocity (m/s)u real velocity (m/s)

Ub superficial bubble velocityUge superficial gas velocity in the emulsion phase (m/s)Ugb superficial gas velocity through bubblesUmb superficial minimum bubbling velocity (m/s)Umf superficial minimum fluidization velocity (m/s)Up superficial particle velocityUpe superficial particle velocity in the emulsion phase (m/s)Us superficial slip velocity between gas and particles (m/s)Usb superficial slip velocity between bubble and emulsion

(m/s)Use superficial slip velocity in the emulsion (m/s)ut terminal velocity (m/s)

Greek symbolsb drag coefficient (kg/m3 s)eg the average voidageee the voidage of emulsion phasees solid volume fractioned voidage at the intersection of the fitting Hd function and 1es,max close packing densityq density (kg m�3)l viscosity (Pa s)

Subscriptsb bubblee emulsion phaseg gas phasemf minimum fluidizationp particles solid phase

Fig. 1. System resolution for bubbling fluidized bed.

150 X. Lv et al. / Chemical Engineering Journal 236 (2014) 149–157

However, with the experimental research progress on the bub-bling fluidized beds, many correlations [13–16] which can effec-tively reflect the relationships among the local structureparameters have been obtained. Mori and Wen [17], Darton et al.[18], Werther [19], Weimer and Quarderer [20], Horio and Nonaka[21] studied on the bubble diameter in different heights of bub-bling fluidized beds and proposed their own correlations basedon their experimental results. Davidson and Harrison [22], Daviesand Taylor [23] correlated bubble velocity with bubble diameter.Besides, through lots of experiments with different gas–solid sys-tems, Abrahamsen and Geldart [24] correlated the average voidagewith the gas velocity inside the emulsion phase. These suggest thatit is possible to solve the local structural parameters by empiricalcorrelations besides momentum conservation equation and massconservation equation.

Therefore, the first objective of this work is to establish the dragmodel based on the local structure using the multi-scale method.Then the local structural parameters are determined by solvingmomentum conservation equation, mass conservation equation,and empirical correlations simultaneously. With the two-fluidmodel in the commercial computational fluid dynamics (CFD) soft-ware, the hydrodynamics of a bubbling fluidized bed with fluid cat-alytic cracking (FCC) particles is simulated to study the axial andradial distributions of the solid concentration.

2. The drag model based on local structure

2.1. Resolution of bubbling fluidized bed

For a bubbling fluidized bed, the gas flow is separated into theemulsion phase and bubble phase. As show in Fig. 1, the gas–solid

bubbling fluidized bed is divided into two sub-regions: emulsionphase and bubble phase. It is assumed that the particles in theemulsion phase are uniformly distributed, and the bubble phaseis only consisting of gas (eb = 1). Seven variables are employed todescribe the local structure of the bubbling fluidized bed, whichare the superficial gas velocity in the emulsion phase Uge, the vol-ume fraction of bubbles fb, the rising velocity of bubble Ub, thediameter of bubble db, the voidage of emulsion phase ee, the bubbleacceleration ab and the superficial gas velocity through bubblesUgb. As an approximation, the emulsion phase is viewed as apseudo-fluid with mean density qe, viscosity le [25] and superficialvelocity Ue, which are interpreted as follows:

X. Lv et al. / Chemical Engineering Journal 236 (2014) 149–157 151

qe ¼ qpð1� eeÞ þ qgee; ð1Þ

le¼lg ½1þ2:5ð1�eeÞþ10:05ð1�eeÞ2þ0:00273expð16:6ð1�eeÞÞ�ð2Þ

Ue ¼qgUge þ qpUpe

qpð1� eeÞ þ qgee: ð3Þ

Upe is superficial particle velocity in the emulsion phase (m/s).

2.2. Model correlation

In a unit volume of gas-particles flow, the drag force FDen offlowing gas on the particles in the emulsion phase is

FDen ¼ð1� fbÞð1� eeÞ

p6 d3

p

FDe; ð4Þ

where the drag force of flowing gas on single particle, FDe, can beevaluated by the following equation:

FDe ¼ CDe12qg

p4

d2pU2

se: ð5Þ

Meanwhile, the drag force FDbn of bubbles on the particles in theemulsion phase is

FDbn ¼fb

p6 d3

b

FDbð1� eeÞqp

qe; ð6Þ

where the drag force of single bubble on the particles, FDb, can beevaluated by the following equation:

FDb ¼ CDb12qe

p4

d2bU2

sb: ð7Þ

In the above equations, CDe is the drag coefficient in the emul-sion phase and Use is superficial slip velocity in the emulsion phase.They are calculated by the equations as follows [26]:

CDe ¼ 200ð1� eeÞlg

e3eqgdpUse

þ 73e3

e; ð8Þ

Use ¼ Uge � Upeee

1� ee: ð9Þ

As an approximation, the amount of entrained particles is as-sumed negligible, thus, Up = 0, Upe = 0. The superficial slip velocitybetween bubble and emulsion phase, Usb, is evaluated as

Usb ¼ ðUb � UeÞð1� fbÞ: ð10Þ

And CDbo is the drag coefficient for single bubble, calculated by[27]

CDbo ¼38Re�1:5

i 0 < Rei 6 1:82:7þ 24

ReiRei > 1:8

( )Rei ¼

qedbUsb

le: ð11Þ

According the drag coefficient proposed by Ishii and Zuber [28],the drag coefficient for multi-bubble CDb can be calculated with thefollowing equation:

CDb ¼ CDboð1� fbÞ�0:5: ð12Þ

Therefore, the total drag force on the particles in a unit volumeof bed is

FD ¼ FDen þ FDbn ¼34

CDeqg

dpð1� fbÞð1� eeÞU2

se þ34

fbð1� eeÞCDbqp

dbU2

sb: ð13Þ

On the other hand, according to the definition of averaged dragcoefficient, the total drag force also can be calculated by

FD ¼34ð1� egÞCD

qg

dpU2

s ; ð14Þ

where eg is the average voidage and Us is the superficial slip velocitybetween gas and particles, calculated by

Us ¼Ug

eg� Up

1� eg

� �eg ¼ Ug � Up

eg

1� eg

� �: ð15Þ

Therefore, the equation for averaged drag coefficient CD can beobtained by comparing (13) and (14)

CD ¼ CDeð1� fbÞð1� eeÞð1� egÞ

Use

Us

� �2

þ CDbfbð1� eeÞð1� egÞ

qp

qg

dp

db

Usb

Us

� �2

ð16Þ

Then, the exchange coefficient of momentum can be calculatedby the following equation:

b ¼ FDeg

ðug � upÞ¼ 3

4ð1� egÞCD

� qg

dpjug � upje3

g : ð17Þ

For comparison, the heterogeneous index can be defined as adimensionless drag coefficient scaled with the Wen & Yu drag coef-ficient, that is,

Hd ¼b

bWen & Yu¼ CD

�

CDoe�4:7g

; ð18Þ

where

bWen & Yu ¼34ð1� egÞeg

dpqg jug � upjCDoe�2:7

g : ð19Þ

3. The local structural parameters model

3.1. Model equations

Seven structural parameters were used to describe the localstructure of the bubbling fluidized beds. For solving these struc-tural parameters, seven independent equations built by mass con-servation equation and empirical correlations, were employed asfollows:

3.1.1. The equation on average voidageAccording the definition of average voidage, we have

eg ¼ eeð1� fbÞ þ fb: ð20Þ

3.1.2. The equation on voidage in emulsion phaseTo correlate the superficial gas velocity in the emulsion phase

(Uge) with the voidage of emulsion phase (ee), the Richardson andZaki relationship [29] was used.

ene ¼

Uge

Utð21Þ

According the terminal Reynolds number, Ret, the exponent n isevaluated from the relation suggested by Kwauk [30] and Ut is theterminal particle velocity.

3.1.3. The equation on gas velocity through bubble phaseWhen the bubble velocity is greater than the gas velocity in the

emulsion phase, the movement of gas is viewed as circulating withthe bubbles, which results no gas through the bubbles. Meanwhile,if the bubble velocity is less than the gas velocity in the emulsionphase, the gas velocity through the bubbles is viewed as equalingto the gas velocity in the emulsion phase.

Fig. 2. The structure parameters: (a) the variation of bubble diameter, velocity with height; (b) the variation of bubble diameter with bubble volume fraction; (c) the variationof gas velocity in the emulsion with height and averaged voidage; and (d) the variation of slip velocity between bubble and emulsion with height and averaged voidage.

152 X. Lv et al. / Chemical Engineering Journal 236 (2014) 149–157

Ugb ¼ Uge Ub <Uge

ee

Ugb ¼ 0 Ub P Uge

ee

8<: ð22Þ

3.1.4. The equation on bubble velocityThe rising velocity of bubbles in the bubbling fluidized beds is

mainly dependent on the bubble diameter. In this work, theDavidson and Harrison [22] equation of bubble velocity wasadopted.

Ub ¼ ðUg � Umf Þ þ 0:711ðgdbÞ0:5 ð23Þ

3.1.5. The equation on bubble diameter [17]As typical meso-scale structure, bubbles play an important role

in the momentum transfer. According to the mechanism of bubblegrowth, the average bubble diameter increases with the bed height(h) above the distributor before the bubbles reaching an equilib-rium size [17–21]. The correlation of Mori and Wen [17] was se-lected for this calculation.

dbðhÞ ¼ dbm � ðdbm � dboÞe�0:3h=D

dbm ¼ 0:65½AðUg � Umf Þ�0:4ð24Þ

The diameter dbm is the maximum bubble diameter that wouldexist in this bubbling fluidized bed. dbo is the initial bubble diam-eter at the distributor, which is calculated by

dbo ¼ 0:0038ðUg � Umf Þ2 ð25Þ

3.1.6. The equation on gas mass conservation

Ug ¼ Ugeð1� fbÞ þ ðUb þ UgbÞfb ð26Þ

3.1.7. The equation on bubble accelerationZhang and VanderHeyden [31], Wilde [32,33] both proposed

correlations to evaluate the added mass force, which are

Fam ¼ Cbð1� eeÞfbqeðab � aeÞ; ð27Þ

Fam ¼ r2ðqp � qgÞg; ð28Þ

where the coefficient of the added mass force, Cb, is written by [34]

Cb ¼ 0:51þ 2f b

1� fb

� �: ð29Þ

And, the variance of local solid concentration fluctuation, r, canbe correlated by [35]:

r2 ¼ð1� egÞ2e4

g

1þ 4ð1� egÞ þ 4ð1� egÞ2 � 4ð1� egÞ3 þ ð1� egÞ4: ð30Þ

Because of the large inertial difference between the gas and par-ticles, the acceleration of emulsion phase ae is negligible (ae = 0).

Therefore, the equation on bubble acceleration can be obtainedby comparing (27) and (28):

ab ¼r2ðqp � qgÞgCbð1� eeÞfbqe

: ð31Þ

3.2. Solution scheme

To solve these nonlinear equations, we adopt the scheme asfollows:

1. For a given system with physical parameters (qg, qp, lg) and thesuperficial gas velocity Ug.

2. Traverse h within [0, 2.464] and calculate db from Eq. (24) andUb from Eq. (23).

3. Traverse eg within [emf, ed], calculate ee from Eqs. (20)–(22) and(26).

Table 1Heterogeneous index of different velocities for Geldart A particles.

Gas velocity (m/s) Heterogeneous index (Hd) Application range (eg)

0.06 Hd ¼ 2:046� 8:546eg þ 0:02172hþ 9:212e2g � 0:0436egh emf < eg < 0:75

0.1 Hd ¼ 0:4547� 2:955eg þ 0:05291hþ 4:432e2g � 0:104egh emf < eg < 0:87

0.2 Hd ¼ �0:1702� 0:4251eg þ 0:07973hþ 1:63e2g � 0:1494egh emf < eg < 1

X. Lv et al. / Chemical Engineering Journal 236 (2014) 149–157 153

4. Calculate Uge from Eq. (21).5. Calculate Ugb from Eq. (22).6. Calculate fb from Eq. (20).7. Calculate ab from Eq. (31).8. Calculate Hd using Eq. (18).

4. Results

4.1. The structure parameters

The discussion of the structure parameters starts from a specificcase of FCC/air system (qp = 1780 kg/m3, dp = 6.5 � 10�5 m,Ug = 0.06 m/s). The data in the interval (emf, ed) are used.

Table 2Governing equations for gas–solid flow.

Continuity equations of gas and solid@ðegqg Þ@t þr � ðegqg~ugÞ ¼ 0; @ðesqsÞ

@t þr � ðesqs~usÞ ¼ 0Conservation of momentum of gas and solid@ðegqg~ug Þ

@t þr � ðegqg~ug~ugÞ ¼ �egrpg þr � ðegsgÞ þ egqgg � bð~ug �~usÞ@ðesqs~usÞ

@t þr � ðesqs~us~usÞ ¼ �esrpg �rps þr � ðesssÞ þ esqsg � bð~ug �~usÞGranular temperature equation32

@ðesqsHs Þ@t þr � ðesqs~usHsÞ

h i¼ ð�p~I þ ssÞ : ðr~usÞ þ r � ðksrHsÞ � cs � 3bHs

Gas phase stress

sg ¼ lg ½r~ug þ ðr~ugÞT � � 23 lgr �~ug

Solid phase stress

ss ¼ ls½r~us þ ðr~usÞT � þ ðks � 23 lsÞr �~us

Radial distribution function

go ¼ 1� eses;max

� �1=3� ��1

Solid pressureps ¼ esqsHs þ 2ð1þ eÞe2

s goqsHs

Bulk solid viscosity

ks ¼ 43 esqsdpgoð1þ eÞ Hs

p� 1=2

Frictional models

Pfr ¼ Fr ðes�es;minÞn

ðes;max�es Þp

Johnson et al. [36]

ls;fr ¼ps sin /

2ffiffiffiffiffiI2D

p

Schaeffer [37]Collisional viscosity [38]

ls;col ¼ 0:8esqsdpð1þ eÞgoHsp� 1

2

Kinetic viscosity [39]

ls;kin ¼ qsdpes

ffiffiffiffiffiffiffiHspp

6ð3�eÞ ½1þ 0:4ð1þ eÞð3e� 1Þesgo�Shear viscosity of solidls ¼ ls;kin þ ls;col þ ls;fr

That is,

ls ¼ qsdpes

ffiffiffiffiffiffiffiHspp

6ð3�eÞ ½1þ 0:4ð1þ eÞð3e� 1Þesgo� þ 0:8esqsdpð1þ eÞgoHsp� 1

2 þ ps sin /

2ffiffiffiffiffiI2D

p

Granular conductivity of fluctuation energy

ks ¼ 150qsdp

ffiffiffiffiffiffiffiHspp

384ð1þeÞgo1þ 6

5 ð1þ eÞesgo

� �2 þ 2e2s qsdpð1þ eÞgo

Hsp� 1

2

Collisional energy dissipation

cs ¼ 3e2s qsgoHsð1� e2Þ 4

dp

Hsp� 1

2

� �Inter-phase drag coefficient

b ¼

34ð1�ef Þef

dpqf juf � upjCDoe�2:7

f eg > ed

34ð1�ef Þef

dpqf juf � upjCDOe�2:7

f Hd emf < eg < ed

150 ð1�ef Þ2lf

ef d2pþ 1:75 ð1�ef Þqf juf�up j

dpeg < emf

8>>><>>>:

From Fig. 2(a), it can be seen that the rising velocity of bubble(Ub) and the diameter of bubble (db) increase with the height. InFig. 2(b), the diameter of bubble (db) decrease with the volumefraction of bubbles (fb). Fig. 2(c) describes the gas velocity in theemulsion phase. It can be seen that the gas velocity in the emulsionphase, Uge, increased with the average voidage (eg). When the aver-age voidage increase to ed, the gas velocity in the emulsion is closeto the gas velocity (Ug), which is consistent with the actual situa-tion. For the reason that the rising velocity of bubble increase withthe height, the slip velocity between bubble and emulsion is great-er in the higher section of the bubbling fluidized bed, which can befound in Fig. 2(d).

4.2. The heterogeneous index

The local structure parameters model is solvable in the interval(emf, ed) and the fitting functions are listed in Table 1. The heteroge-neous index (Hd) is equal to 1.0 when the value of average voidageis ed.

4.3. Simulation results of Geldart A

4.3.1. Simulation stepThe CFD model used in this study is two-fluid model, whose

governing equations are listed in Table 2. The solid stress in the

Fig. 3. Schematic diagrams of the simulated 2D/3D bubbling fluidized beds.

154 X. Lv et al. / Chemical Engineering Journal 236 (2014) 149–157

momentum equations is closed by the kinetic theory of granularflow (KTGF). The heterogeneous index, which is a function of aver-age voidage and bed height, is predicted by using our local struc-tural parameters model and is incorporated into Fluent withUser-Defined Functions (UDF). Therefore, the drag coefficientbased on local structure can be calculated and fed back into theCFD calculation.

The bed simulated here is a lab-scale bed, which is 2.464 m inheight and 0.267 m in inner diameter. For the purpose of savingcomputational cost, its disengaging section at the top of fluidizedbed is neglected. Fig. 3 shows the schematic geometry of the 2Dbubbling fluidized bed used in the simulations. At the top outlet,atmospheric pressure is prescribed and the solid mass flux is mon-itored which is fed back into the bottom inlet with a solid concen-tration of 0.2. The no-slip boundary condition is used to the gasphase while the partial slip boundary is used to the solid phasewith a specularity coefficient of 0.6. At the bottom inlet, the con-stant velocity of gas entering the bed is specified. Simulationslasted for 30 s in physical time and the time-average variableswere obtained over the last 15 s. Detailed simulation parametersare summarized in Table 3.

4.3.2. Simulation resultsAll the experimental date in this article were obtained from

the literature [40], where the detailed local flow structures in-

Table 3Summary of parameters used in the simulation of FCC particles.

dp 6.5 � 10�5 m e 0.9qp 1780 kg/m3 es,max 0.56qg 1.225 kg/m3 Ug 0.06, 0.1, 0.2 m/slg 1.7894 � 10�5 Pa s Dt 0.0005g 9.8 m/s2 H0 1.2

Fig. 4. Comparison of axial solid co

Fig. 5. Comparison of axial solid concentratio

side bubbling fluidized bed of FCC were studied. Fig. 4 showsthe time-averaged axial profiles of solid concentration indifferent grid resolutions with different drag models. The resultsindicate that, for the two-dimensional simulation, a girdnumber of 80 � 400 = 32,000 (or grid size 3.34 mm � 6.16 mm),and, for the three-dimensional simulation, a gridnumber of 26 � 26 � 246 = 166,296 (or grid size10.27 mm � 10.27 mm � 10.02 mm) are sufficient to correctlypredict bed expansion characteristics. For simplicity, such gridresolutions are taken in the later simulations. It also can be seenthat the numerical simulation with the structure-based drag(drag S) gives a good agreement with the available experimentaldata. While the hybrid drag model of Wen & Yu and Ergun(drag A) [41] was used, the axial solid volume fraction is signif-icantly smaller than the experimental values. This is due to thefact that the drag A, in such grid resolution, does not considerthe effects of meso-scale bubbling structure, and over-predictsthe inter-phase drag force. Nikolopoulos et al. [42] who studiedthe time averaged pressure profile of the CFB loop with Gidas-pow model, found the Gidaspow’s predictions without highaccuracy under the grid size of 21 particle diameters. Wanget al. [43] reported that, in order to capture the meso-scalestructures in fluidized bed with drag A, the grid size shouldbe fine to 2–4 particle diameters. However, owing to the limita-tion of computational resource, the simulation of large fluidizedbed with such fine grid is not realistic. Fig. 5 further plots thecomparisons between 2D and 3D simulations of bubbling fluid-ized bed for Ug = 0.1 m/s and Ug = 0.2 m/s and the same trendscan be found. Compared to the 2D bubbling fluidized bed sim-ulation, the axial solid concentration in the 3D simulation ismore consistent with the experimental data in the higher sec-tion of the bubbling fluidized beds. It should be noted that,for the absence of orifices area on the plate, the value of the ini-tial bubble diameter for porous plate distributor is used.

ncentration for Ug = 0.06 m/s.

n for Ug = 0.1 m/s (a) and Ug = 0.2 m/s (b).

Fig. 6. Comparison of time-average radial solid concentration profiles for Ug = 0.06 m/s in 2D bubbling bed.

Fig. 7. Comparison of time-average radial solid concentration profiles for Ug = 0.06 m/s in 3D bubbling bed.

X. Lv et al. / Chemical Engineering Journal 236 (2014) 149–157 155

Fig. 8. Average mean velocity of the solid phase in lower section of bubblingfluidized bed for Ug = 0.06 m/s.

Fig. 9. Transient particle velocity signals obtained at two radial positions forUg = 0.06 m/s.

156 X. Lv et al. / Chemical Engineering Journal 236 (2014) 149–157

Fig. 6 shows the comparison between simulated and experi-mental radial solid concentration profiles at four different heightsfor Ug = 0.06 m/s in 2D simulations of bubbling fluidized bed. It canbe seen that the varying trends of solid concentration are capturedcorrectly with higher solid concentration at the near wall regionand lower at the center, as indicated in the whole bed contour ofmean solid concentration in Fig. 6, and the simulation results with

drag S show reasonable agreements with the experimental radialsolid concentration profiles. Meanwhile, drag A simulated resultsare far from the experimental data.

Fig. 7 shows the radial solid concentration profile in 3D simula-tion for Ug = 0.06 m/s. Through the four plane contours of mean so-lid concentration in Fig. 7, it can be seen that the solidconcentration in plane are also higher near the wall region thanin the center of bed. Meanwhile, it can also be seen that drag S sim-ulated results are closer to the experimental data than the drag Asimulated results.

Werther and Molerus [44] considered that the solid circulationpatterns in a slugging bed were that the solid rose in the centre andsubsequently descended near the wall, which has been observed inlaboratory fluidized beds by Leva [45]. Fig. 8 shows that this kind ofmacroscopic solid circulation pattern is captured by our simulationwith the proposed drag modification. In Fig. 9, these solid movingpatterns are further demonstrated.

5. Conclusion

A structure model proposed is used to solve the local structuralparameters. Then the drag based structure is incorporated intotwo-flow model to simulate the hydrodynamics of Geldart A parti-cles in a bubbling fluidized bed. This method is used to improve thedrag force. Literature reported [46] that the improvements of fric-tional forces also have good predictability. Current results showthat the axial and radial solid concentration profiles and particlevelocity profiles can be well predicted. The reasonable agreementsbetween simulated and experimental results indicate that this ap-proach is one of the promising methods, which are suitable forsimulating the hydrodynamics of Geldart A particles in bubblingfluidized beds.

Acknowledgements

The authors are grateful to the State Key Development Programfor Basic Research of China (973 Program) under Grant No.2009CB219904 and 2013CB632603.

References

[1] D. Kunii, O. Levenspiel, Fluidization Engineering, Wiley, New York, 1991.[2] M.A. van der Hoef, M. van Sint Annaland, N.G. Deen, J.A.M. Kuipers, Numerical

simulation of dense gas–solid fluidized beds: a multiscale modeling strategy,Annu. Rev. Fluid. Mech. 40 (2008) 47–70.

[3] C.Y. Wen, Y.H. Yu, Mechanics of fluidization, Chem. Eng. Prog. Symp. Ser. 62(1966) 100–111.

[4] S. Ergun, Fluid flow through packed columns, Chem. Eng. Prog. 48 (1952) 89–94.

[5] J. Li, M. Kwauk, Multiscal nature of complex fluid-particle systems, Ind. Eng.Chem. Res. 40 (2001) 4227–4237.

[6] J. Li, M. Kwauk, Exploring complex systems in chemical engineering: the multi-scale methodology, Chem. Eng. Sci. 58 (2003) 521–535.

[7] R. Beeststra, M.A. van der Hoef, J.A.M. Kuipers, A lattice-Boltzmann simulationstudy of the drag coefficient of clusters of spheres, Comput. Fluids 35 (2006)966–970.

[8] J. Ma, W. Ge, X. Wang, J. Wang, J. Li, High-resolution simulation of gas–solidsuspension using macro-scale particle methods, Chem. Eng. Sci. 61 (2006)7096–7106.

[9] J. Gao, J. Chang, C. Xu, X. Lan, Y. Yang, CFD simulation of gas solid flow in FCCstripers, Chem. Eng. Sci. 63 (2008) 1827–1841.

[10] Z. Zou, H.Z. Li, Q.S. Zhu, Y.C. Wang, Experimental study and numericalsimulation of bubbling fluidized beds with fine particles in two and threedimensions, Ind. Eng. Chem. Res. (2013), http://dx.doi.org/10.1021/ie303105v.

[11] N. Yang, W. Wang, W. Ge, J. Li, CFD simulation of concurrent-up gas–solid flowin circulating fluidized bed with structure-dependent drag coefficient, Chem.Eng. J. 96 (2003) 71–80.

[12] Z. Shi, W. Wang, J. Li, A bubble-based EMMS model for gas–solid bubblingfluidization, Chem. Eng. Sci. 66 (2011) 5541–5555.

[13] R. Andreux, J. Chaouki, Behaviors of the bubble, cloud, and emulsion phases ina fluidized bed, AlChE J. 54 (2008) 406–414.

[14] O.E.P. Colin Fryer, Experimental investigation of models for fluidized bedcatalytic reactors, AlChE J. 22 (1976) 38–47.

X. Lv et al. / Chemical Engineering Journal 236 (2014) 149–157 157

[15] M.J. Lockett, On the two-phase theory of fluidisation, Chem. Eng. Sci. 22 (1967)1059–1066.

[16] N. Mostoufi, J. Chaouki, On the axial movement of solids in gas–solid fluidizedbeds, Chem. Eng. Res. Des. 78 (2000) 911–920.

[17] S. Mori, C.Y. Wen, Estimation of bubble diameter in gaseous fluidized beds,AlChE J. 21 (1975) 109–115.

[18] R.C. Darton, R.D. Lanauze, J.F. Davidson, D. Harrison, Bubble growth due tocoalescence in fluidized beds, Trans. Inst. Chem. Eng. 55 (1977) 274–280.

[19] J. Werther, Hydrodynamics and mass transfer between the bubble andemulsion phases in fluidized beds of sand and cracking catalyst, in:Fluidization, New York Engineering Foundation, 1983.

[20] A.W. Weimer, G.J. Quarderer, On dense phase voidage and bubble size in highpressure fluidized beds of fine powders, AlChE J. 31 (1985) 1019–1028.

[21] M. Horio, A. Nonaka, A generalized bubble diameter correlation for gas–solidfluidized beds, AlChE J. 33 (1987) 1865–1872.

[22] J.F. Davidson, D. Harrison (Eds.), Fluidized Particles, Cambridge UniversityPress, New York, 1963.

[23] R.M. Davies, G.I. Taylor, The mechanics of large bubbles rising throughextended liquids and through liquids in tubes, Proc. Roy. Soc. Lond. A200(1950) 375–390.

[24] A.R. Abrahamsen, D. Geldart, Behaviour of gas-fluidized beds of fine powders:Part II. Voidage of the dense phase in bubbling beds, Powder Technol. 26(1980) 47–56.

[25] D.G. Thomas, Transport characteristics of suspension: VIII. A note on theviscosity of Newtonian suspensions of uniform spherical particles, J. ColloidInterface Sci. 20 (1965) 267–277.

[26] S. Ergun, Fluid flow through placed columns, Chem. Eng. Prog. 48 (1952) 90–98.

[27] R.C. Darton, D. Harrison, The rise of single gas bubbles in liquid fluidized bed,Trans. Inst. Chem. Eng. 52 (1974) 301–304.

[28] M. Ishii, N. Zuber, Drag coefficient and relative velocity in bubbly, droplet orparticulate flows, AlChE J. 25 (1979) 843–855.

[29] J.F. Richardson, W.N. Zaki, Sedimentation and fluidization, Trans. Inst. Chem.Eng. 32 (1954) 35–53.

[30] M. Kwauk, Generalized fluidization: I. Steady-state motion, Sci. Sin. 12 (1963)587–612.

[31] D.Z. Zhang, W.B. VanderHeyden, The effects of mesoscale structures on themacroscopic momentum equations for two-phase flows, Int. J. MultiphaseFlow 28 (2002) 805–822.

[32] J.D. Wilde, Reformulating and quantifying the generalized added mass infiltered gas–solid flow models, Phys. Fluids 17 (2005) 1–14.

[33] J.D. Wilde, The generalized added mass revised, Phys. Fluids 19 (2007) 1–4.[34] N. Zuber, On the dispersed two-phase flow in the laminar flow regime, Chem.

Eng. Sci. 19 (1964) 897–917.[35] R. Zenit, M.L. Hunt, Solid fraction fluctuations in solid–liquid flows, Int. J.

Multiphase Flow 26 (2000) 763–781.[36] P.C. Johnson, R. Jackson, Frictional–collisional constitutive relations for

granular materials, with application to plane shearing, J. Fluid Mech. 176(1987) 67–93.

[37] D.G. Schaeffer, Instability in the evolution equations describing incompressiblegranular flow, J. Diff. Eq. 66 (1987) 19–50.

[38] D. Gidaspow, R. Bezburuah, J. Ding, Hydrodynamics of circulating fluidizedbeds, kinetic theory approach, in: Fluidization VII, Proceedings of the 7thEngineering Foundation Conference on Fluidization, 1992, pp. 75–82.

[39] M. Syamlal, W. Rogers, T.J. O’Brien, Mfix Documentation Theory Guide. U.S.Department of Energy, Office of Fossil Energy, Technical Note, 1993.

[40] H. Zhu, J. Zhu, G. Li, F. Li, Detailed measurements of flow structure inside adense gas–solid fluidized bed, Powder Technol. 180 (2008) 339–349.

[41] D. Gidaspow, Multiphase Flow and Fluidization: Continuum and KineticTheory Description, Academic Press, New York, 1994.

[42] A. Nikolopoulos, N. Nikolopoulos, A. Charitos, P. Grammelis, E. Kakaras, A.R.Bidwe, G. Varela, High-resolution 3-D full-loop simulation of a CFB carbonatorcold model, Chem. Eng. Sci. 90 (2013) 137–150.

[43] J. Wang, M.A. van der Hoef, J.A.M. Kuipers, Why the two-fluid model fails topredict the bed expansion characteristics of Geldart A particles in gas-fluidizedbeds: a tentative answer, Chem. Eng. Sci. 64 (2009) 622–625.

[44] J. Werther, O. Molerus, The local structure of gas fluidized beds—II. The spatialdistribution of bubbles, Int. J. Multiphase Flow 1 (1973) 123–138.

[45] M. Leva, The use of gas-fluidized systems for blending particulate solids, Symp.interaction between fluids and particles, Inst. Chem. Eng. Lond. (1962) 143–150.

[46] Aristeidis Nikolopoulos, Nikos Nikolopoulos, Nikos Varveris, Sotirios Karellas,Panagiotis Grammelis, Emmanuel Kakaras, Investigation of proper modeling ofvery dense granular flows in the recirculation system of CFBs, Particuology 10(2012) 699–709.