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Investigation of Flow Behaviors and Bubble Characteristics of a PulseFluidized Bed via CFD ModelingZhanyong Li a; Weiguang Su a; Zhonghua Wu ab; Ruifang Wang a; A. S. Mujumdar b

a College of Mechanical Engineering, Tianjin University of Science and Technology, Tianjin, China b

Minerals, Metals, and Materials Technology Centre, Department of Mechnical Engineering, NationalUniversity of Singapore, Singapore

Online publication date: 30 December 2009

To cite this Article Li, Zhanyong, Su, Weiguang, Wu, Zhonghua, Wang, Ruifang and Mujumdar, A. S.(2010) 'Investigationof Flow Behaviors and Bubble Characteristics of a Pulse Fluidized Bed via CFD Modeling', Drying Technology, 28: 1, 78— 93To link to this Article: DOI: 10.1080/07373930903430785URL: http://dx.doi.org/10.1080/07373930903430785

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Investigation of Flow Behaviors and Bubble Characteristicsof a Pulse Fluidized Bed via CFD Modeling

Zhanyong Li,1 Weiguang Su,1 Zhonghua Wu,1,2

Ruifang Wang,1 and A. S. Mujumdar21College of Mechanical Engineering, Tianjin University of Science and Technology, Tianjin, China2Minerals, Metals, and Materials Technology Centre, Department of Mechnical Engineering,National University of Singapore, Singapore

The bubbling flow in a 2D pulsed fluidized bed is simulated usinga developed Eulerian-Eulerian two-fluid modeling approach. Para-metric studies are carried on for pulsed inflow of the gas phase withrectangular and sawtooth patterns where three pulsating frequenciesof 0.4, 4, and 40Hz are used. The flow instabilities are observed todevelop in the pulsed fluidized bed and the mechanisms leading to theinstabilities are discussed. The formation, coalescence, split-up ofair bubbles, and their trailing wakes inside the bed are describedand discussed in detail. Bubble rising velocity and size are obtainedand analyzed. Bed expansion and fluctuation ratio are calculated toevaluate the pulsed fluidization quality. Numerical results improvethe understanding of the pulsed fluidization.

Keywords Bubbling flow; CFD; Fluidization; Pulsed fluidizedbed; Two-fluid model

INTRODUCTION

Pulsed fluidization is an operation in which the fluidiza-tion velocity pulsates with time in the form of regular orirregular patterns; e.g., rectangular pulsation, cosine wave,or any other pattern. Many studies have shown that pulsedfluidization can improve the fluidization quality because itcan eliminate slugs and gas bypassing=channeling, reducethe bubble size, and enhance gas–particle contact in flu-idized beds.[1–5] Besides, some hard-to-fluidize materialscan be readily fluidized when pulsation or vibration isintroduced.[6] Carrot cubes and spent grains from breweriesare examples of materials that are not easily fluidized butcan be dried under pulsed fluidization.[7] Due to these mer-its, many research efforts have been devoted to variousaspects associated to pulsed fluidization; e.g., hydrody-namic characteristics,[8] mass oscillation[9] and transfer,[10]

heat transfer,[11] numerical models[3,12,13] and simula-tion,[6,14] and many other industrial applications such as

drying,[15–20] etc. There have been many experimentalstudies of pulsed fluidized beds, but the theoretical under-standing of pulsed fluidized beds is far from complete.

For a better understanding of pulsed fluidization, it isnecessary to obtain detailed information on the predomi-nate bubbling flow in fluidized beds. In addition to experi-mental work using various probes, topographic, and otherimaging techniques used in the investigation of fluidizedbeds,[8–11,21] in recent years, computational fluid dynamics(CFD) modeling has evolved into a useful tool to obtaindetailed information about the gas–particle flow in spoutedbeds, fluidized beds, etc.[12–14,22–25] Recent CFD modelingfor fluidized beds can be categorized into two methods:one is the so-called two-fluid model (TFM) and the otheris the discrete-element method (DEM). In the DEM model,two models, viz. a soft sphere model and a hard spheremodel, are used in CFD simulation of the fluidized beds.[14]

In the soft sphere model, one can estimate the interactionforces using multiple particle–particle or particle–wall con-tacts. On the other hand, in the hard sphere model, it isonly possible to compute the instantaneous change inparticle velocity through particle collision. In general, thesediscrete models would be superior for their few postula-tions and adjustable parameters compared to TFM.However, the computational demand rises strongly withthe number of traced particles, which limits its applicabil-ity. The TFM is based on the momentum equationspresented by Jackson.[26,27] In the TFM approach, thetwo phases are mathematically treated as interpenetratingcontinua. In TFMs, the conservation equations for eachof the two phases are derived to obtain a set of equationsthat have a similar mathematical structure for both phases,which makes the mathematical manipulation of the systemrelatively easier and minimizes the computation cost.

In this work, a TFM developed early is employed tosimulate a 2D pulsed fluidized bed. The gas–particle flowbehaviors and the bubble characteristics inside the fluidizedbed are discussed based on numerical results. Parametric

Correspondence: Zhanyong Li, College of MechnanicalEngineering, Tianjin Univeristy of Sceience and Technology,1038 South Dagu Road, Tianjin 300222, China; E-mail: [email protected]

Drying Technology, 28: 78–93, 2010

Copyright # 2010 Taylor & Francis Group, LLC

ISSN: 0737-3937 print=1532-2300 online

DOI: 10.1080/07373930903430785

78

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studies are carried on for two pulsed fluidizing jets withrectangular and sawtooth pulsation patterns. The effectof pulsed jets with low frequency of 0.4Hz, intermediatefrequency of 4Hz, and high frequency of 40Hz on pulsedfluidization was also studied.

PHYSICAL MODEL

Figure 1 shows the schematics of the simulated 2D flu-idized bed. The bed has a dimension of 200mm wide and1000mm high. Above the bottom, there is a static bed ofglass beads with a dimension of 200� 300mm. The glassbeads have a uniform diameter of 0.5mm and a densityof 2650 kg=m3. A pulsed air stream is injected from thebed bottom to fluidize the static bed of glass beads. Thepulsed fluidizing air jet is operated under normal pressure,and its velocity is

vg ¼ vg0 þ vgw 0 < ðt� nsÞ � s=2vg ¼ vg0 s=2 < ðt� nsÞ � s

�ð1Þ

where vg, vg0, vgw are total, steady, and pulsate componentsof the fluidizing gas velocity, 0.5, 0.45, and 0.05m=s,respectively; t is time, s; s is one pulsation period, s; andn is pulsation cycles. The pulse frequency, the time ofon-period and off-period of the pulsed jet, is controlledby the graded time lag relay. Table 1 summarizes theoperation parameters and material properties used for thefluidized bed.

There are numerous empirical equations to estimatethe natural frequency of the fluidized beds. According to

the Eq. (2) proposed by Wong and Baird,[3] the natural fre-quency of this fluidized bed was calculated to be 1.8Hz.However, it was recommended that the pulsating frequencyof the inlet air be maintained in the range of slugging fre-quency of 2.7–4.8Hz,[28] and hence the pulsating frequencyof the pulsed air jet is chosen to be 4Hz in this work.

f ¼ 1

p

ffiffiffiffiffiffiffiffiffig

Hmf

rð2Þ

Hmf is the bed height where the fluidization begins.

MATHEMATICAL MODELING

The Eulerian approach is used for both gas and particu-late phases within fluidized beds, taking into account allpossible intra- and interphase interactions. In this work,the gas phase is air and the particulate phase is glass beads.The equations describing mass, momentum, turbulence,and their interactions are shown below. Meanings of thesymbols used are listed in the Nomenclature section.

Governing Conservation Equations

The volume fraction balance equation is (q¼ g, s)

Xnq¼1

aq ¼ 1 ð3Þ

The mass conservation equations for phase (q¼ g, s) is

@

@tðqqaqÞ þ r � ðqqaq~vvqÞ ¼ 0 ð4Þ

The momentum balance for the continuous phase g yields

@

@tðagqg~vvgÞþr�ðagqg~vvg~vvgÞ¼�agrPþr��ssgþagqg~ggþ~RRgs

ð5ÞFIG. 1. Schematic of the simulated 2D fluidized bed.

TABLE 1Properties of glass beads

Parameter Unit Magnitude

Density kg=m3 2650Diameter mm 0.5Minimum fluidization velocity m=s 0.22Viscosity of particle Pa � s 1.7894E-5Loose packing volume fraction — 0.59Particle restitution coefficient — 0.9Initial voidage — 0.404Air density kg=m3 1.225Air viscosity Pa � s 1.8E-5Bed height m 0.3

A PULSED FLUIDIZED BED VIA CFD MODELING 79


Similarly, the momentum balance for particulate phase syields

@

@tðasqs~vvsÞ þ r � ðasqs~vvs~vvsÞ ¼ �asrPs þr � �sss þ asqs~gg� ~RRgs

ð6Þ

Here,~vvgs is the interphase velocity and ~RRgs is an interactionforce between phases. �ss is the qth phase Reynolds stress tensor

�ss ¼ aqlqðr~vvq þr~vvTq Þ þ aq kq �2

3lq

� �r �~vvqI

Here, lq and kq are the shear and bulk viscosity of phase q,respectively, and I is the unit tensor. The solid pressure Ps,solid shear viscosity ls, and solid bulk viscosity ks are calcu-lated from the kinetic–frictional stress model discussed inthe following section.

Kinetic–Frictional Stress Model

For granular flows in the compressible regime (i.e.,where the solids volume fraction is less than its maximumallowed value), a solids pressure is calculated indepen-dently and used for the pressure gradient term, rPs, inthe granular-phase momentum Eq. (6). According to thegranular kinetic theory derived by Lun et al., the kineticenergy of granular mean flow first degrades into the kineticenergy of random particle fluctuations and then dissipatesas heat because of inelastic collisions.[29]

Following Lun et al.’s[29] theory, the kinetic energy offluctuations is accounted for in the granular kinetic theoryby defining a granular temperature h:

h ¼ hc2i=3

where c is the particle fluctuating velocity. Thus, thegranular temperature for the solid phase is proportionalto the kinetic energy of the random motion of the particles.The granular temperature conservation equation is[30]

3

2

@

@tðqsashsÞ þ r � ðqsasvshsÞ

� �¼ ð�PsI þ �sssÞ : r~vvs þr � ðkWrhsÞ � ch

ð7Þ

Here the first term on the right-hand side is the generationof energy by the solid stress tensor, the second term repre-sents diffusion of the energy, and the third term is the colli-sional dissipation of energy. The diffusion coefficient forgranular energy, kh, is given by Gidaspow et al.:[31]

kh ¼25qsds

ffiffiffiffiffiffiffiffiffiffiffiðhspÞ

p64ð1þ essÞg0

1þ 6

5asg0ð1þ essÞ

� �2

þ 2qsa2s dsðaþ essÞg0

ffiffiffiffihsp

r ð8Þ

The collisional dissipation of energy, ch, represents therate of energy dissipation within the solid phase due to

collisions between particles. This term is represented bythe expression derived by Lun et al.:[29]

ch ¼12ð1� e2ssÞg0

dsffiffiffip

p qsa2sh

3=2s ð9Þ

The particle pressure in Eqs. (6) and (7) is given by Lunet al.:[29]

Ps ¼ asqsWs þ 2qsð1þ essÞa2s g0; Ws ð10Þ

Here ess is the coefficient of restitution for particlecollisions and g0 is a radial distribution function. Theradial distribution function can be seen as a measure forthe probability of interparticle contact and expressed by:[32]

g0 ¼ 1� asas;max

� �1=3" #�1

ð11Þ

where as,max is the maximum particle packing.The solids stress tensor incorporates shear and bulk

viscosities arising from particle momentum exchange dueto translation and collision. A frictional component of vis-cosity was also included to account for the viscous–plastictransition that occurs when particles reach their maximumsolid volume fraction. The collisional, kinetic, and frictionalterms are added to give the total solid shear viscosity:

ls ¼ ls;col þ ls;kin þ ls;fr ð12Þ

The collisional contribution of the shear viscosity ismodeled as:[29]

ls;col ¼4

5asqsg0ð1þ essÞ

hsp

� �1=2

ð13Þ

The kinetic part of the shear viscosity is proposed byGidaspow et al.:[31]

ls;kin ¼10qsds

ffiffiffiffiffiffiffihsp

p

96asðaþ essÞg01þ 4

5g0asð1þ essÞ

� �2ð14Þ

In dense flow at low shear, when the secondary volume frac-tion for a solid phase nears the packing limit, the generationof stress is mainly due to friction between particles. The fric-tional part of the shear viscosity is described by Schaeffer’sexpression as[33]

ls;fr ¼Ps sin/

2ffiffiffiffiffiffiffiI2D

p ð15Þ

Here Ps is the solids pressure, / is the angle of internal fric-tion, and I2D is the second invariant of the deviatoric stresstensor. The value of / is taken to be 28.5 degrees for glassbeads.[34]

80 LI ET AL.


The solids bulk viscosity accounts for the resistance ofthe granular particles to compression and expansion. Ithas the following form from Lun et al.:[29]

ks ¼4

3asqsdsg0ð1þ essÞ

ffiffiffiffihsp

rð16Þ

Turbulence Model

The effects of turbulent fluctuations of velocities andscalar quantities in the gas phase are described using thedispersed turbulence model. In this turbulence closuremodel, turbulence predictions for the continuous phaseare obtained using the standard k-e model supplementedwith extra terms dealing with the interphase turbulentmomentum transfer, whereas predictions of the turbulencequantities for the particulate phase are obtained usingTechen theory correlations.[35]

Turbulence in the Continuous Phase

The Reynolds stress tensor for the continuous phase inEq. (5) takes the following form:

�ssg¼�2

3ðqgagkgþqgaglt;gr�~vvgÞ�IIþqgaglt;gðr~vvgþr~vvTg Þ

ð17Þ

The turbulent viscosity lt,g is written in terms of the turbu-lent kinetic energy of phase g as

lt;g ¼ qgClk2geg

ð18Þ

A characteristic time of the energetic turbulent eddies isdefined as

st;g ¼3

2Cl

kgeg

ð19Þ

where eg is the turbulence dissipation rate and Cl¼ 0.09.Turbulence predictions of the continuous phase are

obtained from the following equations of the modifiedk-e model:

@

@tðagqgkgÞ þ r � ðagqg~vvgkgÞ ¼ r � ag

lt;grk

rkg

� �þ agGk;g � agqgeg þ agqgPk;g

ð20Þ

and

@

@tðagqgegÞ þ r � ðagqg~vvgegÞ ¼ r � ag

lt;grk

reg

� �

þ agegkg

ðC1Gk;g � C2qgegÞ

þ agqgPe;g ð21Þ

Here Pk,g and Pe,g represent the influence of the dispersedphase on the continuous phase and are given by:

Pk;g ¼Xmp¼1

Ksg

agqgðksg � 2kg þ~vvsg �~vvdrÞ ð22Þ

Pz;g ¼ C3egkg

Pk;g ð23Þ

Here C3¼ 1.2 and the drift velocity is given by:[35]

~vvdr ¼ � Ds

rgsasras �

Dg

rgsagrag

� �ð24Þ

Here, Ds and Dg are diffusivities, and the dispersionPrandtl number rgs¼ 0.75. The drift velocity results fromturbulent fluctuations in the volume fraction. The produc-tion of turbulence kinetic energy, Gk,g, is computed from

Gk;g ¼ lt;gðr~vvg þ ðr~vvgÞTÞ : r~vvg ð25Þ

Turbulence in the Dispersed Phase

Time and length scales that characterize the motion areused to evaluate the dispersion coefficients, the correlationfunctions, and the turbulent kinetic energy of the particu-late phase. The characteristic particle relaxation timeconnected with the inertial effects acting on a particulatephase is defined as[36]

sF ;sg ¼ asqsb�1sg

qsqg

þ CV

!ð26Þ

The Lagrangian integral timescale calculated along theparticle trajectories, st,sg, is given by[36]

st;sg ¼st;gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1þ Cbn2Þ

q ð27Þ

where Cb¼ 1.8�1.35cos2h and

n ¼ j~vvsgjst;gLt;g

ð28Þ

The length scale of the turbulent eddies is given by

Lt;g ¼ffiffiffi3

2

rCl

k3=2g

egð29Þ

The turbulence quantities for the particulate phase include:

ks ¼ kgb2 þ gsg1þ gsg

!ð30Þ

ks;g ¼ 2kgbþ gsg1þ gsg

!ð31Þ



Dt;sg ¼1

3ksgst;sg ð32Þ

Ds ¼ Dt;sg þ2

3ks � b

1

3ksg

� �sF ;sg ð33Þ

b ¼ ð1þ CvÞqsqg

þ Cv

!�1

ð34Þ

Here the added mass coefficient Cv¼ 0.5.

Drag Model

The drag force acting on a particle in gas–particle sys-tems can be represented by the product of a momentumtransfer coefficient b and the slip velocity ð~vvg �~vvsÞ betweenthe two phases:

~RRgs ¼ bgsð~vvg �~vvsÞ ð35Þ

The correlations for bgs are usually obtained from the pres-sure drop measurements in fixed, fluidized, or settling beds.For the motion of particle swarms, however, it has beenfound that the volume fraction of particle phase has a com-plex and subtle influence on drag force. In 1966, Wen andYu proposed a correlation for high void fraction(as� 0.8).[37]

bErgun ¼3

4CD

asagqgj~vvs �~vvgjds

a�2:65g ð36Þ

In this work, the Gidaspow et al. model,[31] which is a com-bination of the Wen and Yu[37] and Ergun equations,[38]

was used.

b ¼bErgun ¼ 3

4CDasaf qf j~vvs�~vvgj

dsa�2:65f ag > 0:8

bWen�Yu ¼ 150a2slgagd2

sþ 1:75

asqgds

j~vvs �~vvgj ag � 0:8

8<:

ð37Þ

The drag coefficient CD is evaluated using Eq. (38).

CD ¼24

afRes½1þ 0:15ðafResÞ0:687� Res < 1000

0:44 Res > 1000

8<: ð38Þ

The relative Reynolds number, Res, is defined as

Res ¼qdsj~vvs �~vvgj

lgð39Þ

To avoid discontinuity of the two equations, Gidaspowintroduced a switch function that gave a smooth but rapidtransition from one regime to the other[31]:

ugs ¼arctan½150� 1:75ð0:2� asÞ�

pþ 0:5 ð40Þ

Thus, the fluid–particle interaction coefficient is given by

b ¼ ð1� ugsÞbErgun þ ugsbWen�Yu ð41Þ

Boundary Conditions

The boundary conditions for the pulsed fluidized bedmodel equations are as follows:

1. At the entrance, the gas is injected only in the axialdirection and inlet solids velocity is zero; the velocityvalue of the pulsed air jet is specified. For turbulentkinetic energy of the gas phase, k¼ 1.5(uI)2, where I isthe turbulent intensity and set as 10%; for the turbulentdissipation rate in the gas phase, e ¼ c3=4l

k3=2

l , l¼ 0.07Dh;for the solid phase, k¼ 0, because of the fact that theinlet solid velocities are both zero.

2. At the exit outlet, a pressure outlet boundary conditionis specified. That is, P¼ 0 (relative to ambient pressure).

3. Along the wall, a no-slip condition is assumed for thegas and particulate phases.

Solution Procedure

The set of governing equations presented previously wassolved by a finite control volume technique. Gidapow’sdrag model and the unsteady velocity of the pulsed jet wereincorporated into the CFD code (Fluent 6.3, ANSYS Inc.,Canonsburg, PA, USA) using the user defined functions(UDFs) utility. The properties of gas and wet materials suchas specific heat and viscosity are temperature and composi-tion dependent. The pressure–velocity coupling is obtainedusing Semi-Implicit Method for Pressure-Linked EquationsConsistent (SIMPLEC) algorithm. Grid independence istested by varying the mesh numbers and the time step waskept to meet the courant number requirement. The typicalvalues of underrelaxation factors were 0.2–0.4. The conver-gence criterions for the solution were that all variables’residuals such as temperature and velocity were less than1� 10�3 but 1� 10�4 for the mixture’s density.

RESULTS AND DISCUSSION

Model Validation

In our previous work,[20,21] the above CFD model wasused to investigate the gas–particle flow behaviors incylindrical spouted beds[39,40] and a 3D spout-fluidbed.[41,42] The predicted axial and radial velocity distribu-tion, bed pressure drop, mass flow rate, etc., are in goodagreement with the published experimental data. Forexample, Fig. 2 shows the simulated relationship betweenthe pressure drop and the spouting gas velocity in thecylindrical spouted beds. The simulated pressure drop atminimum spouting velocity is 3000 Pa, which also agreeswell with the measured data of 3.0 kPa. The details of othercomparisons between the predicted and experimentalresults can be found in Li et al.[20] and Simons.[21] Hence,the above CFD model was fully validated in this work.

82 LI ET AL.


Gas–Particle Flow Behaviors in the Fluidized Bed

Steady Fluidizing Air Jet

Figure 3 shows the successive particle volume fractiondistributions in the fluidized bed with a steady air jet of0.5m=s in a period of 14–15 s, where the red color meansparticle dense area and the blue color means particle dilutearea. From Fig. 3, it can be seen that gas bubbling flowoccurs inside the bed and the bubble shape and sizes oscil-late with time and position. Surface disturbances are alsoobserved in the upper part of the bed. Thus, the flow be-havior in the fluidized bed is complex and flow instabilitiesdevelop. Such phenomena are often observed in experi-ments about fluid beds and the cause of these instabilities

has not been clarified.[40] However, the simulated resultsmay suggest a possible formation mechanism to explainthe experimental observations. In Fig. 3, it can be seen thatsome small bubbles ‘‘randomly’’ develop near the gas dis-tributor and then lift off. When these small bubbles moveupwards, they combine into=with several big bubbles inthe lower and=or middle part of the bed. The big bubblescontinuously coalesce to grow as they rise, causing largerbubbles to form in the upper part of the bed, as shownin the snapshots of 14.1–14.3 s of Fig. 3. Finally, the largerbubbles reach the freeboard and break, causing the surfacedisturbances and flow instabilities to form. In Fig. 3, it wasalso observed that many small bubbles generate in thelower part of the bed, meaning a high bubble frequency.Few larger bubbles appear in middle=upper part of thebed with low frequency. Thus, bubble frequency decreaseswith increasing height.

Figure 4 shows the transient variation of the bed pres-sure drop fluctuation inside the fluidized bed with a steadyair jet of 0.5m=s. In Fig. 4, the amplitude of the bed pres-sure drop fluctuation is 3800–5800 Pa. In a time period of10–15 s, there are about 9.5 peak–peak pressure oscillationsshown in Fig. 4 and hence the oscillation frequency of thefluidized bed is calculated to be 1.9Hz, which is close to thenatural frequency (1.8Hz) of the fluidized bed predicted byEq. (2). Based on Figs. 3 and 4, it can be speculated thatthe bed pressure fluctuation is mainly caused by the move-ment of air bubbles. In the lower part of the bed, pressureoscillation is caused by the generation of many small bub-bles; in the middle part, pressure oscillation results fromcoalescence, splitting, and diminishing of bubbles; in theupper part, pressure oscillation is due to the breakup ofbubbles. Then, the whole bed pressure drop fluctuation is

FIG. 2. Pressure drop vs. injected gas velocity curves for spouting of

glass beads in the cylindered spouted bed (H¼ 0.325m, qs¼ 2500 kg=m3,

ds¼ 1.41mm, Ums¼ 0.54m=s).

FIG. 3. Particle phase distribution inside the bed with a steady air jet in the period of 14–15 s.



the summary of the local pressure drop fluctuations, whoseamplitude reflects the turbulence intensity of the bubblingflow inside the fluidized bed.

Pulsed Fluidizing Air Jets

It is well known that the bed of glass beads expandshigher with an increasing fluidizing air superficial velocityand collapses when the superficial velocity is being reducedor even stopped. In the pulsed fluidization, the expansionand collapse processes alternatively take place due to thevariation of the pulsed jet velocity and there is a complexinteraction with void fraction fluctuation inside the bed.

In this work, the effect of a pulsed jet on gas–particle flowbehavior inside the fluidized bed is investigated.

Figures 5–7 show the transient particle volume fractiondistributions inside the fluidized bed with a rectangularpulsed jet of different oscillation frequencies. The velocityvariation of this pulsed jet is described by Eq. (1) and itsshape is indicted in Fig. 8. From Figs. 5–7, it can befound that the intensive bubbling flow occurs inside thepulse fluidized bed. Some typical phenomena of the bub-bling flow are also observed: For example, Fig. 5 shows atypical growth process of an air bubble. In the time pe-riod of 14.1–14.8 s, the bubble rises up continuously.Simultaneously, its size increases smoothly. At 14.9 s,the bubble reaches the freeboard and arrives at its maxi-mum size. Then, it breaks up at 15.0 s. The coalescenceof bubbles is found in the snapshots of 14.4–14.7 s ofFig. 6 and in the snapshots of 14.4–14.7 s of Fig. 7, wheretwo separate blue areas are indentified to combine into abig one in the middle=upper part of the fluidized bed. Thedetails of bubble movement are described in the section(Bubble Characteristics in the Fluidized Bed).

Figures 8–10 show the transient bed pressure dropfluctuations inside the fluidized bed with a rectangularpulsed jet of different oscillation frequencies. In Fig. 9,the oscillation amplitude of the bed pressure drop fluctua-tions is 3200–6000 Pa in the case of a 0.4Hz rectangularpulsed jet. The oscillation frequency of the fluidized bed isunstable: in the time period of 11–12.5 s, there is onlyone peak–peak pressure oscillation and the frequency is0.67Hz, whereas in the time period of 12.8–13.8 s, twopeak–peak pressure oscillations are observed and thefrequency is 2Hz. In Fig. 8, a higher amplitude of

FIG. 4. Variation of the bed pressure drop in the fluidized bed with a

steady air jet in the period of 14–15 s.

FIG. 5. Particle phase distribution inside the bed with a 0.4Hz rectangular pulsed jet in the period of 14–15 s.

84 LI ET AL.


3100–6400 Pa is observed in the case of the 4Hz rectangu-lar pulsed jet. In the time period of 10–15 s, there are about17.5 peak–peak pressure oscillations observed in Fig. 8 andthe oscillation frequency of the fluidized bed is calculatedto be 3.5Hz. In Fig. 10, the amplitude of 3100–5400 Pa isobserved in the case of the 40Hz rectangular pulsed jet.The oscillation frequency of the fluidized bed is calculatedto be 1.8Hz, which is close to the natural frequency ofthe bed.

By comparing the three cases, it was found that the bedpressure drop fluctuations are strongest in the case of the4Hz pulsed jet. This intensifying may be caused by the

resonance between the pulsed jet and the fluidized bedbecause the frequency (4Hz) of the pulsed jet is close tothe natural one (1.8Hz) of the fluidized bed. Due to theresonance, the frequency of the fluidized bed is alsochanged to 3.4Hz. A pulsed jet with low frequenciesproduces an unsteady frequency of the fluidized bed. Inthe case of pulsed jets with high frequencies, the fluidizedbed keep its natural frequency of 1.8Hz. Comparing thepulsed jets and the steady jet, the amplitude of bedpressure drop fluctuation is greater in the case of pulsedjets, indicating an intensifying of the turbulent bubblingflow inside the fluidized bed.

FIG. 6. Particle phase distribution inside the bed with a 4Hz rectangular pulsed jet in the period of 14–15 s.

FIG. 7. Particle phase distribution inside the bed with a 40Hz rectangular pulsed jet in the period of 14–15 s.



Besides the pulsed jets with a rectangular velocity shape,the pulsed jet with a sawtooth velocity shape and frequen-cies of 0.4, 4, 40Hz are also investigated in this work. Thesawtooth shape indicates a smoother variation of thepulsed jet velocity compared with the rectangular shapeand its shape is shown in Fig. 11. Figures 11 and 12 showthe transient particle volume fraction distributions and bedpressure drop fluctuation inside the fluidized bed witha 4Hz sawtooth pulsed jet, respectively. In Fig. 11, theoscillation frequency of bed pressure drop fluctuation iscalculated to be 3.4Hz and the amplitude is 3500–6000 Pa. Compared with the 4Hz rectangular pulsed jet,the oscillation frequency of the fluidized bed is similarbut the amplitude of the bed pressure drop fluctuation is

a bit smaller, indicating a smooth intensifying of the turbu-lent bubbling flow in the fluidized bed. A further compar-ison on fluidization quality of the bed is made later.

Bubble Characteristics in the Fluidized Bed

Determination of Bubble Properties

Figures 3, 5–7, and 12 show the successive images ofparticulate phase volume fraction distributions inside thefluidized bed, where gas bubbles are observed clearly. Post-processing of these images helps to quantify some bubbleproperties such as size, shape, and velocity. Taking theimages shown in Fig. 3 as an example, only the contourswhose value equals 0.12 (particle volume fraction) areshown and then we obtain the new images shown in


4Hz rectangular pulsed jet in the period of 14–15 s.


0.4Hz rectangular pulsed jet in the period of 14–15 s.


40Hz rectangular pulsed jet in the period of 14–15 s.


4Hz sawtooth pulsed jet in the period of 14–15 s.

86 LI ET AL.


Fig. 13. From Fig. 13, it can be seen that some contours areclosed circuit. By comparing Figs. 3 and 13, it was foundthat the particle volume fraction is less than 0.12 insidethese closed circuits and this area is regarded as the gasbubble area. On the contrary, the area outside the closedcircuit is regarded as particle phase area. In Fig. 10, these

closed circuits are treated as gas bubbles and marked bythe number for further analysis. Their shapes also indicatethe gas bubble shapes. It should be noted that some gasbubbles appearing in the lower part of the fluidized bedshown in Fig. 3 are filtrated in Fig. 13 due to their smallbubble size. Figures 14 and 15 show the postprocessed

FIG. 13. Gas bubble behaviors in the fluidized bed with a steady air jet in the period of 14–15 s.

FIG. 12. Particle phase distribution inside the bed with a 4Hz sawtooth pulsed jet in the period of 14–15 s.



particulate phase volume fraction distribution in the fluidbed with pulsed jets, corresponding to Figs. 6 and 12,respectively.

The area of bubbles shown in Figs. 13–15 is calculatedby the grid counting method. That is, based on the meshused in the simulation, we integrate the area of these

FIG. 15. Gas bubble behaviors in the fluidized bed with a 4Hz sawtooth pulsed jet in the period of 14–15 s.

FIG. 14. Gas bubble behaviors in the fluidized bed with a 4Hz rectangular pulsed jet in the period of 14–15 s.

88 LI ET AL.


grids that is occupied by the closed circuit and the totalarea is treated as the bubble area. Because gas bubbles haveirregular shapes, the equivalent diameter is used here and itis defined as the diameter of the sphere with the same areaas the 2D bubbles. To calculate the bubble growing rate,ug, the equivalent diameter of a given bubble in two con-secutive images is tracked and its change Ddbq within a timespan (Dt¼ 0.1 s) is calculated. Then the growing rate can bedecided by ug¼Ddbq=Dt. Similarly, the bubble rising veloc-ity, up, is calculated by measuring the height displacement(Dh) of the center of a given bubble with respect to time;i.e., up¼Dh=Dt.

Bubble Characteristics Inside the Fluidized Bed

Figure 16 shows the predicted particle flow patternand particle velocity around a gas bubble in the fluidizedbed with a steady air jet of 0.5m=s. The gas bubble refersto the one marked as ‘‘number one’’ in the image of14.2 s in Fig. 13. From Fig. 16, it can be seen that thebubble has a nonspherical shape and the particulate phasemoves in the area around the gas bubble. In the left side ofthe bubble, particles move downward, with few particlesentering into the bubble. In the top and right side of thebubble, particles move upward and rightward, respectively,causing the upward and rightward movement of the bubbleshown in the consecutive image of 14.3 s in Fig. 13. At thebottom area marked by ‘‘A,’’ particles are observed to beentrained into the bubble and this phenomenon is generallycalled a trailing wake of bubbles, which are often men-tioned in the literature on bubbling fluidized beds. A trail-ing wake was formed due to the negative pressuredifference between the gas bubble and the underlying par-ticulate phase. It is interesting to note that with the liftingof the gas bubble shown in Fig. 13, the trailing wake bringsup particles that initially resided at the low part of thefluidized bed continuously until the bubble reaches the

freeboard. Then, the bubble breaks up and these particlesfall down. During the lifting process, bubbles may coalesce,split up, or diminish and hence increase the uncertainty ofthe movement of particles brought by the trailing wake.From this point of view, the trailing wake increases thecomplexity and intensity of the particulate phase move-ment, which contributes to the heat and mass transferoccurring inside the fluidized bed.

Lare et al. reported that bubble velocity increased withincreasing measuring height.[43] However, Fig. 17 showsa more complex varitions of the bubble rising velocity withthe measuring height. In Fig. 17a, the rising velocity of thesecond gas bubble first increases from 0.17 to 0.69m=s inthe bed height range of 100–200mm and then decreasesgradually to 0.5m=s in the bed height range of 200–260mm, and then quickly to 0m=s in the height range of260–300mm. The first bubble, which is the largest one inthe period of 14–15 s, almost keeps its velocity of about0.45m=s in the height range of 160–250mm and thendecreases quickly to 0m=s in the height range of250–300mm. Thus, it was not easy to describe the bubblevelocity inside the fluidized bed. Comparing the bubblevelocities shown in Figs. 17a and b, it was found that theamplitue of the bubble velocity varation was smaller inthe case of the 4Hz retangular pulsed jet. Similar results

FIG. 17. The relationship of bubble rising velocity and bed height inside

the fluidized bed (a, steady jet; b, 4Hz rectangular pulsed jet).

FIG. 16. Predicted particle flow pattern and solid velocity around a bub-

ble in the fluidized bed with a steady air jet of 0.5m=s at the time of 14.2 s.



were also found in the 4Hz sawtooth pulsed jet. From thispoint of view, the bubble rising velocity was much morestable for the pulsed fluidized bed than the ordinaryfluidized bed.

Figure 18 shows the variation of bubble sizes with thebed height in the fluidized bed. It was found that the bub-ble size increased with increasing height. The results are asexpected and agree well with those of Lare et al.[43] Com-paring the bubble growth process shown in Figs. 18a andb, bubble size grows smoothly with the increasing bedheight in the case of the 4Hz retangular pulsed jet. Similarresults were also found for the 4Hz sawtooth pulsed jet.Hence, a conclusion can be drawn that bubble movementwas more stable in the pulsed fluidized bed.

Fluidization Performance of the Fluid Bed

As described above, gas bubbling flow occurs in thefluidized bed. The formation, growth, and collapse ofbubbles causes erratic bed expansion with intense bed fluc-tuation, which influences the efficiency and the quality ofgas–solid fluidized beds. Here, two important parametersare defined to evaluate the fluidization performance ofthe fluidized bed.

Bed Expansion Ratio

Bed expansion ratio is defined as the ratio of the averageheight of a fluidized bed (H) to initial static bed height

(Hmf) at a flow rate of the fluidizing medium above theminimum fluidizing velocity. The expansion ratio of afluidized bed depends on fluidizing gas velocity, particlesize, and initial bed height. Figure 19 shows the variationsof the bed height with time with=without a pulsed jet in theperiod of 14–15 s. The time-average bed height is 385mm inthe case of the steady air jet, 379.25mm in the 4Hz rectan-gular pulsed jet, and 401.25mm in the 4Hz sawtoothpulsed jet, respectively. Considering the initial static bedheight of 300mm, the expansion ratio of the fluid bedwas calculated to be 1.28mm in the case of the steady jet,1.26mm in the 4Hz rectangular pulsed jet, and 1.34mmin the 4Hz sawtooth pulsed jet, respectively. Because therectangular pulsed jet has a lower average fluidizing gasvelocity (0.475m=s) than the steady air jet (0.5m=s), it iseasy to understand that the rectangular pulsed jet has alower bed expansion ratio than the steady jet. Both thesawtooth and rectangular pulsed jets have the same fluidiz-ing air velocity, but a higher expansion ratio is achieved forthe sawtooth pulsed jet, meaning an improvement on thefluidization quality. This can be explained by the fact thatthe smooth bed height fluctuation in the sawtooth pulsedjet case as shown in Fig. 3 results in a higher average bedheight. The smooth bed height fluctuation may be causedby the smooth variation of fluidizing gas velocity in thecase of the sawtooth pulsed jet.

Bed Fluctuation Ratio

Figure 19 shows the considerable bed fluctuationscaused by the bubbling flow occurring inside the fluidizedbed. Bed fluctuations lead to instability in operation andaffect the fluidization quality adversely. To specify theextent of bed fluctuation, a parameter called the bed fluc-tuation ratio is used here and it is defined as the ratio ofthe maximum bed height to the minimum bed height. FromFig. 19, we can find the values of maximum and minimum

FIG. 18. The relationship of bubble size and bed height inside the

fluidized bed (a, steady jet; b, 4Hz rectangular pulsed jet).

FIG. 19. The transient variation of bed height in the case of steady and

pulsed jets.

90 LI ET AL.


bed heights in the period of 14–15 s under different fluidiz-ing gas velocities. Hence, the bed fluctuation ratio of thefluid bed is calculated to be 1.41 in the case of a steadyjet, 1.36 for the 4Hz rectangular pulsed jet, and 1.24 forthe 4Hz sawtooth pulsed jet, respectively. Both pulsed jetshave a smaller bed fluctuation ratio than the steady jet andthe smallest ratio is found for the 4Hz sawtooth pulsed jet.From this point of view, pulsed jets contribute to inhibitthe bed fluctuations and improve the fluidization quality.Furthermore, the pulsed jet with a continuous, smoothvelocity variation exerts a better performance.

CONCLUSIONS

In this work, the bubbling flow in a 2D pulsed fluidizedbed was simulated using a developed Eulerian-Euleriantwo-fluid modeling approach. Parametric studies were con-ducted for pulsed inflow of the gas phase with rectangularand sawtooth patterns where three pulsating frequencies of0.4, 4, and 40Hz were used. It was found that (1) the flowinstabilities and bed pressure drop inside the fluidized bedare mainly caused by the formation and motion of air bub-bles; (2) the trailing wake of air bubbles continually bringsup particles that initially reside at the bed bottom duringtheir lifting process; (3) a low frequency of 0.4Hz producesan unsteady or intermittent fluidization; intermediatefrequency of 4Hz causes a resonant fluidization and highfrequency of 40Hz results in a normal fluidization; (4)the pulsed fluid bed has a high bed expansion ratio andlow bed fluctuation ratio, accompanied by a better fluidiza-tion quality. Numerical results improve the understandingof the pulsed fluidization.

NOMENCLATURE

A Area (m2)b, Cl, C3, CV, Cb Coefficients in the turbulence modelCD Drag coefficientDc Column diameter (m)Di Orifice diameter (m)Ds, Dt,sg Turbulent quantities for the dispersed

phaseds Particle diameter (m)ess Coefficient of restitution of particlesew Coefficient of restitution for particle–

wall collisions~FFq External body force (N=m3)~FFlift;q Lift force (N=m3)~FFvm;q Virtual mass force (N=m3)f Pulsation frequency of pulsed jetsGk,g Production of turbulence kinetic

energyg Gravitational constant (9.81m=s2)g0 Radial distribution functionH Bed height (m)

Hmf Bed height with initial fluidization (m)k Turbulence kinetic energy (m2=s2)kg Turbulence quantities of gas phase

(m2=s2)ks Turbulence quantities of solid phase

(m2=s2)kh Diffusion coefficient for granular

energyLt,g Length scales (m)M Molar weight (kg=kmol)P Gas pressure (N=m2)Pr Prandtl numberPs Particle pressure (N=m2)R Gas constant (J=mol �K); Particle

radius (m)Re Reynolds numberRes Particle Reynolds number~RRgs Drag force between gas and

particulate phases (N=m3)t Time (s)ug Bubble growth rate (m=s)up Bubble rising rate (m=s)vg Gas velocity (m=s)~vvgs Drift velocity (m=s)vs Particle velocity (m=s)~vvq Velocity of phase q (m=s)x Axial coordinate (m)y Radial coordinate (m)

Greek Letters

ag Gas volume fractionaq Volume fraction of phase qas Particle volume fractionas,max Maximum of particle volume fractionbgs Interphase momentum exchange

coefficient (kg=m3 � s)bErgun Fluid–particle interaction coefficient

of the Ergun equation (kg=m3 � s)bWen-Yu Fluid–particle interaction coefficient

of the Wen-Yu equation (kg=m3 � s)eg Turbulence dissipation rate of

gas phase (m2=s3)es Turbulence dissipation rate of

solid phase (m2=s3)gsg Turbulence quantitieshs Granular temperature (m2=s2)rgs Dispersion Prandtl numberks Solid bulk viscosity (Pa s)lg Gas viscosity (Pa s)ls Solid shear viscosity (Pa s)ls,col Collisional part of particle shear

viscosity (Pa s)ls,kin Kinetic part of particle shear viscosity

(Pa s)



ls,fr Frictional part of particle shearviscosity (Pa s)

lt,q Turbulence viscosity of phase q (Pa s)Pk,g, P,z,g Influence of dispersed phases on

continuous phaseqg Gas density (kg=m3)qq Density of phase q (kg=m3)qs Particle density (kg=m3)s Pulsation period of the pulsed jet (s)sF,sg Characteristic particle relaxation time

connected with inertial effects (s)�ssq Reynolds stress tensor (N=m2)ss Solid stress tensor (N=m2)st,g Characteristic time of the energetic

turbulent eddies (s)st,sg Lagrangian integral timescale (s)ugs Switch function in the Gidaspow

model/ Angle of internal frictionPk,g, P,z,g Influence of dispersed phases on

continuous phase

Subscripts

0 Initialcol Collisionds Dry solideq EquilibriumF Fountainfr Frictiong Gas phasekin Kinetick Turbulence kinetic energymax Maximumq qth phase, p, q¼ g, ss Solid phaset Turbulence; terminal velocity

ACKNOWLEDGMENT

This research was sponsored by the Tianjin MunicipalScience & Technology Commission, Tianjin, China (Pro-ject No. 09JCZDJC24200).

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