bubble formation and dynamics in gas ou liquid ou solid fluidization

Chemical Engineering Science 62 (2007) 2–27www.elsevier.com/locate/ces

Bubble formation and dynamics in gas–liquid–solid fluidization—A review

G.Q. Yang, Bing Du, L.S. Fan∗

Department of Chemical and Biomolecular Engineering, The Ohio State University, Columbus, OH 43210, USA

Available online 25 August 2006

Abstract

Current worldwide commercial activities in converting natural gas to fuels and chemicals, or gas-to-liquids technology use slurry bubblecolumn reactors with column sizes considerable larger than those currently in practice. Such commercial activities have prompted furtherfundamental research interest in fluid and bubble dynamics, transport phenomena and the scale up effects of three-phase fluidization systems.The fundamental behavior of particular relevance to these activities is associated with the elevated temperature and pressure conditions.

This review attempts to summarize the salient characteristics of liquid, bubbles, and particles and their interactive behavior and dynamics inthe process of bubble formation and bubble rising in gas–liquid–solid fluidization systems. Measurement techniques including both intrusivetechniques such as the probes, and non-intrusive techniques such as tomography, that are used to study fluid and bubble properties in gas–liquidand gas–liquid–solid systems, are illustrated. Governing mechanisms of bubble–particle collision and bubble breakup are discussed. The state-of-the-art computational techniques, that consider both the discrete and the continuum approaches for movement of the particle and bubblephases along with the discrete simulation results, are presented. Of particular emphasis is the effect of pressure and temperature on the fluidand bubble dynamics in three-phase fluidization.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Bubble formation; Bubble dynamics; Measurement techniques; Gas–liquid–solid fluidization; Pressure; Computational fluid dynamics (CFD);Bubble–particle collision; Bubble breakup

1. Introduction

Gas–liquid–solid fluidization systems have been appliedextensively in industry for physical, chemical, petrochemicaland biochemical processing (Shah, 1979; L’Homme, 1979;Ramachandran and Chaudhari, 1983; Fan, 1989). Currentworldwide commercial activities in converting natural gas tofuels and chemicals, or gas-to-liquids technology use slurrybubble column reactors with column sizes considerable largerthan those currently in practice (Sookai et al., 2001). Such com-mercial activities have prompted further fundamental researchinterest in fluid and bubble dynamics, transport phenomena,and the effects due to scale up of three-phase fluidizationsystems. The fundamental behavior of particular relevance tothese activities is associated with the elevated temperature andpressure conditions.

In gas–liquid–solid fluidization systems, bubble dynamicsplays a key role in dictating the transport phenomena and

∗ Corresponding author. Tel.: +1 614 688 3262.E-mail address: [email protected] (L.S. Fan).

0009-2509/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2006.08.021

ultimately affects the overall rates of reactions. It has been rec-ognized that the bubble wake, when it is present, is the dominantfactor governing the system hydrodynamics (Fan and Tsuchiya,1990). In general, consideration of the flow associated with thebubble wake near the bubble base, whether laminar or turbu-lent, is essential to characterize the complete behavior of therising bubble, including its motion. Conversely, examining theshape, rise velocity, and motion of a bubble can provide an in-direct understanding of the dynamics of the liquid–solid flowaround the bubble.

Most of the three-phase processes with considerablecommercial interest are conducted under high pressureand high temperature, for example, methanol synthesis (atP = 5.5 MPa and T = 260 ◦C), resid hydrotreating (atP = 5.5–21 MPa and T = 300–425 ◦C), Fischer–Tropsch syn-thesis (at P = 1.5–5.0 MPa and T = 250 ◦C), and benzenehydrogenation (at P = 5.0 MPa and T = 180 ◦C) (Fox, 1990;Jager and Espinoza, 1995; Saxena, 1995; Mills et al., 1996;Peng et al., 1999). Fundamental study of bubble dynamics inthese gas–liquid–solid fluidization systems, particularly underhigh-pressure and high-temperature conditions, is thus crucial.

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G.Q. Yang et al. / Chemical Engineering Science 62 (2007) 2–27 3

This review describes the flow behavior of liquid, bubbles,and particles in gas–liquid–solid fluidization systems. Themechanisms of the bubble formation, bubble instability, andbubble rise dynamics along with pertinent forces governingsuch mechanisms are illustrated. The review also surveys mea-surement techniques that are used to quantify the flow andbubble properties in gas–liquid and gas–liquid–solid systems.These techniques include both the intrusive ones such as theprobes, and non-intrusive ones such as tomography. Salientbubbling phenomena, related to bubble–particle collision andthe hydrodynamic similarity rules, are also discussed. Thestate-of-the-art computational techniques that consider boththe discrete and the continuum approaches for the particle andbubble phases as well as some discrete simulation results arepresented. Of particular emphasis in this review are the pres-sure and temperature effects on the fluid and bubble dynamicproperties in three-phase fluidization.

2. Measurement techniques

The quantification of bubble characteristics in gas–liquid–solid fluidized beds is normally made through direct visualiza-tion or by employing instruments. To visually observe bubblebehavior in three-phase fluidization systems, a two-dimensional(2D) fluidized bed is commonly used (e.g., Chen et al., 1989,1994; Fan and Tsuchiya, 1990; Kim and Kim, 1990; Fan et al.,1992; Tzeng et al., 1993; Kluytmans et al., 2001, 2003; Bech,2005; Vandu et al., 2005; Zaruba et al., 2005). By analyzing thedata obtained by the photography or video images, the dynamicbehavior of bubbles, including bubble shape, bubble wake,bubble size and bubble rise velocity, is quantified. The bubbleflow behavior in a 2D fluidized bed is treated as a vertical sliceof the three-dimensional (3D) system. Chen et al. (1994) foundthat there were some similarities between the flow structuresof 2D and 3D beds. However, the direct visualization onlyprovides limited information regarding bubble dynamics in 3Dsystems.

On the other hand, a large number of measurement tech-niques, including intrusive and non-intrusive methods, havebeen developed to investigate the bubble flow behavior inthe 3D fluidized bed systems. A comprehensive review ofmeasurement techniques in gas–liquid and gas–liquid–solidreactors can be found in Boyer et al. (2002). In the following,some measurement techniques that quantify bubble character-istics, along with recent advances in measurement techniquesfor gas–liquid and gas–liquid–solid fluidization research arediscussed.

2.1. Intrusive techniques

Considerable intrusive techniques have been developed tostudy bubble behavior in gas–liquid and gas–liquid–solid flu-idized systems. These intrusive techniques include impedance(conductivity or resistivity) probes, optical fiber probes, ultra-sound probes, endoscopic probes and hot film anemometry.A brief summary of these techniques is given below.

The impedance probe has been applied to measure the bub-ble volume fraction, bubble length and bubble rise velocity inthree-phase fluidized beds with relatively high liquid conductiv-ity. The method utilizes the difference in conductivity betweenthe liquid and the gas phase. For the three-phase fluidized bedswith low liquid conductivity, the addition of some salts into thesystem is required (Boyer et al., 2002). Hills and Darton (1976)investigated the bubble rising velocity in a bubble column byan impedance probe. Matsuura and Fan (1984) studied the bub-ble size and bubble rise velocity in three-phase fluidized bedsunder three different flow regimes by using a dual electrical re-sistivity probe. Tang and Fan (1989) applied a dual-resistivityprobe to study the bubble size distribution and the axial distri-bution of gas volume fraction. Liu (1993) used a dual-sensorresistivity probe to measure the bubble size, bubble rise velocityand bubble frequency in the bubble column. Chen et al. (1998)applied a dual-resistivity probe to measure the axial and radialdistributions of bubble diameter, bubble rise velocity, bubblefrequency and gas volume fraction in a three-phase fluidizedbed. Zenit et al. (2001) applied a dual impedance probe to studythe gas volume fraction, bubble velocity and bubble collisionin a vertical channel. To obtain accurate bubble volume frac-tion using the impedance probe technique, the interaction be-tween bubbles and the probe must be considered (Zenit et al.,2003). Based on the statistical, fractal, chaos and wavelet anal-yses, the conductivity bubble probe signal can be analyzed todiscern the local flow structure of the three-phase fluidized bed(Briens and Ellis, 2005).

The optical fiber probe utilizes the principle that the lighttransmits in liquid medium and is reflected by the gas mediumor bubbles. The optical probe is not effective, however, whenthe difference in the refraction index between the gas and liq-uid phases is small. Lee et al. (1986) and Lee and De Lasa(1987) measured the local gas volume fraction and bubble fre-quency in a three-phase fluidized bed using the U shape opticalfiber probe. Yu and Kim (1988) applied the U shape opticalfiber probe to study the radial distributions of the bubble size,bubble rise velocity and bubble volume fraction in three-phasefluidized beds. Frijlink (1987) developed a four-point probe toimprove the detection of the direction of the movement and theshape of the bubble. Chabot and de Lasa (1993) measured theaxial and radial distributions of the bubble chord length, bubblerise velocity and gas volume fraction in a bubble column at hightemperature by using the refractive optical probe. Xue et al.(2003) applied the four-point optical fiber probe to investigatethe bubble size and bubble rise velocity in gas–liquid systems.They found that a precise calibration of the probe by a CCDcamera was needed to obtain the accurate measurement on thebubble size and bubble rise velocity. Shoukri et al. (2003) mea-sured the gas volume fraction, bubble size, bubble rise velocity,bubble frequency and interfacial area in a large scale bubblecolumn using a dual optical probe. One of the advantages forthe optic fiber probe technique is that it can also be effectivelyapplied to high-pressure and high-temperature conditions forthe bubble property measurement (Luo et al., 1997, 1998b).

Stolojanu and Prakash (1997) obtained the solids concentra-tion and bubble volume fraction in a three-phase fluidized bed

4 G.Q. Yang et al. / Chemical Engineering Science 62 (2007) 2–27

using the ultrasonic technique. Al-Masry et al. (2005) studiedthe bubble frequency and bubble size distribution in bubblecolumns using the statistical analysis of acoustic signals. Theultrasonic techniques are not suitable for use under high gasholdup conditions because of the significant acoustic attenua-tion due to reflection on gas bubbles (Broering et al., 1991).Peters et al. (1983) studied the particle ejections by the bubbleeruption at the surface of a bubbling gas–solid fluidized bedusing an image system carrying a fiber optic probe or an en-doscopic probe. Such a probe was also used by Kumar et al.(1992) in the study of the solid concentration effects on the heattransfer in bubbly liquid–solid systems. Wang and Ching (2001)measured the multiple bubble velocities in the gas–liquid flowusing a dual-probe hot-film anemometry. However, the hot-film probe is so fragile that it can only be used for low solidsconcentration conditions. Furthermore, hot-film anemometrymeasurement requires uniform temperature distribution in themeasured volume.

The probe measures the point properties. One of the disad-vantages for intrusive probe techniques is the probe interfer-ence with the flow field and hence the bubble dynamics. Thebubble could be disintegrated, accelerated or elongated by theimmersed probe (Rowe and Masson, 1981; Chabot et al., 1992;Kiambi et al., 2003; Zenit et al., 2003; Julia et al., 2005). Al-though reducing the probe size could reduce the interferingeffect, the probe could also easily be damaged. The accurateconversion of the chord length distribution to the bubble sizedistribution is another challenging area as the bubble shape andthe size distribution vary with time and the location.

2.2. Non-intrusive techniques

The non-intrusive technique has the advantage of no mea-surement interference with the flow field. The informationprovided by the non-intrusive techniques varies from the cross-sectional bed density profile to the particle trajectory map(Chaouki et al., 1997; Chen et al., 1999; Seeger et al., 2003;Hubers et al., 2005; Warsito and Fan, 2005). The non-intrusivetechniques that were used to measure the three-phase fluidizedbed properties include pressure transducer, visualization tech-nique, particle image velocimetry (PIV), X-ray, �-ray, positronemission tomography (PET), radioactive particle tracking(RPT), ultrasonic tomography, nuclear magnetic resonanceimaging (NMR), laser techniques and electrical tomography.Some measurement examples in three-phase fluidized beds aredescribed below.

The pressure drop measurement together with the statisti-cal analysis techniques has been used to study the bubble flowbehavior in the bubble column and three-phase fluidized beds(Drahos et al., 1991, 1992; Johnsson et al., 2000; Kluytmanset al., 2001; Briens and Ellis, 2005; Chilekar et al., 2005). Thepressure transducer is usually positioned on the wall of the bed.The pressure fluctuation signal is a reflection of the overall hy-drodynamic behavior in the column. That is, the signals arecontributed from such sources as bubble flow, bubble coales-cence and bubble breakup, bubble burst at the top surface, andbubble formation at the distributor.

Direct visualization is also useful for property measurementin the systems with relatively low gas holdup and solids load-ing. Jiang et al. (1995), Luo et al. (1998a) andYang et al. (2000)utilized the visualization technique to study the bubble charac-teristics and bubble formation behavior in a largely transparentapparatus operated at high-pressures (up to 20 MPa) and high-temperatures (up to 220 ◦C). They also developed visualizationtechniques for in situ measurement of the physical propertiesof reacting or non-reacting fluids, such as dynamic surface ten-sion, and fluid density and viscosity at high pressure and hightemperature (Lin and Fan, 1997).

The X-ray technique has been widely used to investigate thebubble flow behavior including bubble shape, bubble size, bub-ble rise velocity, bubble growth and bubble breakage in thegas–solid or gas–liquid–solid fluidized beds. The X-ray tech-nique consists of the X-ray source to generate the X-ray beamto pass through the fluidized bed, an image intensifier to pro-duce an image, a CCD video camera to capture the image, andthe image analysis system. Fournier and Jeandey (1993) mea-sured the void fraction in the gas–liquid two-phase flow usingthe X-ray attenuation technique. The X-ray computer assistedtomography (CAT) was developed to investigate the fluidizationcharacteristics (Kumar et al., 1997). The X-ray CAT techniqueis able to provide rather high spatial resolution (1%), while itstemporal resolution is low. Seeger et al. (2003) measured thelocal solids velocity and local solids holdup in a three-phasefluidized bed by using the X-ray based particle tracking ve-locimetry (XPTV). Hubers et al. (2005) applied the X-ray CTtechnique to measure the phase holdups in the three-phase flu-idized bed.

The �-ray density gauge technique has been applied to studythe bubble size, bubble frequency and bubble coalescence in afluidized bed. The voidage between the radiation source anddetector in the bed is obtained by relating the ionization ofgas to the amount of radiation received by the detector. Sevilleet al. (1986) studied the jet and bubble behavior above thedistributor of a gas–solid fluidized bed by the �-ray tomogra-phy technique. The system included a �-ray source and a NaIdetector, which rotated along the axis of the fluidized bed. Thetotal scan time was up to 7.5 h. The same measurement tech-nique can be extended to the liquid system. Kumar et al. (1995)measured the voidage distribution in the bubble column usingthe �-ray tomographic scanner. Veera and Joshi (2000) used the�-ray tomography to investigate the radial distributions ofthe gas holdup in a bubble column. Jin et al. (2005) studied thephase holdups in a pressurized bubble column using the �-raydensitometry. Due to long scanning time, the �-ray tomographytechnique is only suitable for studying the time-averaged flowproperties. It is not suitable for the measurement of bubbleformation and bubble dynamics in the bed.

Non-intrusive laser techniques are also widely used to studybubble behavior, including the PIV, laser Doppler anemometry(LDA), phase Doppler anemometry (PDA) and laser Dopplervelocimetry (LDV). Chen and Fan (1992) and Reese et al.(1995, 1996) investigated the bubble characteristics in the slurrybubble column using the 2D and 3D PIV techniques. Lee et al.(1999) studied the bubble size distribution in the bubble column


and slurry bubble column using the gas disengagement tech-nique together with PIV technique. Vial et al. (2001) studiedthe liquid velocity and turbulence in the bubble columns withdifferent distributors by using the LDA technique. Kulkarni etal. (2004) applied the LDA technique to study the bubble sizedistribution in the bubble columns. Braeske et al. (1998) mea-sured the size, velocity and holdup of bubble and solid phases inthe three-phase fluidized beds using the PDA technique. Brennet al. (2002) applied the PDA technique to measure the veloc-ities of liquid and bubbles in the bubble column. Cui and Fan(2004, 2005) investigated the turbulence energy distribution inbubble columns and three-phase fluidized beds by measuringthe liquid velocity using the LDV technique. For all the lasertechniques, the laser beam needs to penetrate the flow system.Thus, the laser techniques limited only to the low gas holdupconditions.

Some other non-intrusive techniques are used for trackingthe particle movement, and/or mapping the instantaneous ortime-averaged, local or cross-sectional averaged, phase holdupsand phase velocities. They include PET (e.g., Bemrose et al.,1988; Stein et al., 2000; Dechsiri et al., 2005; Hoffmann et al.,2005), and RPT (e.g., Cassanello et al., 1995; Larachi et al.,1996, 1997; Chaouki et al., 1997; Chen et al., 1999; Kiaredet al., 1999; Nedeltchev et al., 2003), ultrasonic tomography(e.g., Wolf, 1988; Xu et al., 1997; Warsito et al., 1999; Utomoet al., 2001), nuclear magnetic resonance imaging (NMR orMRI) (e.g., Gladden, 1994, 2003; Chaouki et al., 1997; Leblondet al., 1998; Le Gall et al., 2001; Lim et al., 2004; Sedermanand Gladden, 2005; Gladden et al., 2005), electrical impedancetomography (EIT) (George et al., 2001; West et al., 2001; Kimet al., 2005) and electrical capacitance tomography (ECT)(Warsito and Fan, 2001, 2003). The details of each of thesetechniques and the specific hydrodynamic parameters theymeasure can be found in the corresponding references.

The MRI technique has been widely used in medical appli-cations. This technique, however, has also been used for themeasurement of multiphase flow systems such as the fixed bedsand trickle beds (Gladden, 1994, 2003; Chaouki et al., 1997;Leblond et al., 1998; Le Gall et al., 2001; Lim et al., 2004;Sederman and Gladden, 2005; Gladden et al., 2005). Limet al. (2004) applied the ultra-fast MRI technique to investi-gate the hydrodynamics in the trickle bed reactors. The 2Dimages of the trickle bed can have a higher spatial resolutionof 351 �m × 351 �m with a slow acquisition time of 6.4 s,or a low spatial resolution of 1.4 mm × 2.8 mm with a rel-atively fast acquisition time of 20 ms. The MRI can also beused to quantify the flow field in bubble column systems. Therelatively high cost of the technique and certain fluid prop-erty requirements, however, may hamper widespread usageof the MRI as a process tomography technique. The ECTis developed to image the multiphase media with dielectricproperties. It can be used to quantify the dynamic bubbleflow behavior in the gas–liquid and gas–liquid–solid three-phase fluidized beds. Compared to the CT, the ECT techniquehas a relatively low spatial resolution but a relatively hightemporal resolution. The ECT is suitable for process tomog-raphy applications for various multiphase flow systems. The

recent development of the 3D ECT or electrical capacitancevolume tomography (ECVT) with geometrically configuredsensor design and the neural network image reconstructiontechnique has further advanced the imaging technique and al-lows the dynamic 3D multiphase flow behavior to be captured(Du et al., 2005; Warsito and Fan, 2005). The ECVT with aspatial resolution of 5 × 5 × 5 mm3 and a temporal resolu-tion of 80 Hz has revealed the bubble formation process andbubble dynamics in three-phase fluidized beds (Warsito andFan, 2005). Fig. 1 shows the 3D bubble swarms obtained fromthe ECVT along with corresponding video images. The imagesin the figure represent one cycle of the circular motion of thespiral rising bubble plume, showing the consecutive motionsof the central bubble plume on the image plane. The directionof the circular motion is not constant, and is mixed with a highfrequency back and forth dancing motion of bubble swarm.

3. Single bubble behavior

In the following, the phenomena related to single bubblebehavior are discussed, which include bubble formation froma single orifice, bubble shape, single bubble rise velocity andbubble induced liquid flow. Experimental studies of the singlebubble behavior have been extensively reported, and the the-oretical account and CFD simulation have provided detailedinformation on single bubble properties.

3.1. Bubble formation

The fundamental study of the bubble formation behaviorfrom orifices is important for understanding the bubble sizevariation in the system, particularly for the case of a low gasvelocity or a single orifice gas injector. In these situations, thebehavior of bubble formation from the distributor mainly deter-mines bubble characteristics. There are two typical mechanicalarrangements for bubble formation from a single orifice; thatis with the orifice connected or not connected to a gas cham-ber. For bubble formation from a single orifice without a gaschamber, the gas flow rate through the orifice is always con-stant, which is referred to as constant flow conditions. The phe-nomenon of bubble formation from a single orifice connectedto a gas chamber varies with gas injection conditions, which arecharacterized by the dimensionless capacitance number Nc de-fined as 4Vcg�l/�D2

oP (Kumar and Kuloor, 1970; Tsuge andHibino, 1983). When Nc is smaller than 1, the gas flow ratethrough the orifice is almost constant during the bubble forma-tion process, similar to the first mechanical arrangement. WhenNc is larger than 1, the gas flow rate through the orifice is notconstant, and it is dependent on the pressure difference betweenthe gas chamber and the bubble. Such bubble formation con-ditions are characterized as variable flow conditions by Yanget al. (2000) or as constant pressure and intermediate conditionsby Tsuge and Hibino (1983).

Numerous experimental and modeling studies have been con-ducted over the past decades on bubble formation from a sin-gle orifice or nozzle submerged in liquids, mostly under am-bient conditions (Kupferberg and Jameson, 1969; Kumar and


Fig. 1. Snapshots of the 3D volume images of bubble plumes compared with photographs: Ug = 0.02 m/s (from Warsito and Fan, 2005).

Kuloor, 1970; Azbel, 1981; Lin et al., 1994; Ruzicka et al.,1997; Kulkarni and Joshi, 2005; Zhang et al., 2005; Xiao andTan, 2006). A few studies were conducted at elevated pres-sures (La Nauze and Harris, 1974; Idogawa et al., 1987; Tsugeet al., 1992; Wilkinson and van Dierendonck, 1994). Thesestudies indicated that an increase in gas density reduces the sizeof bubbles formed from the orifice.

Bubble formation in liquids with the presence of particles, asin slurry bubble columns and three-phase fluidized bed systems,is different from that in pure liquids. The experimental dataof Massimilla et al. (1961) in an air–water–glass beads three-phase fluidized bed revealed that the bubbles formed from asingle nozzle in the fluidized bed are larger in size than those inwater, and the initial bubble size increases with the solids con-centration. Yoo et al. (1997) investigated bubble formation inpressurized liquid–solid suspensions. They used aqueous glyc-erol solution and 0.1-mm polystyrene beads as the liquid andsolid phases, respectively. The densities of the liquid and theparticles were identical, and thus, the particles were neutrally

buoyant in the liquid. The results indicated that initial bubblesize decreases inversely with pressure under otherwise constantconditions, that is, gas flow rate, temperature, solids concentra-tion, orifice diameter, and gas chamber volume. Their resultsalso showed that the particle effect on the initial bubble size isinsignificant. The difference in the finding regarding the parti-cle effect on the initial bubble size between Massimilla et al.(1961) and Yoo et al. (1997) is possibly due to the differencein particle density.

Bubble formation in a hydrocarbon liquid and liquid–solidsuspension with significant density difference between the liq-uid and solid phases was investigated by Luo et al. (1998a)and Yang et al. (2000) under various gas injection conditions.A mechanistic model was developed to predict the initial bub-ble size in liquid–solid suspensions at high-pressure conditions.The model considers various forces induced by the particles,and is an extension of the two-stage spherical bubble formationmodel developed by Ramakrishnan et al. (1969) for liquids. Inthe two-stage spherical bubble formation model, bubbles are as-


uo

FI,g

Fσ

FBA

FD

FB

FM

FI,m

FC

ro

Fig. 2. The balance of all the forces acting on a growing bubble (from Luoet al., 1998a).

sumed to be formed in two stages, namely, the expansion stageand the detachment stage. The bubble expands with its base at-tached to the nozzle during the first stage. In the detachmentstage, the bubble base moves away from the nozzle, althoughthe bubble remains connected with the nozzle through the neck.The shape of the bubble is assumed to remain spherical duringthe entire bubble formation process. It is also assumed in thismodel that a liquid film always exists around the bubble. Dur-ing the expansion and detachment stages, particles collide withthe bubble and stay on the liquid film. The particles and theliquid surrounding the bubble are set in motion as the bubblegrows and rises.

The steady-state initial bubble size is of interest and im-portance in the hydrodynamic characterization of the bubbleflow system. The volume of the bubble at the end of the firststage and during the second stage can be described by con-sidering a balance of all the forces acting on the bubble if theinstantaneous gas flow rate, Qo, or the instantaneous gas ve-locity, uo, through the orifice, is known. The forces induced bythe liquid include the upward forces (effective buoyancy force,FB , and gas momentum force, FM), and the downward resis-tance (liquid drag, FD , surface tension force, F�, bubble iner-tial force, Fl,g , and Basset force, FBA) as shown in Fig. 2. It isassumed that the particles affect the bubble formation processonly through two additional downward forces on the bubble,that is, the particle–bubble collision force, FC , and the suspen-sion inertial force, Fl,m. The suspension inertial force is dueto the acceleration of the liquid and particles surrounding thebubble. Therefore, the overall force balance on the bubble in

0.01 0.1 1 100.01

0.1

1

VB

∗ 106 (

m3 )

Qg∗106 (m3/s)

Key P(MPa)0.12.08.0

εs= 0.3

Fig. 3. Effect of pressure on initial bubble volume (from Yoo et al., 1997).

this model can be written as

FB + FM = FD + F� + FBA + FI,g + FC + FI,m. (1)

The expansion stage and the detachment stage follow the sameforce balance equation, although the expression for the sameforce in the two stages may be different due to different bubblemoving velocities in the two stages. The expressions for all theforces under two stages are given in Table 1.

The effect of pressure on the initial bubble volume is shownin Fig. 3. Under relatively low pressures (less than 2.0 MPa),the pressure effect on the initial bubble volume is seen to besignificant; however, under relatively high pressures, the effectof pressure on the initial bubble volume becomes insignificant(Yoo et al., 1997).

The effect of wetting conditions on bubble formation at a verylow gas flow rate was recently investigated by Gnyloskurenkoet al. (2003, 2005). The wetting conditions were varied viacoating the orifice plate by vacuum silicon grease or by paraf-fin. In their experiments, the equilibrium contact angles werechanged from 68◦ to 110◦ (the higher contact angle, the poorerwettability). It was found that the final bubble size detachedfrom the orifice increased significantly as the wetting conditionworsened, i.e., a higher contact angle.

3.2. Bubble shape

The characteristics of a rising bubble can be describedin terms of the shape, rise velocity, and motion of the bub-ble. These rise characteristics are closely associated with thebehavior of the bubble wake, and the flow and physicalproperties of the surrounding medium (mainly viscosity andthe presence/absence of solid particles) as well as the interfa-cial properties of the bubble surface (i.e., presence/absence ofsurfactant). In this section, the bubble shape and the single bub-ble rise velocity in both liquids and liquid–solid suspensionsare described.


Table 1Expressions for the forces involved in the bubble formation process

Forces Expansion stage Detachment stage

FB

�6

d3b (�l − �g)g Same as expansion stage

FM

�4

D2o�gu2

o Same as expansion stage

FD CD

(�4

d2b

) �lu2b

2

(CD = 24

Re

)Same as expansion stage

F� �Do� cos � Same as expansion stage

FI,g

d

dt

[�g

(�6

d3b

)ub

]Same as expansion stage

FBA Not applicable3

2d2b

√��l�l

∫ t

0du/dt√

t−�d�

FC

�4

D2o(1 + e)�s�su

2e

�4

d2b �s�su

2

FI,m

d(∫ ∫ ∫

�mumV )

dt=

d

dt

[�m

(�6

d3b

)ub

]Same as expansion stage

The shape of bubbles moving in Newtonian liquids can gen-erally be identified as spherical, oblate ellipsoidal and spheri-cal/ellipsoidal cap. The observed bubble shape is a result of anintricate balance among forces acting on the rising bubble, in-cluding surface tension, viscous, and buoyancy forces. As thedominant forces change with increasing bubble size, the bubbleshape undergoes changes from spherical, ellipsoidal to spheri-cal cap shape.

When the bubble size, db, is small, e.g., less than 1 mmin water, and the shape is spherical, viscous forces and sur-face tension forces dominate. At low Reynolds numbers (i.e.,Re = �lubdb/�l < 1), the Hadamard–Rybczynski theory andthe Stokes’ theory apply to spherical bubbles of mobile andimmobile (rigid) surfaces, respectively. In practice, especiallywhen bubbles in low viscosity liquids, the Reynolds numbereasily exceeds unity and the inertial term is no longer negligi-ble. Levich (1962) obtained an equation for bubble rise veloc-ity based on the boundary layer theory for spherical bubbles athigher Reynolds numbers. All these theories result in an identi-cal analytical expression with different constants accounting fordifferent bubble surface conditions (Fan and Tsuchiya, 1990).For bubbles of an intermediate size, both the surface tensionand the inertia (buoyancy) force are important in dictating theshape fluctuation and the dynamics of motion of the bubble.The wake forms and undergoes a complex shedding process foran intermediate size of bubbles, which is also the case when thebubble size is large. For large bubbles, the inertia force domi-nates and the effects of the surface tension, viscosity, and purityof the liquid media on bubble dynamics are negligible. A largebubble rising in water is of a spherical cap shape whereas it isof an ellipsoidal shape for an intermediate bubble (Clift et al.,1978; Fan and Tsuchiya, 1990).

In slurry bubble columns and gas–liquid–solid fluidizedbeds of small and light particles, bubble behavior has oftenbeen observed to resemble that in viscous liquids (Stewart andDavidson, 1964; Oestergaard, 1973; Dayan and Zalmanovich,

1982). This resemblance is based on the premise that theliquid–solid mixture in such systems can be regarded as apseudo-homogeneous medium of higher apparent viscositycompared to the liquid. On the other hand, in three-phase flu-idized beds of large and/or heavy particles, gas bubbles behavedifferently.

Although the flattening of the bubble with increasing bubblesize can be explained qualitatively (Fan and Tsuchiya, 1990),quantitative predictions are difficult, especially the variations inthe physical properties of the surrounding medium are involved.In this regard, the state-of-the-art knowledge on bubble shapesin various liquids is established mainly from experimental ob-servations. Three dimensionless groups, the Reynolds number(Re), the Eötvös number (Eo), and the Morton number (Mo),are commonly used to characterize the bubble shape and risebehavior (Grace, 1973; Bhaga and Weber, 1981). They are de-fined as

Re = dbub�l

�l

, (2a)

Eo = g��d2b

�, (2b)

Mo = g��4l

�2l �

3where �� = �l − �g . (2c)

For bubbles in low Mo liquids (Mo < 10−3), Vakhrushev–Efremov’s (1970) formula given below can be used to predictthe bubble aspect ratio:

h/b =

⎧⎪⎨⎪⎩

1, Ta < 1,

{0.81 + 0.206 tanh[2(0.8 1�Ta�39.8,

−log10 Ta)]}3,

0.24, 39.8�Ta.

(3)

Here, Ta(=Re Mo0.23) is the Tadaki number.


The bubble characteristics in 2D systems were studiedby many researchers (e.g., Henriksen and Ostergaard, 1974;Tsuchiya and Fan, 1988; Song, 1989; Tsuchiya et al., 1990).The experimental study in a 2D fluidized bed for the water-glass beads system indicated that, for a given particle size, thevariation in bed voidage (�l = 1 − �s) has a minimal influenceon the bubble aspect ratio over the range 0.5 < �l �1 (Tsuchiyaet al., 1990). The effect of particle size is also not signifi-cant for most particles; however, at higher Reynolds numbers(Re > 3000) the bubbles flatten less for larger particles (1 mmglass beads with a terminal velocity of 0.17 m/s).

The effect of particle wettability on the bubble shape wasinvestigated by Tsutsumi et al. (1991a). Comparing the bub-ble shapes in systems with normal glass beads (wettable) andidentical sizes of Teflon-coated glass beads (non-wettable), it isfound that the effect of particle wettability on the bubble aspectratio is insignificant. The same qualitative dependence of thebubble aspect ratio on the particle size and the solids holdupwas observed for both types of particles.

3.3. Bubble rise velocity

The rise velocity of a single gas bubble depends on its size:for small bubbles, the rise velocity also strongly depends onliquid properties such as surface tension and viscosity; for largebubbles, the rise velocity is insensitive to liquid properties (Fan,1989; Kulkarni and Joshi, 2005). Under limited conditions, therise velocities of single bubbles in liquid–solid suspensionswere found to be similar to those in highly viscous liquids(Massimilla et al., 1961; Darton and Harrison, 1974).Liquid–solid suspensions can thus be characterized as New-tonian homogeneous media, but they often exhibit the non-Newtonian or heterogeneous behavior (Tsuchiya et al., 1997).Differences in fluidizing media, pressure, and temperature leadto different bubble rise characteristics. This section focuseson the bubble rise characteristics in liquids and liquid–solidsuspensions at elevated pressure and temperature.

3.3.1. Single bubble rise velocity in liquidsIn most applications of three-phase fluidization systems, the

liquid phase is a hydrocarbon based medium. The physicalproperties of hydrocarbon liquids (e.g., Paratherm NF heattransfer fluid) vary dramatically with pressure and temperature.Lin and Fan (1997) and Lin et al. (1998) developed variousin situ techniques to measure the physical properties of thereactive and non-reactive fluid at elevated pressures and tem-peratures. The hydrostatic weighing method, the falling-balltechnique, and the emerging-bubble technique were used tomeasure in situ liquid density, viscosity, and surface tension, re-spectively. Based on their measurements of Paratherm NF heattransfer fluid, the liquid and interfacial properties were foundto change significantly with pressure and temperature. For ex-ample, at room temperature, as the pressure increases from 0.1to 20 MPa, the liquid density increases by approximately 5%,the liquid viscosity increases by 65%, and the surface tensiondecreases by 25%. Therefore, it is important to note that vari-

ations of such physical properties, particularly the liquid vis-cosity and surface tension, be included in the analysis of data,empirical correlation, theory and model developments and nu-merical simulations.

Since the bubble rise velocity depends on liquid properties,the pressure and temperature thus affect the bubble rise char-acteristics. Krishna et al. (1994) studied the pressure effect onthe bubble rise velocity and found that the single bubble risevelocity does not depend on the gas density over the range of0.1–30 kg/m3. The conclusion is limited to a narrow range ofpressures. Lin et al. (1998) measured the rise velocity of sin-gle bubbles of known sizes in the Paratherm NF heat transferfluid at various pressures and temperatures. The pressure rangesfrom 0.1 to 19.4 MPa. For a given bubble size, ub tends to de-crease with increasing pressure at both temperatures. The ef-fects of pressure and temperature, or more directly, the effectsof the physical properties of the gas and liquid phases on thevariation of ub with db can be predicted by the Fan–Tsuchiyaequation (Fan and Tsuchiya, 1990), the modified Mendelson’swave-analogy equation (Mendelson, 1967) by Maneri (1995)and a correlation proposed by Tomiyama et al. (1995).

The Fan–Tsuchiya equation, generalized for high-pressuresystems, can be written in a dimensionless form:

u′b = ub

(�l

�g

)1/4

={[

Mo−1/4

Kb

(��

�l

)5/4

d′2b

]−n

+[

2c

d ′b

+(

��

�l

)d ′b

2

]−n/2}−1/n

, (4)

where the dimensionless bubble diameter is given by

d ′b = db(�lg/�)1/2. (5)

Three empirical parameters, i.e., n, c, and Kb in Eq. (4), accountfor the contamination level of the liquid phase, the dynamiceffects of the surface tension, and the viscous nature of thesurrounding medium, respectively.

The modified Mendelson equation is valid only under theinviscid condition and has limited predictive capability atthe low temperature. The Fan–Tsuchiya equation applied forthe given liquid, a pure, multicomponent, and organic solvent,provides good overall predictions except for the sharp peakexisting under the high temperature condition. The equationby Tomiyama et al. (1995) also has good general applicability,especially around the peak point occurring near db = 2 mmat 78 ◦C; however, it tends to underestimate the bubble risevelocity over the rest of the bubble size range (Lin et al., 1998).

3.3.2. Single bubble rise velocity in liquid–solid suspensionsMuch progress has also been made regarding single bubble

rise characteristics in liquid–solid suspensions at high pressureand high temperature. Fig. 4 shows the effect of pressure on thebubble rise velocity in a fluidized bed with Paratherm NF heattransfer fluid and 0.88-mm glass beads at two different tem-peratures (Luo et al., 1997). At both temperatures, the bubblerise velocity decreases with an increase in pressure for a givensolids holdup. The extent of the reduction is as high as 50%


Fig. 4. Effect of pressure on bubble rise velocity in a fluidized bed of 0.88-mmglass beads at (a) 26.5 ◦C and (b) 87.5 ◦C (solids holdups for +, open, andfilled symbols are 0, 0.384, and 0.545, respectively; from Luo et al., 1997).

from 0.1 to 17.3 MPa. A more drastic reduction in ub, however,arises from the addition of solid particles. While the particleeffect is small at a low solids holdup (�s < 0.4), the effect isappreciable at a high solids holdup (�s = 0.545), especially forhigh liquid viscosity [Fig. 4(a)]. A comparison of bubble risevelocity at 26.5 and 87.5 ◦C, for the same solids holdup, indi-cates that the viscosity effect appears to be significant. It wasalso found that the extent of decrease in bubble rise velocitywith increasing solids holdup is much smaller in a fluidizedbed containing smaller particles.

In the presence of solid particles, as a first approximation,it can be assumed that the particles modify only homogeneousproperties of the surrounding medium. Based on this homoge-neous approach, the Fan–Tsuchiya equation, Eq. (4), can be ex-tended to liquid–solid suspensions by replacing the liquid prop-erties, �l and �l , with the effective properties of the liquid–solidsuspension, �m and �m, respectively (Tsuchiya et al., 1997).The effective density can be estimated by

�m = �l (1 − �s) + �s�s . (6)

The effective viscosity of liquid–solid suspensions is estimatedby

�m

�l

= exp

[K�s

1 − (�s/�sc)

], (7)

with two parameters correlated by Luo et al. (1997):

K = 3.1 − 1.4 tanh[0.3(10 − 102ut )]�

(8a)

and

�sc = {1.3 − 0.1 tanh[0.5(10 − 102ut )]}�s0, (8b)

where ut is in units of m/s.The Fan–Tsuchiya equation with constant values of �m esti-

mated from Eq. (7) predicts reasonably well the general trendof bubble rise velocity variation in liquid–solid suspensionsas shown in Fig. 4. However, a detailed match between theexperimental results and predictions appears to be difficultto attain by assigning a constant value of �m for each condition.A more elaborate analysis is required to account for the effectof bubble size on interactions of the bubble with the surround-ing medium (non-Newtonian approach) or with individual par-ticles (heterogeneous approach).

The effect of particle wettability on the bubble rise velocitywas examined by Tsutsumi et al. (1991a,b). Non-wettable parti-cles have a larger contact angle compared to wettable particles;that is non-wettable particles favor contact between bubbles andsolids. Tsutsumi et al. (1991a,b) observed that particle–particleaggregates and particle–bubble aggregates are formed at lowgas velocities with non-wettable particles. Consequently, thebubble rise velocity is smaller than that for wettable particles.Further, the liquid velocity for minimum fluidization is lowerand the liquid velocity for transition to the transport regimeis higher for wettable particles compared to those for non-wettable particles. On the other hand, a negligible effect ofparticle wettability was observed on the rise velocity of largebubbles (db > 15 mm). For large bubbles with a circular capshape, the attachment of particles to the bubbles occurs only atthe bubble base, and was not observed on the bubble roof dueto the fluid shear effects caused by the fast rising bubbles.

3.4. Bubble induced liquid–solid flow

A major factor contributing to the complexity of theliquid–solid flow in the vicinity of the rising bubbles is theinstability induced by the wakes behind the bubbles. Throughthis instability, the wake flow becomes unsteady and interactivewith the external flow. The phenomenon is often characterizedby a cycle of vortex formation and shedding and, for free risingbubbles, it is intimately related to oscillatory bubble motion.Various transport phenomena taking place in gas–liquid–solidsystems are closely associated with the wake flow behavior. Inthe following section, the pressure distribution around a bub-ble in liquids and liquid–solid suspensions is discussed alongwith particle entrainment and drift effect associated with risingbubbles.


3.4.1. Wake pressurePressure distribution in the wake is critical and is closely

associated with the fluid motion, solids concentration in the bulkphase, and the size of the wake, or more generally, the wakestructure. The pressure field around a large circular cap bubblewas studied in water and liquids of different viscosities (Lazarekand Littman, 1974; Bessler and Littman, 1987). The pressuredistribution around the bubble in liquids showed a symmetricminimum in the primary wake. A sharp minimum followedby localized recovery of the pressure immediately beneath thebubble base was also observed, but only for liquids with lowviscosity. In these studies, the bubble wake, even for water, wasclosed laminar in structure in spite of the large bubble size (upto 10 cm wide). The closed laminar wake was possibly causedby the delayed onset of the vortex shedding from the wallstabilized large bubble. Raghunathan et al. (1992) studied thepressure distribution around a bubble in a water-163-�m glassbead suspension. They found that the pressure distribution inthe liquid–solid mixture is qualitatively similar to those in water(Lazarek and Littman, 1974). However, the pressure recoveryis nearly complete in the liquid–solid suspension indicating asmaller secondary wake compared to that in water.

3.5. Particle entrainment and drift effect

The bubble wake behavior directly affects the particle en-trainment in the operation of a three-phase fluidization systemand may be significant if the freeboard region is not suffi-ciently large. The problem appears to be more pertinent forbeds of small and/or light particles than those of large and/orheavy particles. Furthermore, these particles yield apprecia-bly different axial solids holdup distributions in the freeboardregion.

The fundamental mechanisms for particle entrainment andde-entrainment in the transitional region of the freeboard wererevealed by Page and Harrison (1974). The particles are drawnfrom the upper surface of the fluidized bed into the freeboardmainly by the wake behind the bubble and the vortices con-taining the particles are shed from the wake in the freeboard.Particle entrainment normally decreases with a decrease in bub-ble size and bubble frequency, and with an increase in liquidvelocity and particle size. A model, developed by El-Temtamyand Epstein (1980) to predict the solids holdup distribution inthe freeboard, indicates the central roles played by the bubblewake in particle entrainment and the wake shedding in par-ticle de-entrainment. Mechanics and phenomena of the wakeentrainment and particle penetration through the bubble havebeen thoroughly discussed and described by Fan and Tsuchiya(1990).

Some physical insight into the mechanisms of particle en-trainment can be gained by closely following the time evo-lution of the particle flow around a single bubble (Miyaharaet al., 1989). When a single bubble emerges from the free sur-face of a bed, a mantle of particles that cover the roof of thebubble drain away and rush into the near-wake of the bubble.Overall, the particles move upward due to this near-wake cap-ture as well as due to the drift effect (Darwin, 1953). The drift

effect, however, is confined to the vicinity of the bed surface.The drift effect can be distinctly seen when low-density calciumalginate particles (1.2 mm in diameter and 1.02 g/cm3 in den-sity) are used (Tsuchiya et al., 1992). The particle displacementcaused by the drift effect is thus relatively insignificant in thecase of large bubbles and/or heavy particles. In a recent study ofhigh pressure bubble columns, Lau et al. (2005) indicated thatthe liquid entrainment rate increases with system pressures andgas velocities. The same behavior would hold for solid particleentrainment in high pressure three-phase fluidized beds.

4. Multi-bubble behavior

Behavior of multi-bubbles in the three-phase fluidization sys-tem is complex. In this section, several fundamental aspects ofmulti-bubble behavior are presented, which include bubble co-alescence (small bubbles merging into a large bubble), bubblebreakup (a large bubble breaking into small bubbles), and bub-ble size distribution due to dynamic bubble coalescence andbreakup.

4.1. Bubble coalescence

For gas–liquid systems, the experimental results availablein the literature indicate that an increase of pressure retardsthe bubble coalescence (Sagert and Quinn, 1977, 1978). Thereare three steps in the bubble coalescence process (Vrij, 1966;Chaudhari and Hoffmann, 1994): (1) approach of two bubblesto form a thin liquid film between them; (2) thinning of thefilm by the drainage of the liquid under the influence of gravityand suction due to capillary forces; and (3) rupture of the filmat a critical thickness. The second step is the rate-controllingstep in the coalescence process and the bubble coalescence ratecan be approximated by the film-thinning rate (Vrij, 1966). Thefilm thinning velocity can be expressed as (Sagert and Quinn,1977, 1978)

−dl

dt= 32 l3�

3�R2d�ldb

, (9)

where the parameter � is a measure of the surface drag or veloc-ity gradient at the surface due to the adsorbed layer of the gas.It is known that surface tension decreases and liquid viscos-ity increases with increasing pressure. In addition, � increaseswith pressure. As can be seen from Eq. (9), all of these varia-tions contribute to the reduction of the film thinning velocity,and hence the bubble coalescence rate, as pressure increases.As a result, the time required for two bubbles to coalesce islonger and the rate of overall bubble coalescence in the bedis reduced at high pressures. Moreover, the frequency of bub-ble collision decreases with increasing pressure. An importantmechanism for bubble collision is the bubble wake effect (Fanand Tsuchiya, 1990). As the bubble size and the rise velocityreduce at high pressures, the likelihood of small bubbles beingcaught and trapped by the wakes of large bubbles decreases.Therefore, bubble coalescence is suppressed by the increase in


pressure, due to the longer bubble coalescence time and thesmaller bubble collision frequency.

The presence of pulp even at very low pulp consistencies(0.1%) in the column leads to enhanced bubble coalescenceand hence a narrowing of the gas velocity for the dispersedbubble regime as the pulp consistency increases (Reese et al.,1996). Bubble coalescence inhibitors such as inorganic salts(e.g., sodium chloride and sodium phosphate dibasic) and or-ganic compounds (e.g., ethanol, n-pentanol, iso-amyl alcohol,and benzoic acid) can be effectively applied to the liquid atconcentrations up to 200 ppm to inhibit bubble coalescence be-havior in three-phase fluidization (Briens et al., 1999). Withthe addition of the bubble coalescence inhibitor, the bed hydro-dynamics at low gas velocities are significantly different fromthose without the inhibitor and the influence of the gas distrib-utor becomes distinct (Nacef et al., 1995).

4.2. Bubble breakup

Bubble breakup has been investigated theoretically and ex-perimentally. This section will discuss bubble breakup due tobubble–particle collision and bubble instability.

4.2.1. Bubble breakup due to bubble–particle collisionIn a three-phase fluidization system, bubble size variation is

intimately related to bubble–particle collisions. The collisionscan yield two different consequences: the particle is ejectedfrom the bubble surface, or the particle penetrates the bubbleleading to either bubble breakage or non-breakage.

Bubble–particle collisions generate perturbations on the bub-ble surface. After the bubble–particle collision, three factorsbecome crucial in determining the breakage characteristics ofthe bubble (Clift et al., 1978):

(1) shear stress, which depends on the liquid velocity gradientand the relative bubble–particle impact speed, and tends tobreak the bubble;

(2) surface tension force, which tends to stabilize the bubbleand causes it to recover the bubble’s original shape;

(3) viscous force, which slows the growth rate of the surfaceperturbation, and tends to stabilize the bubble.

There are three criteria that are required for particle pene-tration through a bubble. These criteria were developed by ne-glecting the shear effects due to the liquid flow (Chen and Fan,1989). The particle will penetrate the bubble when any of thefollowing three criteria is satisfied. The three criteria are givenas follows:

(1) the acceleration of the particle is downward;(2) the particle velocity relative to the bubble is downward;(3) the particle penetration depth is larger than the deformed

bubble height.

When none of these criteria are satisfied, the particle will beejected from the bubble surface upon contact with the bubble.

By extending Boys’ instability analysis (Boys, 1959), a criterionfor bubble breakage after penetration can be obtained. When theparticle diameter (dp) is larger than the height of the doughnut-shape bubble (Hd), the bubble will breakup, that is,

dp > Hd . (10)

Hong and Fan (1994) conducted experiments to elucidatethe effects of flow field and liquid viscosity on the bubblebreakage. The system in their study included a freely risingbubble colliding with a falling copper ball with a density of8.7 g/cm3. Particles of three different sizes (0.92, 1.22, and1.53 cm in diameter) were used, with the volume equivalentdiameter of a spherical cap bubble fixed at 1.97 cm. Eightywt.% aqueous glycerine solution and pure water were used asthe liquid phase. The surface tensions of these two liquids arecomparable (�solution = 65.9 mN/m vs. �water = 72.6 mN/m);however, the viscosities differ greatly (�solution = 52.9 mPa svs. �water = 1.0 mPa s). Experiments by Hong and Fan (1994)also showed that the bubble breakage mechanism in water andglycerine solution was different. Although the only significantdifference in conditions between these two cases is the liquidviscosity, the consequences of the particle penetration are op-posite: in the glycerine solution, the bubble recovers its originalshape; while in water, the bubble is broken down into pieces.In water the particle Reynolds number is 1.2 × 104 and theliquid flow around the particle is turbulent, while in the glyc-erine solution the particle Reynolds number is less than 500and the flow is laminar. Hence, the bubble in water is disin-tegrated by the surface perturbations induced by the turbulentshear stress when the particle penetrates. The numerical simu-lation by Hong et al. (1999) shows bubble elongation and tinybubble generation upon particle collision, which are in a goodagreement with the experimental observation by Hong and Fan(1994). Their simulation also shows a large resulting pressureoscillation, which could also contribute to bubble surface in-stability leading to its breakage.

4.2.2. Bubble breakup due to bubble instabilityThe second cause of bubble breakup is due to bubble insta-

bility. The upper limit of the bubble size is set by the maximumstable bubble size, Dmax, above which the bubble is subjectedto breakup and hence unstable. Several mechanisms have beenproposed for the bubble breakup phenomenon and based onthese mechanisms theories have been established to predict themaximum bubble size in gas–liquid systems. In this section,the mechanisms of bubble breakup and the theories to predictthe maximum bubble size are discussed.

Hinze (1955) proposed that the bubble breakup is caused bythe dynamic pressure and the shear stresses on the bubble sur-face induced by different liquid flow patterns, e.g., shear flowand turbulence. When the maximum hydrodynamic force in theliquid is larger than the surface tension force, the bubble dis-integrates into smaller bubbles. This mechanism can be quan-tified by the liquid Weber number. When the Weber numberis larger than a critical value, the bubble is not stable and dis-integrates. This theory was adopted to predict the breakup of


bubbles in gas–liquid systems (Walter and Blanch, 1986). Cal-culations by Lin et al. (1998) showed that the theory under-predicts the maximum bubble size obtained by experiments andcould not predict the effect of pressure on the maximum bubblesize.

A maximum stable bubble size exists for bubbles rising freelyin a stagnant liquid without external stresses, e.g., rapid ac-celeration, shear stress, and/or turbulence fluctuations (Graceet al., 1978). The Rayleigh–Taylor instability has been regardedas the mechanism for bubble breakup under such conditions. Ahorizontal interface between two stationary fluids is unstable todisturbances with wavelengths exceeding a critical value if theupper fluid has a higher density than the lower one (Bellmanand Pennington, 1954):

�c = 2�

√�

g(�l − �g). (11)

Grace et al. (1978) modified the Rayleigh–Taylor instabilitytheory by considering the time available for the disturbance togrow and the time required for the disturbance to grow to anadequate amplitude. Batchelor (1987) pointed out that the ob-served size of air bubbles in water was considerably larger thanthat predicted by the model of Grace et al. (1978). Batchelor(1987) further took into account the stabilizing effect of the liq-uid acceleration along the bubble surface and the non-constantgrowth rate of the disturbance. In Batchelor’s model, the magni-tude of the disturbances is required to predict maximum bubblesize; however, the magnitude of the disturbances is not known.The models based on the Rayleigh–Taylor instability predict analmost negligible pressure effect on the maximum bubble size;in fact, Eq. (11) implies that the bubble is more stable whenthe gas density is higher.

The Kelvin–Helmholtz instability is similar to the Rayleigh–Taylor instability, except that the former allows a relative ve-locity between the fluids, ur . Using the same concept of Graceet al. (1978), Kitscha and Kocamustafaogullari (1989) appliedthe Kelvin–Helmholtz instability theory to model the breakupof large bubbles in liquids. Wilkinson and van Dierendonck(1990) applied the critical wavelength to explain the maximumstable bubble size in high-pressure bubble columns. Distur-bances in the liquid with a wavelength larger than the criticalwavelength can break up a bubble. The critical wavelength de-creases with an increase in pressure and therefore bubbles areeasier to break apart by disturbances at higher pressures. How-ever, the critical wavelength concept alone cannot account forthe effect of pressure on the maximum bubble size.

All of the models mentioned above do not account for theinternal circulation of the gas. The internal circulation velocityis of the same order of magnitude as the bubble rise velocity.A centrifugal force is induced by this circulation, pointing out-wards toward the bubble surface. This force can suppress thedisturbances at the gas–liquid interface and thereby stabilizethe interface. Centrifugal force explains the underestimation ofDmax by the model of Grace et al. (1978). On the other hand,the centrifugal force can also break apart the bubble, as it in-creases with an increase in bubble size. Levich (1962) assumed

the centrifugal force to be equal to the dynamic pressure in-duced by the gas moving at the bubble rise velocity, that is,kf �gu

2b/2 (kf ≈ 0.5), and proposed a simple equation to cal-

culate the maximum stable bubble size:

Dmax ≈ 3.63�

u2b

3√

�2l �g

. (12)

Eq. (12) severely under-predicts the maximum bubble size inthe air–water system, although it shows a significant effect ofpressure on the maximum bubble size. Luo et al. (1999) con-sidered that the bubble would break up when the centrifugalforce exceeds the surface tension force, especially at high pres-sures when gas density is high. They arrived at a criterion forbubble instability:

u2bdb � 8 4/3E(

√1 − 2)

0.312

�

�g

. (13)

The criterion leads to a maximum stable bubble size expressedby

Dmax ≈ 2.53

√�

g�g

(for = 0.21) (14a)

in liquids, and

Dmax ≈ 3.27

√�

g�g

(for = 0.3) (14b)

in liquid–solid suspensions. Further, based on the Davies–Taylor equation, the rise velocity of the maximum stablebubble is

umax =(

1.6�g

�g

)1/4

. (15)

The comparison of experimental maximum bubble sizes and thepredictions by various instability theories indicates that bub-ble breakup is governed by the internal circulation mechanismat high pressures over 1.0 MPa, whereas the Rayleigh–Taylorinstability or the Kelvin–Helmholtz instability is the dominantmechanism at low pressure.

4.3. Bubble size distribution

In multi-bubble systems, a mean bubble size is usually usedto describe the system. The mean bubble size is commonlyexpressed through the Sauter, or volume-surface, mean. For agroup of bubbles with measured diameters, the Sauter mean is

dvs =∑

nid3bi∑

nid2bi

, (16)

where ni is the number of bubbles in the class i with its volumeequivalent size dbi .

Some studies have been conducted to investigate pressure orgas density effects on the mean bubble size and bubble size dis-tribution in bubble columns (Idogawa et al., 1986, 1987; Jiang


Fig. 5. Visualization of bubbles emerging from the surface of a three-phasefluidized bed at (a) P = 0.1 MPa; (b) P = 3.5 MPa; (c) P = 6.8 MPa; (d)P = 17.4 MPa (from Fan et al., 1999).

et al., 1995; Soong et al., 1997; Lin et al., 1998) as well asin three-phase fluidized beds (Jiang et al., 1992, 1997). Fig. 5shows bubbles emerging from the surface of the three-phasefluidized bed of Paratherm NF heat transfer fluid and 2.1-mmglass beads over a wide range of operating conditions. As shownin the figure, bubble size is drastically reduced as pressure in-creases. The most fundamental reason for the bubble size re-duction can be attributed to the variation in physical propertiesof the gas and liquid with pressure. According to these exper-imental studies, pressure has a significant effect on the meanbubble diameter. The mean bubble diameter decreases with in-creasing pressure; however, above a certain pressure, the bub-ble size reduction is not significant. The effect of pressure onthe mean bubble size is due to the change of bubble size distri-bution with pressure. At atmospheric pressure, the bubble sizedistribution is broad, while under high pressure, the bubble sizedistribution becomes narrower (Luo et al., 1999). The bubblesize is mainly determined by three factors, that is, bubble for-mation at the gas distributor, bubble coalescence, and bubblebreakup. When the pressure is increased, the bubble size at thedistributor is reduced (Luo et al., 1998a), bubble coalescence issuppressed (Jiang et al., 1995), and large bubbles tend to breakup (Luo et al., 1999). The combination of these three factorscauses the decrease of mean bubble size with increasing pres-sure.

The bubble size distribution can normally be approximatedby a log-normal distribution with its upper limit at the max-imum stable bubble size. The contribution of bubbles of dif-ferent sizes can be examined by analyzing the relationship be-tween overall gas holdup and bubble size distribution. In slurry

bubble columns, the gas holdup can be related to the superfi-cial gas velocity, Ug , and the average bubble rise velocity, ub,(based on bubble volume) by a simple equation:

Ug = �gub. (17)

When the distributions of bubble size and bubble rise velocityare taken into account, ub can be expressed as

ub =∫ db,maxdb,min

Vb(db)f (db)ub(db) ddb∫ db,maxdb,min

Vb(db)f (db) ddb

. (18)

The outcome of Eq. (18) and the gas holdup strongly de-pend on the existence of large bubbles, because of their largevolume and high rise velocity. An experimental study by Leeet al. (1999) revealed that, in the coalesced bubble regime, morethan 70% of the small bubbles are entrained by the wakes oflarge bubbles and consequently have a velocity close to largebubbles. It is clear that the large bubble behavior characterizesthe overall hydrodynamics due to their large volume, their highrise velocity, and their large associated wakes.

5. Other bubble behavior

5.1. Gas jetting

One of important phenomena associated with plenum flowbehavior is gas jetting. Two main gas flow regimes can be iden-tified for gas injection from a single orifice, that is, bubblingand jetting regimes. At low gas velocities, discrete bubbles areformed from the orifice, marking the single bubbling regime. Athigh gas velocities, a coherent gas jet forms with small bubblessplit at the end of the jet, marking the jetting regime. The bub-bles formed from a jet are of a wide size distribution. Betweenthe single bubbling and jetting regimes, a transition regime ex-ists, which is known as doubling or coalesced bubbling regime(Miyahara et al., 1983). In this regime, the bubble that is stillbeing formed at the orifice coalesces with the preceding bub-ble. Bubble coalescence may occur among three or more bub-bles with increasing gas velocity. This section will focus onthe transition from bubbling to jetting regimes. The transitionvelocity from bubbling to jetting regimes is also referred to asthe onset velocity of gas jetting.

The transition from single bubbling or coalesced bubblingto jetting regimes has an important practical implication forthe design and operation of industrial reactors, since industrialreactors are often operated at high gas velocities in order toyield high throughput. Experimental work has been reportedon the transition behavior from bubbling to jetting regimes inliquids under ambient conditions. Some studies (Ozawa andMori, 1983; Chen and Richter, 1997) have indicated that asonic velocity at the exit of the orifice is essential for the onsetof jetting. However, others have noted that a critical value ofthe mass flux of gas or some dimensionless numbers have tobe reached for the onset of jetting. McNallan and King (1982)observed that the transition from bubbling to continuous jet flowis controlled by the mass flux of gas and occurs at a flow rate


of approximately 40 g/cm2 s. Leibson et al. (1956) studied themechanics of bubble formation at high gas flow rates and con-cluded that the regimes of gas injection are dictated by the gasflow conditions through the orifice. When the gas flow throughthe orifice is fully turbulent, that is, the orifice Reynolds num-ber (Reo = �gDouo/�g) is larger than 10,000, a continuousgas jet is formed. Mersmann (1980) found that the transitionfrom bubbling to jetting regimes in the air/water system occursat a critical Weber number of 2, irrespective of orifice diame-ter. The critical Weber number (Wecr) increases with the liq-uid viscosity (Rabiger and Vogelpohl, 1982). Through studyingbubble formation at high gas flow rates with a high-speed cam-era, Miyahara et al. (1983) proposed an empirical correlation topredict the transition point in the liquid using two dimension-less numbers, Weber and Bond numbers. The discrepancy ofthe different criteria used to identify the bubbling–jetting tran-sition in the literature is possibly due to different experimentaland theoretical approaches used and different interpretations ofthe concept of the gas jetting phenomenon.

The system pressure or gas density has a significant effect onthe bubbling–jetting transition. Ozawa and Mori (1986) stud-ied the effects of the physical properties of gas and liquid onthe bubbling–jetting transition in liquids. They found that thebubbling–jetting transition occurs at lower gas velocities whenthey increased the ratio of gas to liquid densities. Idogawaet al. (1987) studied the formation of gas bubbles in a bubblecolumn under system pressures of 0.1–15 MPa. The empiricalcorrelation they proposed indicated that the bubbling–jettingtransition velocity in water and ethanol systems is proportionalto �−0.8

g . Luo et al. (1998b) studied gas jetting and bubbleformation in hydrocarbon liquids and liquid–solid suspensionsat high pressures (up to 17 MPa) using an optic fiber probe.They revealed significant effects of the orifice Reynolds num-ber and the system pressure on the bubbling and jetting phe-nomena. Photographs of the gas flow through an orifice in aParatherm NF heat transfer liquid (�l =30 mPa s, �=29 mN/m,�l = 868 kg/m3 at 0.1 MPa and 25 ◦C) at a high pressurefor various Reo are shown in Fig. 6. At Reo = 1075, singlebubbles are formed from the orifice. When Reo increases to5321, bubbles being formed at the orifice start to interact withthe preceding bubble. Bubble coalescence occurs between twobubbles, sometimes involving more bubbles (Reo = 8809). AtReo = 10, 243, frequent coalescence of successive bubbles isobserved, that is, the beginning of gas jetting. As Reo fur-ther increases, the jetting regime becomes more apparent andbubbles break away from the top of the jet. Moreover, the jetpenetration depth increases with an increase in Reo. The onsetvelocity of gas jetting or the bubbling–jetting transition veloc-ity can be identified based on photographs obtained from thehigh-speed video camera and the analysis of the light intensitysignals from the optic fiber probe (Luo et al., 1998b; Sundar andTan, 1999). The transition velocity was found to decrease withan increase in pressure in both the liquid and the liquid–solidsuspension. In other words, the transition to the jetting regimeis accelerated. The acceleration of the transition to the jettingregime at high pressures is mainly due to an increase in gasmomentum.

5.2. Liquid weeping

Another interesting plenum bubble behavior is liquid weep-ing at the orifice, which can affect the distributor performancesignificantly. Liquid weeping is the downward liquid flow intothe plenum region through the orifice after the bubble detachesfrom the orifice. The liquid leaking into the plenum region cansignificantly affect the pressure drop across the distributor, andthus the performance of the distributor. Therefore, understand-ing the liquid weeping phenomenon is of critical importance tothe improvement of distributor design.

Liquid weeping at orifices is mainly due to the pressure fluc-tuations in the liquid phase above the distributor and in the gaschamber underneath the distributor. To initiate the formation ofa bubble, the pressure in the gas chamber has to be higher thanthe capillary pressure of the orifice. As the bubble grows, thepressure in the chamber decreases. When the bubble detachesfrom the orifice, the pressure in the chamber reaches a minimumvalue. The minimum pressure can be significantly lower thanthe pressure of the liquid phase above the orifice and thus, theliquid weeping occurs immediately. A typical plenum pressurefluctuation is characterized by three basic regimes that makeup the overall bubbling period. A sudden pressure drop is ob-served as the bubble forms above the orifice plate and grows,which is followed by a sudden pressure increase upon bubbledetachment. During this period, weeping may occur as a newgas–liquid interface is reestablished along the underside of theorifice rim. Bridging follows this weeping period and is char-acterized by a slow pressure buildup that leads to another bub-bling cycle. Any factors that affect the bubble formation processand pressure fluctuations in the chamber and the liquid phasewould affect the liquid weeping rate. The factors include orificegas velocity, volume of the gas chamber, geometry of the ori-fice (diameter and thickness), liquid-phase velocity, and physi-cal properties of the liquid. The liquid weeping phenomenon ismore complicated in the multi-orifice plate than in the single-orifice plate. In addition to the factors affecting liquid weepingin the single-orifice case, the liquid weeping rate also dependson the number of orifices, the percentage of plate opening, andthe arrangement of the orifices. In the following, the effects ofvarious factors on the liquid weeping rate will be discussed.

It is commonly accepted that a maximum weeping rate existswhen the orifice gas velocity is increased from zero to the jet-ting velocity (McCann and Prince, 1969; Miyahara et al., 1984;Akagi et al., 1987; Che and Chen, 1990). The weeping rate inthe jetting regime is believed to be negligible. Liquid weepingcan be minimized by adjusting the volume of the gas cham-ber, because the liquid weeping rate decreases as the chambervolume increases, especially for small orifices. The effect ofchamber volume on liquid weeping is related to the pressurefluctuation in the chamber. As the chamber volume increases,the frequency and the magnitude of the pressure fluctuation inthe chamber decrease, resulting in a smaller weeping rate. Thegeometry of the orifice also affects the liquid weeping rate.In general, the liquid weeping flux (weeping rate per unit ori-fice area) increases significantly with an increase in the orificediameter at a constant orifice gas velocity when the orifice


Fig. 6. A series of photographs showing the bubbling–jetting transition at P =4.24 MPa and T =28 ◦C for (a) uo =0.27 m/s and Reo =1075; (b) uo =1.35 m/sand Reo = 5321; (c) uo = 2.23 m/s and Reo = 8809; (d) uo = 2.60 m/s and Reo = 10, 243; (e) uo = 3.99 m/s and Reo = 15, 759; (f) uo = 6.42 m/s andReo = 25, 355 (from Luo et al., 1998b).

diameter is larger than 2 mm. Sometimes, simultaneous liquidweeping and bubbling were observed for large orifices and thevolume of a bubble formed at an orifice with weeping is largerthan that without weeping (Miyahara et al., 1984; Miyaharaand Takahashi, 1984). Liquid weeping is not remarkable fororifices smaller than 2 mm if the liquid is quiescent and thechamber volume is large, because the surface tension force isdominant for small orifices (Klug and Vogelpohl, 1983; Akagiet al., 1987; Che and Chen, 1990).

Liquid weeping is also affected by a superimposed motionof liquid around the orifice. Klug and Vogelpohl (1983) foundthat the velocity of the liquid, flowing cocurrently with thegas, influences the liquid weeping. The weep point, defined asthe condition above which liquid weeping is negligible, movesto higher gas flow rates with liquid motion. Among variousliquid properties, liquid viscosity is the main factor that possiblyaffects flow resistance across the orifice, and thus the weepingrate. Akagi et al. (1987) found that the effect of viscosity onthe weeping rate is insignificant at low gas velocities, and theweeping rate is reduced with increasing liquid viscosity at highgas velocities. In their study, the liquid viscosity varied from0.9 to 17 mPa s.

The operating pressure also has a significant effect on the liq-uid weeping phenomenon (Peng et al., 2002). Higher pressureor gas density contributes to the increased momentum forceand the smaller bubble size. Lower pressure fluctuations acrossthe orifice plate are also observed. This lower pressure dropacross the plate results from smaller bubbles invoking smallerwakes upon bubble detachment and decreases mixing or tur-bulent effects around the orifice. Therefore, the weeping rateis reduced at high pressures. A maximum weeping rate is ob-served at low pressures as the orifice gas velocity increases.The weeping maximum also shifts toward lower gas veloci-ties as the pressure increases. At high pressures, the weepingmaximum disappears. The existence of the maximum weepingrate may be qualitatively explained based on the variations ofbubble growth rate and gas momentum force with orifice gasvelocity (Peng et al., 2002).

5.3. Scaling rule

The scaling rules based on hydrodynamic similarity are use-ful for the purposes of experimental design and reactor designand scale up. For the commercial size of a reactor at 10 m in di-ameter for the Fischer–Tropsch synthesis (Sookai et al., 2001),a scaling rule would undoubtedly be helpful. There are severalhydrodynamic scaling rules reported that have limited usage(Fan et al., 1999; Safoniuk et al., 1999; Krishna et al., 2001;Krishna and van Baten, 2001, 2003; van Baten and Krishna,2004; Zhang and Zhao, 2006). Fan et al. (1999) studied thehydrodynamics and transport phenomena in the high-pressurebubble columns and slurry bubble columns. They proposed thehydrodynamic similarity based on the gas holdup matching andarrived at the following dimensionless groups, Ug/umax, Mom,and �g/�m. Safoniuk et al. (1999) arrived at the following di-mensionless groups, Mo, Eo, Re, �g/�L, and Ug/UL by con-sidering parameters affecting the equilibrium bubble size andshape. The scale-up of effects were also elucidated by the useof computational fluid dynamics (Krishna et al., 2001; Krishnaand van Baten, 2001, 2003; van Baten and Krishna, 2004). Asthe scale up is a complex and important issue and all the ap-proaches indicated above require further work.

6. Computational fluid dynamics

The CFD is a viable means for describing the fluid dy-namic and transport behavior of gas–liquid or gas–liquid–solidflow systems. There are three basic approaches commonlyemployed in the CFD for the study of multiphase flows: theEulerian–Eulerian (E–E) method, the Eulerian–Lagrangian(E–L) method, and direct numerical simulation (DNS) method.In the E–E method (Anderson and Jackson, 1967; Joseph andLundgren, 1990; Sokolichin and Eigenberger, 1994, 1999;Zhang and Prosperetti, 1994, 2003; Mudde and Simonin,1999; Matonis et al., 2002), both the continuous phase and thedispersed phase, such as particles and bubbles, are treated as


interpenetrating continuous media, occupying the same spacewith different velocities and volume fractions for each phase.In this method, the closure relationships such as the stress andviscosity of the particle phase need to be formulated. In theE–L method, or discrete particle method (DPM) (e.g., Tsujiet al., 1993; Lapin and Lübbert, 1994; Hoomans et al., 1996;Delnoij et al., 1997), the continuous fluid phase is formulatedin the Eulerian mode, while the position and the velocity ofthe dispersed phase, particles or bubbles, is traced in the La-grangian mode by solving Lagrangian motion equations. Thegrid size used in the computation for the continuous phaseequations is typically much larger than the object size of thedispersed phase, and the object in the dispersed phase is treatedas a point source in the computational cell. With this method,the coupling of the continuous phase and the dispersion phasecan be made using the particle-source-in-cell method (Croweet al., 1977). The closure relationship for the interaction forcesbetween phases is required to be provided in the E–L method.

In the DNS (Unverdi and Tryggvason, 1992; Feng et al.,1994; Sethian and Smereka, 2003), the grid size is commonlymuch smaller than the object size of the dispersed phase, andthe moving interface can be represented by implicit or explicitschemes in the computational domain. The velocity fields ofthe fluid phase are obtained by solving the Navier–Stokes equa-tion, considering the interfacial forces such as surface tensionforce or solid–fluid interaction force. The motion of the ob-ject of the dispersed phase is represented in terms of a time-dependent initial-value problem. With the rapid advances in thespeed and memory capacity of the computer, the DNS approachhas became important in characterizing details of the complexmultiphase flow field.

In the following, the front capturing and front tracking meth-ods that highlight the DNS are described. It is followed byan account of application examples using the level-set method(Sussman et al., 1994) for the 3D DNS of gas–liquid–solid flu-idization.

6.1. Front capturing and front tracking methods

In the DNS of multiphase flow problems, there are variousmethods available for predicting interface position and move-ment, such as the moving-grid method, the grid-free method(Scardovelli and Zeleski, 1999) and the fixed-grid front track-ing/front capturing method. In the moving-grid method, whichis also known as the discontinuous-interface method, the in-terface is a boundary between two sub-domains of the grid(Dandy and Leal, 1989). The grid may be structured or unstruc-tured and even near-orthogonal, moving with the interface (Hirtet al., 1974). It treats the system as two distinct flows separatedby a surface. When the interface moves or undergoes deforma-tion, new, geometrically adapted grids need to be generated orre-meshed. The re-meshing can be a very complicated, time-consuming process, especially when it involves a significanttopology change, and/or a 3D flow. Methods in which gridsare not required include the marker particle method (Harlowand Welch, 1965) and the smoothed particle hydrodynamicsmethod (Monaghan, 1994).

The fixed-grid method, which is also known as thecontinuous-interface method, employs structured or unstruc-tured grids with the interface cutting across the fixed grids. Ittreats the system as a single flow with the density and viscos-ity varying smoothly across a finite-thickness of the interface.The numerical techniques used to solve the moving interfaceproblem with fixed, regular grids can be categorized by twobasic approaches: the front tracking method (e.g., Harlow andWelch, 1965; Peskin, 1977; Unverdi and Tryggvason, 1992;Fukai et al., 1995) and the front capturing method (e.g., Osherand Sethian, 1988; Sussman et al., 1994; Kothe and Rider,1995; Bussmann et al., 1999). For a 3D multiphase flow prob-lem, the fixed-grid method is the most frequently used due toits efficiency and relative ease in programming.

The front tracking method explicitly tracks the locationof the interface by the advection of the Lagrangian markerson a fixed, regular grid. The marker-and-cell (MAC) methoddeveloped by Harlow and Welch (1965) was the first fronttracking technique applied in DNS. The front tracking methoddeveloped by Unverdi and Tryggvason (1992) and Tryggvasonet al. (2001) leads to many applications in the simulation ofdroplet or bubble flow. In this method, the location of theinterface is expressed by discrete surface-marker particles.High-order interpolation polynomials are employed to ensurea high degree of accuracy in the representation of the interface.An unstructured surface grid connecting the surface-markerparticles is introduced within a volumetric grid to track the bub-ble front within the computational domain. Thus, discretizationof the field equations is carried out on two sets of embeddedmeshes: (a) the Eulerian fluid grid, which is 3D, cubical, stag-gered structured, and non-adaptive; and (b) the Lagrangianfront grid, which is 2D, triangular, unstructured, and adaptive(Unverdi and Tryggvason, 1992). The infinitely thin boundarycan be approximated by a smooth distribution function of a fi-nite thickness of about three to four grid spacing. The variabledensity Navier–Stokes equations can then be solved by conven-tional Eulerian techniques (Unverdi and Tryggvason, 1992).This method can be numerically stiff as the density ratio of thetwo fluids increases, and may pose difficulties when the appear-ance, the connection, the detachment, and the disappearance ofthe gas–liquid interface are encountered. Such interface behav-ior occurs in the coalescence, breakup or formation of bubblesand droplets in an unsteady flow. The front tracking methodis therefore computationally intensive. Agresar et al. (1998)extended the front tracking method with adaptive refined gridsnear the interface to simulate the deformable circulation cell.Sato and Richardson (1994) developed a finite-element methodto simulate the moving free surface of a polymeric liquid. Theimmersed boundary method (IBM) proposed by Peskin (1977)in studying the blood flow through heart valves and the cardiacmechanics also belongs to the class of front tracking tech-niques. In the IBM method, the simulation of the fluid flowwith complex geometry was carried out using a Cartesian grid,and a novel procedure was formulated to impose the boundarycondition at the interface. Some variants and modifications ofthis method were proposed in simulating various multiphaseflow problems (Mittal and Iaccarino, 2005). An example of the


Fig. 7. Simulation results of air bubble formation from a single nozzle in water (nozzle size 0.4 cm I.D. and nozzle gas velocity 10 cm/s) (from Ge and Fan, 2006).

front tracking method as applied to three-phase fluidization ispresented by van Sint Annaland et al. (2005).

The front capturing method, on the other hand, is the Eu-lerian treatment of the interface, in which the moving inter-face is implicitly represented by a scalar–indicator functiondefined on a fixed, regular mesh point. The movement of theinterface is captured by solving the advection equation of thescalar–indicator function. At every time step, the interface isgenerated by piecewise segments (2D) or patches (3D) recon-structed by this scalar function. In this method, the interfacialforce, such as the surface tension force, is incorporated into theflow momentum equation as a source term using the contin-uum surface force (CSF) method (Brackbill et al., 1992). Thistechnique includes the volume of fluid (VOF) method (Hirtand Nichols, 1981; Kothe and Rider, 1995), the marker densityfunction (MDF) (Kanai and Miyata, 1995), and the level-setmethod (Osher and Sethian, 1988; Sussman et al., 1994).

In the VOF method, an indicator function is defined as: 0 fora cell with pure gas, 1 for a cell with pure liquid, and 0 to 1 for acell with a mixture of gas and liquid. An interface exists in thosecells that give a VOF value of neither 0 nor 1. Since the indica-tor function is not explicitly associated with a particular frontgrid, an algorithm is needed to reconstruct the interface. Thisis not an easy task, especially for a complex dynamic interface

requiring 3D calculation. The VOF method was used to sim-ulate the liquid droplet collision with solid surface (e.g., Karlet al., 1996; Bussmann et al., 1999, 2000; Harvie and Fletcher,2001a,b). Mehdi-Nejad et al. (2003) also used the VOF methodto simulate the bubble entrapment behavior in a droplet whenit impacts a solid surface. The VOF was also used in the 2Ddiscrete phase simulation of bubble motion in gas–liquid andgas–liquid–solid flows (Li et al., 1999, 2001; Zhang et al., 2000;Dijkhuizen et al., 2005). In the approach of Li et al. (1999), theEulerian volume-averaged method, the Lagrangian DPM, andthe VOF volume-tracking method were employed to describethe motion of liquid, solid particles, and gas bubbles, respec-tively. A bubble induced force (BIF) model, a CSF model, andNewton’s third law are applied to account for the couplingsof particle–bubble, bubble–liquid, and particle–liquid interac-tions, respectively. A close distance interaction (CDI) model isincluded in the particle–particle collision analysis, which con-siders the liquid interstitial effect between colliding particles(Zhang et al., 1999). Other front capturing methods include theconstrained interpolation profile (CIP) method (Yabe, 1997),and the phase-field method (Jamet et al., 2001).

In the level-set method, the moving interface is implicitly rep-resented by a smooth level-set function (Sethian and Smereka,2003). The level-set method has proved capable of handling


Fig. 8. Simulation results of bubble formation and rising in Paratherm NF heat transfer fluid with and without particles (nozzle size 0.4 cm I.D., liquid velocity0 cm/s, gas velocity 10 cm/s, and particle density 0.896 g/cm3): (a) no particle; (b) 2000 particles; (c) 8000 particles; (d) 8000 particles (from Ge and Fan, 2006).


problems in which the interface moving speed is sensitive to thefront curvature and normal direction. A significant advantageof the level-set method is that it is effective in 3D simulation ofthe conditions with large topological changes, such as bubblebreaking and merging, droplet-surface collisions with evapora-tion. While similar to the VOF method, the level-set methodalso uses an indicator function to track the gas–liquid interfaceon the Eulerian grid. Instead of using the marker particles orpoints to describe the interface, a smooth level-set function isdefined in the flow field (Sussman et al., 1994). An example ofthe front capturing method as applied to three-phase fluidiza-tion is presented by Chen and Fan (2004).

6.2. Simulation examples

The flows in a gas–liquid or gas–liquid–solid fluidized bedare represented by the bubble dispersed and bubble coalescedregimes. In the bubble dispersed regime, there is little variationof bubble sizes. The flow structure in the bubble coalescedregime is complex due to substantial coalescence and breakupof bubbles. Both the E–E and the E–L methods have provento be more effective in modeling the bubble dispersed regimethan the bubble coalesced regime of gas–liquid flow. In thesimulation of the bubble coalesced regime of gas–liquid flowsusing either the E–E or the E–L method, the challenge lies inthe establishment of an accurate closure relationship for theinterphase momentum exchange. The interphase momentumexchange is induced through the drag force that liquid exertson the bubble surface, the virtual mass force due to the bubbleand liquid inertial motion, and the lift force caused by theshear flows around the bubbles. In the following simulationexamples, deformable bubble behavior in the gas–liquid bub-ble columns and gas–liquid–solid fluidized systems are illus-trated based on the level-set method. Newton’s second law andthe locally averaged Navier–Stokes equations (Anderson andJackson, 1967; Jackson, 2000) are used to describe the particlemotion and the liquid phase flow, respectively. The detailedmodel formulation is given in Chen and Fan (2004). In the fol-lowing, the simulation results for bubble formation from a sin-gle nozzle in liquids and liquid–solid suspensions are presented(Ge and Fan, 2006).

6.2.1. Bubble formation in liquidFig. 7 shows the simulated air bubble formation and rising

behavior in water. For the first three bubbles, the formationprocess is characterized by three distinct stages of expansion,detachment and deformation. In comparison with the bubbleformation in the air–hydrocarbon fluid (Paratherm) system, thecoalescence of the first two bubbles occurs much earlier in theair–water system. This is due to the fact that, compared to thatin the air–Paratherm system, the first bubble in the air–watersystem is much larger in size and hence higher in rise velocityleading to a shorter time for its coalescence with the secondbubble. Beginning with the third bubble, the formation and ris-ing behavior of air bubbles in water shows strongly asymmetricbehavior. As is evident from the figure, the bubble rises in aspiral path or a zigzag path.

6.2.2. Bubble formation in liquid–solid suspensionThe air–Paratherm–solid fluidized bed system with solid par-

ticles of 0.08 cm in diameter and 0.896 g/cm3 in density is sim-ulated. The solid particle density is very close to the liquid den-sity (0.868 g/cm3). The boundary condition for the gas phaseis inflow and outflow for the bottom and the top walls, respec-tively. Particles are initially distributed in the liquid medium inwhich no flows for the liquid and particles are allowed throughthe bottom and top walls. Free slip boundary conditions areimposed on the four side walls. Specific simulation conditionsfor the particles are given as follows: case (b) 2000 particlesrandomly placed in a 4 × 4 × 8 cm3 column; case (c) 8000particles randomly placed in a 4 × 4 × 8 cm3 column; andcase (d) 8000 particles randomly placed in the lower half ofthe 4 × 4 × 8 cm3 column. The solids volume fractions are0.42%, 1.68% and 3.35%, respectively, for cases (b), (c) and (d).Case (a) is free of particles.

The bubble formation process at different solids concentra-tions is shown in Figs. 8(b)–(d) and is compared with thatwithout particles as shown in Fig. 8(a). For the first 0.3 s, lit-tle change is observed in the bubble formation process for thethree solids concentrations used in this simulation. After 0.4 s,however, significant changes can be found for the cases withhigh solids concentrations. This can be seen from the first bub-ble in each case. When the solid concentration is low or nosolids are present, the first bubble grows on the orifice and con-nects to the second bubble. For the high solids concentrationcases, the first bubble is not well connected to the second bub-ble. This is particularly true for case (d) when the bubble risesinto the solids-free region or freeboard region of the bed. Thesolid particle entrainment is clearly observed in case (d).

7. Summary

Bubble dynamics in gas–liquid–solid three-phase fluidizedbed systems have been studied extensively over the past years.Considerable recent research in this field has been focusing onthe effects of pressures and temperatures due to the renewedinterest in the Fischer–Tropsch synthesis in the commercial de-velopment of large-scale slurry reactors for gas-to-liquids tech-nology. Further, the industrial three-phase fluidization systemsare commonly operated under elevated pressure and tempera-ture conditions. The temperature and pressure play the key rolein dictating the physical properties of the gas and liquid phasesand hence the bubble behavior. The high pressure decreasesthe bubble size, delays the flow regime transition from the dis-persed to the coalesced regime, and reduces liquid weepingfrom the distributor. The liquid–solid suspension through whichgas bubbles rise can be characterized as a pseudo-homogenousmedium under some operating conditions. Most distinct phe-nomena associated with the bubbly flow include bubble insta-bility, bubble wake enhanced transport properties and coherentlarge-scale vortical structure.

The results from the computational fluid dynamics providevaluable insights into the complex flow field of three-phasefluidization systems. The direct numerical simulation has beenattempted in the literature. Numerical examples based on the


level-set interface tracking approach are presented, which ac-count for the bubble, particle and liquid interactive behaviorand the flow structure in three-phase flows. Challenges on CFDremain, however, with respect to improving computational ac-curacy, as well as incorporating reaction kinetics and transportphenomena properties into fluid dynamics calculations, and themicro-scale flow properties into simulation of a large scalethree-phase reactor. The formulation of closure relationshipsfor the Eulerian computation is also in need of further work.

Notation

b bubble breadthc parameter in Fan–Tsuchiya equation reflect-

ing surface tension effectCD drag coefficientdb volume equivalent bubble diameterd ′b dimensionless bubble diameter

dp particle diameterdvs Sauter mean bubble diameterDmax maximum stable bubble sizeDo orifice diametere restitution coefficientE(

√1 − 2) complete second kind Elliptic integral

Eo Eötvös number based on bubble diameterf (db) probability density function of bubble sizeFB effective buoyancy force; buoyancy forceFBA Basset forceFC particle–bubble collision forceFD liquid drag forceFI,g bubble inertial forceFI,m liquid–solid suspension inertial forceFM gas momentum forceF� surface tension forceg gravitational accelerationh bubble height; particle penetration depthHd height of a doughnut-shape bubbleK proportionality constant for calculating the

effective viscosity of liquid–solid suspen-sions

Kb parameter in Fan–Tsuchiya equation reflect-ing viscous nature of surrounding medium

l thickness of the liquid film between two co-alescing bubbles

Mo Morton number based on liquid propertiesn parameter in Fan–Tsuchiya equation reflect-

ing system purityni number of bubblesNc dimensionless capacitance numberP system pressureQg volumetric gas flow rate into the gas chamberQ0 volumetric gas flow rate through the orificero radius of orificeRd radius of acontacting circle between two bub-

blesRe bubble Reynolds number based on liquid

properties

Reo orifice Reynolds numbert timeT temperatureTa Tadaki numberu rise velocity of bubble baseub bubble rise velocity relative to the liquid phaseu′

b dimensionless bubble rise velocity; observed ab-solute bubble rise velocity

ue bubble expansion velocityum suspension velocityumax rise velocity of maximum stable bubbleuo superficial gas velocity through the orificeut particle terminal velocity in liquidub average bubble rise velocityUg superficial gas velocityUl superficial liquid velocityUp0 initial descending velocity of a particleVb bubble volumeVc volume of gas chamberWecr critical Weber number

Greek letters

aspect ratio of bubble� contact angle between bubble and orifice sur-

face; heat capacity ratio�� density difference between liquid and gas phases�g gas holdup�l liquid holdup�s solids holdup�sc critical solids holdup�s0 solids holdup at incipient fluidization coefficient of suspension inertial force�c critical wavelength for bubble breakup�g gas viscosity�l liquid viscosity�m effective viscosity of liquid–solid suspension�g gas density�l liquid density�m density of liquid–solid suspension�s solids density� surface tension� dimensionless form of time� particle sphericity; parameter reflecting the sur-

face drag in the equation calculating film thin-ning velocity

Acknowledgment

The work was supported by the National Science FoundationGrant CTS-0207068.

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bubble formation and dynamics in gas ou liquid ou solid fluidization

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