Chemical Engineering Science 58 (2003) 4159–4172www.elsevier.com/locate/ces

Axial dispersion of particles in a slugging column—the role of thelaminar wake of the bubbles

O. N. Cardoso, T. Sotto Mayor, A. M. F. R. Pinto, J. B. L. M. Campos∗

Centro de Estudos de Fen�omenos de Transporte, Departamento de Engenharia Qu��mica, Faculdade de Engenharia da Universidade do Porto,Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

Received 2 December 2002; received in revised form 18 June 2003; accepted 27 June 2003

Abstract

Axial solid dispersion promoted by Taylor bubbles in a batch liquid column was studied. A mechanistic model was developed topredict the axial solid dispersion. The model is based on the upward transport of particles inside closed wakes of non-interacting Taylorbubbles. The model predictions are compared with experimental data. The experimental data were obtained in a test tube of 32 mminternal diameter. The particle volumetric distribution was measured by several di:erential pressure transducers placed along the column.Two classes of glass beads, mean diameter 180 and 280 �m, were suspended in aqueous glycerol solutions, with glycerol percentageranging from 40% (v/v) to 100% (v/v). The amount of particles in the column was such that the volumetric particle fractions were0.1, 0.2 and 0.3, supposing homogeneous liquid–solid suspension. The air <ow rate ranged from 90 × 10−6 to 250 × 10−6 m3=s atPTN conditions. The obtained experimental data are in good agreement with the model predictions for laminar wakes, i.e., closed wakeswith internal recirculation and without vortex shedding. The experimental data show a higher upward particle transport for wakes in thetransition laminar-turbulent regime; closed wakes with internal recirculation and vortex shedding. The upward particle transport is higherfor increasing air <ow rate, decreasing particle diameter and increasing amount of particles in the column.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Fluid mechanics; Multiphase <ow; Slurries; Suspension; Slug <ow; Laminar wakes

1. Introduction

Three-phase <ows are widely used in several industrialcatalytic processes, and more recently in biological wastew-ater treatment and bacterial leaching processes. The bubblephase promotes the suspension and dispersion of the solidsin the liquid phase. In suspension, the activity of the solidsincreases, either acting as a catalyst or undergoing a chemi-cal reaction. Numerous investigations about solid phase dis-persion in three phase <ows have been carried out, and along list of works is supplied by Abraham, Khare, Sawant,and Joshi (1992). They also describe the measurement tech-niques used and the correlations proposed to quantify theaxial solid phase dispersion. According to these correlations,the relevant parameters to quantify the axial solid dispersionare: liquid properties such as density, viscosity and surfacetension; particle properties such as density and size; and gasproperties such as bubble velocity and bubble frequency.

∗ Corresponding author. Tel.: +351-225081692; fax: +351-225081649.E-mail address: jmc@fe.up.pt (J. B. L. M. Campos).

The so-called axial dispersion–sedimentation model hasbeen widely used to quantify the axial solid dispersion inthree-phase batch columns. The model contains two basicparameters: a particle settling velocity, vP , and a particleaxial dispersion coeEcient, DP . The formulation is phe-nomenological and is based on the concept of a particledispersion <ux, described by a di:usion-type equation, to-gether with a particle <ux due to the gravitational settling.Jean, Tang, and Fan (1989) summarized the di:erent ver-sions of the dispersion–sedimentation model, as well as thelarge number of empirical correlations proposed in the lit-erature to quantify vP and DP .

Mechanistic models describing the solid dispersion inbubbling columns have also been developed. In thesemodels, the transport in the bubble wake is the mech-anism responsible for the upward motion of the parti-cles. Dayan and Zalmanovich (1982) proposed a fullymechanistic model to quantify the axial solid dispersion.The model was developed for spherical cap bubbles andthe wakes of the bubbles were taken as spherical andclosed.

0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0009-2509(03)00305-1

4160 O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172

Coppus, Rietema, and Ottengraf (1977), Komasawa,Otake, and Kamojima (1980), Bhaga and Weber (1981),Campos and Guedes de Carvalho (1988) and Fan andTsuchiya (1990) published works about bubble wake dy-namics, most of them concerning spherical cap bubbles.

Slug <ow is one of the several gas–liquid <ow regimesoccurring inside pipes over a wide range of gas and liquid<ow rates. In a vertical pipe, this <ow pattern is character-ized by long bullet-shaped bubbles, called Taylor bubblesor gas slugs, rising and almost Jlling the pipe cross section.Campos and Guedes de Carvalho (1988) investigated thewake <ow patterns of gas slugs rising through stagnant liq-uids. They identify three <ow patterns: a closed and axisym-metric wake with internal recirculatory <ow (laminar wakeregime); a closed and unaxisymmetric wake with recircula-tory <ow and wake shedding (transition wake regime), andan open and perfectly mixed wake (turbulent wake regime).Pinto, Coelho Pinheiro, and Campos (1998) studied thewake <ow patterns in a co-current gas–liquid slugging col-umn and observed the same regimes.

In the present work, a simpliJed mechanistic model isdeveloped to predict the particle axial dispersion in a verti-cal slugging column. The particles are transported upwardinside closed wakes and, during the bubble path, some par-ticles are expelled due to gravitational and inertial e:ects.The upward wake displacement induces a downward <owof the suspension. This downward <ow and the particle set-tling velocity are responsible for the downward transport ofparticles. The model predictions are compared with experi-mental data.

2. Experimental work

2.1. Basis of the experimental method

The experimental method used to measure the axial parti-cle distribution is based on the response of low-range di:er-ential pressure transducers (±8 cm H2O) placed along thebubbling tube.

In the experimental set-up, the plastic tubes connectingthe transducers to the pressure taps, as well as the pres-sure cavities, were Jlled with the liquid in the bubblingtube. The di:erence in piezometric pressure is then given by(Fig. 1)

LP = P+ − P− = (P2 − �Lgh2) − (P1 + �Lgh1)

= (P2 − P1) − �Lgd1–2; (1)

where P is the actual pressure, h1 and h2 are the verti-cal distances from a reference datum level, d1–2 is the dis-tance between pressure taps 1 and 2 and �L is the liquiddensity.

When the test tube is Jlled between taps 1 and 2 withliquid–solid suspension, the pressure di:erence, P2 − P1,

∆P = P+- P-

P -P +

2

h2

h1

1

P2

P1

d1-2

Fig. 1. Schematic representation of a transducer and connecting tubes.

is given by

P2 − P1 = �PgSd1–2 + �Lg(1 − S)d1–2

= [�PS + �L(1 − S)]gd1–2; (2)

where �P is the particle density and S is the volumetricparticle fraction.

The di:erence in piezometric pressure is obtained by com-bining Eqs. (1) and (2) and depends on S :

LP = (�P − �L)Sgd1–2: (3)

For the low-range di:erential pressure transducers used(±8 cm H2O), the uncertainty associated to the LP valuesis lower than 1 mm H2O. According to Eq. (3) and theexperimental conditions, the uncertainty associated to theS values is lower than ±0:01.

2.2. Experimental set-up

The experimental facility is schematically shown in Fig. 2.Experiments were performed in an acrylic tube with 32 mminternal diameter and 7:5 m height. Five di:erential pressuretransducers (Validyne P305D) were used and their locationalong the tube is shown in the Jgure. The distance betweenpressure taps, d1–2, was 80 mm.

The signals from the pressure transducers were ac-quired at a frequency of 200 Hz by a computer with ananalogical-digital board and recorded for later processing.Two sizes of glass beads, 125–200 and 225–325 �m, wereused. The particle size distribution was measured by a lasergranulometer, and the mean diameter of each class was,respectively, 180 and 280 �m.

Experiments were carried out with aqueous glycerol so-lutions, from 40% (v/v) to 100% (v/v) in glycerol, and theliquid temperature was continuously measured by two ther-mocouples placed at the top and the bottom of the tube. Thetemperature di:erence between the top and the bottom ofthe column was always less than 0:5◦C. The liquid viscositywas measured with a rotating BrookJeld viscometer.

O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172 4161

Data AcquisitionSystem

0.08 m

+T1

-

+T2

-

+T3

-

+T4

-

+T5

-2.75 m

2.35 m

1.62 m

0.83 m

0.46 m

Fig. 2. Experimental set-up.

The amount of particles supplied was pre-determined sup-posing homogeneous solid–liquid suspension; the homoge-neous volumetric liquid–solid fractions, 0

S , were 0.1, 0.2and 0.3.

The air was axially injected at the bottom of the columnthrough an oriJce with 5 mm of internal diameter. The gas<ow rate was measured by calibrated rotameters and at PTNconditions ranging from 90 × 10−6 to 250 × 10−6 m3=s.

2.3. Experimental procedure

The liquid was poured into the column and some prelim-inary data acquisitions were done to conJrm the zero of thepressure transducers. After these tests, the desired amountof particles was fed into the column. The height of liquidand particles was always around 3:0 m.

The bubbling air was turned on, sometimes, before thesettling of the particles at the bottom of the tube, other timesafter the settling of all the particles, depending on the typeof experiment desired. In the second type of experimentthe settling time of 90% of the particles, initially dispersedin homogeneous suspensions with the following character-istics, 280 �m, 0

S = 0:2 and 70% (v/v) in glycerol, and280 �m, 0

S = 0:3 and 80% (v/v) in glycerol, was about 40and 170 min, respectively.

After a pre-deJned bubbling time, the air was turned o:and the volumetric particle fraction in the unareated suspen-sion was instantaneously measured in Jve locations alongthe tube. The values measured were the mean values be-tween pressure taps (8:0 cm). After a short reading time,

about 3 min, the bubbling air was turned on again andthe bubbling time increased before new data acquisition.This procedure was repeated until identical distribution wasobtained in three consecutive acquisitions. Following thisprocedure, the steady state was attained. In the most severecondition, high liquid viscosity, low gas-<ow rate, high par-ticle diameter and high particle volumetric fraction, the bub-bling time needed to reach steady state was less than 1 h.

3. Flow patterns in a slugging three-phase column

3.1. Physical properties

In the literature, there are several studies concerningphysical properties of liquid–solid suspensions. Accordingto Barnes, Hutton, and Walters (1989), suspensions withvolumetric particle fractions less than 0.3 behave like aNewtonian <uid. At very low and very high shear rates, thesuspension viscosity is essentially the same.

The e:ective Newtonian viscosity of a liquid–solid sus-pension, S , can be evaluated by Ball and Richmond’s(1980) equation:

S = L(1 − kS)−5=2k ; (4)

where L is the dynamic liquid viscosity.This equation assumes that the e:ect of all the particles

in the suspension is equal to the sum of the e:ects of theparticles added in sequence. The parameter k accounts forthe so-called “crowding” e:ect and 1=k is identiJed withthe maximum solid fraction. For a random closed packingof monodisperse spheres, the value of k is 0.637.

The e:ective density of the liquid–solid suspension, �S ,can be evaluated by

�S = �L(1 − S) + S�P: (5)

3.2. Slug bubble velocity

If a liquid–solid suspension behaves like a Newtonian liq-uid, the velocity, UB, of a slug bubble rising in a suspensioncan be evaluated by the following equation:

UB = CUG + UB∞ ; (6)

where UB∞ is the velocity of an individual slug bubble, UG

the superJcial gas velocity and C an empirical coeEcient,which depends on the <ow regime in front of the bubble“nose”.

For gas–liquid slug <ow, Nicklin, Wilkes, and Davidson(1962) and Collins, De Moraes, Davidson, and Harrison(1978) suggested the value of C = 1:2 for turbulent regimeand Collins et al. (1978) the value of C = 2:0 for lam-inar regime. The value of the parameter C can be eval-uated measuring the expansion of an aerated suspension.

4162 O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172

Reδ

0.00

0.05

0.10

0.15

0.20

0.25

0 20 40 60 80 100

laminar regimetransition regime

eq.8 (� = 1)

UB∞

(ms-1

)

Fig. 3. Experimental data for the velocity of individual bubbles risingin liquid–solid suspension versus Re�. The non-dashed symbols refer to0S = 0:2, right and left dashed symbols to 0

S = 0:1 and 0.3, respectively.The open symbols refer to particle size 180 �m and the full symbols to280 �m. The circles refer to 60% (v/v) in glycerol, the squares to 70%,the lozenges to 80% and the triangles to 90%.

According to Matsen, Hovmand, and Davidson (1969) andGrace, Krochmalnek, Clift, and Farkas (1971), the expan-sion of liquids in slugging columns is given by the followingexpression:

Hmax − H0

H0=

UG

V=

UG

(C − 1)UG + UB∞

=x

(C − 1) x + 1; (7)

where Hmax is the maximum liquid height in the aeratedcolumn, H0 the liquid height before slugging, V the relativevelocity between the slug and the liquid <owing ahead theslug nose and x = UG=UB∞ .

This equation can be applied to a three-phase column ifthe velocity of an individual slug bubble and the maximumsuspension height are measured. The velocity of individ-ual slug bubbles was measured by the pressure transducersplaced along the test tube. This technique and the respectivedata processing are described by Pinto and Campos (1996).Fig. 3 shows the results obtained for a large range of exper-imental conditions: particles with 180 and 280 �m of diam-eter, aqueous glycerol solutions ranging from 70% to 90%in glycerol and 0

S =0:2 and 0.3 . The experiments were per-formed injecting one single bubble just after the addition ofan homogeneous suspension to the column (prepared out-side of the column). The Reynolds number represented inthe Jgure is given by Eq. (10) with UG = 0. According tothe Jgure, UB∞ in the suspensions is similar to UB∞ in aliquid, which, according to White and Beardmore (1962), isgiven by

UB∞ = 0:35�√

gD; (8)

where D is the internal tube diameter and � = 1.The technique used to measure the maximum suspension

height is described by Coelho Pinheiro, Pinto, and Campos

UB∞

UG

C = 2.0

C = 1.2

(Hm

ax-H

0)

H0

0.8

0.6

0.4

0.2

0.00.0 0.5 1.0 1.5 2.0

Fig. 4. Expansion of a three-phase bubble column.

(2000). In Fig. 4, Eq. (7) and experimental data of (Hmax −H0)=H0 versus x are represented. In most of the experimentsperformed, the value of the constant C is around 2.0, thevalue suggested by Collins et al. (1978) for gas–liquid slug<ow in laminar regime.

3.3. Wake ;ow regime

Campos and Guedes de Carvalho (1988) studied the wake<ow regime for individual long slug bubbles rising in a stag-nant liquid. Pinto et al. (1998) extended the work to individ-ual slug bubbles rising in a co-current <owing liquid. Bothworks concluded that the <ow regime in the wake bubbledepends on the Reynolds number of the liquid Jlm, Re�, atthe bottom of the bubble, i.e., before the liquid expansion:

Re� =v���

=(UB − UG)D

4�× 1

1 − �=D; (9)

where � is the kinematic viscosity, v� the mean velocity inthe stabilized liquid Jlm relatively to a referential movingwith the bubble and � the thickness of the stabilized Jlm.

Most of the times, the liquid Jlm is very thin, �=D¡¡ 1,and the previous equation can be simpliJed as

Re� =v���

=(UB − UG)D

4�: (10)

According to Pinto et al. (1998), the <ow wake regimeis laminar, with liquid in recirculation in a toroidal vortexwhen Re� ¡ 50, is transitional, with vortex shedding when50¡Re� ¡ 90 and is turbulent when Re� ¿ 90.

Visualization experiments, with particle suspensions,were performed to characterize the bubble wake regime in athree-phase <ow. A suspension of inert particles (liquid andparticles with the same density) colored by a strong red dyewas placed in the lower part of an acrylic tube of 32 mm ofinternal diameter. At the top of this suspension, there was aball valve. With this ball valve closed, the upper section ofthe tube was Jlled with the same suspension but colorless.After the opening of the valve, individual slug bubbles wereinjected at the bottom of the tube, and their images rising

O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172 4163

Fig. 5. (a, b) Images of bubble wakes rising in inert suspensions; (a)laminar wake, Re� = 40; (b) transition wake, Re� = 53.

in the colorless suspension recorded by a fast CCD camera.For laminar and transition wakes, the images show a de-Jned red region attached to the bottom of the bubble and ris-ing at the same velocity. These images were digitalized andthe volume and length of the wakes determined by a soft-ware developed. This visualization technique is describedin detail, for gas–liquid slug <ow, by Campos and Guedesde Carvalho (1988). Fig. 5 shows images of laminar andtransition wakes, Re� = 40 and Re� = 53, respectively. The<ow patterns and the corresponding ranges of Re� were iden-tical to those cited for gas–liquid slug <ow. The values ofRe� were determined with the e:ective viscosity of the sus-pension.

3.4. Length and volume of the wake

The visualization technique described was used byCampos and Guedes de Carvalho (1988) to determine thelength, lw, and volume, Vw, of closed wakes in gas–liquid

0

0.1

0.2

0.3

0.4

0.5

0.6

0 10 20 30 40 50 60 70

Vw

/ D

3

laminar regime

transition regime

eq. (12)

Reδ

0.0

0.5

1.0

1.5

2.0

0 10 20 30 40 50 60 70

Dl w

transition regimelaminar regime

eq. (11)

Reδ

(a)

(b)

Fig. 6. (a, b) Wake volume (a) and wake length (b) versus Re�. Thesymbols represent data obtained in inert suspensions. The lines representEqs. (11) and (12).

slug <ow. The following equations were established:

lwD

= 0:30 + 1:4 × 10−2Re�; 15¡Re� ¡ 90; (11)

Vw

D3 = 8:6 × 10−3Re�; 15¡Re� ¡ 90: (12)

From the visualization experiments, values for the volumeand length of closed wakes were determined. These data arein agreement with Eqs. (11) and (12), as can be seen inFigs. 6a and b. The values of Re� were determined using thee:ective viscosity of the suspension.

3.5. Wake shape

The images in Fig. 5 suggest that the wake shape is simi-lar to the shape of a half-ellipsoid with the largest axis ver-tical. To prove this similarity, the volume of a half-ellipsoidwas calculated and compared with the wake volume fromEq. (12). To calculate the volume of the half-ellipsoid, thevertical axis was taken equal to two times the wake length,lw, and the horizontal axis was taken equal to two times thedi:erence between the tube radius and the Jlm thickness,R− �.

Brown (1965) deduced an expression to estimate thethickness of a stabilized laminar liquid Jlm, in gas–liquid

4164 O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172

0.0

0.5

1.0

1.5

2.0

0 20 40 60 80 100 120

laminar regimetransition regime

Vw

Vel

lip

Reδ

Fig. 7. Comparison between the wake volume and the volume of a halfellipsoid versus Re�.

slug <ow. Taking the physical properties of the suspension,Brown’s equation is written as

� =

[3�SD

4g× 1

1 − �=R

[(1 − �

R

)2

UB − UG

]]1=3

: (13)

The comparison between the volume of a half-ellipsoid andthe wake volume is shown in Fig. 7. The relative di:erencebetween volume increases with Re�, and in the transitionregime has a maximum value around 25%.

3.6. Velocity pro>le in the liquid >lm at the bottom ofthe slug bubble

Brown (1965) presented the velocity proJle, referenceframe attached to the bubble, for a stabilized liquid Jlm.Taking the physical properties of the suspension, the velocityproJle is given by

v(r) =g�S

[R2 − r2

4− (R− �)2

2ln

Rr

]+ UB: (14)

The velocity at the suspension–bubble interface, i.e. themaximum velocity in the Jlm, is given by

v(r = R− �) =g�S

×[R2−(R−�)2

4− (R−�)2

2ln

RR−�

]+ UB: (15)

4. Mechanistic dispersion model

A mechanistic model was developed to predict the axialparticle dispersion in a slugging vertical column. Accord-ing to the model, the particles are transported upwardinside closed wakes and, during the bubble path, theyare expelled due to gravitational and inertial e:ects. Theupward wake displacement induces a downward <ow ofthe suspension. This downward <ow and the particle set-tling velocity are responsible for the downward transportof particles. This simpliJed model was developed under

Fd

Fg

Fc

r (�)

Gas Bubble

Particle

Film

�

�

Fig. 8. Schematic representation of a laminar wake with a half ellip-soidal shape. The trajectories and the forces acting on the particles arerepresented.

several assumptions:

• the bubbles do not interact during their rise;• the bubbles are long enough to generate a stabilized sus-

pension Jlm <owing around;• the wake regime is laminar; the wake shedding observed

in the transition regime cannot be modelled;• when a bubble enters into the column, all the bubbles and

suspension above are displaced in a volume equal to thevolume of the bubble. The <ow regime in the suspensionbetween the bubbles is laminar and so the particles placednear the axis of the column su:er a higher displacementthan those near the wall. There is an e:ective upwardtransport of particles. However, during the bubble rise, thesuspension <ows down around the bubble. In this down-ward <ow, once again, the particles at the axis of the col-umn travel a higher distance than those near the wall (dueto the non-slip condition). The model assumes that the ef-fective particle transport promoted by the bubble inlet intothe column (upward transport) and by the <ow of the sus-pension around the bubble (downward transport) is muchsmaller than the transport induced by the wakes. This lastassumption is controversial and will be a matter of dis-cussion when the results of the experiments are analyzed.

4.1. Forces acting on a particle in recirculation in thewake

In a closed wake, the particles are in intense recirculationin trajectories around the so-called toroidal vortex-ring (ref-erence frame attached to the bubble). A closed wake, witha half-ellipsoidal shape, and the trajectories of the particlesare schematically represented in Fig. 8. The particles have

O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172 4165

tangential (vt) and radial (vr) velocity components. In theradial direction, the slip velocity is the result of the com-bined action of gravitational, centrifugal and drag forces.The forces acting in a particle recirculating in a trajectorynear the boundary of the wake are represented in Fig. 8.

Assuming Stokes’ law, the drag force is expressed by

Fd = 3# SW vrdP; (16)

where dP is the particle diameter and SW is the viscosityof the suspension inside the wake.

In the radial direction, the apparent weight of a particleis given by

Fg = (�P − �SW )#6

d3Pg sin %; (17)

where the angle % is shown in Fig. 8 and �SW is the densityof the suspension inside the wake.

The centrifugal force is given by

Fc = mPv2t

r(%)= (�P − �SW )

#6

d3P

v2t

r(%); (18)

where mP is the mass of a particle and r(%) the radius of thetrajectory, coincident with the radius of the wake.

Assuming a constant radial velocity, the forces acting onthe particle are in balance. The radial particle velocity isevaluated from Eqs. (16)–(18):

vr = vIP

(sin % +

1g

v2t

r(%)

); (19)

where vIP represents the settling velocity of an individual

particle deJned by

vIP =

[g(�P − �SW )

118 SW

d2P

]: (20)

In laminar wake conditions, the value of vIP is very low

and so, according to Eq. (19), vt ¿¿vr . This means that,in a short interval of time, only particles recirculating intrajectories near the boundary of the wake have any chanceto exit.

4.2. Tangential velocity of a particle

In a reference frame attached to the bubble, the tangen-tial velocity of a particle recirculating in the vicinity of thewake boundary is equal to the velocity of the suspension<owing around the wake. Therefore, the tangential velocityof a particle recirculating at % = 0 (Fig. 8) is equal to thevelocity of the suspension Jlm at R− �, Eq. (15):

vt (% = 0) =g�S

×[R2 − (R− �)2

4− (R− �)2

2ln

RR− �

]+ UB: (15)

Along the wake–suspension boundary, the tangential veloc-ity of a particle in recirculation is given by

vt(%) = vt(% = 0) cos %: (21)

At %=0, the tangential velocity has the direction of the tubeaxis and at the rear of the wake, % = #=2, its value is zero.The particle radial velocity is obtained combining Eqs. (19)and (21).

4.3. Concentration pro>le

The mass of particles leaving the wake through a inJnites-imal element of area �A in the interval of time �t, �m, isgiven by

�m = �Pwvr�A; (22)

where w is the particle volumetric fraction inside the wake.The inJnitesimal area of an ellipsoidal element in polar co-ordinates is given by

�A = 2#r2(%) cos %�%: (23)

The integration of Eq. (22) between % = 0 and % = #=2gives the mass <ow rate of particles leaving the wake:

m(t) = 2#�Pw(t)vIP

(I1 +

v2t (% = 0)

gI2

); (24)

where I1 and I2 are the following integrals:

I1 =∫ #=2

0r2(%) cos (%) sin (%) d%; (25)

I2 =∫ #=2

0r(%) cos3 (%) d%: (26)

The relation between r(%) and % is given by the deJnitionequation of an ellipse in polar coordinates:

r(%) =lw√

1 − e2√1 − e2 sin2(%)

; (27)

where e is the sphericity of the ellipse.Eq. (24) can be rewritten assuming, at any instant, uni-

form particle concentration inside the wake:

m(t) = −dm(t)dt

= 2#m(t)Vw

v IP

[I1 +

v2t (% = 0)

gI2

]; (28)

where m(t) is the mass of particles inside the wake at a giveninstant t and Vw is the wake volume.

The integration of Eq. (28) between the bubble departure(t = 0; z = 0) and instant t(=z=UB) gives

m(z)m0

= e− 2#

UBVwvIP

[I1+

v2t (%=0)

g I2

]z; (29)

4166 O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172

where m0 represents the mass of particles inside the wakeafter the bubble formation (z = 0).

The height of the bubbling suspension z can be relatedto the height of the unaerated suspension z∗. If the bubbledistribution along the column is supposed uniform, z and z∗

are related by

z∗ = z(

1 − UG

UB

): (30)

The mass of particles inside the wake at a level z∗ is thengiven by

m(z∗)m0

= exp− 2#

(UB−UG)VwvIP

[I1+

v2t (%=0)

g I2

]z∗: (31)

The di:erence between the mass inside the wake at levelsz∗1 and z∗2 gives the mass of particles expelled from the wakebetween these two levels.

The downward mean particle velocity, vS(z), is inducedby two mechanisms: (i) the settling velocity of the particlesin the suspension, and (ii) the downward suspension velocitycaused by the upward displacement of suspension inside thewakes:

vS(z) = vSP(z) +

VwfA

; (32)

where vSP(z) represents the settling velocity of the particles

in the suspension and f the slug bubble frequency.The settling velocity of small particles in high viscous sus-

pensions is very small. This velocity was determined fromKhan and Richardson’s (1989) equation, and the conclusionis that vS

P(z)6 0:01Vwf=A for all the experimental condi-tions studied. The bubbling frequency was experimentallymeasured and the wake volume is given by Eq. (12). There-fore, the contribution of the settling velocity to the down-ward transport can be neglected and the downward velocityexpressed only by

vS(z) ∼= VwfA

: (33)

At this point, the model can be described in a straightforwardway. There is an upward transport of particles inside thewakes and some of them are expelled during the wake rise.The particles still inside the wake at the top of the column aredeposited. In counterpart, there is a downward <ow causedby the upward displacement of the wakes. An elucidativeimage is a little bag taken from the bottom, some of thepieces inside the bag are lost during the rise, and the bagis deposited at the top with the pieces still inside it. Thebag taken from the bottom to the top, creates a “hole” atthe bottom and so all the suspension moves down along adistance equal to the height of the bag to Jll the “hole”.

Based on this model, the particle dispersion along thecolumn was numerically simulated. In the next section, thenumerical procedure developed is brie<y described.

5. Numerical simulation

The test tube was divided in a Jne horizontal grid. Thedistance between horizontal lines was lower than 1 mm,corresponding to more than 3000 horizontal grid lines. Theparticle dispersion was simulated, bubble after bubble. Twoinitial particle distributions were considered: (i) a distribu-tion in step, with particles packed from the bottom until acertain level and liquid free of particles from there until thetop; (ii) a uniform particle distribution along all the test tube.

The volume of the wake j=1, and of the following wakes,was estimated by Eq. (12), using the physical properties ofthe suspension at the bottom of the column; to be more pre-cise, the mean properties in a distance equal to the equiva-lent height of the wake, from z = 0 to Vw=A (this calculationneeds an iterative process). The volumetric fraction in thewake at the bottom was the mean volumetric fraction fromz = 0 to Vw=A. In the upward motion, the mass lost by thewake j=1 inside mesh i∗, between grids i−1 and i, was de-termined by subtracting the mass inside the wake, Eq. (31),at grid levels i − 1 and i.

The mean volumetric fraction inside mesh i∗, after thepassage of the wake j = 1, is given by

US)i∗ ;1 = US)i∗ ;0 +mi−1 − mi

�PAhi; (34)

where US)i∗ ;0 is the volumetric particle fraction inside meshi∗ before the passage of the wake and hi is the height of themesh i∗.

At the arrival of wake j = 1 at the top, the suspensionstill inside the wake is deposited in a distance equal to theequivalent height of the wake. The downward <ow was sim-ulated bringing down all the suspension in a distance equalto the equivalent height of the wake.

The particle distribution before the simulation of wakej=2, or any wake j, was known, and the simulation processdescribed for wake j=1 was successively applied. The sim-ulation was repeated the number of times (Taylor bubbles)required to reach a steady or quasi-steady particle distribu-tion. The simulation stopped when, after 100 consecutivebubbles, the volumetric particle fraction did not change inany point of the grid more than 1%.

6. Experimental results

Same of the experiments performed and the correspond-ing operation conditions are listed in Table 1. The valuesof the Reynolds number in the liquid Jlm, Re�, and of theReynolds number in the suspension <owing in front of thenose of the bubbles, ReUL , are also listed. In most of the ex-periments, the wake regime is laminar or in transition fromlaminar to turbulent. In both regimes, there is recirculationinside the closed wakes, respectively, without and with vor-tex shedding. According to ReUL values, the regime in the

O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172 4167

Table 1Experimental conditions

Reference dp 0S L UG Re� Wake ReUL Liquid

number (�m) (Pa s) (m/s) regime regime

A 280 0.2 0.017 0.187 152 T 300 LB 280 0.1 0.039 0.248 90 L/T 210 LC 280 0.2 0.038 0.111 50 L 80 LD 280 0.2 0.038 0.187 68 L/T 135 LE 280 0.2 0.038 0.248 79 L/T 180 LF 280 0.2 0.038 0.311 90 L/T 220 LG 180 0.2 0.036 0.111 60 L/T 85 LH 180 0.2 0.036 0.187 73 L/T 140 LI 180 0.2 0.036 0.248 84 L/T 190 LJ 280 0.3 0.038 0.111 44 L 65 LK 280 0.3 0.037 0.248 66 L/T 150 LL 180 0.3 0.037 0.111 46 L 65 LM 280 0.2 0.069 0.111 31 L 45 LN 280 0.2 0.069 0.187 38 L 75 LO 280 0.2 0.069 0.248 44 L 100 LP 280 0.2 0.069 0.311 50 L 125 LQ 280 0.3 0.071 0.111 24 L 35 LR 280 0.3 0.071 0.187 30 L 60 LS 280 0.3 0.071 0.248 34 L 80 LT 280 0.3 0.071 0.311 40 L 100 LU 280 0.3 0.071 0.343 42 L 110 LV 280 0.2 0.216 0.187 12 L 25 LY 280 0.2 1.45 0.187 2 L 4 LW 180 0.2 0.076 0.111 28 L 40 LX 180 0.2 0.076 0.187 35 L 70 LZ 180 0.2 0.076 0.248 40 L 90 L

suspension <owing ahead the bubbles is always laminar,supporting the values found for the coeEcient C (Fig. 4).

6.1. Reproducibility, initial distribution e@ect and steadystate

Several preliminary experiments were done to study thereproducibility of the results, the bubbling time needed toreach the steady state, and the e:ect on the particle dispersionof the initial particle distribution.

The reproducibility was very good, in most of the casesthe deviations observed were less than the experimental un-certainty. The steady state was reached, in the most adverseconditions, 1 h after the beginning of the bubbling process.The e:ect of the initial particle distribution on the axialdispersion was studied performing experiments with di:er-ent initial distributions: a packed bed at the bottom of thecolumn (bubbling turned o:), and a homogeneous distri-bution (bubbling turned on); this homogeneous suspensionwas prepared outside of the column. The steady-state parti-cle distribution was the same, whatever the initial condition.

6.2. The transport in the wake

The data obtained with increasing liquid viscosity are plot-ted in Fig. 9 (each run is associated to a letter according to

0.0

0.5

1.0

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0z*

A

Y

V

N

D

1.5

� s /

� s0

Fig. 9. Experimental axial dispersion data for di:erent liquid viscosities.

Table 1). For the lower liquid viscosity, run A, the wakeregime is not closed, the axial particle transport is poor,and the volumetric particle fraction at the top is practicallyzero. With a liquid of viscosity two times greater, run D, thewake is closed and there is vortex shedding along the wakerise (transition regime). The axial dispersion increases re-markably and at the top, the particle volumetric fraction isalmost one-third of the value at the bottom. With a liquidof viscosity two times greater, run N , the wake regime islaminar, there is no vortex shedding, but the wake volumeis lower than in the previous run. The axial dispersion is

4168 O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

QRST

z*

� s /

� s0

Fig. 10. Experimental axial dispersion data for di:erent superJcial gasvelocities.

identical to that observed in run D. With a liquid of viscos-ity two times greater, run V, the wake regime continues tobe laminar, but the wake volume is much lower. There isan evident decrease in the axial transport. The same trendis observed in the last data, run Y, a smaller wake volumeinduces a smaller transport.

The previous results are relevant to demonstrate the im-portance of the wake in the axial particle transport. TheJgure also shows that the wake transport overlaps the com-bined particle transport due to bubble inlet and bubble rise(assumption of the model). In all the experiments, the <owregime in the suspension between bubbles is laminar (A, D,N, V and Y - Table 1), and so the upward transport due tobubble inlet should be identical for all these runs. Campos(1987) studied the distortion su:ered by a “horizontal lineof <uid”, initially placed in a cross section above the bubblenose, due to the <ow around a long slug bubble. Accordingto this study and taking on account the properties of the sus-pensions represented in the Jgure, it can be concluded thatthe downward transport is identical for all of them. Moreprecisely, the downward displacement of 90% of each sus-pension is identical for all the runs, and only 10%, placednear the wall, su:ers a higher downward transport in the lessviscosity runs. If the combined particle transport due to bub-ble inlet and bubble rise was a relevant mechanism, identi-cal dispersions should be observed in all the runs, but Fig.10 shows the opposite. Therefore, according to the data, thetransport in the wake overlaps the particle transport inducedby this combined mechanism. It should be referred that thehigh bubbling frequency can reinforce this behavior since,along the column, most part of the suspension is inside thewakes and only a small part is <owing between and alongbubbles.

6.3. The e@ect of the super>cial gas velocity

Data obtained with increasing superJcial gas velocityare plotted in Fig. 10, runs Q to T. In all the experiments,the bubbles were long slugs. For the lower superJcial gasvelocity, the mean bubble length was higher than 140 mm;

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0� s

/ � s

z*

MW

Fig. 11. Experimental axial dispersion data for di:erent particle diameters.

this value was obtained reading the surface height beforeand after the bubbles burst at the top of the column. Thewake regime is laminar (Table 1), and the axial disper-sion increases with increasing superJcial gas velocity. Thisbehavior is partially explained by the decrease of the resi-dence time of the bubbles, which induces a lower amountof particles expelled from the wake during the wake rise.The number of bubbles is not an important parametersince the steady state is reached for di:erent bubblingtimes.

6.4. The e@ect of the particle diameter

Data obtained in experiments with particles of di:er-ent size are plotted in Fig. 11, runs M and W. The wakeregime is laminar, and the axial dispersion is higher forthe smaller particles. This behavior is partially explainedby the less amount of particles expelled during the wakerise due to the lower centrifugal force acting on the smallerparticles. In both experiments, the settling velocity of theparticles can be neglected when compared with the down-ward velocity of the suspension promoted by the upwarddisplacement of the wakes (Eq. (32)). If the previous re-ferred combined particle transport due to bubble inlet andbubble rise was the relevant transport mechanism, the ax-ial particle dispersion should be independent of the particlediameter.

6.5. The e@ect of the particle charge

Data obtained with di:erent particle charges are plot-ted in Fig. 12, runs B, E and K. The axial dispersionincreases fairly with increasing values of particle charge.This behavior is partially explained by the increase ofthe e:ective viscosity of the suspension, which inducesa decrease in the centrifugal force. This e:ect overlapsthe lower transport due to a small decrease in the wakevolume.

O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172 4169

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0� s

/ � s

z*

B

E

K

Fig. 12. Experimental axial dispersion data for di:erent particle charges.

7. Numerical results and comparison with experimentaldata

The axial particle dispersion was numerically simulatedfor some of the operation conditions in Table 1. The sim-ulation was based on the laminar wake model developed.The experiments with open wakes, run A, were not sim-ulated, as well as the experiments with an Re� ¡ 15, runsV and Y. From the visualization experiments, it is evidentthat for very low Reynolds number, there is a quick ex-pansion of the suspension coming from the Jlm <owingaround the bubble. The wake volume decreases markedly,the shape is quite di:erent from ellipsoidal and the wakelength and volume are no more well represented by Eqs. (11)and (12).

7.1. Initial distribution e@ect and steady state

Figs. 13a and b show the simulated particle distributionfor increasing number of slug bubbles, run N. The simu-lation was done for two initial conditions: a homogeneoussuspension and a packed bed at the bottom of the column.Two important conclusions can be taken, both in agreementwith the experimental data:

• after a large number of bubbles, the axial distributiontends to be constant, i.e., the suspension tends to a steadystate;

• the steady state does not depend on the initial distribution.

7.2. Laminar wake regime

For laminar wakes, experimental and simulated data arecompared in Figs. 14–16 the symbols represent experimen-tal data and the solid lines simulated data. The data in Fig.14 refer to experiments with the larger particles and di:er-ent superJcial gas velocities, runs M–P (0

S = 0:2). The datain Fig. 15 refer to experiments with the smaller particles anddi:erent superJcial gas velocities, runs W to Z (0

S = 0:2).The data in Fig. 16 refer to experiments with both particle

0.0

2.0

4.0

Steady state distribution

Initial distribution

j = 10 j = 130 j = 710

0.0

2.0

4.0

6.0

8.0

0.0 2.0z* (m)

Steady state distribution

Initial distribution

j = 980j = 390j = 20

1.0 3.0

0.0 2.0z* (m)

1.0 3.0

� s /

� s� s

/ � s0

0

Fig. 13. Simulated axial dispersion data for two initial particle distribu-tions: a homogeneous suspension (a), and a suspension in step (b).

0.0

0.5

1.0

1.5

2.0

0.0 0.5 3.0z*

M

N

O

P

1.0 1.5 2.0 2.5

� s /

� s0

Fig. 14. Comparison, for laminar wake regime, between experimentaland simulated axial dispersion data for di:erent superJcial gas velocities(larger particles). Lines represent simulated data and symbols experimentaldata according to Table 1.

sizes and 0S = 0:3, runs J and L. In all the conditions, the

agreement between experimental and simulated data is quitegood. The e:ects in the particle dispersion of the superJ-cial gas velocity, particle size and particle charge are wellpredicted by the model.

4170 O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172

W

X

Z

z*

3.02.52.01.51.00.50.0

2.0

1.5

1.0

0.5

0.0

� s /

� s0

Fig. 15. Comparison, for laminar wake regime, between experimentaland simulated axial dispersion data for di:erent superJcial gas velocities(smaller particles). Lines represent simulated data and symbols experi-mental data according to Table 1.

J

L

3.02.52.01.51.00.50.0

z*

2.5

2.0

1.5

1.0

0.5

0.0

� s /

� s0

Fig. 16. Comparison, for laminar wake regime, between experimental andsimulated axial dispersion data for di:erent particle sizes and 0

S = 0:3.Lines represent simulated data and symbols experimental data accordingto Table 1.

z*

G

H

I

2.0

1.5

1.0

0.5

0.00.0

� s /

� s0

3.02.52.01.51.00.5

Fig. 17. Comparison, for transition wake regime, between experimentaland simulated axial dispersion data for di:erent superJcial gas velocities(particle size 180 �m). Lines represent the simulated data and symbolsthe experimental data according to Table 1.

7.3. Transition wake regime

Data referring to experiments in the transition wakeregime are plotted in Figs. 17 and 18; runs G–I (smallerparticles) and runs C–F (larger particles). The agreementbetween experimental (identical to that observe in laminar

C

D

E

F

z*3.02.52.01.51.00.50.0

3.0

2.0

1.0

0.0

� s /

� s0

Fig. 18. Comparison, for transition wake regime, between experimentaland simulated axial dispersion data for di:erent superJcial gas veloci-ties (particle size 280 �m). Lines represent simulated data and symbolsexperimental data according to Table 1.

wake regime) and simulated data is poor, as was expecteddue to the assumptions of the model. However, a lowerexperimental dispersion was previsible due to the appear-ance of the vortex shedding phenomena, which goes alongwith a continuous oscillation of the bottom of the bubbleand the wake. During the bubble rise, the fraction of thewake falling out by vortex shedding is replaced by sus-pension with a lower particle concentration, resulting, apriori, in a lower transport. However, the vortex sheddingfrequency is not known (how many times it happens insidethe column?), and also the continuous oscillation of thebubble bottom and wake modiJes the tangential velocityof the suspension <owing along the wake boundary. Theconsequences of these <ow modiJcations are physically un-foreseeable. A study employing particle image velocimetry(PIV) technique is presently underway to know more aboutthe <ow pattern in the transition wake in a gas–liquid slug<ow.

Another possible justiJcation for the high particle trans-port in the transition regime can be found in the interac-tions between the bubbles <owing continuously along thecolumn. According to Pinto et al. (1998), there is a mini-mum distance between two bubbles for which the <ow be-hind the leading bubble does not interfere in the velocity ofthe preceding one (increasing the velocity). This distance isstrongly dependent on the <ow (velocity proJle) emergingfrom the wake region of the leading bubble and increaseswith Re�. Therefore, a high minimum distance for bubblesinteraction is expected in the transition wake regime. If thevelocity of the bubbles increases during their rise, the resi-dence time of the bubbles in the column decreases and theaxial transport increases.

8. Conclusions

The importance of the bubble wake in the upward trans-port of particles in a three-phase reactor was highlighted

O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172 4171

in the present work. For di:erent wake <ow patterns,di:erential pressure transducers measured the axial particledistribution. For laminar wakes, a simple model was devel-oped to predict the axial particle dispersion. The model pre-dictions are in good agreement with the experimental data.For wakes in the transition regime, the experimental trans-port is higher than the laminar model predictions. Thecontinuous oscillation of the bubble bottom induces vortexshedding and modiJes the <ow patterns around the wake.This phenomena can be responsible by the higher transportobserved. According to the model predictions and to the ex-perimental data, the upward transport of particles increaseswith increasing particle charge, increasing gas <ow rate anddecreasing particle diameter.

Notation

A cross-sectional area of the tube, m2

C empirical coeEcient, Eq. (6)dP particle diameter, md1–2 distance between pressure taps, mD internal pipe diameter, mDP axial dispersion coeEcient, m2=se sphericity of the ellipsoid Jtted to the wakef slug bubble frequency, s−1

Fc centrifugal force, NFd drag force, NFg apparent weight, Ng acceleration due to gravity, m=s2

h1; h2 vertical distances from a reference datum level, mhi height of the mesh i∗, mHmax maximum liquid height in the aerated column, mH0 liquid height in the column before slugging, mI1; I2 integrals deJned in Eqs. (25) and (26)lw wake length, mj number of the iterationk constant, Eq. (4)m(t) mass <owrate of particles leaving the wake at a

given instant t, kg/sm(t) mass of particles inside the wake at a given instant

t, kgm0 mass of particles inside the wake after bubble for-

mation, kgmP mass of a particle, kgN number of slug bubblesP pressure, PaP1 static pressure at tap 1, PaP2 static pressure at tap 2, Par(%) radius of the particle trajectory, mR internal radius of the pipe, mt time, sUB slug bubble velocity, m/sUB∞ velocity of an individual slug bubble rising in a

stagnant liquid, m/s

UG superJcial gas velocity, m/svP settling velocity of the particles, m/svIP settling velocity of an individual particle, m/s

vSP settling velocity of the particles in the suspension,

m/sv(r) velocity in the stabilized Jlm, m/svr radial velocity of a particle inside the wake, m/svS(z) downward mean particle velocity in the suspen-

sion, m/svt tangential velocity of a particle at the wake bound-

ary relatively to a referential moving with the bub-ble, m/s

v� mean velocity in the stabilized Jlm relatively toa referential moving with the bubble, m/s

V relative (to the bubble) mean velocity of the liquid<owing ahead of the slug nose, m/s

Vw wake volume, m3

x =UG=UB∞ in Eq. (7)z axial distance in the bubbling column, mz∗ axial distance in the column free of bubbles, m

Dimensionless groups

Re� Reynolds number in the Jlm <owing around thebubble (=v��=�S)

ReUL Reynolds number in the suspension ahead thebubble (=UGD=�S)

Greek letters

� parameter in Eq. (8)� thickness of the stabilized Jlm <owing around a

slug bubble, mLP di:erence in piezometric pressure, PaS volumetric particle fraction in the solid–liquid

suspension0S volumetric particle fraction supposing homoge-

neous suspensionw volumetric particle fraction in the wakeUS)i∗ ;1 mean volumetric fraction inside mesh i∗ after the

passage of wake j = 1US)i∗ ;0 mean volumetric fraction inside mesh i∗ before

the passage of wake j = 1% angle represented in Fig. 8, rad L dynamic liquid viscosity, Pa s S e:ective dynamic viscosity of the suspension, Pa s SW e:ective dynamic viscosity of the suspension in-

side the wake, Pa s� kinematic viscosity, m2=s�S e:ective kinematic viscosity of the suspension,

m2=s�L liquid density, kg=m3

�P particle density, kg=m3

�S e:ective density of the suspension, kg=m3

�SW e:ective density of the suspension inside thewake, kg=m3

4172 O. N. Cardoso et al. / Chemical Engineering Science 58 (2003) 4159–4172

Acknowledgements

The authors would like to acknowledge the Jnancial sup-port given by Funda[cao da Ciencia e Tecnologia—projectPOCTI 33761/99.

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