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Analysis of breakage and coalescence models for bubble columns

 J.F. Mitre, R.S.M. Takahashi, C.P. Ribeiro Jr.1, P.L.C. Lage Ã

Programa de Engenharia Quımica — COPPE, Universidade Federal do Rio de Janeiro, PO Box 68502, Rio de Janeiro, RJ 21941-972, Brazil

a r t i c l e i n f o

 Article history:

Received 30 January 2010

Received in revised form

11 July 2010

Accepted 20 August 2010Available online 27 August 2010

Keywords:

Bubble

Simulation

Population balance

Bubble columns

Coalescence

Breakage

a b s t r a c t

This work was aimed at evaluating the performance of bubble breakage and coalescence models in

bubble column simulations. A total of five different models were considered (two for breakage and

three for coalescence). These selected models had their parameters estimated using experimental data

of bubble size distributions for the air–water system in an isothermal bubble column. Bubble sizes were

measured with a photographic technique for two gas superficial velocities at three bubbling heights.

Model parameters were estimated using the maximum likelihood technique applied to a one-

dimensional population balance model, which was solved by the method of classes. The results allowed

to conclude, within the experimental uncertainty, that it was necessary to adjust the parameters of 

some of the models according to the operational conditions. For other models, the same parameter

values could be used to represent different operational conditions.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Several unit operations in chemical engineering involve the

contact of two or more phases to promote mass and heat transferwith or without chemical reactions. Distillation towers, direct-

contact evaporators, bubble columns and multiphase reactors are

some examples (Ribeiro and Lage, 2004a; Chen et al., 2005; Joshi,

2001).

A multiphase flow is characterized by the existence of at least

two phases. The spatial distribution of these phases characterizes

the flow. A two-phase flow is called disperse when one of these

phases is distributed as small elements. These may be fluid or

solid particles, whereas the other phase must be a fluid.

When the particles in a two-phase dispersed flow can be

differentiated by their own properties (size, temperature, species

concentration, etc.), the flow is called polydisperse. These proper-

ties are called internal variables (Ramkrishna, 2000). One of the

most important internal variables is the particle size, which can beexpressed by mass, diameter, volume or other extensive particle

property.

Particles in a dispersed flow can interact among themselves.

They can aggregate to produce larger particles and they can also

break to generate smaller ones. These processes are important in

chemical engineering applications because they imply the

exchange of relevant properties, like energy or mass, among

mother and daughter particles.

The population balance is a modeling framework that

incorporates the interaction among particles in a dispersed flow(Ramkrishna, 2000). It is based on an equation that states the

conservation of particle number density which is called the

population balance equation (PBE). In order to include particle

interactions, the particle breakage and aggregation phenomena

must be modeled.

In turbulent flows, the main cause of fluid particle breakage is

the particle–turbulence interaction. Basically, the existing break-

age models can be divided into two groups. The first group

comprises statistical models where breakage occurs due to

collisions between particles and turbulent eddies that carry

enough energy to promote particle deformations sufficiently

intense to produce an immediate breakup. The second kind of 

models assumes pure kinematic arguments based on cohesive and

disruptive tensions to establish the necessary conditions forparticle rupture (Lasheras et al., 2002).

Aggregation of fluid particles is called coalescence. Coalescence is

assumed to occur in three stages. First, there is a collision between

two particles that are forced against each other by the turbulent

dynamic pressure, thereby forming a thin film between their

interfaces. Then, the fluid within this film is drained until it reaches

a critical thickness at which molecular attractive forces dominate,

quickly promoting the film rupture and leading to coalescence

(Marrucci, 1969; Coulaloglou and Tavlarides, 1977; Prince and

Blanch, 1990; Chesters, 1991; Chaudhari and Hofmann, 1994).

There are several models in the literature for the breakage

and coalescence of fluid particles (Coulaloglou and Tavlarides,

Contents lists available at ScienceDirect

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

Chemical Engineering Science

0009-2509/$- see front matter& 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ces.2010.08.023

à Corresponding author. Tel.: +55 21 25628346; fax: 55 21 25628300.

E-mail address: paulo@peq.coppe.ufrj.br (P.L.C. Lage).

URL: http://www.peq.coppe.ufrj.br/areas/tfd/ (P.L.C. Lage).1 Current address: Center for Energy and Environmental Resources, University

of Texas at Austin, 10100 Burnet Road, building 133, Austin, 78758 TX, USA.

Chemical Engineering Science 65 (2010) 6089–6100

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1977; Prince and Blanch, 1990; Martınez-Bazan et al. 1999a,b;

Luo and Svendsen, 1996; Kamp et al., 2001; Wang et al., 2003,

2005b; Andersson and Andersson, 2006). Unfortunately, most of 

these models were developed using complex flows, like aerated

stirred tanks or bubble columns, in which the turbulent field is

not well characterized. It is not only inhomogeneous, but also

anisotropic along several length scales (Martınez-Bazan et al.,

1999a).

Chesters (1991) and Chaudhari and Hofmann (1994) reviewedthe existing coalescence models and concluded that they are quite

similar to each other being also based on the same hypotheses.

There are a few models that have been published after these

reviews, but the main conclusion remains. Lasheras et al. (2002)

compared several breakage models and showed the existence of 

inconsistencies in almost all of them.

There have been a few studies on the influence of several

bubble coalescence and breakage models on the evolution of the

particle size distribution function (Mitre et al., 2004; Wang et al.,

2005a; Bordel et al., 2006). Wang et al. (2005a) concluded that the

predicted bubble size distributions were quite different when

different bubble coalescence and breakup models were used. They

also concluded that it is necessary to take into account bubble

coalescence and breakup due to different mechanisms. However,

some of the models employed by Wang et al. (2005a) have some

theoretical inconsistencies, like the asymmetry of the coalescence

frequency function generalized from the Hibiki and Ishii (2000)

coalescence model (Mitre, 2006). This could have affected their

results.

In a previous work, Mitre et al. (2004) used bubble size

distribution data obtained in a direct-contact evaporator to test

the bubble breakage model of  Luo and Svendsen (1996) and the

coalescence model of  Prince and Blanch (1990). These authors

concluded that it was necessary to fit parameters in these models

to obtain a good agreement with the experimental data.

In the specific case of bubble columns, Bordel et al. (2006)

compared the performance of different bubble coalescence and

breakage models in the prediction of the evolution of bubble size

distributions under different experimental conditions. However,only 100 bubbles were analyzed at each experimental condition,

which does not guarantee a statistically representative sample. In

addition, the adopted gas superficial velocities were very low

(below 1.0 cm/s), for which appreciable rates of bubble breakage

and coalescence are not expected.

A somewhat recent trend in the literature is the coupling of the

PBE with CFD codes to model polydispersed multiphase systems,

the so-called PB-CFD simulation (Chen et al., 2005; Jakobsen and

Dorao, 2005; Podila et al., 2007; Frank et al., 2007; Bhole et al.,

2008). Some of these works focused on evaluating breakage and

coalescence models.

Chen et al. (2005) used two-dimensional axisymmetric

Eulerian models for simulating bubble column flows with several

breakage and coalescence closures using the FLUENT software. Inorder to simplify the model, it was assumed that all bubbles share

the same velocity field. For breakage, the models of  Luo and

Svendsen (1996) and Martınez-Bazan et al. (1999a,b) were

chosen, whereas in the case of coalescence the models of  Prince

and Blanch (1990), Luo (1993) and Chesters (1991) were tested.

The PBE was solved by the Kumar and Ramkrishna (1996) method

of classes. It was concluded that the solution including the

population balance is much better than the one using a single

bubble size even though bubble breakage had to be 10 times more

intense to match the experimental data. The choice of different

breakage and coalescence models did not impact the simulated

results for the flow velocity field, gas hold-up and volume-based

particle distribution function. However, no experimental data for

bubble size distribution were used to validate the simulations.

Furthermore, the simulation results might have been affected by

the use of  Miyahara et al. (1983) model to predict the bubble

mean diameter at the gas sparger and by the turbulence closure

model employed.

Podila et al. (2007) analyzed the gas–liquid flow in tubular

reactors using axially symmetric two-dimensional simulations

coupled with population balance modeling of the bubble sizes in

FLUENT. It was shown that the employed turbulence closure

adequately predicts the turbulent intensity field. Similar to Chenet al. (2005), Podila et al. (2007) also assumed that all bubbles

share the same velocity field. The breakage models of  Luo and

Svendsen (1996) and Lehr et al. (2002), as well as the coalescence

models of  Prince and Blanch (1990), Luo (1993) and Lehr et al.

(2002), were employed. Experimental data on gas hold-up,

turbulent intensity and Sauter mean diameter were used for

comparison. The results were somewhat dependent on the

breakage and coalescence models. Reasonably good predictions

were obtained using the Lehr et al. (2002) breakage model, the

Prince and Blanch (1990) coalescence model, and the Lehr et al.

(2002) coalescence model with a different coalescence efficiency

constant.

Bhole et al. (2008) simulated the axisymmetric steady-state

gas–liquid flow in bubble columns including the PBE solution by

the Kumar and Ramkrishna (1996) method. In this case, a

different velocity field for each bubble class and an algebraic slip

model were considered. A modification of the Prince and Blanch

(1990) coalescence model to correct the fluctuating bubble

velocity was proposed, which led to a decrease in the coalescence

rate, in complete agreement with the well-known characteristic

of the original model to overestimate it. Bubble breakage was

represented by the model of  Luo and Svendsen (1996), while the

model of  Miyahara et al. (1983) was utilized to predict the initial

bubble sizes. Bhole et al. (2008) concluded that the assumption of 

the same velocity field for all bubbles is inadequate. Comparison

of simulations with experimental data for the radial profiles of 

mean bubble diameter, gas hold-up and axial liquid velocity

showed a good agreement.

It seems reasonable to assume that if the simulated flowcharacteristics in the multiphase system matched the experi-

mental results, then one would have an indirect evidence of the

adequacy of the breakage and coalescence models used in the

simulation. However, as breakage and coalescence are phenom-

ena with opposite trends, their models might be in error even if 

the main simulated flow variables agree with their experimental

data. Therefore, although much has been made on advancing the

modeling of gas–liquid two-phase flow, no work has actually

validated the breakage and coalescence models used in PB-CFD

simulations because, as highlighted by Bhole et al. (2008), this

requires experimental data on local bubble size distributions.

Besides, other sources of errors do exist, such as the sparger

(Miyahara et al., 1983), turbulence and interface force modeling.

Therefore, the evaluation of breakage and coalescence modelsrequires simple models, with much less modeling uncertainties,

but considering different velocities for different bubble sizes

(Bhole et al., 2008).

In this work, we evaluated the application of some breakage

and coalescence models to bubble columns. Instead of a PB-CFD

simulation, a one-dimensional steady-state population balance

model with different bubble velocities was used. Such model is

more adequate to represent the experimental data on bubble size

distributions obtained at different heights by the image analysis

technique. Experiments were performed with the air–water

system in a cylindrical column operating in semi-batch mode at

two distinct gas superficial velocities. The model was solved by

the method of classes of Kumar and Ramkrishna (1996) using the

bubble size distribution data at the lowest height as initial

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condition. The experimental data were interpreted using different

combinations of the selected breakage and coalescence models. In

order to verify model consistency, parameters were introduced to

correct the breakage and coalescence rates and the coalescence

efficiency. The values of such parameters were estimated by the

maximum likelihood technique. Experimental data uncertainty

was taken into account in evaluating the parameters values and

their standard deviations.

2. Material and methods

Experimental bubble size distributions in a bubble column

were obtained using the same photographic technique previously

employed by Silva and Lage (2000) and Ribeiro and Lage (2004a).

In the following, the apparatus and experimental conditions are

described.

 2.1. Experimental set-up

The experimental set-up is shown in Fig. 1. It consisted of a

cylindrical glass column, 7.3cm in inner diameter and 2 m in

height, at the bottom of which the sparger was placed. The gasdistribution system was composed of a ceramic plate containing

89 orifices, 0.5mm in diameter, arranged in a quadrangular

pattern with a pitch of 6.3mm. Beneath this ceramic plate, a

stainless steel porous plate was used to reduce the gas velocity

fluctuations through the orifices and thereby keep bubble

formation in the constant flow rate regime (Gaddis and Vogelpohl,

1986). The distribution plates were fixed between the two flanged

Nylon sections that form the gas distribution system. The system

was sealed with Vitons O-Rings. Three parallel rotameters were

available for flow rate measurement, whose error is estimated to

be about 5%. Liquid temperature during operation was monitored

with the aid of a 1.8-m-long platinum resistance thermometer

(PT-100).

 2.2. Experimental procedure

The column was operated in semi-batch, isothermal (251C)

mode using distilled water as the continuous phase and

compressed air as the disperse phase. A constant gauge pressure

of 4 bar was kept at the gas distribution chamber. Air was

considered to be an ideal gas. Due to its sensitivity to impurities

and its importance in the coalescence and breakage phenomena,

the surface tension of the employed distilled water was measured

to be 0.067 N/m in a KSV Sigma 70 tensiometer. Other properties

of air and water came from the literature.

A photographic technique was employed to measure statisti-

cally significant bubble size distributions under different operat-

ing conditions. All photos were taken with a SONY digital camera

(model MAVICAs MVC-FD91) adjusting the shutter speed to 1ms.

The camera flash light provided the required illumination.Aiming at determining the bubble size distribution, five

pictures were periodically taken at three different column heights.

The camera was focused at a ruler located outside the column but

at its middle plane. Therefore, the focus was approximately at the

column center. The focal depth was experimentally determined to

be around 1.5 cm. Thus, the photograph method captures bubbles

at the center of the column and near the column wall, as long as

they are within the focused slab. Due to axisymmetric behaviour

of the mean flow, this does not affect the results if a statistically

relevant number of bubbles are measured. The definition of the

height of a picture was made considering the center of the photo.

The field of view of each photo is approximately 6 cm in height.

The bubble size distributions were obtained for two values of gas

superficial velocity. These conditions are listed in Table 1.

A correction for the bubble volume during its ascension along

the column was computed considering isothermal operation. This

correction accounts for the gas expansion caused by the reduction

in the hydrostatic pressure along the column height. The area of 

each bubble was evaluated using a graduate scale fixed to the side

of the column and located at its central plane, on which the

camera was focused for all experiments.

All photos were analyzed using the software Tnimages, which

can calculate the number of pixels associated with any delimited

area in a picture. Thus, in each photo, different regions with the

same area were initially selected in the image of the graduate

scale, so as to obtain a relationship between the areas in pixels

and in square millimeters. Next, the contours of the bubble on

focus were manually traced, the area of each bubble was

determined, and then the diameter of the circle with the samearea was computed. Based on a hypothesis of random positioning

of the bubble in the photographic plane, this diameter was

considered to be equivalent to the one related to the sphere with

the same volume. A minimum of 500 bubbles was analyzed for

each experimental condition in order to guarantee the statistical

significance of the determined size distributions (Ribeiro and

Lage, 2004a). The same photos were analyzed by two different

experimenters, but no significant difference in the obtained

bubble size distributions was noticed.

Due to the curved surface of the glass column and to the

considerable differences between the refractive indices of glass

and air, there is a systematic parallax error in the measurement of 

the bubble areas by the photographic technique. In order to

quantify this error and correct the measurements, Ribeiro andLage (2004a) described a procedure that was used in the present

work. This procedure applies a single correction factor value of 

0:6970:02 to all bubble diameter measurements.

Fig. 1. Schematic diagram of the experimental set-up: (1) flow meter (2) camera

and (3) bubble column. The arrows indicate the gas flow direction. The operation

was in semi-batch mode.

 Table 1

Operating conditions associated with experimental data for bubble size

distributions.

uG (cm/s) Distance from sparger (cm) f d30 (mm)

Base

height

Middle

height

Top

height

2.0 70.2 11.0 64.6 142.0 0.09570.002 4.1 70.2

3.8 70.4 11.0 80.1 132.4 0.10670.002 4.9 70.2

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3. Mathematical formulation

 3.1. Population balance model

The number of particles per unit volume of the state variable

space is the particle number distribution function, f . Considering

only one internal variable, the particle mass, f (t ,m,z) is called the

particle size distribution function (PSDF). In order to predict the

evolution of the PSDF, it is necessary to solve the populationbalance equation (PBE). Considering breakage and coalescence of 

particles and neglecting nucleation and particle growth, the PBE is

given by Ramkrishna (2000) as

@ f 

@t þr Á ðUf Þ ¼ H  ð1Þ

where U ¼U (t ,m,z) is the velocity of the particle with mass m at

spatial position z and at instant t. H (t ,m,z, y ) is the particle net

production rate, which can be written as

H ¼ BBÀDB þBC ÀDC  ð2Þwhere BB, DB, BC  and DC  represent, respectively, the particle

production (birth) and the destruction (death) by breakage and

the birth and the death by coalescence. For the monovariate

distribution on particle mass, these terms are given by Ram-krishna (2000) as

BBðt ,m,z, y Þ ¼Z 1

mu

zðmu, y ÞP ðmjmu, y Þbðmu, y Þ f ðt ,mu,zÞdmu ð3Þ

DBðt ,m,z, y Þ ¼ bðm, y Þ f ðt ,m,zÞ ð4Þ

BC ðt ,m,z, y Þ ¼ 1

2

Z m

0 f ðt ,mÀmu,zÞ f ðt ,m,zÞQ ðmÀmu,mu, y Þdmu ð5Þ

DC ðt ,m,z, y Þ ¼Z 1

0

 f ðt ,m,zÞ f ðt ,mu,zÞQ ðm,mu, y Þdmu ð6Þ

In all these terms, it was assumed that they are affected by ( t ,z)

only through y . From now on, in order to simplify the notation, y 

will be omitted in all the equations.

In the present case, the conditional probability P ðmujmÞ has

three important properties (Ramkrishna, 2000). First, none of the

daughter particles can be bigger than the mother particle. This leads

to the second property: the probability of a daughter particle being 

smaller than the mother particle is equal to one. Finally, if there is

mass conservation, then the sum of the masses of all daughter 

 particles was equal to the mother particle mass. This last property is

represented by Eq. (7).

zðmÞZ m

0

P ðmujmÞdmu¼ m ð7Þ

4. A simple PBE model for a bubble column

Considering that the liquid phase is stagnant (no liquid

circulation), the column height is sufficiently small to neglect

the effect of gas expansion on the bubble ascension velocities, f 

varies only along the axial direction of the column, and also

assuming steady-state conditions, Eq. (1) simplifies to

U  z df 

dz ¼ H  ð8Þ

where U  z  is the bubble ascension velocity. In our model, U  z  was

assumed equal to the terminal velocity, which is a function of the

the bubble diameter. The value of  U  z  was calculated from the

following balance between the drag and the buoyancy forces:

p

6d3 g 

ðrÀr g 

ÞÀF D

¼0

ð9Þ

where the drag force was given by

F D ¼ p

8d2rðU  z Þ2C D

dh

d

2

ð10Þ

The ratio between the hydraulic bubble diameter, dh, and the

equivalent bubble diameter (diameter of the sphere with the

bubble volume), d, was estimated using the correlation recom-

mended by Clift et al. (1978). The drag coefficient was given byC D ¼ C D,isoð f 1 f 2ÞÀ2 ð11Þwhere C D,iso is the drag coefficient for an isolated bubble,

calculated by the drag correlation of  Karamanev (1994), f 1 is the

correction factor for the population effect given by Behring

(1936), and f 2 is the wall correction factor given by Clift et al.

(1978). Further details can be found elsewhere (Ribeiro and Lage,

2004b,c; Ribeiro et al., 2005).

The breakage model consists of defining the functions zðmÞ,P ðmujmÞ and b(m). In some works, OðmujmÞ zðmÞP ðmujmÞbðmÞ is

modeled instead, being defined as the specific rate of production

of daughters of mass mu by the breakage of particles with mass m.

From the P ðmujmÞ properties, it is possible to write that

bðmÞ ¼ ½1=zðmÞ R m0 OðmujmÞdm

u. In order to model the coalescence,it is only necessary to define Q ðm,muÞ.

4.1. Breakage models

All the analyzed models assumed zðmÞ ¼ 2, although the

Martınez-Bazan et al. (1999a,b) model can be generalized to

zðmÞ42.

The breakage models were based on two theories. Breakage

occurs due to collisions between particles and turbulent eddies or

due to deformation induced by interaction with the fluid flow

(Lasheras et al., 2002).

The first theory encloses the breakup models of  Coulaloglou

and Tavlarides (1977), Prince and Blanch (1990), Luo and

Svendsen (1996), Hagesaether (2002), Lehr et al. (2002), Wanget al. (2003), Andersson and Andersson (2006) and others.

Coulaloglou and Tavlarides (1977), Prince and Blanch (1990)

and Andersson and Andersson (2006) only modeled the breakage

frequency of particles. Coulaloglou and Tavlarides (1977) and

Prince and Blanch (1990) suggested statistical models forP ðmjmu, y Þ. Luo and Svendsen (1996), Hagesaether (2002), Lehr

et al. (2002) and Wang et al. (2003) models suffer from the

hypothesis associated with the eddy sizes which are considered to

promote particle rupture. However, the model of  Luo and

Svendsen (1996) is still used in the literature (Podila et al.,

2007; Frank et al., 2007; Bhole et al., 2008) and is the only

breakage model implemented in CFX s 12 (ANSYS Inc., 2009).

For comparison with experimental data, only the models of 

Luo and Svendsen (1996) and Martınez-Bazan et al. (1999a,b)were considered. The former was included due to historical

reasons and the latter was considered the most consistent model

available (Lasheras et al., 2002).

The model of  Luo and Svendsen (1996) is given by

bðmÞ ¼ K 10:923ð1ÀfÞ e

d2

1=3Z 1

xmin

ð1þxÞ2

x11=3

Z 1=2

0expðÀwc Þ d f v dx

ð12Þand

P ðm1jmÞ ¼

1

2m

R 1xmin

ð1þxÞ2

x11=3expðÀwc Þ dx

R 1xmin

ð1þxÞ2

x11=3 R 1=2

0 exp

ðÀwc 

Þd f v dx

ð13Þ

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with

wc  ¼12C  f s

bre2=3d5=3x11=3

ð14Þ

C  f  ¼ f 2=3

v þð1À f vÞ2=3À1 ð15Þ

xmin ¼C x

l

d ð16Þwhere K 1 is the correction factor for the breakage rate, which has

to be estimated. In this model, K 1 represents a correction for the

bubble-eddy collision rate and it would be equal to 1 if this model

were perfect.

Martınez-Bazan et al. (1999a,b) defined the breakage fre-

quency as

bðmÞ ¼ K 1

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibðedÞ2=3À12

s

rd

r d

ð17Þ

where K 1 is the correction factor for the breakage rate, whose

value was experimentally determined by Martınez-Bazan et al.

(1999a) to be equal to 0:2570:03 in the case of air bubbles in

water, and

P ðm1jmÞ ¼

D1

3m1½D2=3

1 ÀD5=3c  ½ð1ÀD3

1Þ2=9ÀD5=3c  R Dmax

Dmin½D2=3ÀD

5=3c  ½ð1ÀD3Þ2=9ÀD

5=3c  dD

, D1A½Dmin,Dmax

0, D1=2½Dmin,Dmax

8>>>><>>>>:

ð18Þwhere D1 ¼ d1/d, Dmin ¼ dmin=d, Dmax ¼ dmax=d and Dc  ¼ dc /d,

being d the diameter of the mother bubble, d1 the diameter of the

smallest daughter and dc  is the critical diameter, which is given by

dc  ¼ 12s

br

3=5

eÀ2=5 ð19Þ

The minimum diameter of a daughter bubble, dmin is given by

dmin ¼ ð12s=ðbrdÞÞ3=2eÀ1 ð20Þ

Therefore, the maximum diameter of a daughter bubble, dmax, is

dmax ¼ d 1À dmin

d

3" #1=3

ð21Þ

4.2. Coalescence models

The coalescence frequency of the particles with mass m and mu

is usually decomposed into two independent functions: thecollision frequency, y, and the coalescence efficiency (or coales-

cence probability), Z. Considering these, Q ðmu,mÞ could be

expressed by

Q ðmu,mÞ ¼ yðmu,mÞZðmu,mÞ ð22Þ

The coalescence efficiency is defined as the conditional

probability of occurring coalescence considering that a collision

has already occurred. Basically, it depends on the result of a two-

stage process: the draining of liquid film between the two

particles and the rupture of this film that occurs when its

thickness reaches a critical limit. If the collision time between the

particles is not long enough to allow the film thickness to reach

this limit, coalescence does not occur and the particles bounce.

4.2.1. Collision frequency

Basically, there are three different sources of relative bubble

motion: motion induced by turbulence, motion induced by mean-

velocity gradients and body-force-induced motion, arising from

different bubble slip velocities or wake interactions.

The motion induced by turbulence in the continuous phase is

always present in bubble columns as long as turbulent flow

prevails.

In contrast, the motion induced by mean-velocity gradients issignificant only in small-diameter bubble columns (Kamp et al.,

2001). The only coalescence model for this type of bubble collision

is the one given by Prince and Blanch (1990). However, according

to Prince and Blanch (1990), collisions due to laminar shear occur

if the bubble column operates in the heterogeneous regime, for

which liquid circulation does exist. Since the heterogeneous

regime is associated with gas superficial velocities larger than

4 cm/s for the air–water system, this mechanism was not

considered in the present work.

Finally, the motion induced by body forces has been con-

sidered in the past by two different reasonings. Prince and Blanch

(1990) and Lehr and Mewes (2001) considered that buoyancy is

responsible for the relative bubble motion. However, their models

are not correct because they consider the same pair of particles

twice disregarding the magnitude of the speed of each particle

(Mitre, 2006). Besides, it seems that the relative bubble motion

that generates more bubble coalescence is not caused by buoy-

ancy. In fact, bubble wake interactions are considered to be

responsible for it (Otake et al., 1997; Wu et al., 1998; Colella et al.,

1999; Hibiki and Ishii, 2000; Rafique and Dudukovic, 2004; Wang

et al., 2005b).

Wu et al. (1998) and Hibiki and Ishii (2000) developed models

for the collision frequency due to wake interactions. However,

theses models cannot be directly applied to population balance

models because they were developed for interfacial area models

with one or two bubble sizes. Wang et al. (2005b) generalized the

model of  Wu et al. (1998), but the resulting model has a serious

inconsistency because it violates the principle of symmetry of the

coalescence frequency (Ramkrishna, 2000). Therefore, there is stillno appropriate model for coalescence due to collisions induced by

buoyancy. Moreover, this type of bubble interaction is only

relevant in the heterogeneous regime. Thus, it was not considered

in the present work either.

There are two main models of turbulence-induced collisions in

the literature: those given by Prince and Blanch (1990) and Kamp

et al. (2001).

Prince and Blanch (1990) stated the following expression for

the turbulent-induced collision rate:

yðmu,mÞ ¼ K 2S uðdsÞ ð23Þ

where S  is the cross-section collision area, S ¼ ðp=4ÞðdþduÞ2, and

K 

2

is a correction factor, theoretically equal to 1. Originally, Princeand Blanch (1990) mistakenly wrote S  using the radius instead of 

the diameters, which was properly corrected here. The fluctuation

velocity, u(ds) is given by

uðdsÞ ¼ ffiffiffi

2p 

½ðedÞ2=3 þðeduÞ2=31=2 ð24Þ

For Kamp et al. (2001), u(ds) is a characteristic velocity

between two points separated by the distance ds, which is the

mean diameter, dm ¼ ðdþduÞ=2. Therefore, uðdsÞ ¼ ðedmÞ1=3.

In the model of Kamp et al. (2001) the collision frequency was

given by

y

ðmu,m

Þ ¼2

8

3p

1=2 C t  ffiffiffiffiffiffiffiffiffiffi1:61p 

S 

ðedm

Þ1=3

ð25

Þ

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where the parameter C t  considers the difference between the

fluctuation velocity of each phase. For the air–water system it was

given by

C 2t  ¼ 9þ43:2n=ðe1=3d4=3m Þ

1þ43:2n=ðe1=3d4=3m Þ

ð26Þ

and K 2 is a correction factor, theoretically equal to 1.

4.2.2. Coalescence efficiency

In general, bubble sizes are assumed to be in the inertial

subrange of the turbulence spectrum and the film drainage model

assumes that bubbles are either rigid or deformable fluid particles

with fully mobile interfaces. Taking this into consideration, the

coalescence efficiency for the air–water system in the absence of 

any solute can be generalized by the expression (Chesters, 1991):

Z¼ exp½ÀK E ðWeÞ0:5 ð27Þwhere We is a relevant Weber number. Three models of 

coalescence efficiency were selected: Prince and Blanch (1990),

Kamp et al. (2001) and Luo (1993) models. All these models are

quite similar and could be expressed in the form of Eq. (27) with

different expressions for K E 

and We.For rigid particles, Prince and Blanch (1990) stated

Z¼ exp ÀK 3

ðdeq=2Þ3eð2=3Þr16s

!1=2

lnðh0=h f Þ

ðdeq=2Þ2=3

2666664

3777775

¼ exp ÀK 3lnðh0=h f Þ

3:5636

We

2

0:5" #

ð28Þ

where h0 is the initial thickness of the film formed between the

bubbles, h0¼10À4 m, (Kirkpatrick and Lockett, 1974), h f  is the

final thickness of the film formed between bubbles, h f ¼

10À8 m,

(Kim and Lee, 1987) and We is defined as We rðedeqÞ2=3deq=2s,where deq is the equivalent diameter. Luo (1993) defined the

coalescence efficiency as

Z¼ exp ÀK 3½0:75ð1þB2Þð1þB3Þ1=2

ðrd=rþ0:5Þ1=2ð1þB3ÞðWeÞ0:5

" #ð29Þ

where the Weber number is defined as above.

Kamp et al. (2001) stated that

Z¼ exp ÀK 3WeK 

C vm

0:5" #

ð30Þ

where WeK  is the Weber number based on the relative velocity of 

the bubbles at the onset of deformation, V 0, WeK  rV 20 deq=2s,

given by V 0 ¼ ðC t = ffiffiffiffiffiffiffiffiffiffi

1:61p  Þ½eðdþduÞ=21=3

and C vm is the virtual masscoefficient given by

C vm ¼ LN ÀM 2

LÀ2M þN 

1

d3eq

ð31Þ

where the coefficients L, M  and N  are geometrical parameters

given by convergent series of the diameters of the particles (Kamp

et al., 2001).

In all of these models, K 3 is the correction factor for the

coalescence efficiency, being an experimental parameter of unity

order.

It is important to emphasize that each model of coalescence

efficiency was used only with its respective model of collision

frequency, except the model of  Luo (1993) that also employs the

Prince and Blanch (1990) collision frequency model.

5. Numerical procedure

Several numerical methods have been developed for the

solution of the population balance equation. Among them, two

types are commonly employed: the moment based methods

(McGraw, 1997; Marchisio and Fox, 2005) and the sectional

methods (Ramkrishna, 2000; Campos and Lage, 2003). This work

used the method of classes of  Kumar and Ramkrishna (1996). By

this method, only two population properties can be exactlyconserved, which are usually taken as the zeroth and first order

moments.

The sectional moment N k includes all the particles with

properties between mk and mk+1 that are represented by the

pivot xk ðmko xkomk þ1Þ and is defined by

N kð z Þ ¼Z mk þ 1

mk

 f ðm, z Þdm, k ¼ 1 . . . n ð32Þ

where n is the number of classes.

Following Kumar and Ramkrishna (1996), the method of 

classes was applied to (8) leading to the following expression:

U  z k

dN k

dz  ¼ XiZ j

i, j xkÀ1 r ð xi þ x j Þr xk þ 1ð

1

À0:5dij

ÞCijkQ i, jN iN  j

ÀN k X

n

i ¼ 1

N iQ i,k

þXn

i ¼ k

jk,ibiN iÀN kbk ð33Þ

where Ci, j,k and jk,i are defined as

Ci, j,k ¼

 xk þ 1Àð x j þ xiÞ xk þ 1À xk

for xkrð x j þ xiÞr xk þ 1

ð x j þ xiÞÀ xkÀ1

 xkÀ xkÀ1for xkÀ1rð x j þ xiÞr xk

8>>><>>>:

ð34Þ

jk,i ¼Z xk þ 1

 xk

 xk þ1Àm

 xk þ 1À xkP ðmj xiÞ dmþ

Z xk

 xkÀ1

mÀ xkÀ1

 xkÀ xkÀ1P ðmj xiÞ dm ð35Þ

Further details about these equations can be found in Ramkrishna(2000) and Campos and Lage (2003).

The breakage models included one empirical parameter, K 1,

whereas the coalescence models have two empirical parameters,

K 2 and K 3. These parameters were estimated by means of the

maximum likelihood method (Bard, 1974; Anderson et al., 1978).

According to this method, the experimental error of all measured

variables is assumed to follow the normal distribution, and the

estimated parameters are those which minimize the objective

function Z  given by

 Z ¼XNE i ¼ 1

ð$mi À$iÞT V À1

i ð$mi À$iÞ ð36Þ

where V i is the matrix of covariance of the measured variables intest i, $mi and $i are the values calculated by the population

balance model. The minimization problem was solved by the

particle swarm method (Kennedy and Eberhart, 1995) followed by

a direct Newton method for refining the solution utilizing the

software ESTIMA (Pinto et al., 1990; Pinto, 1999; Schwaab et al.,

2008).

6. Results

6.1. Experimental results

After measuring the size of each particle and its respective

error, it was necessary to express these data in terms of the

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relative frequency, G, and its corresponding error. This was done

following the procedure detailed by Ribeiro and Lage (2004a).

At each experimental condition, two data sets consisting

of around 500 bubble sizes with their measurement error

were generated. Since bubble contours were manually traced,

each data set was obtained with a different operator to properly

check for reproducibility. The bubble sizes were discretized into

five classes. When a larger number of classes was used, the errors

in relative frequency increased too much to allow modeldiscrimination.

The reproducibility of the replicate experiments is illustrated

in Fig. 2 by the individual data sets for uG ¼ 2.0 cm/s at the lowest

height. Similar agreement between the replicates was observed

for all the other experimental conditions. Considering that the

two data sets for each experimental condition actually represent

different estimates of the same quantity, the bubble size values

used in their determination were combined to obtain a single data

set based on a higher number of bubbles (above 1000 bubbles).

These results are shown in Fig. 3 and were used in the parameter

estimation procedure.

Since the experimental results were for relative frequency, it

was necessary to relate this frequency with the sectional moment.

This relation is given by

N i ¼ N T Gi ð37aÞ

N T  ¼f

p

6d

3

30

ð37bÞ

d3

30 ¼Z 1

0d3GðdÞddffi

Xn

i ¼ 1

d3i Gi ð37cÞ

where N T  is the total particle number density, f is the global gas

hold-up and d30 is the volumetric mean bubble diameter. At each

gas superficial velocity, the value of f was measured by the height

expansion method and it is given in Table 1. The referred table

also includes the values of  d30 obtained from the bubble size

distributions at the lowest column height. It is clear that the mean

bubble size increased 20% for a 90%-increase in the gas superficial

velocity. If the Gaddis and Vogelpohl (1986) model for bubble

formation at submerged orifices is used to predict the mean

bubble size at the gas distributor, the estimated bubble diameters

are 3.6 and 4.3 mm for uG ¼ 2.0 and 3.8 cm/s, respectively, which

also implies a 20% increase with the change in gas superficial

velocity. These diameters are different from those listed in Table 1,

indicating that bubble coalescence takes place between the gas

distributor and the first measurement height.

6.2. Simulation results and parameter estimation

6.2.1. Additional considerations about the simulation process

The simulations were carried out by using the size distribution

at the smallest column height as the initial condition of the

problem.

The model consists of Eqs. (33)–(35) and the selected models

represented by

Eqs. (12) and (13) for the breakage model of  Luo and Svendsen

(1996),

Eqs. (17) and (18) for the breakage model of  Martınez-Bazan

et al. (1999a,b),

Fig. 2. Relative frequency of bubble diameters for uG ¼ 2.0 cm/s at the base height.

Different symbols refer to replicates.

Fig. 3. Relative frequency of bubble diameters with five classes: (a) with

uG¼2.0 cm/s and (b) with uG¼3.8 cm/s. More than 1000 bubbles were analyzed

for each height.

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Eqs. (22), (23) and (28) for the coalescence model of  Prince and

Blanch (1990),

Eqs. (22), (25) and (30) for the coalescence model of  Kamp

et al. (2001) and

Eqs. (22), (23) and (29) for the coalescence model of  Luo

(1993).

The evolution of the bubble size distribution along the height

was simulated and compared with the experimental data at thetwo highest column positions.

In the simulations, the global variables f and e were considered

instead of the local variables required by the models. The specific

rate of dissipation of turbulent energy, e, was assumed to be the

input power to the small height bubble column, as given

approximately by e¼ uG g  (Bhavaraju et al., 1978).

In the parameter estimation procedure, the errors in uG, f andN i were all considered.

6.2.2. Results of parameter estimation

The lower superficial velocity was chosen to be small enough

to render unlikely bubble breakage by turbulence. This was

confirmed by the Martınez-Bazan et al. (1999a,b) model that

predicts that all existing bubble sizes are stable under suchconditions. Therefore, only coalescence needed to be considered

for the data associated with uG ¼ 2.0 cm/s, and accordingly, only

the two parameters K 2 and K 3 were estimated for these data. The

experimental data for the two highest column heights were used

first alone and then together to fit these model parameters.

Parameter K 3 could not be obtained with statistical signifi-

cance for the present experimental data. In other words, K 3 was

always estimated with errors that were larger than the K 3 value

itself. This is due to the high coalescence efficiency (generally

higher than 0.9) under these experimental conditions. Since the

coalescence efficiency is close to its limiting value of one, any

value of  K 3 in the order of magnitude of its standard value can

lead to a successful data regression when both K 2 and K 3 are

estimated. Therefore, K 3 was set to its standard value in each

model and kept constant for the other parameter estimation

attempts.Using the values estimated for K 2 of each model, all simulated

results are very similar to each other because all models are

capable of representing the experimental data with fitted para-

meters. For instance, Fig. 4 shows the results for the normalized

bubble distribution function obtained using the coalescence model

of Prince and Blanch (1990) with K 2 fitted considering the data for

the two highest column heights simultaneously, which gives the

worst agreement with the experimental data. This figure repro-

duces the trends observed for all comparisons between calculated

and experimental data in this work.

Since all analyzed models could represent the data with

appropriate parameter values, it was necessary to analyze the

results statistically to draw conclusions regarding the perfor-

mance of these models. Table 2 shows the fitted K 2 values. The

objective function, Z , was almost constant for the parameter

estimation of a given data set. This means that it was not possible

to judge the adequacy of each model simply by comparing these

simulated results with the experimental data. This is an evidence

of the good representation of the experimental results by all

models.

Table 2 shows that, contrary to the model of  Kamp et al. (2001)

and Luo (1993), the Prince and Blanch (1990) model behaved

inconsistently, since the K 2 values fitted using the base-top height

and base-both heights data did not agree within the error ranges.

According to Prince and Blanch (1990), K 2 is close to one.

Nevertheless, the estimates were all in the 10À2–10À3 range.

Using the data for the higher gas superficial velocity, uG¼3.8

cm/s, parameter estimation was carried out for the parameters K 2

(in the coalescence models) and K 1 (in the breakage models). Thevalues obtained for K 2 and K 1 in this case are, respectively, listed

in Tables 3 and 4.

It is possible to observe from Table 3 that the Prince and

Blanch (1990) coalescence model has indeed low internal

consistency. For both breakage models, the K 2 value fitted

using base-middle height and base-top height data in Table 3

did not agree within the error range. This lack of agreement also

occurred for the other two coalescence models but only when

they were used with the Luo and Svendsen (1996) model.

When the Martınez-Bazan et al. (1999a,b) breakage model was

combined with either Kamp et al. (2001) or Luo (1993) models,

the six fitted K 2 values for the two values of  uG using the

three combinations of the data sets (base-middle height, base-top

height, base-both heights) agreed within the error range.Thus, when using the model of  Luo and Svendsen (1996), Table 3

shows that all coalescence models had estimates for K 2 using

the base-middle height and base-top height data that did not

Fig. 4. Comparison between experiment (uG ¼ 2.0 cm/s) and simulation using the

model of  Prince and Blanch (1990) with parameters fitted considering the two

heights simultaneously. The error bars for both experimental and simulated data

are shown.

 Table 2

Estimates for K 2 for the coalescence models using the data for uG¼2.0 cm/s.

Coalescence models Base-middle height Base-top height Base-both heights

ðK 27eÞ Z  ðK 27eÞ Z  ðK 27eÞ Z 

Kamp et al. (2001) ð3:072:8Þ Â 10À3 48 ð7:075:8Þ Â 10À3 22 ð5:074:5Þ Â 10À3 28

Luo (1993) ð1:570:7Þ Â 10À3 49 ð3:072:1Þ Â 10À3 24 ð3:573:0Þ Â 10À3 29

Prince and Blanch (1990)

ð8:475:6

Þ Â10À3 48

ð2:071:2

Þ Â10À2 22

ð13:076:0

Þ Â10À3 28

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agree within their error ranges, indicating that this breakage

model is inadequate.

From Table 4, it can be seen that similarly to the K 2 values, the

estimate for K 1 for the model of  Luo and Svendsen (1996) was

much smaller than the expected value, being in the ð1À2Þ Â 10À3

range. Unfortunately, for uG¼3.8 cm/s data, the model of 

Martınez-Bazan et al. (1999a,b) predicts that there would be

breakage only for the largest bubble class. Therefore, small errors

in the relative frequency of this bubble class can lead to large

errors in the parameter estimate. Despite this limitation, most K 1

estimates for this model agree within their error ranges with the

value of 0:2570:03 reported by Martınez-Bazan et al. (1999a,b).

Furthermore, when using the Martınez-Bazan et al. (1999a,b)

model together with either Kamp et al. (2001) or Luo (1993)

models, the K 1 values fitted for base-middle height and base-top

height data did not agree in their error ranges because of the

small intensity of the breakage phenomena.

Finally, the values of both parameters (K 1 and K 2) were

adjusted using the experimental data obtained with uG¼2.0 and

3.8 cm/s simultaneously. The fitted values of  K 2 and K 1 are given

in Tables 5 and 6, respectively.

Tables 7 and 8 show some statistical functions related to the

estimates of  K 1 and K 2 using only the data for uG

¼3.8cm/s and

the data for both superficial velocities, respectively.The correlation coefficient, a, is defined as the ratio of the

covariance between two parameters to the product of the

standard deviation of both parameters. The lower the value of 

the correlation coefficient, the lower is the correlation between

the two parameters. There is a 95% probability of the parameters

be uncorrelated if the value of the correlation coefficient is below

0.1. Tables 7 and 8 show that there is no linear correlation

between the parameters and that the objective function has

always small values. This means that the parameters are

independent and relevant to the model.

The mean value of  K 1 for the model of  Luo and Svendsen

(1996) does not present significant variations in Tables 4 and 6,

but its errors in the latter are very large, being equal to or larger

than the parameter value itself.

 Table 4

Estimates for K 1 using the data for uG¼3.8 cm/s.

Using these coalescence models Base-middle height Base-top height Base-both heights

K 17e K 17e K 17e

Considering the breakage model of  Luo and Svendsen (1996)

Kamp et al. (2001)

ð1:670:6

Þ Â10À3

ð2:170:4

Þ Â10À3

ð1:670:4

Þ Â10À3

Luo (1993) ð1:170:4Þ Â 10À3 ð1:970:4Þ Â 10À3 ð1:870:4Þ Â 10À3

Prince and Blanch (1990) ð1:170:5Þ Â 10À3 ð1:870:4Þ Â 10À3 ð1:870:4Þ Â 10À3

Considering the breakage model of  Martınez-Bazan et al. (1999a,b)

Kamp et al. (2001) 0.6070.30 0.1670.09 0.4270.30

Luo (1993) 0.5770.25 0.1670.11 0.4170.21

Prince and Blanch (1990) 0.5270.30 0.7670.40 0.5870.32

 Table 5

Estimates for K 2 using the data for uG¼2.0 and 3.8 cm/s.

Coalescence models Base-middle height Base-top height Base-both heights

K 27e K 27e K 27e

Considering the breakage model of  Luo and Svendsen (1996)

Kamp et al. (2001) ð3:370:2Þ Â 10À3 ð3:971:0Þ Â 10À3 ð4:171:1Þ Â 10À3

Luo (1993) ð1:570:4Þ Â 10À3 ð1:571:1Þ Â 10À3 ð1:572:0Þ Â 10À3

Prince and Blanch (1990) ð9:174:0Þ Â 10À3 ð1278Þ Â 10À3 ð1277Þ Â 10À3

Considering the breakage model of  Martınez-Bazan et al. (1999a,b)

Kamp et al. (2001) ð3:670:2Þ Â 10À3 ð3:970:7Þ Â 10À3 ð4:071:0Þ Â 10À3

Luo (1993) ð2:170:4Þ Â 10À3 ð2:471:5Þ Â 10À3 ð372Þ Â 10À3

Prince and Blanch (1990) ð774Þ Â 10À3 ð1074Þ Â 10À3 ð874Þ Â 10À3

 Table 3

Estimates for K 2 using the data for uG¼3.8 cm/s.

Coalescence models Base-middle height Base-top height Base-both heights

K 27e K 27e K 27e

Considering the breakage model of  Luo and Svendsen (1996)

Kamp et al. (2001) ð3:770:2Þ Â 10À3 ð1:670:7Þ Â 10À3 ð2:870:9Þ Â 10À3

Luo (1993) ð1:570:4Þ Â 10À3 ð6:571:4Þ Â 10À3 ð4:071:8Þ Â 10À3

Prince and Blanch (1990)ð9:874

:0Þ Â

10À3

ð4:270

:8Þ Â

10À3

ð6:170

:7Þ Â

10À3

Considering the breakage model of  Martınez-Bazan et al. (1999a,b)

Kamp et al. (2001) ð4:270:7Þ Â 10À3 ð2:970:8Þ Â 10À3 ð3:670:7Þ Â 10À3

Luo (1993) ð2:770:9Þ Â 10À3 ð1:870:5Þ Â 10À3 ð2:370:7Þ Â 10À3

Prince and Blanch (1990) ð3:670:8Þ Â 10À3 ð5:970:5Þ Â 10À3 ð4:070:5Þ Â 10À3

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The parameter of the model of Martınez-Bazan et al. (1999a,b)

does not present significant differences when its values shown in

Tables 4 and 6 are compared. This was expected because thismodel does not predict bubble breakage for uG ¼ 2.0 cm/s.

From Table 5, it can be seen that all six estimates of  K 2 for all

coalescence models agree within their error ranges. However,

their relative errors are quite different. Luo (1993) model showsK 2 relative errors that range from 20% to 133%, while Prince and

Blanch (1990) model shows relative errors around 40–70%.

Finally, Kamp et al. (2001) model shows relative errors smaller

than 27%.

7. Conclusions

This work presents a critical analysis by parameter estimation

of some breakage and coalescence models commonly applied to

bubble columns. Experimental data were obtained in a bubble

column for two different gas superficial velocity at three height

positions. The largest superficial velocity is almost at the end of 

the homogeneous bubbling regime for the air–water system.

The coalescence models of  Prince and Blanch (1990), Luo

(1993) and Kamp et al. (2001) and the breakage models of Luo and

Svendsen (1996) and Martınez-Bazan et al. (1999a,b) were

analyzed.

Among the coalescence models, Kamp et al. (2001) model wasfound to be more consistent. Its parameter was shown to be

independent of the experimental conditions and its relative error

was the smallest among the parameter errors of all coalescence

models.

Although the evaluation of the breakage models was impaired

by the little intensity of breakage in the experimental conditions,

the Martınez-Bazan et al. (1999a,b) model behaved consistently

when all the available data were used in the parameter

estimation. Besides, its parameter value was in agreement with

the value supplied by these authors in very different experimental

conditions.

Although the experimental data were for a limited range of the

gas superficial velocity, the coalescence model of  Kamp et al.

(2001) with K 2

¼ ð471Þ Â 10À3

and the breakage model of Martınez-Bazan et al. (1999a,b) with K 1 ¼ 0:470:3 are recom-

mended for bubble columns operating with air–water in the

homogeneous bubbling regime.

Nomenclature

b(m) breakage frequency of the particle with mass m, sÀ1

BB production of particles by breakage, mÀ3kgÀ1 sÀ1

BC  production of particles by coalescence, mÀ3 kgÀ1 sÀ1

C x parameter between 11.4 and 31.4 (21.4 was used in

the present work), dimensionlessC D drag coefficient, dimensionless

C  f  coefficient associated with the increase in area duethe breakage process, dimensionless

C t  parameter that considers the difference between the

fluctuation velocity of each phase, dimensionless

C vm virtual mass coefficient, dimensionlessd equivalent bubble diameter (diameter of the sphere

with the same volume of the bubble), m

d30volumetric mean diameter, m

dh hydraulic bubble diameter, md1 diameter of the smallest daughter particle, m

deq equivalent diameter, 1=deq ¼ 1=duþ1=d, m

D1 diameter of the smallest daughter, D1¼d1/d,

dimensionlessDB destruction of particles by breakage, mÀ3 kgÀ1 sÀ1

DC  destruction of particles by coalescence, mÀ3

kgÀ1

sÀ1

 Table 6

Estimates for K 1 using the data for uG¼2.0 and 3.8cm/s.

Using these coalescence models Base-middle height Base-top height Base-both heights

K 17e K 17e K 17e

Considering the breakage model of  Luo and Svendsen (1996)

Kamp et al. (2001) ð1:070:6Þ Â 10À3 ð1:072:0Þ Â 10À3 ð1:072:0Þ Â 10À3

Luo (1993) ð1:070:9Þ Â 10À3 ð1:071:0Þ Â 10À3 ð1:072:0Þ Â 10À3

Prince and Blanch (1990)ð1:071

:0Þ Â

10À3

ð2:072

:0Þ Â

10À3

ð1:071

:0Þ Â

10À3

Considering the breakage model of  Martınez-Bazan et al. (1999a,b)

Kamp et al. (2001) 0:670:3 0:370:1 0:470:3

Luo (1993) 0:570:2 0:370:1 0:470:2

Prince and Blanch (1990) 0:570:3 0:270:1 0:570:3

 Table 7

Objective function, Z , and correlation coefficient, a, in the parameter adjustment

using the data for uG¼3.8 cm/s.

Coalescence models Base-middle

height

Base-top

height

Base-both

heights

a Z  a Z  a Z 

Considering the breakage model of  Luo and Svendsen (1996)

Kamp et al. (2001) 0.03 2 0.02 5 0.02 8

Luo (1993) 0.02 1 0.03 5 0.03 9

Prince and Blanch (1990) 0.02 1 0.03 6 0.03 9

Considering the breakage model of  Martınez-Bazan et al. (1999a,b)

Kamp et al. (2001) 0.03 1 0.02 5 0.02 8

Luo (1993) 0.02 1 0.03 5 0.03 9

Prince and Blanch (1990) 0.02 1 0.03 5 0.02 8

 Table 8

Objective function, Z , and correlation coefficient, a, in the parameter adjustment

using the data for uG¼2.0 and 3.8cm/s.

Coalescence models Base-middle

height

Base-top

height

Base-both

heights

a Z  a Z  a Z 

Considering the breakage model of  Luo and Svendsen (1996)

Kamp et al. (2001) 0.03 7 0.01 11 0.02 10

Luo (1993) 0.02 5 0.03 12 0.01 11

Prince and Blanch (1990) 0.03 6 0.03 10 0.02 10

Considering the breakage model of  Martınez-Bazan et al. (1999a,b)

Kamp et al. (2001) 0.03 2 0.02 5 0.02 8

Luo (1993) 0.02 1 0.03 5 0.03 9

Prince and Blanch (1990) 0.02 1 0.03 5 0.02 8

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Dc  critical diameter, Dc ¼dc /d, dimensionlessdm mean diameter, dm ¼ ðdþduÞ=2, mDmax maximum size of a daughter, Dmax ¼ dmax=d,

dimensionlessDmin minimum size of a daughter, Dmin ¼ dmin=d,

dimensionless

e error for a 95% confidence interval of a given variable,

dimensionless

 f  particle number distribution function, mÀ3

kgÀ1

 f 1 drag coefficient correction for the population effect,

dimensionless f 2 drag coefficient correction for the wall effect,

dimensionless f v volume fraction of the smallest daughter particle,

 f v¼(d1/d)3, dimensionlessF D drag force, kgm sÀ2

 g  acceleration of gravity, m sÀ2

G relative frequency, dimensionlessH  particle net production term in the population

balance equation, mÀ3 kgÀ1 sÀ1

h f  final thickness of the film formed between bubbles, mh0 initial thickness of the film formed between bubbles, m

K 

1

correction factor for the breakage rate, dimensionlessK 2 correction factor for the bubble collision rate,

dimensionlessK 3 correction parameter for coalescence efficiency

models, dimensionless

l eddy size, mm particle mass, kgm1 mass of the smallest daughter particle,

m1 ¼ r g ðp=6Þd31, kg

N  sectional zeroth-order moment, mÀ3

n number of classes, dimensionless

Q ðm,muÞ aggregation frequency of the particles with mass m

and mu, m3 sÀ1

Re Reynolds number, dimensionless

S  cross-section collision area into two particles,S ¼ ðp=4ÞðdþduÞ2, m2

t  time, sU  particle velocity, m sÀ1

u(ds) characteristic velocity between two points separated

by distance ds, m sÀ1

U  z  bubble ascension velocity, m sÀ1

uG gas superficial velocity, m sÀ1

We Weber number, dimensionless y  vector of all continuous phase variables that

influence the particle processes

z spatial position vector, m

 Z  objective function, dimensionless

Greek letters

a correlation coefficient, dimensionless

b constant equal to 8.2, dimensionless

wc  critical eddy energy density, dimensionless

e dissipation rate of energy per unit mass, m2 sÀ3

Z coalescence efficiency, dimensionless

r density of continuous phase, kg mÀ3

r g  density of dispersed phase, kg mÀ3

s surface tension, kg sÀ2

B diameter ratio given by B¼ d=du, dimensionless

f gas hold-up, dimensionless

y collision frequency, sÀ1

x size of the eddy, dimensionless

xmin minimum size of the eddy, dimensionless

z mean number of particles daughters, dimensionless

 Acknowledgments

The authors would like to acknowledge the financial support from

CNPq (Grant nos. 485817/2007-1, 301672/2008-3), CAPES, FAPERJ

(Grant no. E-26/152704/2006), FINEP (Grant no. 01.06.1004.00-Ref.

2524/06) and ESSS.

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