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Chemical Engineering Science 61 (2006) 5196 5203

www.elsevier.com/locate/ces

On the influence of the liquid physical properties on bubble volumes andgeneration times

Mariano Martn, Francisco J. Montes, Miguel A. GalnDepartamento de Ingeniera Qumica y Textil, Universidad de Salamanca, Pza. de los Cados 15, 37008 Salamanca, Spain

Received 13 July 2005; received in revised form 29 November 2005; accepted 14 March 2006

Available online 22 March 2006

Abstract

This work presents both theoretical and experimental studies about the specific influence of viscosity, surface tension and density in theformation of a gas bubble. The theoretical model includes both bubble formation and free rising and extends some previous work. As the

previous bibliography provides rather scattered data regarding the effect of the liquid properties on bubble generation times, the results can

be considered a step forward in the understanding of the subject because they separately describe the influence of viscosity, surface tension

and density. The model yields satisfactory results of bubble shapes and formation times when compared to the experimental high-speed video

observations obtained using non-Newtonian solutions of carboxymetyl cellulose in water at different concentrations (0.41.6% w/w). The

complete experimental study also includes a range of different gas flow rates (110 cm3/s) and orifice diameters (1.5 and 2 mm).

2006 Elsevier Ltd. All rights reserved.

Keywords: Bubble phenomena; Bubble volume; Liquid properties; Two-stage model

1. Introduction

Bubble columns and sieve plate reactors are some of the most

typical aeration devices used in the chemical and biochemical

industries. If the bioprocess involves a cell culture that is sen-

sitive to high hydrodynamic stresses (Davis et al., 1986; Hua et

al., 1993), bubble columns and sieve plate towers are the most

adequate aeration devices. During the actual operation in a fer-

menter, most fluid properties change with time. This change is

slow compared to the time of formation of a single gas bubble,

so when modeling the formation and rising of a single bubble,

the fluid properties can be taken as constants. In contrast, when

modeling the formation of a series of bubbles during a largeperiod of time, the variation of the fluid properties with time

must be considered.

One of the most important parameters regarding the

gasliquid mass transfer process is the volume of the gener-

ated bubble. Bubble volume together with the exposed bubble

surface determines the specific area involved in the transfer of

gas to the liquid phase in a bubbling system.

Corresponding author. Tel.: +34 923294479; fax: +34 923294574.E-mail address: [email protected] (F.J. Montes).

0009-2509/$- see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ces.2006.03.027

There are numerous studies regarding the influence of vis-

cosity, surface tension and density in the formation of a gas

bubble (Ramakrishnan et al., 1969; Satyanarayan et al., 1969).

However, the results are a little bit ambiguous. Undoubtedly,

the experimental study of the influence of the fluid properties

in the bubble volume is difficult, because it is impossible to

modify one fluid property without affecting another or even

all the rest. Moreover, the effects of these properties on the

bubble volume are small and, sometimes, opposed to each

other. In this context, the main objective of this work is to

clearly determine the influence of the fluid properties on the

bubble volume and, consequently, on the time of formation

of such bubble. Since this goal is practically unfeasible froman experimental point of view, it was decided to tackle the

problem form a theoretical point of view and later compared

the results with several experiments. In this way, the effect on

the bubble volume of three of the most basic properties of a

fluid, density, viscosity and surface tension was studied sepa-

rately. The theoretical results should clearly point out which

fluid property is responsible for a given variation of the vari-

able under study (generation time-bubble volume), because the

simulations are carried out keeping all fluid properties con-

stant but one. Bubble volume and generation time are directly
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M. Martn et al. / Chemical Engineering Science 61 (2006) 5196 5203 5197

related when the experiments are performed under constant gas

flow rate.

In some fluids, as the value of viscosity increases the flow

behavior changes from Newtonian to non-Newtonian. This fact

is also taken care of in the theoretical approach, in which the

terms corresponding to interfacial friction are modified from a

previous article (Martin et al., 2006) in both the modeling ofbubble expansion while attached to the orifice and the modeling

of free bubble rising. The present model extends some previous

work (Terasaka and Tsuge, 1991, 2001; Li, 1999; Li et al., 2002)

improving the robustness of the differential equation solver and

including the free rising stage.

This paper presents a model including both bubble formation

and free rising in Newtonian and non-Newtonian fluids. The

model yields satisfactory results regarding bubble shapes and

formation times when compared to the experimental high-speed

video observations obtained using water and non-Newtonian

solutions of carboxymetyl cellulose (CMC) in water at dif-

ferent concentrations (0.41.6% w/w). The complete experi-

mental study also includes a range of different gas flow rates

(110 cm3/s) and orifice diameters (1.5 and 2 mm).

2. Theory

The theoretical approach is divided in two parts. First, a

model that applies to Newtonian fluids will reveal the sepa-

rated effects of each of the fluid properties (density, viscosity

and surface tension) on the formation time of the bubble, which

is directly related to the bubble volume when the experiment

is performed at constant flow rate. On top of that, an analysis

based on dimensionless variables will be performed. Second,

a revision of the previous model will extend its applicabilityto the behavior of non-Newtonian fluids. The combination of

both parts mostly covers the model of a process such as the

formation of bubbles in a fermentation broth, where the fluid

properties can change in an appreciable manner as the fermen-

tation process progresses.

2.1. Newtonian model

A Newtonian model intended to predict bubble shapes and

detachment times has already been developed (Martn et al.,

2006). The model consists of two different stages. The first

corresponds to the generation of a gas bubble in a liquid whilestill attached to the orifice. The second corresponds to the period

of free rising previous to the onset of the bubble oscillation.

Both stages are modeled by means of both a momentum bal-

ance, and a force balance. The momentum balance determines

bubble expansion taking into account surface tension, which

is responsible for the stability of the shape of the bubble, and

also considering the viscosity of the liquid, which offers some

degree of resistance to the expansion of the bubble.

The second balance, the force balance, determines the as-

censorial movement of the bubble, considering buoyancy, drag

forces, inertial forces due to the gas flow rate across the orifice

along with those due to the liquid dragged by the bubble as it

rises and gasliquidplate interactions.

Once the bubble has detached, inertial forces due to the ori-

fice and those terms related to the presence of the plate are

modified.

2.2. Non-Newtonian model

2.2.1. Growing stageThe non-Newtonian model is based on a previous Newtonian

model (Martn et al., 2006), being the viscous terms modified

in order to cope with the rheological behavior of the fluid.

Momentum balance has to be reconsidered including the power

law behavior, while drag forces in the force balance have to be

calculated in other terms.

Although it is referred to as the growing stage, both grow-

ing and expansion take place during this phase. Eq. (1) models

the bubble expansion from the orifice using a momentum bal-

ance:

Ljurjt + ur

jur

jr= jPjr + [div ]r , (1)

where

ur (r, t) =

R

r

2 dRdt

. (2)

Substituting Eq. (2) into Eq. (1) and integrating Eq. (1) from

the surface of the bubble to infinity, it yields

L

R

d2R

dt2+ 3

2

dR

dt

2= (PL P)

R

[div]r dr. (3)

The divergence of the shear stress can be described in termsof its three normal components. As the liquid is considered to

be incompressible, the only active component of the stress will

be the radial component. Besides, the pressure at infinity is

equal to the hydrostatic pressure. Under these considerations,

Eq. (3) becomes

L

R

d2R

dt2+ 3

2

dR

dt

2= (PL PH) + rr |r=R

3

R

rr

rdr . (4)

At this stage, bubble expansion is modeled assuming that the

fluid follows the potential law (Li et al., 2002). The shear stress

tensor for the non-Newtonian fluid surrounding the bubble takes

the form

: =

4 R2dR/dt

r 3

3+ 2

2

R2dR/dt

r 3

2

= 24

R2dR/dt

r3

2(5)

and the main component of the shear stress is

rr = 4mf(2

3)

n

1R2dR/dt

r 3

n1R2dR/dt

r3 . (6)
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5198 M. Martn et al. / Chemical Engineering Science 61 (2006) 51965203

Integrating the expression for the shear stress:

3

R

rr

rdr = 4mf(2

3)n1

n

dR/dt

R

n. (7)

So the equation of the momentum balance is

PB PH = L

R d2R

dt2+ 3

2

dRdt

2

+ 2R

+ 4mf(2

3)n1

n

1

R

dR

dt

n. (8)

The hydrostatic pressure (PH) is given by

PH = Patm + Lg(HL z). (9)The pressure inside the bubble (PB ) is time-dependent and

it is determined assuming that the gas is ideal:

PB=

m(t)RgT

PairM VB. (10)

The time change of the mass of gas inside the bubble is cal-

culated using a combination of Bernoullis equation and conti-

nuity throughout the orifice where the bubble is generated:

dm

dt= Cs At

1 (At/A1)2G

PB PC . (11)

The discharge coefficient (Cs ) is equal to 0.6 for orifice plates

(Fox and McDonald, 1995).

The volumetric flow rate entering the bubble is related to the

rate of mass entering the bubble by

dVBdt

= dmdt

1G

. (12)

In order to calculate the pressure in the gas chamber below

the orifice, a mass balance is stated for the chamber:

dPC

dt= PC

VC

QC

dVB

dt

, (13)

where QC is the experimental gas flow rate. The polytropic gas

coefficient for air is = 1.4. Initially, m (t= 0) is equal to themass of a semi-spherical bubble of radius equal to Do/2. The

pressure in the gas chamber at t= 0 is

PC = PB = Patm + LgHL + 4Do

. (14)

It is convenient to transform Eq. (8) into a dimensionless

expression, so the results can be analyzed with more rational-

ity. Then, using the definitions given by Eq. (15), Eq. (16) is

obtained

P = P(/R), t = tDo/u0, R = RDo, (15)

(PB PH)R

= W e

Rd2R

d(t)2+ 3

2

dR

dt

2+ 2

R

+4W e

Re 1

RdR

dtn, (16)

where We andRe are the Weber and Reynolds numbers. Eq. (16)

is solved to obtained R(t), considering that, initially, a semi-spherical bubble of radius equal to Do/2 exists. Previously, Eqs.

(9)(13) are used in order to calculate, PH, PB (t),m(t),VB (t )

and PC (t ). The gas density (G) is considered to be known

from the previous time step.

The growing phenomenon occurring while the bubble staysattached to the orifice is governed by a force balance given by

d

dt

M

dz

dt

= (L G)VB gfc

1

2CD L

dz

dt

2D2max

4

+ 4GD2o

dVB

dt

2(17)

being

M = (G + CL)VB , (18)

where C and fc are equal to 11

16

and 0.005, respectively (Martn

et al., 2006). Regarding the drag under non-Newtonian condi-

tions, some authors use a theoretical result (Terasaka and Tsuge,

1991) while other references rely on a experimental relation

(Kelessidis, 2003) based on the drag coefficient (CD), which

is included in Eq. (17). The latter approach proved to be more

appropriate for this work:

CD = [2.25 Re0.31B + 0.36Re0.06B ]3.45, (19)

ReB =LDmax(Uz)

2n

m.

Considering G>L and noticing that the inertial terms are

negligible, the dimensionless form of Eq. (17) is

d

dt

CVB

dz

dt

= Bo

W eVB fc

1

2CD

dz

dt

2 D2max4

, (20)

where Bo is the Bond number and

Dmax = DmaxDo, VB = VB D3o . (21)

Solution of Eq. (20) provides, dz/dt which is then used toupdate the coordinates of the gas bubble, as it is explained in

detail elsewhere (Martn et al., 2006).

2.2.2. Rising stage

Once the bubble has detached from the orifice, the free rising

stage is modeled based on Eq. (20), but now fully applying the

buoyancy term (fc = 1). Eq. (18) provides M, being C = 0.5in this case (Oguz and Prosperetti, 1993). Eq. (20), uses the

definition of CD given by Eq. (19). Eq. (16) is also used at

this stage to account for the changes in bubble shape due to

variations of pressure.

The scheme for the solution of the non-Newtonian model

follows the same structure than that of the Newtonian model

described elsewhere (Martn et al., 2006). Both, the momen-

tum (Eq. (16)) and the force balances (Eq. (20) with fc

=1),

together with the calculation of the pressure drop across the

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orifice (Eq. (11)) and the mass balance to the gas chamber

in terms of pressure (Eq. (13)) conform a system of coupled

ODEs. The equations are implemented and solved using a com-

bination of routines written in Matlab. As a result, the bub-

ble surface is calculated coupling the effects of expansion and

rising for the r- and z- axis, providing dR/dt and dz/dt

dR/dt is the velocity of the radial expansion of the bubbleand dz/dt is the only component (vertical) of the bubble ris-ing while still attached to the orifice. Then, the global vertical

and horizontal components of the velocity for the first stage of

bubble formation are:

Vertical component:

dzreal

dt= dR

dtabs(cos()) + dz

dt. (22)

Horizontal component:

dr

dt= dR

dtsin(). (23)

3. Materials and methods

Fig. 1 shows the experimental setup used in this work. The

entire structure is built on top of an optical table. The bubble

column is a hand-made small glass cell of 15 15 20 (cm,L WH). There are 8 cm of liquid above the plate where thebubble is generated. The walls of the cell are flat so the video

recordings are not distorted. The volume of the gas chamber be-

low the orifice is 8 cm3. The images are obtained using a Red-

lake Motionscope high-speed video camera, which records

at speeds up to 1000 frames per second. Three different gas

flow rates (1, 5 and 10 cm3

/s) and two orifice diameters (1.5and 2 mm) were used during the experiments. The rheological

properties of the non-Newtonian solutions of CMC (Sigma

C-5678) were characterized experimentally at 20 C. Density is

Fig. 1. Experimental setup: (1) optical table; (2) fiber optics; (3) supporting

structure for the bubble column; (4) magic arm holding the camera; (5)

high-speed video camera; (6) bubble column; (7) air cylinder; (8) rotameter;

(9) valve; (10) illumination source.

Table 1

Physical properties of the aqueous solutions of carboxymetyl cellulose

CMC (% w/w) (kg/m3) (N/m) mf n

1.2 1003 0.068 0.459 0.740

1.4 1003 0.067 0.502 0.750

1.6 1004 0.066 0.581 0.757

measured by picnometry. Surface tension is calculated using a

stalagmometer calibrated with deionized water. Viscosity was

measured using a rotational viscosimeter from Fungilab S.A.

Five different solutions of CMC in water (0.4, 0.8, 1.0, 1.2,

1.4 and 1.6% w/w) were used along the experiments. Just the

three most concentrated solutions showed non-Newtonian be-

havior, verifying the potential law. The physical properties of

those solutions and the coefficients of the potential law are

schematized in Table 1. The two most diluted solutions showed

an intermediate behavior between a Newtonian fluid and non-

Newtonian fluid. That is the reason why the results obtained

from these solutions are not reported in Table 1.

4. Results and discussion

4.1. Newtonian fluids

Table 2 shows the values of the different liquid properties,

orifice diameters and gas flow rates used during the theoretical

simulations of bubbles growing in a Newtonian fluid. In the

simulations, the fluid properties are changed one at a time.

Density has been reported not having any effect on bubble

volumes (Quigley et al., 1955; Siemes and Kaufmann, 1956)

or to slightly decrease bubble volume when density increases

Table 2

Values of the different liquid properties and gas flow rates used during the

theoretical simulations of bubble generation times using a Newtonian fluid

Do (mm) QC (cm3/s) (kg/m3) (mN/m) (mPas)

1.5 750 18.2 1

900 36.5 5

1 998 54.7 10

1125 73.0 25

1250 91.2 50109.5 100

1

18.2 5

750 36.5 10

5 998 73.0 25

1250 109.5 50

100

2.0 1

18.2 5

750 36.5 10

1 998 73.0 25

5 1250 109.5 50

100
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Fig. 2. Effect of density, viscosity and surface tension on bubble generation times. QC = 5 cm3/s and Do = 2mm.

(Coppock and Meiklejohn, 1951; Davidson and Schler, 1960;

Benzing and Myers, 1955). Looking at Fig. 2 it can be con-

cluded that there is a small but positive variation of bubble vol-

ume when density increases. Obviously, the buoyancy effect

increases as density increases but, on the other hand, the mass

of liquid that moves with the bubble during the rising affects

in the opposite way.

The effect of viscosity on bubble volume is almost negligible

when the values of viscosity are close to those of water. When

using values 10 times bigger than the viscosity of water, the

effect starts to be important. That is the reason why the results

of different authors disagree with each other. Some authors

(Coppock and Meiklejohn, 1951; Benzing and Myers, 1955)

report no variations with viscosity. Meanwhile, other references

(Datta et al., 1950) show a negative variation, and some others

(Quigley et al., 1955) a small positive variation or even a large

positive variation (Siemes and Kaufmann, 1956; Davidson and

Schler, 1960).

The theoretical simulations showed in Fig. 2 predict that

for viscosity values close to that of water, the variation is

small but positive. When using values of viscosity 25 times

the viscosity of water at 20 C or larger, the variation of bub-ble volume and generation time is largely positive. Physically,

this effect can be explained observing the flow patterns of a

Fig. 3. Flow patterns inside and outside a growing bubble.

growing bubble while still attached to the orifice as it is shown

in Fig. 3. The streamlines indicate that the constriction or bot-

tleneck that eventually will lead to the detachment of the bubble
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is formed as a consequence of the direction of the movement

of the gasliquid interface. The velocity of the liquid layer that

stays in contact with the gas must equal the velocity of the gas

in its outer layer. This layer of liquid is directed from the top

of the bubble to the orifice, surrounding the shape of the bub-

ble. The dynamic pressure exerted by this streamline of fluid

when it reaches the gasliquidplate point (three phase point)is responsible for the formation of the bottleneck of the bubble.

As viscosity increases, the velocity of the particles contained

in this streamline of liquid will be smaller, allowing the bub-

ble to grow further before the dynamic pressure on the three

phase point is high enough to cut the bubble neck. This is the

reason why, for high-viscosity liquids, long cylindrical-shaped

bubbles are obtained (Terasaka and Tsuge, 2001).

The effect of surface tension presents more complexity. The

model predicts that decreasing surface tension will increase

bubble volume. This is a logical result when thinking that the

bubble tries to minimize its superficial energy when growing

from the orifice. At high surface tensions, the maximum reach-

able bubble surface is quite small. Meanwhile, at low values of

surface tension, the bubble is allowed to grow further. The ex-

perimental results from bibliography are once again scattered.

Some authors report no influence (Quigley et al., 1955; Siemes

and Kaufmann, 1956; Davidson and Schler, 1960). Other au-

thors report increase in bubble volume when increasing surface

tension (Datta et al., 1950; Coppock and Meiklejohn, 1951;

Benzing and Myers, 1955). The main difficulty when analyzing

the effect of surface tension is due to fact that the influence of

some other experimental variables (orifice diameter, plate ma-

terial, gas flow rate) can modulate the influence of surface ten-

sion. For example, at low gas flow rates, the effect of surface

tension is very high, showing an increase of bubble volumewhen surface tension increases. However, at higher gas flow

rates, bubble volume is almost not affected by surface tension

(Ramakrishnan et al., 1969).

Table 3 shows a comparison of experimental and theoretical

detachment times for bubbles generated from different orifice

Table 3

Comparison of experimental and theoretical detachment times for bubbles

generated from different orifice diameters at different gas flow rates in deion-

ized water (20 C, = 998kg/m3, = 0.073N/m and = 0.001037 Pa s)

Do (mm) QC (cm3/s) Experimental

detachment

Theoretical

detachment

time (ms) 1 ms time (ms)1.5 1 20 20

5 20 19

10 19 19

2.0 1 32 30

5 30 30

10 30 29

2.5 1 40 41

5 39 39

10 39 38

3.0 1 53 52

5 47 48

10 50 50

diameters at different gas flow rates in deonized water (20 C, = 998kg/m3, = 0.073N/m and = 0.001037Pa s). Theagreement between the obtained theoretical results and the ex-

perimental data is notorious.

The task of explaining the experimental influence of the liq-

uid physical properties on the generation of a bubble at a sieve

plate is quite complex if one tries to tackle the problem fromthe dimensional point of view. The dimensionless approach in-

troduced in the theoretical section allows to explain the results

of the model based on the Weber and Reynolds numbers, which

constitutes a more rational and straightforward way of looking

at the results.

Reynolds number defines the relationship between inertial

and viscous forces, meanwhile Weber number identifies the

ratio between inertial and surface forces as measurement for

the deformability of the bubble. Fig. 4 shows the influence

of the Reynolds, Weber and Bond numbers on the dimension-

less formation times. Fig. 4 indicates that at low values of

the Reynolds number, there is an appreciable influence on the

dimensionless formation time, decreasing this value when in-

creasing the Reynolds number. For turbulent Reynolds num-

bers, this influence becomes very small. Regarding the Weber

and Bond numbers, the results were plotted against a com-

bination (the product) of those numbers. At low values of

W e Bo, an increase of this number results in a big incrementin the dimensionless formation time. This increase becomes

less sharp at higher values of W e Bo Obviously, the prod-uct W e Bo decreases as the surface tension increases, whichcauses a reduction in the bubble formation time because high

surface tensions strongly constrain the expansion of the growing

bubble.

Fig. 4. Influence of the Reynolds, Weber and Bond numbers on the dimen-

sionless bubble formation time.
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Fig. 5. Theoretical (above) and experimental (below) bubble shapes and generation times of bubbles growing in a non-Newtonian fluid (1.2% aqueous solution

of CMC) with QC = 1 cm3/s and Do = 2mm.

4.2. Non-Newtonian fluids

Both visualization experiments and theoretical simulations of

bubbles generated in non-Newtonian aqueous solutions of CMC

(1.2, 1.6 and 1.8% w/w) were carried out using orifice diametersof 1.5 and 2 mm and gas flow rates of 1, 5 and 10cm3/s.

The experimental results obtained with the solutions containing

0.4 and 0.8% w/w of CMC are just used for determining the

transition of bubble generation times when the fluid evolves

from Newtonian rheology to non-Newtonian rheology. Neither

the Newtonian nor the non-Newtonian model can completely

simulate the experimental bubble shapes obtained with these

two CMC solutions.

Theoretical and experimental results of bubbles shapes and

generation times obtained using the three most concentrated

solutions of CMC present magnificent agreement, as it is shown

in Fig. 5, which was obtained using a 1.2% w/w CMC aqueous

solution, an orifice diameter of 2 mm and a gas flow rate of1 cm3/s.

In general, generation times and bubble volumes increase as

the coefficients of the potential law (n and mf) increase. Re-

garding the orifice diameter and the gas flow rate, the effect of

orifice diameter is small for high gas flow rates but is appre-

ciable for low gas flow rates. A compilation of these results is

shown in Table 4. Both density and surface tension of all CMC

solutions are quite similar, so the variation in bubble volume

at equal values of orifice diameter and gas flow rate is mainly

due to the effect of viscosity.

5. Conclusions

Considering that bibliography provides rather scattered data

regarding the effect of density, viscosity and surface tension on

bubble generation times, the results obtained by the theoretical

model presented here can be considered a step forward in the

understanding of this matter because it allows to describe in

a rational manner the influence of the different liquid proper-

ties agrupated in dimensionless numbers on bubble generation

times.

The results of bubble generation times as a function of the

Reynolds, Weber and Bond number indicate that at low val-

ues of the Reynolds number the dimensionless formation time

Table 4

Theoretical and experimental detachment times of air bubbles generated

in different aqueous solutions of CMC for different orifice diameters and

gas flow rates

Do (mm) QC (cm3/s1) CMC Experimental Theoretical

(% w/w) detachment detachment

time (ms) 1ms time (ms)

1.5 1 0.4 29

0.8 29

1.2 29 27

1.4 32 29

1.6 33 33

5 0.4 29

0.8 30

1.2 29 26

1.4 31 28

1.6 32 32

10 0.4 30

0.8 30

1.2 30 27

1.4 31 30

1.6 31 32

2.0 1 0.4 33

0.8 34

1.2 34 34

1.4 37 36

1.6 38 39

5 0.4 33

0.8 33

1.2 34 32

1.4 36 36

1.6 37 38

10 0.4 33

0.8 33

1.2 33 32

1.4 36 35

1.6 36 36

decreases when increasing the Reynolds number. For turbu-

lent Reynolds numbers, this influence is small. Moreover, at

low values of the product W e Bo, an increase of this num-ber results in a big increment in the dimensionless formation

time. This increase becomes less sharp at higher values of

W e Bo. For very high values of viscosity, real fluids behave as

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non-Newtonian, so the model is modified including different

terms for the drag coefficient and the shear stress following

the potential law. The non-Newtonian model describes bubble

shapes and generation times with pretty good approximation,

concluding that the factor fc is mostly characteristic of the plate

and it is not influenced by the type of fluid.

Notation

At orifice area, m2

A1 flow area of the gas chamber, m2

C constant in Eq. (18)

Cs discharge coefficient

CD drag coefficient

Do orifice diameter, m

fc experimental factor related to buoyancy

g gravity m/s2

HL height of the liquid in the column, m

m mass of gas entering the bubble, kgmf fluid constant in the power law

M virtual mass, kgn fluid exponent in the power law

Patm atmospheric pressure, Pa

PB bubble pressure, Pa

PC chamber pressure, Pa

PH hydrostatic pressure, Pa

PL pressure at the gasliquid interface, Pa

PairM molecular weight (air), (kg/mol)

QB gas flow rate entering the bubble, m3/s

QC gas flow rate entering the chamber, m3/s

r radial coordinate, m

R bubble radius, m

Rg universal gas constant, J mol/K

Reb bubble Reynolds number: LDmax(Uz)2n/mf

Re Reynolds number, Newtonian fluids L u0 Do/L,Non-Newtonian fluids L u2n0 Do/mf

t time, s

T temperature, K

u velocity, m/s

VB bubble volume, m3

VC volume of the gas chamber, m3

V0 initial bubble volume, m3

We Weber number: L

u20

Do/

z apparent height of a point on the bubble surface, mGreek letters

exponent relating fluid density and bubble genera-

tion time

L liquid fraction in the column

exponent relating surface tension and bubble gener-

ation time

exponent relating fluid viscosity and bubble gener-

ation time

polytropic gas coefficient

liquid viscosity, Pa s

liquid density, kg/m3

surface tension, N/m

shear stress, N/m2

Acknowledgments

The support of the Ministry of Education and Science of

Spain providing a F.P.U. fellowship to Mariano Martn and re-

search funds (project PPQ2000-0097-P4-02) are greatly appre-

ciated.

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