59

Chapter 4: One group interfacial

area transport equation

4.1. Introduction

When fluid particles of various shapes and size present simultaneously, their transport

mechanisms can be significantly different. In such cases, it may be necessary to employ

multiple transport equations to describe the fluid particle transport. In view of this, we

first consider the two-phase flow system of the dispersed bubbles in a continuous liquid

medium (namely, bubbly flow), where all the present bubbles can be categorized as one

group. In such flow conditions, it is assumed that the bubbles are spherical in their

shapes, and they are subject to the similar characteristic drag on their transport

phenomena. Hence, accounting for the spherical shape in the one-group transport, can

be approximated by:

for dispersed bubbles because the bubble Sauter mean

diameter is approximately equal to the volume-equivalent diameter. This isn’t entirely

true in our case, since we find ourselves in the distorted bubble regime, which implies

that certain corrections should be made, as it was seen in the lift force section in the

previous chapter.

If we consider our bubbles possible shape and all its implication, it means that:

Eq.4. 1

May not be simplified for the one group interfacial area transport equation under certain

circumstances, since the mean Sauter diameter and the equivalent diameter will not

have the same value, the equivalent diameter should satisfy:

Corrections should be made in the source and sink terms for the interfacial area

transport equation that will be seen in this chapter. This will be left to future works, and

the terms were modeled in CFX as they were modeled by the corresponding authors in

the form introduced this chapter.

Furthermore, noting that critical bubble size due to nucleation is much smaller

compared to the average bubble Sauter mean diameter, we may assume

(

60

Also since can be approximated as:

Eq.4. 2

The interfacial area transport equation for the dispersed bubbles, or the one group

interfacial area transport equation, is given by:

Eq.4. 3

About the constitutive relations needed for Eq.4. 3, the number source and sink should

be established through mechanistic modeling of the major particle interactions that

contribute to the change in the interfacial area concentration. Accounting for the wide

range of gas liquid two phase flow, the major bubble interaction mechanisms that lead

to the particle coalescence or disintegration was summarized as follows by [Ishii and

Hibiki(2006)].

• Random Collision ( ): coalescence through random collision driven by turbulent

eddies;

• Wake Entrainment ( ): coalescence through collision due to acceleration of the

following particle in the wake of the preceding particle;

• Turbulent Impact ( ): disintegration upon impact of turbulent eddies;

• Shearing-off ( ): shearing-off around the base rim of the cap bubble;

• Surface Instability ( ): break-up of large cap bubble due to surface instability;

• Rise Velocity ( ): collision due to the difference in the bubble rise velocity;

• Laminar Shear ( ): breakup due to the laminar shear in viscous fluid,

• Velocity Gradient ( ): collision due to the velocity gradient.

As far as the adiabatic bubbly flow is concerned, the effects of nucleation and interfacial

heat and mass transfers are out of consideration, thus the coalescence and breakup

effects due to the interactions among bubbles and between bubbles and turbulent eddies

have been the subject of more attention. [Wu et al. (1997)] have considered five

mechanisms responsible for bubbles coalescence and breakup: (1) coalescence due to

random collisions driven by turbulence, (2) coalescence due to wake entrainment, (3)

breakup due to the impact of turbulent eddies, (4) shearing-off of small bubbles from

larger cap bubbles, (5) breakup of large cap bubbles due to interfacial instabilities. In

the case of low void fraction conditions where no cap bubbles are present, the authors

have simplified their model by considering only one bubble size and the first three

coalescence and breakup mechanisms.

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The volumetric interfacial area transport equation written by [Ishii and Hibiki(2006)]

writes:

Eq.4. 4

If, besides, we consider steady state conditions, which are normally the conditions given

in our particular object of study, it can be reduced to:

Eq.4. 5

The first term on the right-hand side represents the effects of the variation in bubble

volume, or gas expansion term. Which will be taken into account by some models

([Hibiki and Ishii (1999)] and [Ishii and Kim(2000)]), but not by others [Wu et

al.(1997)], although the first investigation groups have shown that it may contribute

significantly to the total variation of volumetric interfacial area. As said, closure laws

for the interaction mechanisms are needed and will be seen.

4.2 Interaction mechanisms and their modeling:

The interaction mechanisms and the transport phenomena depend strongly on the type

of bubble. The cross sectional area also influences the interaction mechanisms. Hibiki

and co-workers reported that the dominant coalescence mechanisms are different in

small diameter pipes and in big diameter pipes.

Fig.4. 1: From left to right, coalescence and breakup example images.

62

a) Coalescence mechanisms:

Coalescence refers the generation of new bubbles thanks to existing bubbles joining

together.

Coalescence due to random collisions driven by turbulence: This mechanism can be

described by two consecutive processes, consisting on the drainage of the fluid film

separating the bubbles, and the rupture of this film. A really important parameter is the

relative velocity between bubbles, the velocity at which they approximate each other,

because, if it’s really big, the film may not be drained on time and the bubbles would

rebound unaffected. [Kirkpatrick and Lockett(1974)].This is often modeled through a

collision frequency between bubbles, and a coalescence efficiency. It is normally

considered the frequency as a function of the required time to complete the coalescence

process, and of the contact time in turbulent flow. This will be seen deeply ahead in our

work. This kind of models, based on the film drainage theory, are just an option to

analyze the process, through a different concept, [Stewart(1995)] establishes that the

coalescence is a binary process, occurring in bubbles of similar size, and in really short

time, being the penetration in the interface, or the breaking of the same, immediate. A

third analysis criteria may be considering the drainage theory from a point of view

similar to Stewart’s, like [Kim(1999)] did, that means, considering that the interaction is

binary and between bubbles of similar size, being the collision frequency proportional

to the fluctuation of the turbulent velocity , and the maximum packing value required to

compact the control volume.

Coalescence due to wake entrainment: This mechanism

appears when a bubble provokes a depression in the wake

generating the suction of smaller bubbles. There´s a critical

length from which the suction appears, as described by

[Nevers (1977)]. Stewart in his work observed that this

interaction may be binary, including a group of bubbles with

values more adjusted to this critical length. On the other

hand, it exists a different formulation, supposing that the

bubble is spherical and calculating the wake this sphere

would create, a collision frequency depending on the bubbles

relative velocity can be determined [Kim (1999)].

This will be seen deeply in the next chapters. It may be

interesting to comment, that some experimental results [Stewart(1995)], [Otake et

al.(1977)] show that the wake entrainment results in coalescence primarily between

pairs of large cap bubbles in fluid sufficiently viscous to keep their wake laminar;

whereas small spherical or ellipsoidal bubbles tend to repel each other. In addition, in

Fig.4. 2: Coalescence due

to wake entrainment.

63

low viscous liquids such as water, the turbulent wake has the tendency to break trailing

bubbles because of its intermittency and irregularity. [Hibiki and Ishii (1999)] also

pointed out that the wake entrainment induced coalescence results in minor contribution

to the volumetric interfacial area variation in the bubbly flow with high flow rate

because a bubble captured in the wake region can leave the region easily due to liquid

turbulence, even though it may play an important role in the bubbly to slug flow

transition. But this doesn´t conclude this matter, since, although the wake entrainment

effect can be omitted for Hibiki, it appears dominant in Wu’s calculation cases, as it was

correctly pointed by [Yao and Morel(2004)].

b) Breakup mechanisms:

The breakup provokes an increase of the interfacial area concentration, therefore

incrementing the transfer rate between the system´s phases. There are some models that

consider a binary breakup where bubbles of the same size are created [Prince and

Blanch(1990)], although others are more accurate considering them of different sizes

[Wang et al(2003)].

Fig.4.3:Breakup Mechanisms

Fig.4.3: From left to right: breakup due to the impact of turbulent eddies, shearing-off of small bubbles

from larger cap bubbles, breakup of large cap bubbles due to interfacial instabilities

As said, for us only the breakup due to turbulent eddies impact will be of importance.

64

Breakup due to impact of turbulent eddies: When the turbulent eddies inertia is

bigger than the surface tension of a bubble it collides with, the bubble breaks. It is

possible to characterize the breakup condition as a function of a critical Weber number.

Later some studies proposed a breakup frequency depending on the particle size, surface

tension, both phases viscosity, void fraction and turbulence eddy dissipation

[Coulaloglou and Tavlaridès (1977)]. This breakup frequency begins supposing that the

turbulent kinetic energy transferred by a turbulent impact is the exact sufficient energy

to break the bubble. In a similar way break frequency considering also the eddies with

more energy than the required for breakup.

Afterwards a breakup frequency was formulated by [Prince and Blanch (1990)]

considering bubbles of any size, and eddies of any scale present in the liquid.

[Kim(1999)] got to similar models from a binary conception of breakup, which may not

be acceptable in lots of cases, since a breakup creates a bubble distribution with

different sizes that may be described statistically determining the probability density

function of the produced bubbles. [Martinez Bazan1999a]

This really complicates even more this matter, since all possible interaction mechanisms

shall appear for the new bubbles.

4.3 Analysis of the considered Models

As said, we´ll focus on the coalescence through random collision, wake entrainment,

and breakup through turbulent impact. The models, namely those of Hibiki, Ishii, Wu,

Wang, and Yao and Morel, will be analyzed and then tested. In addition, it may be

interesting to say that, although the coefficients they used for their respective models are

fitted from experimental data and can give reasonable one-dimensional calculation

results, the values presented by the different authors show significant differences.

Recently, [Delhaye (2001)] gave a systematic comparison and detailed analysis of their

work.

4.3.1 General ideas

As pointed in [Yao and Morel(2004)] coalescence and breakup terms induced by

turbulence can be written in the following general forms:

Eq.4. 6

Eq.4. 7

65

where and are the coalescence and breakup times of a single bubble, and are

the coalescence and breakup efficiencies, and n is the bubble number per unit volume.

The Weber number is defined by Eq.2.9 and is a constant designated as the

critical Weber number ([Wu et al.(1997)]). In a turbulent flow, if the bubble size is in

the inertial subrange [Kolmogorov (1949)], [Risso(2000)], the square of the velocity

that appeared before is considered by Yao and Morel like :

Eq.4. 8

Where is the (Sauter mean) bubble diameter and is the turbulent kinetic energy

dissipation rate per unit mass of the continuous liquid phase. Therefore, we have:

Eq.4. 9

4.3.2 Differences among models:

[Wu et al.(1997)] and [Hibiki and Ishii (1999)] models are based on different methods

to evaluate the time required for the bubbles breakup and coalescence (the model

proposed by [Ishii and Kim(2000)] is similar to the one proposed by Wu et al. for the

turbulence induced coalescence and breakup).

For the breakup term, Wu et al. evaluate the breakup time from a simplified momentum

equation, considering only the interaction of a breaking bubble with a turbulent eddy of

the same size. Therefore, this first time is hereby called the “interaction time”. Hibiki

and Ishii evaluate the breakup time as the time necessary for a given bubble to be

collided by a turbulent eddy. We will call this second time the “free travelling time”. It

should be noted that the breakup time proposed by Hibiki and Ishii is proportional to

where is the “maximum packing” value of the void fraction

[Ishii(1990)], therefore reaching an infinite breakup frequency when . In fact,

the breakup frequency should be zero when because there is almost no liquid

between the bubbles, and hence no turbulent eddies. The breakup model proposed by

Wu et al. does not consider the necessary free travelling time for a bubble to encounter a

liquid eddy. The proposed model of [Yao and Morel(2004)] does take into account the

“free traveling time” and the “interaction time” separately.

For the coalescence term, Wu et al. and Hibiki and Ishii derived different characteristic

times for binary collisions between bubbles from considerations about mean distance

between neighboring bubbles. Another important difference between their models is the

coalescence efficiency retained by the different authors. Hibiki and Ishii choose to

model this coalescence efficiency as in [Coulaloglou and Tavlaridès (1977)] for

66

liquid/liquid dispersions, although Wu et al. assume in their work the coalescence

efficiency to be constant. The time for liquid film drainage between two interacting

bubbles can be quite long, giving smaller values for the coalescence efficiency. But this

drainage time is also the time during which the two bubbles interact before to coalesce

or to be separated by the turbulent eddies. On the other hand, Yao and Morel model take

into account, as for the breakup term, the time necessary for collision and the interaction

time separately.

Another shortcoming in Wu et al. and Ishii and Kim models is that the breakup model

they propose can´t be used for , due to the dependence in

(This

term will be seen in the next part of our chapter)

These authors therefore assume that the bubble breakup is likely to occur only if:

.

The significance of the previous condition is that only a turbulent eddy with sufficient

energy to overcome the surface energy of the interacting bubble can break this bubble.

However, as pointed out by [Risso(2000)], turbulence acts on each bubble as a

succession of colliding eddies, with random arrival times and intensities, instead of a

single isolated eddy. If the frequencies of these successive eddies are close to the natural

frequency of the bubble, the bubble shape may become to oscillate and these

oscillations can break the bubble, even if . Yao and Morel tried to consider

this resonance mechanism.

For the boiling two-phase flow conditions, although it isn´t of our concern, and there´s

few available literature about the topic, we will only say that [Kocamustafaogullari and

Ishii (1983)] simply added the volumetric bubble number variation due to the nucleation

in the corresponding balance equation.

4.3.3 Yao and Morel [Yao and Morel (2004)]:

First of all, it must be said, that the IATE considered by Yao and Morel, takes the

following form:

Eq.4. 10

Where, as it can be appreciated, the pressure term is included, and so is also the

nucleation term, although in our case it won’t be, given our conditions of application.

Yao and Morel did not consider the coalescence due to wake entrainment.

67

As this is the first case, the modeling of the turbulent impact and the random collision

term will be analyzed.

1. Coalescence due to turbulence induced random collision

In a turbulent flow, small bubble motions are driven randomly by turbulent eddies. In

the case where , where L is the average distance between two neighbouring

bubbles, the probability of collision between two bubbles is larger than the one among

three or more bubbles. Assuming only binary coalescence events, [Prince and

Blanch(1990)] gave the following expression for the collision frequency between two

bubbles of different groups induced by turbulence:

Eq.4. 11

Where the effective collision cross-sectional area is given by:

Eq.4. 12

It is interesting to emphasize, that the cross sectional area of Yao and Morel, that is

defined as Prince and Blanch did, doesn´t agree with the general formulations of the

kinetic theory of gases(e.g.[Loeb(1927)], where it´s modeled as:

Eq.4. 13

That would mean in their case they consider it is multiplied by a 0.25 factor.

In the case of a single bubble size, given by , the total collision frequency between

bubbles in a unit volume can be simplified into:

Eq.4. 14

Where . It should be noted that Yao and Morel have divided the collision

frequency by 2 for the calculation of the total collision frequency, in order to avoid

double counting of the same collision events between bubble pairs [Prince and

Blanch(1990)]. Accordingly the average free travelling time of one bubble writes:

Eq.4. 15

68

It can be considered that when the void fraction reaches a certain value , as said,

called “maximum packing value” [Ishii (1990)], the bubbles touch each other and the

average free travelling time becomes zero. In the case of a single bubble size, this

limiting value of the void fraction was considered to have the same value proposed by

[Hibiki and Ishii (1999)]:

Eq.4. 16

Therefore, the following modification factor is modeled and included:

Eq.4. 17

This factor considers the effect of the void fraction on the average free travelling time.

When the void fraction is very small, , which corresponds to the case of a dilute

two-phase flow where no correction is needed. If the void fraction reaches its limit

value, and the free travelling time becomes nil. At the end, the free travelling

time can be expressed by the following relation:

Eq.4. 18

In the modeling of the interaction between two interacting bubbles, they adopted the

film thinning model ([Kirkpatrick and Lockett(1974)]). This model assumes that two

bubbles will coalesce if the contact time between them, depending on the surrounding

turbulent eddies, is larger than the liquid film drainage time. [Prince and Blanch(1990)]

gave the following expression for the film drainage time:

Eq.4. 19

69

Where the initial film thickness and the critical film thickness are suggested to

be 104 m [Kirkpatrick and Lockett(1974)] and 108 m in [Kim et al.(1987)] (for an air–

water system). Therefore, this can be written:

Eq.4. 20

This time is used as the interaction time before coalescence of two bubbles:

The contact time is given by the characteristic time of the eddies having the same size

than the bubbles [Prince and Blanch(1990)]:

Eq.4. 21

In addition, an exponential relation was assumed by [Kim et al.(1987)] to estimate the

collision efficiency:

Eq.4. 22

For Yao and Morel, the coalescence time contains two parts: the free travelling time and

the interaction time, i.e.:

Eq.4. 23

Finally, the bubble coalescence frequency can be expressed as:

Eq.4. 24

Where the factor one-half has been put to avoid double counting of the same

coalescence events between bubble pairs [Prince and Blanch(1990)]. Substituting from

the above relations, the final form writes:

Eq.4. 25

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Which means:

Eq.4. 26

If we introduce the following value of the critical Weber number [Yao and

Morel(2004)]: the coefficients will be , ,

.

It may be interesting to say that the ratio of the free travelling time and the interaction

time is:

Eq.4. 27

And therefore decreases when the Weber number increases or the void fraction

increases.

2. Breakup due impact of turbulent eddies:

For analyzing the collisions between bubbles and eddies using the equation for the

collision frequency of two bubbles, the information of eddies number per unit volume is

needed. According to [Azbel and Athanasios (1983)]:

Eq.4. 28

Where is the number of eddies of wave number k per unit volume of the fluid

(

where is the diameter of the eddy).

Considering the void fraction effect, the number of eddies per unit volume in the two

phase mixture is given by Hibiki and Ishii:

Eq.4. 29

Which can be rewritten in terms of eddy diameter:

Eq.4. 30

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The collision frequency between bubbles characterized by their volumetric number n

and their diameter and eddies whose diameters are comprised between two fixed

values and is:

Eq.4. 31

Assuming that only the eddies with a size comparable to the bubble diameter can break

the bubbles, and numerically integrating this equation from to

(the value 0.65 was chosen by them in order to obtain a good agreement on the bubble

diameter profile in comparison to the DEBORA experiment, this means, although their

model is said to be universal, it does contain parameters) we obtain:

Eq.4. 32

Where

(

)

Therefore, the average free travelling time of bubbles can be written as:

Where

Eq.4. 33

As pointed out previously, the breakup mechanism is assumed to be due to the

resonance of bubble oscillations with turbulent eddies, especially in the conditions of

low Weber number. The natural frequency of the nth order mode of the oscillating

bubble is given by [Risso(2000)] or [Kirkpatrick and Lockett(1974)].

Eq.4. 34

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For the second mode oscillation, and if is assumed for bubbly flow, this

equation gives:

Eq.4. 35

The breakup characteristic time describes the increase of the most unstable oscillation

mode:

Eq.4. 36

This characteristic time is used as the interaction time between bubbles and eddies. As

for the coalescence, we assume that the breakup time is the sum of the free travelling

time and the interaction time:

Eq.4. 37

In addition, the breakup efficiency can be expressed as:

Eq.4. 38

Finally, the bubble breakup frequency should be written as:

Substituting from the above relations, the final form is obtained:

Eq.4. 39

Where and .

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3. Modeling of the nucleation induced source term in the volumetric interfacial

area transport equation:

Since, the nucleation generates many small newborn bubbles which give quite different

contribution to the interfacial area concentration in comparison to the interfacial mass

transfer through the surface of the existing bubbles, an important question is: how to

model the nucleation effect on the volumetric interfacial area properly? There are two

ways of doing it, derivation from the bubble number density equation, or derivation

from the so-called Liouville equation. But, since in our particular case of study the

nucleation effect doesn´t have to be considered, we won´t deepen into this matter, for

more information regarding it, please go to [Yao and Morel(2004)].

4.3.4 Hibiki and Ishii [Hibiki and Ishii (1999)]:

The one group interfacial area transport equation considered by Hibiki and Ishii

includes the pressure term representing the variation in bubble volume, being therefore

the same considered by Yao and Morel, including the nucleation term, which once again

won’t be taken into account by us.

1. Coalescence due to turbulence induced random collision

For the estimation of bubble-bubble collision frequency, they assumed that the

movement of bubbles behaves like ideal gas molecules [Coulaloglou and Tavlaridès

(1977)]. Hibiki and Ishii considered, that following the kinetic theory of gases [(Loeb

(1927)] the bubble random collision frequency can be expressed by assuming the

same velocity of bubbles as a function of surface available to the collision , and

volume available to the collision :

As they considered the average bubble velocity, taking account of the relative motion

between bubbles, to be:

Eq.4. 40

Where is a constant and the turbulence eddy dissipation is simply obtained from the

mechanical energy equation:

Eq.4. 41

Where

denote the mixture volumetric flux, the mixture density,

and the gradient of the frictional pressure loss along the flow direction, respectively.

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The collision frequency will increase to infinity, as the void fraction approaches to

maximum void fraction calculated by closed packing condition. Following [Taitel et

al.(1980)], the maximum allowable void fraction is determined to be 0.52, which

gives the finely dispersed bubbly to slug flow transition boundary. Finally, one obtains:

Eq.4. 42

Where is an adjustable parameter. [Coulaloglou and Tavlaridès (1977)] gave an

expression for the coalescence efficiency as a function of a time required for

coalescence of bubbles and a contact time for the two bubbles , the same

considered by [Kim (1987)] having the form of equation Eq.4. 22.

What is finally modeled as:

Where

Eq.4. 43

Finally:

Eq.4. 44

Where is an adjustable valuable, which is determined experimentally to be 0.188 for

bubbly flow.

2. Breakup due to impact of turbulent eddies:

The bubble breakup is considered to occur due to the collision of the turbulent Eddy

with the bubble. For the estimation of bubble-eddy collision frequency, it is assumed

that the movement of eddies and bubbles behaves like ideal gas [Coulaloglou and

Tavlaridès (1977)]. Furthermore, the following assumptions were made for the

modeling of the bubble-eddy collision rate by [Prince and Blanch(1990)]: (1) the

turbulence is isotropic; (2) the eddy size de of interest lies in the inertial subrange; (3)

the eddy with the size from to can break up the bubble with the size of ,

since larger eddies have the tendency to transport the bubble rather than to break it and

smaller eddies do not have enough energy to break it.

To avoid deepening into too many details, from now on only the final expressions will

be seen:

75

Eq.4. 45

Where ´s value for the average breakup efficiency is calculated by setting

to be 1.37.

4.3.5 Wu et al. [Wu et al.(1997)] :

If the general one group transport equation has the already shown form of Eq.4. 5

The first term on the right-hand side, as it has already been said, represents the effects of

the variation in bubble volume, as it has been commented, Wu et al. assumed the gas

phase to be incompressible without phase change, from the gas phase continuity

equation, this term is zero.

Eq.4. 46

1. Coalescence mechanisms

On one side the coalescence due to random collisions, which is modeled as:

Eq.4. 47

Where and are adjustable parameters, depending on the properties of the

fluid, and set to values of 0.016 (experimentally) and 3 respectively.

Nevertheless, the constant coalescence efficiency is only an approximation, and

further efforts are needed to model the efficiency mechanistically. The

remaining unknowns are the maximum void fraction and the mean bubble

fluctuating velocity. By definition, the dense packing limit of void fraction

when the coalescence rate approaches infinity. A rational choice of should

be approximately.75-0.8 at the transition point from slug to annular flow [Wallis

(1969)], [Wu et al.(1997)]. The turbulence eddy dissipation appearing in the

equation comes from the mean bubble fluctuating velocity, ,,that is

proportional to the root-mean-square liquid fluctuating velocity difference

between two points of length scale D, and is given by [Ishii and

Kojasoy (1993)].

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And on the other side we have the coalescence term due to wake entrainment:

Eq.4. 48

Where is an adjustable constant mainly determined by the ratio of the

effective wake length to the bubble size and the coalescence efficiency which is

determined experimentally to be 0.0076.. A proper choice for should yield an

effective wake length roughly between 5 and 7 as they consider. The bubble

terminal velocity, is a function of the bubble diameter and local time-average

void fraction. Based on the balance between the buoyancy force and drag force

in a two-phase bubbly flow, [Ishii and Chawla (1979)] applied a drag-similarity

criterion with the mixture-viscosity concept to obtain the following expression

for the relative velocity:

Eq.4. 49

Eq.4. 50

Eq.4. 51

2. Breakup mechanisms:

Eq.4. 52

Again, the adjustable parameters and should be evaluated with experimental

data, having the first one a value of 0.17. It is slightly different from other models,

because the breakup rate equals zero when the Weber number is less than . This

unique feature permits the decoupling of the bubble coalescence and breakup processes.

At a low liquid flow rate with small void fraction, the turbulent fluctuation is small and

thus no breakup would be counted, which allows the fine-tuning of the adjustable

parameters in the coalescence terms, independent of the bubble breakage. They used the

value suggested by [Prince and Blanch(1990)] for air-water system, .

77

4.3.6 Ishii and Kim [Ishii and Kim(2000)]:

The equation as considered by Ishii et al is the general equation of Eq.4. 5.Where the

effect due to change in bubble volume is considered, and also the disintegration due to

turbulent impact (TI), coalescence through random collision driven by turbulent eddies

(RC), and coalescence due to the acceleration of the following bubble in the wake of the

preceding bubble (WE).Also the phase change effect was taken into account by them, as

said, not by us.

Eq.4. 53

When

Eq.4. 54

Eq.4. 55

Here, the , , and are coefficients to be determined through experiments, and

and are the critical Weber number over which the bubble disintegrates and

the maximum packing limit, respectively. For them, the values that showed the best

agreement with the experimental results were:

; ;

Eq.4. 56

4.3.7 Wang [Wang (2010)]:

The interfacial area transport equation considered by [Wang (2010)] in her PhD

dissertation has the general form considered by [Ishii and Kim(2000)] and [Wu et

al.(1997)] that has already been shown, as well as the same basic models for the bubble

interaction mechanisms.

To start with the discussions on the bubble interaction mechanisms, the Weber number

(We ),as we know, is first defined as the ratio of the particle turbulent inertial energy to

the surface energy as: , where and are the turbulent

velocity of the liquid phase and average bubble diameter, respectively. The critical

value, , is used to describe a condition where the cohesive and disruptive forces

balance.

78

The breakup due to turbulent impact is modeled as:

Eq.4. 57

The bubble collision is assumed to take place in an isotropic turbulence system. This

phenomenon is therefore similar to the intermolecular collisions in an ideal gas, and the

collision frequency is calculated using the kinetic theory of gas molecules. The two

colliding bubbles are assumed to have the same size. The constitutive relation of is

given as:

Eq.4. 58

In the above closure model, is the void fraction at the bubble maximum packing.

The adjustable coefficient is used to account for the coalescence efficiency instead

of the complicated model applying film drainage theory.

The other mechanism of the bubble coalescences in bubbly flows is due to the

entrainment of the following bubbles in the wake region of a preceding bubble,

resulting in the loss of the interfacial area concentration. A critical distance has been

discovered between the following and leading bubbles, within which the following

bubbles will make collision with the leading bubble without exception, if time is

allowed. The critical distance ( ) is generally proposed as where is the

diameter of the leading bubble. The mechanism, , is proportional to the relative

velocity between the liquid (continuous) and gas (dispersed) phases, , and was

modeled by Wang like:

Eq.4. 59

where is the drag coefficient.

In these formulations and are model coefficients determined

experimentally by comparing the values of the interfacial area concentration obtained

from numerical calculations to those from the experimental data.

The coefficients of , C, and were assumed the same as the values suggested

by [Ishii et al.(2002)]. And an iterative process was followed for identifying the other

model coefficients. The following coefficients yield the best fit for his particular case:

Turbulent impact: = 0.005, = 6.0;

Wake entrainment: = 0.006; Random collision: = 0.013, C = 3.0, = 0.