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l
2003
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I Michigan State ’
University

This is to certify that the
thesis entitled

NUMERICAL APPROACHES TO PREDICT BUBBLE
GROWTH IN VISCOUS LIQUIDS USING POLYNOMIAL {
PROFILES

presented by

PRAVEEN K VASAM

has been accepted towards fulfillment
of the requirements for the

 

 

 

 

 

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NUMERICAL APPROACHES TO PREDICT BUBBLE GROWTH IN
VISCOUS LIQUIDS USING POLYNOMIAL PROFILES

By

Praveen K Vasam

A THESIS
Submitted to
Michigan State University

In partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Mechanical Engineering

2003

ABSTRACT
NUMERICAL APPROACHES TO PREDICT BUBBLE GROWTH IN VISCOUS
LIQUIDS USING POLYNOMIAL PROFILES
By
Praveen K Vasam
Modeling the phenomenon of bubble growth in liquids is important to ﬁelds such as
boiling, cavitation, bubble removal during glass making, micro-gravity studies,
composite materials, manufacturing void-free high performance materials, measuring the
elongational viscosity mid polymer processing. The main objective of this thesis is to
develop improved models to predict bubble growth in a liquid shell using the previously
developed models based on approximate polynomial proﬁles. Earlier models
concentrated on obtaining solutions for growth of a bubble surrounded by an inﬁnite
amount of liquid. These models do not represent the correct physics of the problem as the
bubbles are in fact surrounded by a ﬁnite shell of liquid and there is a ﬁnite amount of
gas dissolved in the liquid. To incorporate the correct physics in the problem the new
V models are developed. In these new models the previously used numerical technique of
‘Integral method’ was retained. This method allows us to formulate approximate
solutions for bubble growth. Since the polynomial proﬁles approximations give
reasonable results during initial times, they were retained to develop a better solution for
the bubble growth. The new models developed were solved and compared with the
approximate solutions for bubble growth in an inﬁnite medium and also the solutions
obtained by solving the full diffusion equation. After comparing the results it was found

that the new models predict the bubble growth behavior reasonably well.

Dedicated to my parents

iii

ACKNOWLEDGEMENT

I am grateful to my thesis advisor Dr. Andre Benard for his valuable academic and non-
academic assistance, continuous guidance, and motivation. He helped me a great deal in
the formulations discussed in this work. His support and encouragement were crucial to
the completion of this work. The constructive suggestions that he made during the

reading of the manuscript of this thesis helped in improving it a great deal.

I would like to thank Dr. Indrek Wichman for the suggestions he made during the course
of my research work. I want to acknowledge Dr. Craig Somerton’s and Dr. Indrek

Wichman’s help for serving on my thesis defense committee.

I would like to thank all my instructors at Michigan State University (MSU) and Indian
Institute of Technology (IIT), Madras who taught and encouraged me to reach this point.
I would like to recognize the support and encouragement provided by my friends. Finally,

a very special thanks to my parents and sisters for their love and moral support.

iv

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. vii
LIST OF FIGURES .......................................................................................................... viii
CHAPTER 1 ....................................................................................................................... 1
INTRODUCTION .............................................................................................................. 1
Issues ............................................................................................................................... 2
Motivation ....................................................................................................................... 4
CHAPTER 2 ....................................................................................................................... 5
LITERATURE REVIEW ................................................................................................... 5
Bubble Nucleation in Liquids ......................................................................................... 5
Bubble Growth in Inviscid liquids .................................................................................. 6
Bubble Growth in Viscous liquids .................................................................................. 8
Summary ....................................................................................................................... 10
CHAPTER 3 ..................................................................................................................... 12
THEORY OF SINGLE BUBBLE GROWTH .................................................................. 12
Growth of a Bubble in an Inﬁnite Newtonian Medium ................................................ 12
Hydrodynamics of the grth process ......................................................................... 13
Mass diffusion ............................................................................................................... l7
Integral method ............................................................................................................. 19
Summary ....................................................................................................................... 25
CHAPTER 4 ..................................................................................................................... 26
GROWTH OF A GAS BUBBLE IN A LIMITED BODY OF NEWTONIAN LIQUID 26
Hydrodynamics of the growth in a shell of Newtonian liquid ...................................... 27
Diffusion of mass in the liquid shell ............................................................................. 29
First approach ................................................................................................................ 31
Alternate simpliﬁcation ................................................................................................ 36
Second approach ........................................................................................................... 37
Third approach .............................................................................................................. 40
Complete solution ......................................................................................................... 41
Numerical evaluation .................................................................................................... 44
Equilibrium calculations ............................................................................................... 46
Comparison of proﬁles ................................................................................................. 47

CHAPTER 5 ..................................................................................................................... 59

SUMMARY AND CONCLUSIONS ............................................................................... 59

Further Study ................................................................................................................ 62
APPENDIX A ................................................................................................................... 64
DERIVATION OF INTEGRAL DIFFUSION EQUATION ........................................... 64
APPENDD( B ................................................................................................................... 66
MATHEMATICA FILE ................................................................................................... 66
REFERENCES ................................................................................................................. 70

vi

LIST OF TABLES

Table 3.1: Expressions for the constants C2, C3, C4, C5, C6 and C7 .................................. 25
Table 4.1: Methods adopted to predict bubble growth behavior in ﬁnite shell ................ 32
Table 4.2: Lower and upper bounds for dimensionless variables C2 , C 3 and C4 ........... 45

vii

LIST OF FIGURES

Figure 4.1: Schematic of a bubble in a shell of viscous liquid containing dissolved gas. 28

Figure 4.2: Comparison between various methods of evaluating the radius proﬁles for
low thickness shell (C2 = 104 ), C3 = 0.4, C4 = 102 (average gas diffusivity) and initial gas

pressure PEG = 10 . ............................................................................................................. 50

Figure 4.3: Comparison between approximate and accurate numerical solutions for
medium thickness shell (c2 = 10‘), c3 = 0.4, c, = 102 (average gas diffusivity) and

Pg}, =10 ............................................................................................................................ 51

Figure 4.4: Comparison between approximate and accurate numerical solutions for high
thickness shell (C2 = 108 ), C3 = 0.4, C4 = 102 (average gas diffusivity) and PEG =10 . .. 52

Figure 4.5: Comparison between second, third order proﬁle solutions using ‘method 1’
and the complete solution obtained by Arefmanesh [1] for medium thickness shell (C2 =

106 ), C3 = 0.4, C4 = 10 (average gas diffusivity) and initial gas pressure Pgo =10 . ....... 53

Figure 4.6: Comparison between second order, third order polynomial proﬁle solutions
for inﬁnite medium and complete solution obtained by Arefmanesh [1] for medium
thickness shell (C2 = 106 ), C3 = 0.4, C4 = 10 (average gas diffusivity) and initial gas

pressure Pgo = 10. ............................................................................................................. 54

Figure 4.7: Comparison between approximate and accurate numerical solutions for
medium thickness shell (c2 = 106 ), c3 = 0.4, c, = 104 (High gas diffusivity) and initial

gas pressure PEG =10. ....................................................................................................... 55

Figure 4.8: Comparison between various methods of evaluating the radius proﬁles for
medium thickness shell (c2 = 10‘ ), 03 = 0.4, c. = 10 (low gas diffusivity) and initial gas

pressure Pgo = 10. ............................................................................................................. 56

Figure 4.9: Comparison between approximate and accurate numerical solutions for
medium thickness shell (C2 = 106 ), C3 = 0.4, C4 = 102 (average gas diffusivity) and low

initial gas pressure P30 = 5. .............................................................................................. 57
Figure 4.10: Comparison between various methods of evaluating the radius proﬁles for

medium thickness shell (C2 = 106 ), c3 = 0.4, c4 = 102 (average gas diffusivity) and initial
gas pressure Pgo =15 . ....................................................................................................... 58

viii

CHAPTER 1

INTRODUCTION

Modeling the phenomenon of bubble growth in liquids is important to processes
involving boiling, cavitation, bubble removal during glass making, micro-gravity studies,
composite materials, manufacturing void-free high performance materials, measuring the
elongation viscosity and polymer processing. The objective in foam processing is to
manufacture lightweight articles with densities lower than the density of the polymer
without signiﬁcantly compromising the mechanical and physical properties of the
products. Lower densities can be achieved by introducing a gas in the form of bubbles
into the polymer. Gases may be introduced into a polymer by two different methods. In
the ﬁrst method, the polymeric pellets are compounded with an organic substance
(blowing agent) in the pellet of powdered form. In the second method, an inert gas is
dissolved in the polymer either in the solid stage (micro cellular foam) or in the molten
stage under high pressures (foam injection molding). Pressure is then reduced carefully to

allow the formation of bubbles in a controlled manner.

Three important phenomena govern the growth or collapse of bubbles in liquids, namely
mass, momentum and energy transfer between the bubbles and the liquid. The transfer of
mass can be either due to the phase change at the bubble interface or due to the diffusion
of mass from the liquid into the bubble or both. In addition the rheological characteristics
of the liquid surrounding the bubble will affect the growth process. The subject of this

study is to improve on current models to investigate the bubble growth phenomenon in

highly viscous fluids. This work is particularly relevant to bubble growth associated with

the processing of foams.

Boiling and cavitation studies motivated the early studies on bubble growth in liquids. In
these studies, the growth of a single bubble was assumed to occur in a large body of
liquid with the main mechanisms for the growth of the bubble being the momentum
transfer between the bubble and the surrounding liquid, and the heat transfer-induced
mass diffusion due to the phase change. Viscous effects in these cases were not important
compared to the inertia and surface tension was ignored. There are practical cases where
viscous effects during the bubble growth or collapse are important and cannot be ignored.
Bubble growth or collapse during polymer processing and glass making are among such

C3888.

Polymer processing in general and foam processing in particular occur under non-
isotherrnal conditions with heat transfer to the surrounding through the mold walls. The
heat transfer has an impact on the properties of the gas-polymer system, some which,
such as the melt viscosity, change by an order of magnitude during the non-isothermal
process. The heat transfer does not change the qualitative features of the bubble growth

process but it does effect the quantitative predictions.

Issues
Earlier studies to solve for the evolution of the bubble radius involved generally an

inﬁnite medium around the bubble and an inﬁnite amount of gas is available for diffusion

into the bubble. Thus we obtain a bubble that is ever growing, but the reality is that in
polymer foam processing there is no inﬁnite shell of fluid surrounding the bubble and
hence the amount of gas available for diffusion into the bubble is also limited. When a
ﬁnite thickness shell of liquid surrounds the bubble, the size of the bubble reaches a
maximum value and should stop growing unless there are other modes of increasing the

bubble size.

Earlier models of bubble growth involved using polynomial proﬁle approximations for
the concentration of gas inside the liquid surrounding the bubble. The concentration of
gas inside liquid surrounding the bubble was assumed to be invariant outside a
concentration boundary layer of ﬁnite thickness. In this way the three coupled equations
i.e. mass, momentum and diffusion equations were simpliﬁed to two coupled equations.
The assumption of a polynomial concentration proﬁle simpliﬁed the coupled equations
and allowed for easily solving the coupled equations numerically. The polynomial
proﬁles seem to match the complete solutions obtained by Arefmanesh et al., under
certain conditions [1]. Interestingly the solutions matched at initial times, but the growth
rate projected starts diverging after a certain point. This leads one to believe that the
theory does not take into account the point when the concentration boundary layer hits
the outer shell of the ﬂuid. The solutions obtained seem to diverge when the
concentration boundary layer hits the outer shell. The reason is that the boundary
conditions change when the concentration boundary layer hits the outer shell. Moreover

the theory also does not incorporate the ﬁniteness of the shell into the solution.

Motivation

The main objective of this analysis is to develop an improved model for bubble growth in
a liquid shell using the previously developed models based on approximate polynomial
proﬁles. The usage of polynomial proﬁles not only results in simplicity and less
numerical complexity, but they also give results in good agreement with the results
obtained by Arefmanesh et al. [1]. Thus to take into account the case of bubble growth in
a ﬁnite shell of liquid this study was taken up. Since the polynomial proﬁles give
reasonable results during initial times, they were retained to develop a better solution for
the bubble growth. Using polynomial proﬁles also decreases the numerical complexity in

arriving at the solution.

CHAPTER 2

LITERATURE REVIEW

Literature on bubble growth can be divided into two main categories, studies on the
bubble nucleation and the studies on the bubble growth. Nucleation of bubbles in single
component systems, such as boiling of water, is relatively well understood in simple
systems and can be described by the classical nucleation theory. On the contrary,
nucleation of bubbles in multi—component systems, such as bubble nucleation during
polymer processing, is not well understood. The existing theories of bubble nucleation in
such systems are mainly modiﬁcations to the classical nucleation theory. On the other
hand, there exists a rich body of literature on the growth or collapse of bubbles in liquids.

The growth process of a bubble begins with the nucleation of a bubble.

Bubble Nucleation in Liquids

Nucleation is a physical phenomenon during which the initial fragments of a new and
more stable phase are generated within a metastable phase, developing spontaneously
into gross fragments of the stable phase. An example is boiling, which is due to
nucleation and growth of vapor bubbles in superheated water. There are different
mechanisms for bubble nucleation in liquids, namely, homogeneous, heterogeneous and
mixed mode. In the homogeneous nucleation a vapor embryo of the critical radius can be
formed by the normal statistical ﬂuctuations of the liquid. In the heterogeneous mode,
nucleation occurs at an interface between the volatile liquid and the other phase in

contact. The mixed mode allows for both homogeneous and heterogeneous nucleation.

The nucleation phenomenon in polymer processing is far more difﬁcult than the bubble
nucleation in water, which can be described by classical nucleation theory. Due to lack of
a reliable theory, the nucleation is generally assumed to be heterogeneous and
instantaneous. It is often described with empirical relationships tailored to speciﬁc

systems.

Bubble Growth in Inviscid liquids
Before addressing bubble growth in viscous liquids, early developments of bubble growth
in inviscid liquids are brieﬂy introduced to distinguish the phenomenon and the driving

mechanisms from viscous liquids.

Early studies on bubble dynamics in liquids were motivated by problems such as boiling
and cavitation in water. In these single component systems, the bubble growth is
controlled by two distinct mechanisms, namely, the momentum transfer and the mass
transfer due to evaporation and diffusion of heat. Under these conditions, the viscous
term in the momentum equation is negligible compared to the inertia and the surface

tension.

Rayleigh in his pioneering work investigated the bubble growth controlled by inertia
forces [3]. Later, Plesset et al. [4,5], Froster et al. [6], Dergarabedian [7] and Birkhoff et
al. [8], investigated the heat diffusion-induced growth of a spherical vapor bubble in a
large body of superheated liquid and obtained analytical solution for asymptotic cases.

Scriven investigated the spherically symmetric growth of a vapor bubble in an unlimited

body of superheated liquid when momentum transfer and diffusion of heat controlled the

growth [9].

Rosner and Epstein investigated the effects of surface tension and the non-zero initial
bubble size on the growth of a bubble controlled by both mass diffusion and momentum
transfer [10]. They considered the cases in which the bubble grth rate departed from
the parabolic growth rate predicted by Scriven. They assumed that the gas concentration
gradient in the liquid was restricted to a small boundary layer close to the bubble
interface and used the integral method to solve the diffusion equation. Their approach has
been extensively used in solving diffusion-induced growth of gas bubbles in inviscid

liquids.

In studying the bubble growth in liquids, there are cases where a number of gases diffuse
into the bubble simultaneously. Cable et al. [11,12], and Ward et al. [13] investigated

such cases.

The driving mechanisms for the bubble grth in inviscid liquids are the momentum
transfer, the diffusion of mass due to evaporation, and the diffusion of dissolved gases
present in the liquid. In inviscid ﬂuids, the viscous effects are negligible as compared to
the inertia, pressure and surface tension. On the contrary, the driving mechanisms for the
bubble growth in viscous liquids are the momentum transfer and the gas diffusion. The
viscous effects, in this case, are dominant and the inertia terms in the momentum

equation are negligible due to the low Reynolds numbers.

Bubble Growth in Viscous liquids

In the bubble phenomenon, during polymer processing in general and foam molding in
particular, the simultaneous mass, momentum and energy transfer between the bubble
and the polymer are important. There are also other issues in foam processing, such as
concentration of the dissolved gas in the polymeric melt, the pressure distribution in the
mold and the diffusivity of the gas in the polymer, which have signiﬁcant effects on the
bubble growth. Barlow et al. [14], Longlois [15], Szekely et al. [16], Yung et al. [17] and
Patel [18] studied the diffusion-induced growth of a spherical gas bubble in a large body
of Newtonian liquid containing a dissolved gas under isothermal conditions. Szekely et
al. and Patel obtained numerical solutions by assuming polynomial proﬁles for the
concentration of dissolved gas in the ﬂuid around the bubble while Barlow et al. obtained
analytical solutions for the asymptotic stages of the growth by assuming that the gas
concentration was limited to a thin boundary layer close to the bubble interface (thin shell
approximation) outside of which the concentration remained equal to the initial
concentration. This enabled them to eliminate the convective term in the diffusion

equation. Their results showed that initially the growth was a linear function of time.

However, in the ﬁnal asymptotic stage it followed the well—known parabolic behavior.
Qiu et al. investigated the dynamics of a single bubble during pool boiling and under low
gravity conditions [19]. They obtained experimental results on the growth and
detachment mechanisms of a single bubble on a heated surface conducted during the

parabolic ﬂights of aircrafts.

Prodanovic et al. investigated the bubble behavior in sub-cooled ﬂow boiling of water at
low pressures and low ﬂow [20]. Prodanovic et al. obtained experimental data by varying

the pressure, bulk liquid velocities and sub-cooling temperature.

Robinson and Judd investigated the bubble growth in a uniform and spatially distributed
temperature ﬁeld [21]. A theory was developed which was shown to predict experimental
bubble growth data for both spherical growth in an unbounded liquid and hemispherical

growth at a heated plane surface in micro-gravity.

Miyatake et al. obtained a simple universal equation for vapor bubble growth in
uniformly superheated pure liquids and in binary solutions with a non-volatile solute
[22]. There are three regimes of growth process of the bubble: nucleation of the bubble,
inertia controlled growth and heat transfer controlled growth. Once again this theory is
limited by use in the case of inﬁnite shell of liquid surrounding the bubble. Moreover

Miyatake et al. did not consider growth process through diffusion.

Venerus et al. developed transport models for diffusion induced bubble growth in
viscoelastic liquids and evaluated those models [23]. They took into consideration the
convective and diffusive mass transport as well as surface tension and inertial effects.
Venerus et al. developed a rigorous model for the transport analysis of diffusion induced
bubble growth and collapse in viscous liquids [24]. They showed that other models could

be derived from their model using different mathematical models and dimensional

analysis. This model is limited by its applicability in the case of a bubble surrounded by

an inﬁnite amount of liquid only.

Venerus also developed models for bubble growth in viscous liquids of ﬁnite and inﬁnite
extent [25]. These models are similar to that obtained Arefmanesh et al. Both these
methods for evaluating the bubble growth are difﬁcult to solve numerically due to

coupled equations.

Summary

To summarize the contribution of relevant literature to the subject of investigation in this
thesis, as the nucleation of the bubbles in polymeric liquids is concerned, at present there
does not exist a reliable theory, which can predict the bubble nucleation accurately. On
the other hand, the bubble growth phenomenon has been extensively studied both in
inviscid and viscous liquids with emphasis on different aspects of the phenomenon. There
have also been experimental as well as numerical studies on the bubble growth during
polymer processing covering issues such as mass transfer, momentum transfer and
solubility and diffusivity of the dissolved gases in the polymer. These studies are either
restricted to the growth of a bubble in large body of polymer melt or in the case of foam
molding, where bubbles are separated by thin ﬁlms of the polymeric melt, they are
conﬁned to the growth of an average bubble as a representative of all the bubbles in the

mold.

10

The use of polynomial proﬁles to solve for the growth rate of the bubble simpliﬁes the
equations and also the numerical complexity. But their usage has been limited to the case
of inﬁnite medium of liquid surrounding the bubble. This restricts the practical usability
of the model. Therefore the polynomial proﬁles have been used in this thesis work to
solve for bubble growth rates fora ﬁnite shell of liquid surrounding the bubble. The
reasons for the disharmony in the results obtained by polynomial proﬁle solutions and

those obtained by Arefmanesh et al. are also investigated.

11

CHAPTER 3

THEORY OF SINGLE BUBBLE GROWTH

This chapter contains a review of theory of a single bubble growth in viscous Newtonian
liquids. The bubble is assumed to be surrounded by an inﬁnite amount of liquid. The
equations governing the growth of a single bubble in a large body of liquid are stated.
Their modiﬁcations to bubble growth in a liquid shell with ﬁnite thickness and a limited

amount of dissolved gas are presented in the next chapter.

Growth of a Bubble in an Infinite Newtonian Medium

Governing equations for bubble growth in highly viscous liquids such as polymer melts
are derived. The Reynolds number for these liquids is much smaller than one. Therefore,
the inertia terms in the momentum equation are neglected. It is assumed that the
concentration of the dissolved gas at the bubble interface is related to the gas pressure
inside the bubble through the Henry’s law. Henry’s law is a special case of the Raoult’s
law and is valid for low concentrations [27]. This study focuses on liquids with low gas
concentration. It is also assumed that the gas inside the bubble behaves as an ideal gas.
The bubble growth considered in this chapter is due to both mass and momentum transfer
which are coupled through the movement of the bubble interface. In the following
sections, the equations governing these phenomena in Newtonian liquids are presented.
In order to simplify the analysis the whole process is assumed to occur under isothermal

conditions.

12

Hydrodynamics of the growth process
It is assumed that nucleation has occurred and a bubble with radius R0 that is greater

than the critical radius necessary for growth has been produced. Consider the growth of
this spherical bubble in an unlimited body of viscous Newtonian liquid. To study the
spherically symmetric growth of the bubble, a spherical coordinate system with its origin
at the center of the bubble is chosen. Figure (3.1) shows a schematic of the bubble in a
large body of liquid. The expansion of the bubble generates an extensional ﬂow ﬁeld in
the liquid surrounding the bubble. The equation of continuity for the liquid in the

spherical coordinate system is given by

 

 

358+ :li'gr—UZVYU l i(pve sin 9)+

3
rsinOBO 3‘; v¢)—O (3'1)

rﬂnO
where p is the density of the liquid and vr , v9 , and v¢are the components of the

velocity. The assumption of spherical symmetry allows one to simplify the problem. The
ﬂow ﬁeld around the bubble is purely extensional and the only non-zero component of

the velocity is the non-radial component (vr ). Hence the continuity equation reduces to

the following
3p 1 a 2
— + —— R v = 0 3.2
at ,2 3J0 ’) ( )
The boundary condition for the above equation is given by

vr = R (3.3)

where R is the rate of change of the bubble radius with respect to time.

13

 

—<<1

 

 

 

    
   

 

Gas Bubble

 

 

 

 

Viscous Liquid

 

 

 

Figure 3.1: Schematic of a spherical bubble growing in an inﬁnite medium

Since the density of the liquid is only a function of temperature, equation (3.2) can be
easily integrated to yield the velocity ﬁeld. Integrating equation (3.2) and using the

boundary condition at the interface (equation (3.3)) results in the following velocity ﬁeld:

 

 

3 3 ' 2 '
R - R R
vr = 2’ 2+ 2 (3.4)
3r 9 r

14

where p is the derivative of the liquid density with respect to time. Under isothermal

conditions, the density of the liquid does not change with respect to time and hence the
ﬁrst term on the right hand side of equation (3.4) can be neglected. After simpliﬁcation

the velocity ﬁeld reduces to the following well-known equation

 

(3.5)

The conservation of momentum for the liquid around the bubble in the radial direction in

terms of stresses is given by

av av 3p 13 2 teams,
.r m)--— >———-—

—— 1:
8r r2 8r “
where P is the liquid pressure, and 1:rr , 199 and 14» represent the stresses in the liquid.
1:99 is equal to 1W due to spherical symmetry. For a Newtonian liquid and the velocity

ﬁeld given by equation (3.5), the stresses are readily evaluated in terms of the bubble
radius. Using the results in equation (3.6) reduces the momentum equation to the

following

 

 

(3.7)

where p is the density of the liquid and R is the second derivative of the bubble radius
with respect to time. To relate the gas pressure inside the bubble to the bubble radius and
the applied pressure in the liquid, equation (3.7) is integrated with respect to the radial

coordinate from the bubble interface R to inﬁnity [12].

15

The integrated momentum equation is given by

p[-:—R2+ RR] = p(R) - P(oo) + 1,, (R) —t,, (co) +4? (3.8)

P(R) and P(oo) are the liquid pressures at the bubble interface and inﬁnity respectively.

n is the liquid viscosity. The following relation gives the condition of stress continuity at

the bubble-liquid interface:
—Pg sui— = —P(R)+t,,(R) (3.9)
where P8 is the gas pressure inside the bubble and a is the surface tension. Substituting

the expression for the liquid pressure at the interface (P(R)) from equation (3.9) into

equation (3.8) yields the following equation for the momentum transfer in a large body of

Newtonian liquid

3 '2 “ R 20’
—R +RR +4 —+——+P,.,—P =0 3.10
p[2 J 11R R g ( )

where P0° is the applied pressure at inﬁnity. The momentum equation relates the gas
pressure inside the bubble to the bubble radius and the applied pressure. The ﬁrst three
terms in this equation represent the inertia, viscous and surface tension effects. In general,
for highly viscous liquids such as polymer melts the inertia terms are negligible and the

viscous forces play a signiﬁcant role. In this study the inertia terms are neglected.

l6

Mass diffusion

To determine the gas pressure inside the bubble, the principle of conservation of mass is
applied to the gas inside the bubble, which results in the following ordinary differential
equation [12]

f}. 3 _ 2 3CA)
c“(pgR )-3pDR (Tr r=R (3.11)

where pg is the density of the gas inside the bubble, D is the diffusion coefﬁcient of the
. . . . 3C A . . . .
dlssolved gas in the llqu1d, and Y 13 the concentration gradient of the dlssolved
r=R
gas at the bubble interface. The left hand side of equation (3.11) is the rate of
accumulation of mass inside the bubble and the right hand side is the rate of diffusion of

gas from the liquid into the bubble.

At this point two assumptions pertaining to the gas inside the bubble are made. First, it is
assumed that the gas inside the bubble behaves as an ideal gas. Considering that the gas
pressure inside the bubble is relatively low, the ideal gas assumption is reasonable [28].
Second, it is assumed that the concentration of the dissolved gas at the bubble interface is

related to the gas pressure inside the bubble through Henry’s law [12]

Ca(R.t)=Cw(t)=Kth(t) (3.12)
where Cw is the concentration of the dissolved gas at the bubble interface and Kh is the

Henry’s constant. Henry’s law is in general valid for glassy polymers at elevated

temperatures. It is assumed that the dissolved gas is initially uniformly distributed in the

17

liquid around the bubble with concentration C0 . This corresponds to an initial gas
pressure of Pgo , which is related to the initial gas concentration through the Henry’s law.

However, the pressure inside the bubble decreases as the bubble expands in accordance
with the ideal gas law. Consequently, the concentration of the dissolved gas at the
interface also decreases in accordance with Henry’s law. This will create a concentration
gradient in the liquid around the bubble and invokes the diffusion of the dissolved gas
into the bubble. The diffusion of the dissolved gas from the liquid into the bubble is

governed by the Fick’s law of diffusion in the spherical coordinate system

acA acA_ la 23cA

where D is the diffusivity of the gas in the polymer melt, and C A is the weight fraction

of the gas dissolved in the melt.

To solve equation (3.11) for the gas pressure inside the bubble, one needs to know the
concentration gradient at the interface. Therefore it is necessary to solve equation (3.13)
to obtain the concentration proﬁle ﬁrst. There are different approaches for solving the
diffusion equation. Either it may be written in difference form and solved in its entirety or
approximate solutions may be sought. The approximate solutions have certain advantages
over the more difﬁcult to solve complete equation set. One of the advantages is its

simplicity and their reasonable accuracy in the results obtained.

18

Integral method

The integral method is the most widely used approximate method in phase change
problems. This approximate method of solving partial differential equations dates back to
von Karman and Pohlhausen who used the method to solve the boundary layer equations
[29]. There are four basic steps in applying the integral method to the diffusion equation
[30]. First the diffusion equation is integrated over a phenomenological distance

8(t) called the diffusion boundary layer. Second a polynomial proﬁle is chosen for the

concentration distribution in the boundary layer. The coefﬁcients of the polynomial are
evaluated in terms of the boundary layer thickness by utilizing the boundary conditions.
Third the polynomial proﬁle is introduced in the integral equation and an ordinary

differential equation (ODE) for 5(t) is obtained. Finally the solution of the ODE yields

the concentration proﬁle in the liquid.

In employing the integral method, it is assumed that the concentration gradients are
limited to a small boundary layer close to the bubble interface, outside of which the gas
concentration remains virtually undisturbed and equal to the initial concentration [6].
This assumption may be reasonable under special circumstances such as bubble growth in
a large body of liquid and early stages of growth. However, it is not in general valid and
may overestimate the ﬁnal bubble radius. To obtain approximate solutions, second order

and third order proﬁles are assumed in this study.

The momentum and diffusion equations are coupled and as the bubble grows, the bubble

gas pressure Pg will decrease, giving rise to a concentration gradient at the bubble wall,

19

which in turn affects the mass transfer process. The concentration on the wall of the

bubble Cw (t) can be described by the Henry’s Law
Ca(R.t)=Cw(t)=KnPg(t> (3.14)

Assuming that the gas follows the ideal gas law,

Rnggtt)

Pg(t)= M

(3.15)

where M and pg are the molecular mass and density of the gas respectively. Combining

equations (3.14) and (3.15) we have

Cw (t)M
t = —— 3.16
pgt) RgTKh ( >
The conservation of mass at the bubble interface is from equation (3.11)
d 3 2 3C A J
— R = 3 DR — 3.17

In order to solve the equations (3.10, 3.13 and 3.17) simultaneously the integral method is

adopted. In this method, the diffusion equation is multiplied with r2 and integrated from
with respect to r from r = R tor = R + 5 , in which 8 is the concentration boundary layer.

After integrating and using the following relationships:
rzv, =R2R (3.18)

CA(R+6)=C0, CA(R)=Cw and 33%?” at r=R+8 (3.19)

O
whereR is the radius R differentiated with respect to time, which is the velocity of the

bubble surface. We then obtain

20

R+8
3— ] (CA — Co)r2dr = -Daz(ﬂ) (3.20)
(II: R at r=R

Using the mass ﬂux equation (equation (3.11)) we obtain

R‘I‘O _ R3 _ R3
3p I (SQ—9L}2dr -_- pg ng 0 (321)
R CO —cW c0 —Cw

where pgo is the density of the gas bubble at t = O, and Rois the initial bubble radius.

Now the concentration proﬁle is assumed to be a polynomial. If a second-degree

polynomial is assumed then we have:

 

2
ELI—Eisep—(r-RD forR<r<R+5 (3.22)
co-cW 8
—C—9—_—C—A—=O for r>R+8 (3-23)
C0-Cw

and substitution of equations (3.22) and (3.23) in the left hand side of equation (3.21)

results in

 

 

3 3
1 8 1 5 1 8 p R -9 0R
39l—(—)3 +—(—)2 +-(—)]R3 = g 3 0 (3.24)
30 R 6 R 3 R C0-Cw
Neglecting the terms containing (8/ R)2 and higher order terms we have
R3 - R3
8: pg pgo 0 (3.25)

p(C0 - Cw )R 2
Substituting equation (3.25) into the right hand side of equation (3.18) and eliminating 5

from the resulting expression we obtain

21

_ 6p2D(C0 -Cw)R4
pgR3 —pg0R3

 

_d_ 3
dt (pgR ) (3.26)

If (8/ R)2 and lower order terms are also considered we get

spam/122 +3; (3.27)

= pgR3 _ ngR?)

 

 

(3.28)
P(Co-Cw)
we obtain
d 6 DC —c R2
—(pgR3)= p ( 0 f”) (3.29)
C“ 213 (-)

2
R + — 2 — R
( R )
If a third order proﬁle is used for the concentration in the liquid and neglecting the higher
order terms i.e. (5/ R)2 and higher, we get

_ 27p2D(C0 -Cw)R4
4(th3 -pgoR3)

 

fl 3
dt (PgR ) (3.30)

To obtain the second order proﬁle, the gas concentration was assumed to be equal to the
initial gas concentration and gradient of the concentration was assumed to be zero outside
the boundary layer. To obtain equation (3.30), second derivative of the gas concentration

was also assumed to be zero outside the boundary layer.
Equation (3.10) together with equation (3.26) or equation (3.30) constitute the governing

equations for the grth of a bubble in a large body of Newtonian liquid with the

assumption of a second order or a third order proﬁle for the gas concentration. The

22

unknowns in these equations are the bubble radius and the gas pressure or the gas density
inside the bubble. The initial conditions for these equations are

R(t = 0) = R0
(3.31)
Pg (t = O) = PgO

The solution to these equations yields the bubble radius and the gas pressure inside the
bubble as a function of time. In order to solve the equations and provide the results in a
more effective manner the equations are made dimensionless and solved for the

dimensionless parameters.

The momentum equation (equation (3.10)) along with equations (3.26) or (3.30) are made

dimensionless using the following parameters

 

P; = ‘E‘ (3.32)
P; = gi- (3.33)
a
R” = f; (3.34)
t” = EL—f (3.35)
5 = PgR3 —P§0R3 (3.36)
P80Ro

23

Dimensional analysis shows that tref , which is related to the momentum transfer, is equal

to n/Pa. The momentum equation after inserting the dimensionless parameters is given

by

 

if: hi0 —r>f — (3.37)
4"'dt

Equation (3.26) is converted to dimensionless form using the parameters in equations

(3.32), (3.33), (3.34), (3.35) and (3.36). We get

*
‘15“ =C4S7 (R*3-§*—l)2 12 (3.38)
dt E R

 

 

Equation (3.38) is singular at t* = O , therefore the following transformation is used.

C" =32. C*(0)=0 (3.39)
Substituting equation (3.39) in equation (3.38) the reduced mass-balance equation

becomes

dt;*

2
(13” JCT—1)
.. = 2C4C7

dt R"'2

 

 

(3.40)

where the constants C3 , C4 and C7 are given in Table (3.1). C7 depends on the type of
polynomial proﬁle approximation being used. The expressions for C7 , for second and

third order polynomial proﬁles are also shown in Table (3.1). The physical signiﬁcance
of these constants will be explained in chapter 4. The boundary conditions for these sets

of equations are
R*(o)=1 (3.41)

§*(o)=o . (3.42)

Summary

In this chapter the governing equations for the growth of a single bubble in viscous
Newtonian liquids were reviewed. The Integral method was used to modify the diffusion
equation. The integrated diffusion equation was also modiﬁed using the interfacial mass
balance. Then second and third order polynomial concentration proﬁles were substituted
into the modiﬁed diffusion equation (equation (3.21)). The resulting diffusion equation
was then solved along with the momentum equation using the boundary conditions. The
results are then plotted and compared to various other methods of obtaining bubble

growth proﬁles. The plots are shown in the next chapter.

Table 3.1: Expressions for the constants C2, C3, C4, C5, C6 and C7

 

 

 

 

 

 

 

 

 

 

 

 

 

3
S
C2 —3 -
R0
20‘
C3 ROP a
Du
C 2
4 R0P a
pR T
C5 Pg
a
C6 tha
2 2 .
C 6p kh (R gT)2 Second order polynomral proﬁle
7
27p2k121(RgT)2 _ .
4 Third order polynorrual proﬁle

 

25

CHAPTER 4
GROWTH OF A GAS BUBBLE IN A LIMITED BODY OF NEWTONIAN

LIQUID

In this chapter, the equations governing the growth of a gas bubble in a limited body of
Newtonian liquid are presented. This is representative of a situation when a large number
of bubbles are separated by small distances grow in the liquid. In the earlier approach,
using a polynomial proﬁle, the gas concentration far from the bubble was assumed to be
equal to the initial concentration at all times. Hence although the diffusion process slows
down and its rate reduces with time, it will never be zero. This will result in over-
prediction of the ﬁnal bubble radius. Therefore it is necessary to solve the equation in a
different manner by taking into account the ﬁnite thickness of the liquid shell
surrounding the bubble. The assumptions of Henry’s law at the gas-bubble interface are

still valid though.

Three different techniques of predicting the bubble growth behavior in a ﬁnite thickness
shell are developed. These results are then compared with the bubble growth behavior in
an inﬁnite medium predicted using approximate polynomial proﬁles for the gas
concentration and the results obtained by Arefmanesh [1]. Arefmanesh obtained the
bubble growth behavior by solving the full diffusion equation. Then the regimes under
which the assumption of polynomial proﬁle breaks down are investigated by comparing
the approximate solutions with the solutions obtained by solving the full diffusion

equation.

26

Hydrodynamics of the growth in a shell of Newtonian liquid

To analyze the growth of a gas bubble in a ﬁnite body of liquid, a spherical gas bubble
surrounded concentrically by a liquid shell with ﬁnite thickness and constant mass is
considered. This conﬁguration, which ensures spherical symmetry, is shown in ﬁg (4.1).
In this ﬁgure, R(t) is the radius of the bubble and S(t) is the radius of the outer shell. It is
assumed that the bubble growth occurs under isothermal conditions. Therefore, the
volume of the liquid shell remains constant throughout the process. The kinematics of the
ﬂow in this case remains the same as before and the velocity ﬁeld is given by the
equation (3.5). The momentum equation is given by eq (3.7). However in this case the
momentum equation is integrated between the bubble interface (R) and the outer radius

of the shell (S). The integrated momentum equation is given by

. .. . )
-l—RZR4 i—J— — RR2+2R2R (Li) =
2 s4 R4 S R

 

I (4.1)
' )
—P(S)+P(R)+rn(S)—r,,(R)+4r|RR2[—l3--—l§-
S R J

The condition of stress continuity (equation (3.9)) at the bubble interface is applied again

here. Equation (4.1) can then be written as
1 .4. . i Q. .
p — v3 -1RR2— 313 -1 R2R+2RR2 =
2 (4.2)

(Pg - Pf )R — 2e+ 4n(w -1)R

27

9C_A=0 CAUJ)
at

 

4 Cw = 1(th Gas Bubble

 

 

 

 

 

R (t)

S (t)

 

Viscous Liquid + Dissolved Gas

V

Figure 4.1: Schematic of a bubble in a shell of viscous liquid containing dissolved gas

 

 

 

3

Wherew = , V is equal to S3 — R3 and is proportional to the volume of the liquid

 

V+R3
surrounding the bubble and Pf is the applied pressure at the outer boundary of the liquid

shell. For a high viscosity liquid i.e. when the viscous forces are dominant over inertia

forces then the left hand side of the above equation can be neglected as shown in chapter

3.

(Pg --l>f )R — 20+ 4n(tp -1)R = 0 (4.3)

28

Converting it to dimensionless form

4(C21:1C'2R*3)R.* = (1;::2_)>;2_P;R* —C3 (4.4)

Equation (4.4) governs the momentum transfer between the bubble and the liquid shell. If
the volume of the liquid in the shell is much greater than the bubble volume, this equation

reduces to the momentum equation (equation (3.37)) derived previously.

Diffusion of mass in the liquid shell

The conservation of mass for the dissolved gas in the liquid shell is described by the
equation (3.11). The diffusion of gas from the liquid shell into the bubble is governed by
the equation (3.13). However in this case the radial distance is conﬁned between the
bubble radius and the outer radius of the spherical shell. The main difference between the
current and the previous case is that the amount of liquid surrounding the bubble is
limited now. Therefore, the amount of gas will also be limited now. This issue is directly
related to the boundary condition for the diffusion equation at the outer radius of the
liquid shell. The correct boundary condition, which expresses the physics of the problem,
is that the gradient of the gas concentration at the outer boundary of the liquid should be
equal to zero. Earlier it was assumed the concentration outside the boundary layer is
always equal to the initial concentration. Hence the gas concentration at the outer
boundary always equals the initial concentration. This implies that all the derivatives of
the concentration are equal to zero outside the boundary layer and at the outer boundary
of the shell. However for a bubble surrounded by a limited body of liquid, the

concentration at the outer boundary of the liquid is not equal to the initial concentration

29

but decreases as more and more gas diffuses into the bubble. However as the liquid in the
shell around the bubble does not exchange mass with the surroundings, all derivatives of
the concentration are equal to zero at the outer boundary of the liquid. This is also the
case at the line of symmetry between the two bubbles.

The diffusion equation in spherical coordinate system

BCA BCA l a 28CA)
-— V—-——=D—— — .
at + r [r2 (r (45)

In order to solve the set of equations the Integral Method adopted for inﬁnite medium,

will be used. The diffusion equation is multiplied with r2 and integrated from with
respect to r from r = R to r = S, where S is the outer radius of the shell. After integrating

and using the following relationships
rzv, = 1?.2 R (4.6)
3CA
CA(S)=CS,CA(R)=Cw,and-—ar—=0atr=S (4.7)

Where the ﬁrst condition is obtained from the continuity equation as shown earlier in
chapter 3. The second and third conditions mean that CS and Cw give the concentration
of the gas at the outer shell and the bubble-liquid interface respectively. The fourth
condition means that the radial concentration gradient is zero at the outer end of the

liquid shell.

We get

8 S
3 I (CA -Cs)r2dr+ jtz%m=—DR2(3Ca—f—] (4.7)

30

3C5 _ dCs

at (It (4.8)

Equation (4.8) means that CS is a function of time only. At this juncture the equation

(4.7) needs to be looked into to make any good simpliﬁcations so that it can be evaluated
easily.
There are different approaches that can be adopted at this juncture:
Using momentum equation along with

l) Diffusion equation and interfacial mass balance

2) Diffusion equation, interfacial mass balance and overall mass balance

3) Interfacial mass balance and overall mass balance
The various strategies to solve the equations are also shown in Table (4.1). Interfacial
mass balance needs to be used because it simpliﬁes the Right Hand Side of the integrated
diffusion equation (4.7). Simplifying assumptions need to be made in order to solve the
equations. The ﬁrst approach makes simplifying assumptions in the second term in
equation (4.7). The second method uses all the three governing equations to make
simpliﬁcations. In the third method, the Fick’s Law of diffusion equation is not used

explicitly.

First approach
In this method of evaluation of the proﬁle the following assumption is made

83 —R3

 

= Constant (4.9)

This is a good assumption if the bubble radius of the same order of magnitude as the

outer radius of the liquid shell. This will be a good assumption after the bubble has grown

31

to a certain size. There is another assumption that is similar to the one stated above and is
similar in analogy. It can be assumed that the velocity of the liquid in the shell can be
assumed to be negligible as the liquid is highly viscous. It will be shown that both these

assumptions are not very much different from each other.

Table 4.1: Methods adopted to predict bubble growth behavior in ﬁnite shell

 

 

bubble growth
Momentum equation along with
Diffusion equation and Interfacial
Method 1 mass balance equation

 

Diffusion equation, Interfacial mass
Method 2 balance equation and Overall mass
balance equation

 

Interfacial mass balance equation and
Method 3: Overall mass balance equation

 

 

 

 

Integrating equation (4.7) with respect to time from t = 0 to t = t and utilizing the

boundary equations, we get

3 _ _ 3_ 3 R3_ R3
J‘ CA CS rzdr_ C5 CO 8 R zipg ng 0 (410)
Cs —Cw 3 3p Cs --CW

Han and Y00 [26] derived equation (4.11) for gas bubble growing in an inﬁnite medium

as shown in chapter 3.

R+8 3 3
_ R - R
3p j [thZm-pg 93° 0 (4.11)

R C0-Cw C0-Cw

32

A second order polynomial proﬁle is assumed for the concentration proﬁle and the

boundary conditions substituted in it. This results in the following expression

2
———CA_CS =[s") (4.12)
Cw—CS S—R

 

Substituting equation (4.12) in the ﬁrst term on the left side of equation (4.10) and

 

 

simplifying we get
5 2 3 4 5
___ 3p _S___SR +SR _R (4.13)
(s - R)2 30 3 2 5
Substituting equation (4.13) in the diffusion equation (equation (4.10)) we get
pgR3 —pgoR3 +Mcw -pcoh3 -133)
C3 = . . (4.14)

 

M-pIS3-R3)

The interfacial mass balance equation after substitution of expression of a second order

polynomial concentration proﬁle becomes

2
d 3): 6"“ ———(CS— Cw) (4.15)

E (s- R)
Converting the expression for M into dimensionless form using the dimensionless

parameters shown in Table (3.1) we have

 

 

3 R3 4:5 4:2 4:3 4:4 *5
___ p s _s R +s"'R _R (4.16)
M8111”? 30 3 2 5
using
PR3-P R3
E“: 8 go 0 (4.17)

 

3
PSORO

C S in dimensionless form can be written as

33

 

R,3) P R3P‘ g"o§
R ,1“
,3) (4.18)

* * *
MKhPan -pKhP,PgoR3(s 3 -
C3 =

 

Iii-911363 -

Substituting parameters from Table (3.1) into the mass balance equation (equation (4.15))
we get

d5 _ 6RgTupD ‘ R'”2
dt" Rgpfpgo (s*-R

 

 

.)(Cs -Cw) (4.19)

Using the transformation C“ = Efz we get

{
P Rog"

RsT-(M -89R 36”- 113+»

dt’ ’-R")R *3(M-— pR0(s"3- Rj)
_PKhPaRSIS - ’3 _Ce +5“
( (M-pR(3,(s"'3—R"3 R’3 J

The constants C2, C3 , C4, C5 and C6 have a physical signiﬁcance attached to them.

 

 

 

 

 

 

C2 expresses the ratio of volume of the liquid to the initial bubble volume. C3is the
dimensionless surface tension. C4 represents the ratio of the time scale for the

momentum transfer to the time scale of the gas diffusion. C5 is the ratio of liquid to the

gas density when the latter is evaluated at atmospheric pressure and C6 is a dimensionless

Henry’s constant.

If a third order polynomial proﬁle is used instead of second order proﬁle we have

34

3
___CA ‘CS =(__S”] (4.21)
Cw -Cs S—R

Substituting equation (4.21) in the ﬁrst term on the left side of equation (4.10) and

 

 

 

 

 

simplifying we get
6 6 5 4 2 3 3
M: 3p 3 _S_+R _3R 8+3RS _RS (422)
(s _ R) 60 6 5 4 3
Substituting equation (4.22) in the diffusion equation (equation (4.10)) we get
R3- R3 MC - c (83 -R3)
Pg Pgo o + w P 0
C5 = . 3 3* (4.23)
M - p(s - R )
The interfacial mass balance equation becomes after substituting equation (4.21)
3 R3)= ngRZ (Cs -Cw) (4.24)
dt 3 (S -— R)

Converting the expression for M into dimensionless form using the dimensionless

parameters shown in Table (3.1) we have

 

 

M _ 39R?) [8’6 _ s"'-”R"'3 38’2R*4 _ 3R*58* + R"6

(s*—R’)3 60 3 + 4 5 6

Substituting parameters from Table (3.1) equation (4.24) becomes

J (4.25)

dig" 9RgTupD 11'"2
dt" Rgpfpgo E—R

 

*)(cs -cw) (4.26)

The rest of the procedure for modifying the equations is the same as that shown for

second order proﬁle solution.

35

Alternate simplification

Another approach to simplifying the second term in equation (4.7) is to assume that the

velocity of the liquid inside the shell (vr ) is negligibly small.

 

3 3
.21. CSR =§__dCS +r2viCs (4.27)
dt 3 3 dt

If we let vr = 0 during this part of the time we have

3 3
2. CSR =L__dCs (4.23)
dt 3 3 dt

 

Substituting in the diffusion equation expression we have

s 3 3 3

S C R -C R 1
i(CA“Cs)f2dr+—3—(CS‘C0)‘ S 0 0 =‘3_p(’8R3‘980R3) (4°29)
R

 

3

dividing equation (4.29) by Cw -— C S , which is a function of time only we get

(CA 419,2st3 (Cs -Co) _CsR3 -CoR3 = ya“ week?)

(Cw-Cs) —3—(Cw-Cs) 3(CW‘CS) 3P (CS"CW)

 

(4.30)

 

”5—501

Evaluating the above expression after substitution and extracting the expression for C S

pgR3 -pgoRg +MCW -OC0(83 -R3)

M—p(S3 —R3) (4.31)

CS:

 

The only difference between the equation (4.13) and equation (4.31) is that in the last

term of the numerator. Equation (4.31) has R0 in the last term of the numerator, but

equation (4.13) has the instantaneous bubble radius R. Converting equation (4.31) into

dimensionless form we get

36

t *
PaRgpgog

* t 3 #3
MKhPan—pKhPanORO(S -1)+ R r
8

CS = M—pR3(S*3 -R*3)

 

(4.32)

The rest of the procedure to solve the equations is the same that used when the

s3 R3

assumption of

 

= constant, was used. The boundary conditions for this set of

equations are
R’(0) = 1.0 (4.33)
i;’ (0) = o (4.34)
The momentum equation (equation (4.4)) and the reduced mass balance equation

(equation (4.19)) need to be solved simultaneously subject to the boundary equations.
Either equation (4.18) or equation (4.32) can be used to substitute for CS in equation

(4.19).

Second approach
In this approach global mass balance is also instead of just the interfacial mass balance.

The global mass balance at any instant of time can be written as:
Mgb + Mgs = Mginitial (4.35)
Where

M gb = Mass of gas inside the bubble
M gs = Mass of gas inside the liquid shell

Mginitial = Mass of gas initially i.e., at time t = 0

37

4

Mgiuitiall = 31433 -R3kop (4.36)
4

Mgb = 31:11 — (4.37)

At this point the concentration of gas inside the liquid shell can be approximated in two

different ways

1) If Cs is assumed as the concentration inside the shell we get
M - 1:433 4(3): (4 38)
gs - 3 o Sp -
Substituting in the mass balance equation and extracting Cs we get

S3 -R3 R3 Cw

cS =————c0 -
s3 -R3 s3 -R3 KhAp

(4.39)

 

2) If the average of CS, Cw is assumed as the concentration of the gas inside the shell.

We get
_ 4 3 3 CS + Cw
Mgs —§"E(SO —R )—2——p (4.40)
. . . . . CS + Cw .
Equation (4.40) is obtained by substituting ——2— for the concentration of the gas

inside the shell. Substituting in the mass balance equation and extracting CS we get

_ 2C0(S3 4‘3)

 

3
3 3 2CW3R 3--cW (4.41)
s -R KhAp(S -R )

Instead if the exact expression for mass of gas inside the shell is used we have

S
Mgs = 41tp j C Arzdr (4.42)
R

38

Substituting in the mass balance equation (equation (4.35)) and extracting CS we get

 

 

 

S3-Rf;C CWR3 cw g_szk3+SR4_R_5
3 ° 3KhAp (S-R)2 3o 3 2 3

 

 

 

 

 

 

CS = (4.43)
s3—R3_ 1 _sj_szn3+SR4 -3:
3 (S _ R)2 30 3 2 3
From equation (4.14) we have
3 _ 3 _ 63 -R3)
pgR ngR0 + MCW PCO 0
Cs = . 3 3* (4.44)
M — p(s — R )
Equating equation (4.43) and equation (4.44) we get an expression for Cw
R3 s3 - R3
C ——9—+ s3 —R3 +2———£(M— s3 —R3
O[KhA p( 0) 83— R3 p( ))
Cw = (4.45)
3 3 3 3
R 2R (M-p pgs f) »+2M—pE3-R3)
KhA+ K hAp(S —R)
The momentum equation in dimensionless form is
_ *3 o I. *
4(C2 +1 R )R = CWR _pg'R“ —c3 (4.46)

 

 

l + C2 KhPa
Cw can be substituted from equation (4.45). We have just one equation (equation (4.46))

involving the dependent variable, the non-dimensional radius of the bubble. This

equation can be solved using the boundary condition:
R" (0) =1 (4.47)
Only one boundary condition is required, as the momentum equation has been simpliﬁed

to a ﬁrst order equation in R * .

39

Third approach
This approach uses the momentum equation along with the mass balance equations, both
interfacial and overall, to evaluate the proﬁles. This approach does not use the Fick’s law

of diffusion equation explicitly. Using the average of C3 , Cw as the concentration of the

gas inside the shell we obtain the following expression for C S

c _2c0(s3—R3)_ 2CwR3
S _ s3-R3 KhAp(S3—R3

 

 

)-cw (4.48)

The interfacial mass-balance equation involves the gradient of concentration, C A at the
bubble wall, which can be expressed as

3C». = _2(Cw “(38) (4 49)
3r r=R S-R '

 

Equation (4.49) is obtained by differentiating equation (4.21) with respect to the radial
coordinate r. Substituting equation (4.49) in the interfacial mass-balance (equation (3.17))
and converting into dimensionless form using parameters from Table (3.1), we have

*3 ‘3
2c 3 -1 zc R
cS—cw= *g( *3)- 3X3 *34-2cw (4.50)
s —R KhApS -R

 

Because of non-linearity encountered in the differential equation when Cw is used, we

use the transformation

 

(4.51)

:1- at
Pg0[l + g )KhPa
Cw = *3

R

d1? 6RgTupD R'“2

_ C -C (4.52)
dt* R(szango (S*-R*)( S W)

 

Equation (4.52) can also be written as

40

 

 

* *2
“‘9, =2\/E(6C4C5{ *R
S ...

CS —Cw can be substituted from the equation (4.50) into equation (4.53). Now

...](Cs -Cw) (4.53)

equations (4.53) can be solved along with the momentum equation (equation (4.4)) for a
ﬁnite shell.

The boundary conditions for this set of equations are
R*(0) =1 (4.54)

I;* (0) = 0 (4.55)
Complete solution
Arefmanesh obtained complete proﬁle by solving the diffusion, momentum and mass-
balance equation in its entirety [l]. The procedure used by Arefmanesh is shown in this
section. To solve for the concentration proﬁle, the diffusion equation (equation (3.13)) is
transferred to the Lagrangian coordinate which eliminates the convective term. The

Lagrangian coordinate transformation, which is widely used in phase phenomenon, is

deﬁned as

y = r3 —R3(t) (4.56)
t =t (4.57)
where (r,t) and (y,t') are the old and new coordinates respectively. To transfer the
diffusion equation, time and special derivatives are needed in the new coordinate system.

Using the chain rule of differentiation, the following equation are obtained

“5191 3.25
_ By a: + are: (4.58)

2.
8r

 

(4.59)

111.41
3y8t+'dt3t

Substituting these expressions into the above equations, results in the following

i
at

transformations

’3

  
 

3 2

a: ’ 8y ( )
3 2 6
3r 3r 8y ( )

Using the above transformation, equation (3.13) is transferred to the Lagrangian

coordinate system. The diffusion equation in the new coordinate system is given by

, 4 .
301 = 9D-Q-[(y +113); 9&1] (4.62)

 

at 33’

It is assumed that there is no gas diffusion at the outer boundary of the liquid shell.
Mathematically, this implies that the concentration gradient at the outer boundary of the
shell is equal to zero at all times. Hence, the boundary conditions for equation (4.62)

written in terms of the new coordinate system are as follows

c3.)y=0 = 1.1,(Pg 480) (4.63)
ac'A _ 0
_ (4.64)
.. (......s)

and as the initial concentration is equal to C0 ,
C'A M = 0 (4.65)

At the early stage of the bubble growth, the concentration gradient close to the interface

is quite large which makes the task of obtaining numerical solution to equation (4.62)

42

difﬁcult. To overcome this difﬁculty, a vector ﬁeld 9 with the components CA and

 

4 I
9D(y + R3); 3:; in the y - t plane is introduced. Using equation (4.62), it can be shown

that such a vector ﬁeld is irrotational. Therefore, following the properties of the
irrotational ﬁelds, there exists a potential function <I>(y,t) such that

$2 = W) (4-66)
This allows one to recast the diffusion equation (equation (4.62)) in terms of the potential

function. The resulting equation is given by

4 2
a<1> _ 3 3 ch

where 53% is equal to Ck. The initial and boundary conditions given by equations (4.63),

(4.64) and (4.65) are also written in terms of the new dependent variable (<I>) as follows

<I>(y.0) = 0 (4.68)
<1>(y = 0, t) = Kh (Pg — Ps0) (4.69)
(byy(y=S3 -R3,t)=0 (4.70)

The variable <I>(y,t) is the integral of the gas concentration with respect to the y
coordinate and varies more smoothly near the interface. Hence the solution of <I>(y,t) in

the transformed equation (equation (4.67)) poses less numerical difﬁculties and is more
efﬁcient during the early stages of the bubble growth when there is a large variation in

the concentration gradient [1].

43

Equations (3.17), (4.2), (4.67), (4.68), (4.69) and (4.70) constitute the system of the
governing equations and the boundary conditions for the growth of a spherical bubble
surrounded by a shell of Newtonian liquid ﬁnite thickness. Initial conditions for
equations (3.17) and (4.2) are

Pg (t = 0) = Pgo (4.71)

R(t = 0) = R0 (4.72)
Then these set of equations were solved using Fourth-order Runge-Kutta method for

solving differential equations [1].

Numerical evaluation
The results presented in this study are conﬁned to highly viscous liquids such as polymer
melt. To estimate the magnitude of the inertia terms in the momentum equation (4.2), an

order of magnitude analysis was conducted on this equation. To make the terms of the

order of one, the equilibrium radius (Re) and the initial gas pressure were used to non-

2

dimensionalize equation (4.2). This resulted in the dimensionless number[
1]

pRngO]

which can be regarded as a Reynolds number. For typical values of p =103kg/m3 , Re =
1 mm, Pgo =106N/m2 andn =103Ns/m2 , the magnitude of this number is of the

order of 10"3 , which is much smaller than unity. Therefore, for highly viscous liquids, the

role of inertia terms in the growth process can be safely neglected.

The system of differential equations governing the bubble growth is non-linear and
coupled for which there does not exist any known analytical solution. Mathematica 4.1
was used to solve the coupled differential equations. A fourth-order Runge-Kutta method

was used to solve the differential equations. In some of the evaluations there was a

problem in getting the solver started due to singularity of C at time t = 0. Therefore a
non-zero value of C was speciﬁed for examplelO—G. When a non-zero value of C was
chosen the solution was obtained. In order to check the dependency of C on the initial

value chosen, the initial values were varied from 10'3 tolO—7. No signiﬁcant change in

the results was obtained by choosing any value between 10'3 to 10"7 . Arefmanesh
solved the complete concentration proﬁle by solving the three coupled governing
equations (3.17, 4.2 and 4.67) using a combination of the Runge-Kutta method and the
explicit ﬁnite difference scheme [1].

The numerical simulation was used to investigate the effect of various dimensionless

parameters such as C2 ,C3 , C4 and the initial gas concentration on the bubble growth. A

large range of values was used in the simulation. The lower and upper bound of these

parameters are given in Table (4.2).

Table 4.2: Lower and upper bounds for dimensionless variables C2 , C 3 and C4

 

 

 

 

Lower Bound Upper Bound
c2 104 108
C3 0.3 0.9
c4 10 104

 

 

 

 

 

45

Equilibrium calculations

The equilibrium bubble radius can be evaluated by applying a mass balance to the
dissolved gas in the liquid around the bubble. The calculated equilibrium radius may then
be compared to the predicted value from the numerical solution of the governing
equations to verify the accuracy of the numerical predictions. The conservation of mass
for the dissolved gas in the liquid and the ideal gas law for the gas inside the bubble are

used to evaluate the equilibrium bubble radius. The bubble radius Re and the gas
pressure inside the bubble Pge at the equilibrium are related through the following

equation

ch = P. +1-25— (4.73)

C

The equilibrium gas concentration C.: is related to the equilibrium gas pressure through
Henry’s law ( Ce =Kthe ). The ideal gas law for the gas inside the bubble at the
equilibrium gives

chVe = MgcRgTe (4.74)
where V6 is the bubble volume at the equilibrium and is equal to gnRg. Rg is the ideal

gas constant. Tc is the equilibrium temperature of the gas inside the bubble, which is

assumed to be the same as the polymer temperature. M ge is the mass of the gas inside the

bubble at the equilibrium and is equal to (4/3)n(sg - R3)u(co - cc). Substituting these

expressions into the ideal gas equation (equation (4.74)), results in the following

expression for the equilibrium bubble radius

46

26 4 3 _ 4 3 3 20'
[Pf + EJ(§TCR¢) — 3713(80 "' R0 })[Co - Kb [Pf + R_]]R ng (4.75)

e
The equilibrium bubble radius is obtained by solving equation (4.75). The equilibrium
radius obtained by solving the coupled differential equations matches well with the

equilibrium radius obtained from equation (4.7 5).

Comparison of profiles
A variety of cases have been studied by varying the parameters C2 ,C4 and P30. C2
represents the thickness of the liquid shell. The larger the value of C2 , the bigger the

liquid shell and the inﬁnite shell approximation become more valid. The parameter C4 is

indicative of the diffusion coefﬁcient of the gas in the liquid. The higher the value of C4 ,

the higher the diffusion rate. The initial gas pressure Pgo was also varied and the radius

proﬁles studied. The value of the initial gas pressure Pgo was limited to 15. Higher initial

gas pressures were not chosen because they are seldom observed in practical situations

and may also violate ideal gas law.

Figures (4.2), (4.3) and (4.4) show the effect of thickness of the liquid shell
(parameter C2) enveloping the bubble on the bubble growth. The three different values of
C2 (104, 106 and 108) correspond to thin, medium and thick shells. The results show that

as the thickness of the liquid shell enveloping the bubble increases, the time to reach
equilibrium increases. Also, since the mass of the gas within the shell increases with its

thickness, the value of the equilibrium radius will be larger. At the onset of the growth,

47

the various solutions match well with the complete solution obtained by Arefmanesh [1].
More so the ‘method 3’ method matches very well with the complete solution. The
equilibrium radii predicted by the various methods are varying slightly from the complete
solution. It can be seen that the ‘method 1’ predicts the equilibrium radius very well
compared to the other methods. The ‘method 1’, ‘method 3’ under-predict the initial
grth rate, but the growth rate picks up ﬁnally and the ‘method 1’ predicts the ﬁnal
equilibrium radius very closely to the complete equilibrium radius. The ‘method 3’,
‘method 2’ and ‘Inﬁnite medium’ solutions over-predict the ﬁnal equilibrium radius by
about 10-15 %. The polynomial proﬁle solutions for inﬁnite medium of liquid
surrounding the bubble severely over-predict the ﬁnal equilibrium radius. Also from
Figures (4.5) and (4.6), there is no appreciable difference between the second order and
third order polynomial proﬁle solutions as far as the ﬁnal equilibrium radius is predicted.
But as the order of the proﬁle increases the accuracy increases slightly. The relative error

between the ﬁnal equilibrium radius decreases as the shell thickness increases.

Figures (4.2), (4.7), and (4.8) demonstrate the effects of the diffusion coefﬁcient on the
growth dynamics by changing the value of the parameter C4. An increase in diffusion

coefﬁcient accelerates the diffusion process. Therefore, the bubble radius attains its
equilibrium faster. For the case of using polynomial proﬁles for an inﬁnite medium,
increasing the diffusion coefﬁcient results in a higher grth rate throughout the period.
Therefore they overestimate the growth rate signiﬁcantly as seen from the ﬁgures. The
error decreases as the diffusion coefﬁcient is decreased. Comparing the polynomial

proﬁle solutions for the case of ﬁnite thickness shells we can see that the error decrease

48

with increasing diffusion coefﬁcient. All the proﬁles match well with the complete

solution for C4 =104 . However the proﬁles are slightly different for the case of

C4 =102. All the ﬁnite shell methods predict the ﬁnal equilibrium radius within 13 % of

that predicted by complete method. The ‘method 3’ predicts the bubble growth behavior

well for times t* less than two-thirds the time taken to reach equilibrium. At the end of
the growth process the radius proﬁle for complete and ‘method 3’ diverges and reach

different equilibrium radius.

The inﬂuence of initial gas concentration PEG on the growth is depicted in Figs. (4.2), (4.4)

and (4.10). The results are for three different values of the dimensionless initial gas
pressure of 5, 10, and 15, which are related to the initial gas concentration through
Henry’s Law. Higher initial gas pressures were not chosen because they may violate the
ideal gas law and because they are seldom observed in practical situations. Higher initial

gas pressure results in higher growth rate and also a higher ﬁnal equilibrium radius.

From the results it can be seen that the proﬁle solutions for high initial gas pressure and
high diffusivity give good results. It can also be seen that the models do not work well for
low gas pressure, low gas diffusivity and for thick shells. However the models developed
using ‘method 1’ and ‘method 3’ predict the bubble growth better than the inﬁnite
medium solution. The reasons for better proﬁle prediction during high initial gas pressure
and high diffusivity are that the concentration boundary layer reaches the outer shell very

fast, and proﬁle switching is not necessary. However for thick shells, low initial gas

49

pressure and low gas diffusivity the concentration boundary layer takes considerable time
to reach the outer edge. Therefore we need to use proﬁle switching in these cases to
better predict the bubble growth. This proﬁle switching can be done using “Penetration

Dep ” techniques [31].

 

30 f V v f r fﬁ v v v v r v 1 v v v 2
4

 

so
U1
\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a? I
a: 20.
E: .
a L
E 15 . -— Method 1 _
a i 3
o 1 ------ Method2 «
‘g . .
'3 10 7
g 1 - - - ' Method3 .
b 1 L
5 q .’ — — Inf-Medium .
0 5 10 15 20

Dimensionless Time ( t / tref)

Figure 4.2: Comparison between various methods of evaluating the radius proﬁles for

low thickness shell (C2 = 10" ), C3 = 0.4, C4 = 102 (average gas diffusivity) and initial gas

pressure Pgo = 10.

50

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

100 . 1
l
’ /
/
/
A 80 , """"" '—’ --------- r ———————— ———- 7 ___-
I ./
é . I, / (___—___,— ./ _ +
\ 1 / / / {
a ’ : / "“4 """" .
.3 60 5 r‘ .4; r’ .4
'8 l ' / x" """""""" Method 1
g ,' / /
U2 1 / / -
8 1 .' / / / , ------ Method 2
g 40 : '1 /’ .4" _
"-1 ’ , / / x".
g : ’/ - ‘ - Method 3
I i ,u
'3 , = 4’ . .
' // — — Inf - Medium
20 _f / 4 _—
. //
1 / / Complete
:
r J '

Dimensionless Time ( t / tref)

Figure 4.3: Comparison between approximate and accurate numerical solutions for

medium thickness shell (C2 = 10° ), c3 = 0.4, c. = 102 (average gas diffusivity) and

Pg}, =10.

51

 

 

 

 

 

 

 

 

   
  

 

 

 

 

 

 

 

 

 

 

 

 

400’ I . . . T . . . . . . . . . . . 1 1 . ,
L I
r 1 ---------------------------------------- :‘Jv— ———————————— 1
I- I /
350 I /
. 1 /
A b | / 42"“,-
a? 300 t / ’4
.- I z."
\ ' I / /'.¢" 4
m . I / / /‘/.4‘
v I 1 1 4'
g 250. : /, / .
:8 I / / / x / 1
I '.
a 1 / 3 / / .
3 2m. I 41/ / NI". // ‘
8 ~ I // / """""" Method 1 .
'3; 150_ T T —
g ; : ------ Method2
b ' ‘
' I
9100»: “" Method3 T
r I .
1- I 4
. 1 — ‘— .. ' 1
50. : Inf Medium M
I
. . —- Complete
1' I
‘ '4' . . - . . 1 . . . . . . . . 1 1 . . . .
O 200 400 600 800 1000 1200

Dimensionless Time ( t I tref)

Figure 4.4: Comparison between approximate and accurate numerical solutions for high

thickness shell (C2 = 108 ), C3 = 0.4, C4 = 102 (average gas diffusivity) and Pgo =10.

52

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100 6 . . . - r . . . . . . . . . . T
r .
A 80
a? 4 f _
a: / ~ — — ,: ...—___ ————————
v ‘ / ,”
.3 so /« __
52 ------- Second Order 1
3 40 ~
g " — — — Third Order
E _
Q 20 1
_ — Complete
40 60 80 100

 

Dimensionless Time ( t/ tref)

Figure 4.5: Comparison between second, third order proﬁle solutions using ‘method 1’

and the complete solution obtained by Arefmanesh [1] for medium thickness shell (C2 =

106 ), C3 = 0.4, C; = 10 (average gas diffusivity) and initial gas pressure PEG =10.

53

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100 . . . 4 1. . . . . . - . . , - .
A 80 T. -
a? . _ 1
:4 4 .
V L T
U)
5 —4
13 6°. .
a Second Order.
3 ‘ .
'5. 4° "
'g 1 Third Order 1
E I 1
20_ — Complete 2
0 20 40 60 80 100

Dimensionless Time ( t / tref)

Figure 4.6: Comparison between second order, third order polynomial proﬁle solutions
for inﬁnite medium and complete solution obtained by Arefmanesh [l] for medium

thickness shell (C2 = 106 ), C3 = 0.4, C4 = 10 (average gas diffusivity) and initial gas

pressure Pgo = 10.

54

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

175? .
150’ --------- Method 1
A .
a? ----- Method 2 / ’
\ ' /
9417-5. --- Method3 /
T; ’ / 1
.5 1 — —’ Inf - Medium /
g 100 / '4
:12) E -— Complete /
. ____-,;-;_3,-__/.--_-_._- _________
'E 75 ,44‘"";4/ " f /
.9. I ,/’ 4’ ”-.-wa --------------------------- 1"; 1
VJ , / “3"
a " I’ / ”ﬂ
g 50» f / / / /
b ’ I, / . "’I / :
r ,l ..." / 1
25 ’ ’(I / [’31, / / 4
. ”I /’-'/’j:' / j
4,447 i
0 2 4 6 8 10

Dimensionless Time ( t / tref)

Figure 4.7: Comparison between approximate and accurate numerical solutions for

medium thickness shell (C2 = 106 ), C3 = 0.4, C4 = 10" (High gas diffusivity) and initial

gas pressure Pgo =10 .

55

Dimensionless Radius ( R I R0)

100

80

20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Dimensionless Time ( t / tmf)

I // ’ /

: // (rt/i. ‘
l / / / H
I / / V

E l’ / / — Methodl

: / / /

l 7; / , ------ Method2 _
l V

j //

i /// / ___ Method3

E 1 / / / H
1 / ,’ / — — Inf-Medium
1///

0 100 200 300 400 500 600 700

Figure 4.8: Comparison between various methods of evaluating the radius proﬁles for

medium thickness shell (C2 = 106 ), C3 = 0.4, C4 = 10 (low gas diffusivity) and initial gas

pressure PEG = 10.

56

Dimensionless Radius ( R / R0)

 

 

\l
o
l
I
I
I
I
I
I
I
l
u
I
I

 

‘
\
\
\
a
o \
‘
I
\
\
'\
‘
I
\
L A A A

 

VI
0
-h‘
\
\

 

 

 

 

. ‘5

I 1
' ' l 1
I J..' 1
I ,x i
I 7 ~ .o‘
I / ,.-’ 4
D I / ’a"' / i ‘
. : / .""'. /

L 1 1' A
I .’ / ”/7 ---------- Method 1

i Il / / ﬁx

I / / r", ______

. / / Method 2

1

I

I

I

I

I

1

I

I

 

 

 

 

 

 

 

 

 

 

30
I .
20 . / / A / / "u’A‘
: / / // / ,"xfx'ﬁ — — — MCthOd 3
10 / f’ 4 — — Inf-Medium
: ' {/‘:_,ao“"T
'15-" ''''' " -— Complete
0 20 40 60 80 100

Dimensionless Time ( t / tref)

Figure 4.9: Comparison between approximate and accurate numerical solutions for

medium thickness shell (C2 = 106 ), C3 = 0.4, c4 = 102 (average gas diffusivity) and low

initial gas pressure Pgo = 5 .

57

 

 

 

 

 

 

 

 

 

 

 

100 r r * . v v r
I ________________ 7 ____________ _z__________________-
/
_ I I J
I' / /
80 . / /‘
"‘ . f / ------------------- J— --------------------------------------------- L J
a? , x ...;
\ ' ,' / //,//
M I j I / .'
TX 60 f I / —‘
'5 P I J/ / """""""" MethOd 1 I
a I If I"! 1
.,, » : 47 —————— Method 2 I
g 40 .' " a
"g ' : — — — Method 3 ‘
U) | “
8 [ I ’ , J
a 20 i — — Inf-Medium
O , f
. — Complete
F I
I I 1

 

 

   

 

 

 

 

 

O 20 4O 6O 80 100

Dimensionless Time ( t / tref)

Figure 4.10: Comparison between various methods of evaluating the radius proﬁles for

medium thickness shell (C2 = 106 ), C3 = 0.4, C4 = 102 (average gas diffusivity) and initial

gas pressure PEG =15.

58

CHAPTER 5

SUMMARY AND CONCLUSIONS

This thesis involves the study of bubble growth in highly viscous liquids. Numerical
models for predicting bubble growth in viscous liquids were developed and presented.
Numerous researchers have used polynomial proﬁles approximations for the
concentration of the dissolved gas in the liquid to obtain solutions. The same approach
has been adopted in this thesis work as polynomial proﬁle approximation lead to easy

solutions to predict bubble growth with reasonable accuracy.

The growth of a gas bubble in a liquid containing dissolved gas is due to the simultaneous
mass and momentum transfer between the bubble and the liquid. The diffusion of the
dissolved gas from the liquid into the bubble is governed by the convection—diffusion
equation. A common method to solve the diffusion equation is to assume that
concentration gradient is limited to a small boundary close to the bubble interface and use
polynomial proﬁles to describe the gas concentration in the boundary layer. In the ﬁrst
part of the thesis, the governing equations for the growth of a single bubble surrounded
by inﬁnite amount of liquid are presented. This approach was ﬁrst developed by Han and
Y00 [26]. In this approach the Integral method was adopted to modify the diffusion
equation. The diffusion equation is converted into an integral form. Polynomial proﬁles
are used to approximate the gas concentration in the liquid surrounding the bubble.
Second and third order polynomial proﬁles were used to study the bubble growth.

Outside the concentration boundary layer the concentration is assumed to be invariant

59

and equal to the initial concentration. All derivatives of the concentration are also
assumed to be zero outside the boundary layer. This represents the correct physics of the
phenomenon for a bubble growing in a shell of liquid with ﬁnite amount of dissolved gas
only at the initial stage of the bubble growth process. This will depend on the shell
thickness, the rapidity at which the growth process occurs, and the initial radius. Near the
end of the growth process, when the concentration boundary layer has reached the outer
shell, all spatial derivatives of the concentration at the outer boundary of the shell should
be equal to zero. However, using this approach the governing equations are reduced to
equations (3.10) and (3.26). Then these equations were converted into dimensionless

form. These coupled equations were solved using the appropriate boundary conditions.

To incorporate the correct physics of the problem, the bubble growth in a shell of liquid
with ﬁnite amount of dissolved gas needs to be considered. The governing equations for
bubble growth in a shell of liquid were developed. The governing equations are the
diffusion equation, conservation of mass equation and momentum transfer equation. The
diffusion equation was modiﬁed using the ‘Integral method’. This involves multiplying
the equation with r2 and integrating over the domain. This technique results in a modiﬁed
integral diffusion equation. At this point three different approaches were adopted to

simplify the diffusion equation.
In the ﬁrst method it is assumed that the velocity of the liquid surrounding the bubble is

negligible. This simpliﬁes the diffusion equation, and assumptions regarding the

concentration proﬁle of gas surrounding the bubble were made. The concentration

60

proﬁles are assumed to be polynomials. In this study second order and third order
polynomial proﬁles were assumed. The proﬁle equations were substituted into the
integrated diffusion equation. The momentum equation and the reduced integral diffusion
equation were then converted to dimensionless form and solved using the boundary

conditions.

In the second method, apart from the interfacial mass balance equation, the global mass
balance equation is also used. Two different expressions for C3 are developed and then
equated. This resulted in an expression for Cw, which was substituted into the momentum
equation. The momentum equation was then converted to dimensionless form and solved

using the appropriate boundary condition.

In the third method the diffusion equation is not used explicitly. The governing equations
then are the momentum equation and the interfacial mass balance and global mass
balance. The mass balance equations are then combined into one equation and solved
along with the momentum equation after converting them into dimensionless form. After
these methods the exact method of Arefmanesh [l] was shown. Arefmanesh solved the

full diffusion equation without using any approximations for the concentration proﬁle.

A parametric study was conducted and the inﬂuence of various dimensionless parameters
on the dynamics of the growth was investigated. The results in each case were compared
with the approximate solutions obtained using polynomial proﬁles for gas concentration

in inﬁnite medium. They were also compared with the solutions obtained by solving the

61

full diffusion equation. The parametric study revealed that polynomial proﬁle
approximations for gas concentration, to predict the bubble growth behavior are
reasonably accurate. These methods also predict the ﬁnal equilibrium radius very
accurately. The approximated solutions using polynomial proﬁles for bubble growing in
an inﬁnite medium over-predict the ﬁnal equilibrium radius by three to four orders of

magnitude.

Further Study

Other ways to solve this kind of problem involve using the “Volumetric Rise” and
“Penetration Depth” approaches [31]. In the Volumetric Rise approach the unsteady
solution for the flow problem is solved by using scaling. After evaluating the steady state
solution using analytical techniques, the unsteady solution is obtained by introducing a
scale factor. The scale factor is a function of time only. The unsteady solution is
expressed as the product of the steady solution and the scale factor. The success of this

method depends on obtaining the steady state solution for the flow problem.

In the Penetration Depth approach the time domain of the problem is divided into two
domains. The ﬁrst time domain is the domain during which the diffusion of mass or
momentum has not reached the boundary of the problem. So during this time domain, the
diffusion of mass or momentum does not fully affect the whole geometry of the problem.
In the second time domain, the diffusion of mass or momentum fully affects the whole
geometry of the problem. The solution strategy is to use polynomial proﬁles which

satisfy the boundary conditions of the problem. Then these solution proﬁles are

62

introduced in the governing equations solved in the two time domains. At the domain
interface the solutions are forced to match. This approach of obtaining solutions to
predict the bubble growth behavior is very promising. The solution proﬁles seem to
match in initial stages with the complete solution. This suggests that we need to use the

Penetration Depth approach, so that the better proﬁles can be obtained during the latter

stages of the growth.

63

APPENDIX A

DERIVATION OF INTEGRAL DIFFUSION EQUATION

The diffusion equation in cylindrical coordinate system is

acA Va_c__A 1_d_[zacA)
— =D — .
at V’ at [25? at (A1)

multiplying with r2 and integrating with respect to r from r = R to r = S

 

rS= r=S
[R rzaCAdr+ [8 r2v,aCAdr=D j d[rZ-a—CA] (A2)
81'
rR= r=R r=R
as RZR is a function of time only, it can be brought outside the integral and the

expression integrated.
r=S

j rzﬂm+R2§(CS-cw)=—Dkz[ﬂ] (A.3)
r=R at at r=R

using the following transformation

R3
y _ _3_ (AA)
we get
y=s3/3 .
j a—CidyI-Rz R(CS —cW )= —DR2[aC—A (A5)
3 at at _
y=R /3 r—
Leibnitz’s Rule:
b(z) 3f 313(2)
j 52-dx+f(b(z), z)——f(a (2,2) )—=— am x ,z)dx (A.6)

a(z) aza(z)

Consider the following expression

d r=sdy=s313
— I(—CA -Cs)r2 dr=— I(cA-cs)dy
(it (it 3
I: y=R /3
Applying Leibnitz’s Rule
y=s3/3 y=S3/3

5—i- j(cA—Cs)dy= I 3(Ca-Cs)dy-(Cw-Cs)RzlE
dt at

y=R3 I3 y=R3 l3
Reverting to the original coordinates
d 1'

dt CA- cs)r2dr+j— 3C3 r2=dr l:8aC——"‘—dr+(cS—- cw)R2R

R at

'-—-.II
AU)

ll
W

1'

Comparing equation (A.9) to equation (A.3) we get

65

(A.7)

(A.8)

(A.9)

(A.10)

APPENDIX B

MATHEMATICA FILE

(* This program evaluates the radius and concentration proﬁle for a thick shell *)
(* APPROXIMATE NETHOD *)

Needs["Graphics‘Legend"']

$TextStyle ={ FontFamily-) "Times",FontSize-) 15 };

P =1000 ;

Kh = 0.639*10"(-9) ;

Gascon=83 14/30 ;

T = 300. ;

Pa=1*10"5 ;

A=Gascon*T; (*A=Rg*T*)

RO=1/10"6 ; (*initial critical mdius*)

C2=10"8 ; (*indicative of the size of the shell*)

S = (l+CZ)"(l/3) ; (* Size of the liquid shell in N-D form *)

C3 = 0.4 ; (* Slightly varying with C3 decreases R with increasing C3 *)
C4 =100; (* Zero variation with C4 *)

C5 = (p *A)/Pa ;

C6 = Kh*Pa ;

C7 = 6* p “2*Kh"2*(Gascon*T)"2 ;

Pf = Pal 10"5 ; (* Pf is related to Pa, cannot change individually *)

Pg =10 ; (* initial gas Pressure in N-D form *)

66

C0=Kh*Pg*lO"5;

(* c: 802 *) (* xm = c *)

q (* Deﬁnition of M *)

M = (3*p*R0"3)/(S-R[t])"2*(S"5/30-(S"2*R[t]"3)/3+(S*R[t]"4)l2-R[t]"5/5);
(*Mass Balance/Diffusion Equation“)
Terml=(Pa*RO"3*Sqrt[X[t]])/(A*(M-p*RO"3*(SA3-R[t]"3)));
Terrn2=(M*Kh*Pa)*(l+Sqrt[X[t]])/(R[t]"3*(M-p*RO"3*(S"3-R[t]"3)));
Term3=-(p*Kh*Pa*R0"3)*(SA3eR[t]"3)/(M-p*RO"3*(S"3-R[t]"3));
Tennk-C6*(1+(SqHIXItll))/R[tl"3;
f1=2*Sqrt[X[t]]*(6*C4*C5)*(R[t]"2/(S-R[t]))*(Terml+Term2+Term3+Term4);
(*Momentum equation“)

f2 = (l+C2)/(4*(C2+l-R[t]"3))*(((Sqrt[X[t]]+l)*Pg)l(R[t]"2)-Pf"‘R[t]-C3);

soll = NDSolve[{Derivative[l][X] [t] = = f1,Derivative[1][R][t] = = f2, R[0] = = 1.0, KM
= = 0.000001 ),(X[t],R[t]}.{t.0.10000}];

Cwalll = ((Sqrt[Bvaluate[{X[t] }/.soll]]+1)*(Pg*lO"5*Kh))/ Evaluate[{R[t] }/.soll]"3 ;
Pressure=((l+Sqrt[Evaluate[{X[t] }/.sol1]])*Pg)/(Evaluate[{R[t]/.soll m3);

(* All three equations *)

(* This method uses the momentum + interfacial diffusion + overall mass balance *)
Mdash = (3* p)l(S-R[t])"2*(SA5/30-(S“2*R[t]"3)/3+(S*R[t]"4)/2-R[t]"S/5) ;
Cwa112 = CO*(1/(Kh*A)+ p (8"3-1)+2*((S"3-l)/(S"3-R[t]"3))*(Mdash- p (s03-
R[t]"3)))l(R[t]"3/(Kh*A)+(2*R[t]"3*(Mdash- p (SA3-R[t]"3))l(Kh*A* p *(sas—
thl"3)))+2*Mdash- p (8"3-thl"3)) ;

f2 = (l+C2)/(4*(C2+l-R[t]"3))*(((Sqrt[X[t]]+l)*Pg)/(R[t]"2)—PP"R[t]-C3);

67

so12 = NDSolve[{Derivative[l][X][t] = = f1,Derivative[l][R][t] = = f2, R[0] = = 1.0, X[O]
= = 0-000001}.{X[t].R[t]}.{t.0.10000}];

(* This method uses (1) momentum (2) interfacial mass balance (3) Overall mass
balance only and no diffusion equation explicitly *)

Cwall3 = (Kh*Pa*Pg*(Sqrt[X[t]] + l))/R[t]"3; .

Y = -2*Cwall3 + (2*C0*(S"3 - 1))/(303 - R[t]"3) - (2*Cwall3*R[t]"3)/(Kh*A* p *(SA3 -
thl‘3));

fl = (l2*Sqrt[X[t]]*C4*C5*R[t]"2*Y)/(Pg*(S — R[t]));

r2 = ((C2 + 1)*(-c3 - Print] + ((Sqrt[X[t]] + 1)*Pg)/R[t]"2))/(4*(-R[t]"3 + c2 + 1));
sol3 = NDSolve[{Derivative[l][X] [t] = fl, Derivative[l][R] [t] = f2, R[0] == 1., mm
= 1-0*"(-8)}. {XltL thl}. It. 0. 10000”;

(* Inﬁnite medium *)

fl = (2.*C4*C7*(R[t]"3 - Sqrt[X[t]] - l)"2)/R[t]"2;

f2 = -(c3/4) - (1/4)*PP“R[t] + ((Sqrt[X[t]] + l)*Pg)/(4*R[t]"2);

sol = NDSolve[{Derivative[l][X] [t] = fl, Derivative[1][R][t] = f2, R[0] = 1., X[0]
= 0}. {XltL thl}. It. 0.10000”;

Cwall4 = ((Sqrt[Evaluate[{X[t]} I. sol]] + 1)*(Pg*lO"5*Kh))/Evaluate[{R[t] } I. sol]"3;
Pressure = ((1+Sqrt[Evaluate[{X[t] }l.sol]])*Pg)/(Evaluate[{R[t]/.sol}]"3);

(* Plotting the concentration and radius proﬁle *)

Plot[{Evaluate[{Cwalll} /. soll], Evaluate[{CwallZ} /. so12], Evaluate[{Cwall3} I. sol3],
Evaluate[{Cwall4} /. 501]}, {t, 0. ,40}, PlotStyle 9 {RGBColor[1, 0, 0], RGBColor[0, l,

0], RGBColor[0, 0, l], RGBColor[0, 0, 0]}, PlotRange 9 {0, 0.0007}, GridLines -)

68

Automatic, Frame 9 True, FrameLabel 9 {"Dimensionless Time (t \ tref)",

"Dimensionless Radius (R\Ro)}];

Complete = 5.64017003303139 + 0.41736873621895043*t - 0.0005544333726034403*
t"2 + 5.317464912426975*10"(-7)* t"3 - 1.90588641250408*10"(-10)*t"4 ;
Plot[{Evaluate[{R[t]} l. soll], Evaluate[{R[t]} /. s012], Evaluate[{R[t]} I. sol3],
Evaluate[{R[t]} l. sol], Complete}, {t, 0. , 1200}, PlotStyle 9 {AbsoluteDashing[{2}],
AbsoluteDashing[{4 }], AbsoluteDashing[{ 8}], AbsoluteDashing[{ 13 H,
AbsoluteDashing[{0}]}, PlotLegend 9 {"Method 1", "Method 2", "Method 3", "Inﬁnite-
Medium", "Complete"}, LegendPosition 9 {0.43, -0.50}, LegendShadow 9 {0, 0},
LegendSize 9 {0.53, 0.49}, PlotStyle 9 {Thickness[100] }, PlotRange 9 {0, 400},

GridLines 9 Automatic, Frame 9 True];

69

[l]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

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