. “a: n:
$3., , .. _.

.
v

. . s.
3153!:

. 35.... ‘
0 .4 .l .n:. 1
l i 4
v... max»:
,I- o H.-

7)
V 55.33.98
L .21.; 3...!

 

L5... . ‘3‘;

I...
«iiiflss.

I:n:a....xx?
13:0:2: t E
~

; 1.3

gum r—kh

his :

23935:: ....:}..._..v

i: S... anvfzumwwéuflfimg ..

: .53 awvmmfi.

LL;

.3»
. :22... ,
.. ..
.2! . .
.11.; 4.701.:
.. z.

x? .
hittr. . a
Ifinxnl’:
1.3.0.. 2 v. if. 5
‘ 2A..

V :Zi:
.1... a #1... «24:11
an v2. {E‘s z .513...
3'1): .3?) 5 xx“... 2“? I‘
. x. t! 5:.
"Pol... A! .3}!

figé

. \ xx .oaxou:

. . 2
.x... air.

I!

u. ‘
, “a.
3.1: :5
fly... a- : Ma.
.33....3. ‘ .. _ . 4

I... II—
.fifl . 2v .3.

7 mm
M. ..

QR
‘1“. I n W?
V I}: 1 .
. 31.? p!

. 5. :9 hfluuvfla

it.
5.5. 1.99;...a
‘ 51.53%“. :1! .
.v ’33:.
7...! bl;

. . Sin???
3. El... r.
.131: Kazan}.

‘ .C.
..- 32.2.13 c in.
.......:fi:. 11!??? I.
.11... 7 , :
is i; +¢
T

l
a... c. v . :b

v«..w.:£..!..¢.15.
- .1fi3s?4us#uq.ly
{a .5. .3... z
3|..4)v5’..| .3

.qun...»
‘ V 213-52....
3:».

 

!OOi

 

LIBRARY
Michigan State
University

-W C. rnflw—o— J

 

 

This is to certify that the

dissertation entitled

Nucleation of Gas-Supersaturated Liquids

presented by

Xiaobing Liu

has been accepted towards fulfillment
of the requirements for

 

 

Ph. D. degree in Mechanical Engineering

Date 6%? inf/1.4% 20“”

MS U is an Affirmative Action/ Equal Opportunity Institution 0- 12771

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJCIHC/DateDuep65-9. 15

 

NUCLEATION OF GAS-SUPERSATURATED LIQUIDS
BY
XIAOBING LIU

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
DEPARTMENT OF MECHANICAL ENGINEERING
2001

Nucleation of Gas-Supersaturated Liquids
IBy

Xiaobing Liu

AN ABSTRACT OF A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering
2001

Professor Giles Brereton

ABSTRACT

NUCLEATION OF GAS-SUPERSATURATED LIQUIDS
By

Xiaobing Liu

Classical nucleation theory relates nucleation rate to the reversible work required
to form a bubble of critical size, regardless of any irreversible effects. It has been suc-
cessfully applied to predict superheat nucleation thresholds of pure liquids, but fails
to predict nucleation thresholds of gas-supersaturated liquids and does not account
for the effect of volumetric confinement on nucleation. In this study, the equations
for dissipation of energy due to viscous and diffusive effects during bubble growing
processes were derived and it was shown that, for many nucleation processes, the
dissipation of energy due to diffusion may not be neglected as in the classical nu-
cleation theory. Therefore, the classical nucleation rate equation was modified to
include a term of dissipation of energy due to diffusion. This modified theory can
predict nucleation thresholds of liquids supersaturated with different gases in good
agreement with experimental data. For nucleation of gas-supersaturated liquids in
small capillary tubes, a dual dependence model originally proposed by Brereton et
al. was refined to account for the effect of volumetric confinement on heterogeneous
nucleation thresholds. The new model can predict supersaturation nucleation thresh—
olds of gas-supersaturated water in capillary tubes well. It was also found that if
the shape factor for nucleation in small capillary tubes is assumed inversely propor-
tional to the tube diameter, a simpler heterogeneous model can predict accurately
the supersaturation nucleation thresholds of water supersaturated with different gas
species, as well as the superheat nucleation limits of pure water in small capillary

tubes.

This dissertation is dedicated to my parents

Yunyao Liu
&

Zixiu Tang

for their love and their support, encouragement,
and understanding during my years in academia.

iii

ACKNOWLEDGEMENTS

I would like to acknowledge all the help from Dr. Giles Brereton, from the initial
ideas for the foundation of this thesis, to the proof reading of the final draft. Without
him, this would not have been possible.

Also I would like to thank Dr. Abraham Engeda, Dr. Simon Garrett, and Dr.

John McGrath for being on my committee, and providing guidance to my study.

iv

TABLE OF CONTENTS

LIST OF TABLES viii
LIST OF FIGURES x
1 INTRODUCTION 1
1.1 Applications of Nucleation Theory .................... 1
1.2 Literature Review ............................. 3
1.2.1 Classical Nucleation Theory ................... 3
1.2.2 Nucleation in Gas—Liquid Solutions ............... 5
1.2.3 Nucleation in Confined Volumes ................. 10
1.2.4 Studies of Bubble Dynamics ................... 11
1.3 Objectives ................................. 12
THE THEORY OF NUCLEATION 14
2.1 The Equilibrium State .......................... 14
2.1.1 A Pure Liquid in Equilibrium with its Vapor .......... 14
2.1.2 Equilibrium in Gas-Liquid Solutions .............. 16
2.2 The Stability of a Bubble in a Liquid .................. 19
2.2.1 The Stability of a Vapor Bubble in a Pure Liquid ....... 20
2.2.2 The Stability of a Bubble in a Gas-Liquid Solution ...... 23
2.2.3 The Critical Radius and Gas Concentration Gradients 26
2.3 The Nucleation Rate Equation ...................... 28
2.4 Effects of P, T and Concentration on Nucleation Rate ........ 32
2.5 Discussions on the Applicability of Henry’s Law and the Ideal-Gas Law
at High Pressures ............................. 35
2.5.1 The Applicability of Henry’s Law ................ 35
2.5.2 Effects of the Departure from Henry’s Law on Nucleation Rate 38
2.5.3 The Applicability of the Ideal-Gas Law ............. 4O
BUBBLE DYNAMICS 43
3.1 Statement of the Problem ........................ 43
3.2 Governing Equations ........................... 43
3.3 Analytical Solution ............................ 46
3.3.1 Bubble Dissolution in an Undersaturated Solution ....... 49
3.3.2 Bubble Growth in a Supersaturated Solution .......... 51
3.4 Numerical Approach ........................... 53
3.4.1 Dimensionless Form ........................ 54
3.4.2 Discretization ........................... 55
3.4.3 Stability Criterion ........................ 55
3.4.4 Moving Grids ........................... 56
3.4.5 Numerical Results ........................ 57
3.4.6 Comparison of Analytical and Numerical Results ....... 60

4 TARGET DATA FOR TESTING THEORIES OF NUCLEATION IN GAS-

SUPERSATURATED SOLUTIONS 62
4.1 Nucleation of Supersaturated COg—Water Solutions in Large-Scale Sys-
tems .................................... 62
4.2 Experiments on Formation of Gas-Vapor Bubbles in Supersaturated
Solutions of Gases in Water ....................... 64
4.2.1 Experimental Setup and Results ................. 65
4.2.2 Problems with Finkelstein and Tamir’s Equation on Nucleation
of Liquids Supersaturated with Gases .............. 67
4.2.3 Inconsistencies between F inkelstein and Tamir’s Results and
Classical Nucleation Theory ................... 67
4.3 Experiments on Nucleation of Gas-Supersaturated Solutions in Small
Capillary Tubes ................ . .............. 71
4.3.1 Experimental Setup and Procedures ............... 71
4.3.2 Experimental Results for Nucleation of Gas-Supersaturated So-
lutions in Capillary Tubes .................... 73
4.3.3 Interpretations of Experimental Data Describing Nucleation in
Capillary Tubes .......................... 75
4.4 Some Shortcomings of the Classical Nucleation Theory ........ 76
5 MODIFICATIONS TO THE CLASSICAL NUCLEATION THEORY 78
5.1 Prediction of Nucleation Thresholds of Liquids Supersaturated with
Different Gases .............................. 78

5.1.1 Dissipation of Energy through Diffusion and Viscous Effects . 80
5.1.2 The Significance of Diffusion and Viscous Effects to Nucleation 83
5.1.3 A New Nucleation Model for Liquids Supersaturated with Dif-
ferent Gases ........................... 89
5.1.4 Calculation Results of the New Nucleation Model for Water
Supersaturated with Different Gases at a Constant Temperature 92
5.1.5 Temperature Dependence of Nucleation Thresholds of Gas-

Supersaturated Liquids ...................... 97

5.2 Prediction of Nucleation Thresholds of Gas—Liquid Solutions in Small
Capillary Tubes .............................. 99

5.2.1 Dual Dependence Model to Account for Effects of Volumetric
Confinement on Nucleation ................... 99

5.2.2 A Preliminary Model to Account for Effects of Volumetric Con-
finement on Nucleation ...................... 103
6 CONCLUDING REMARKS AND RECOMMENDATIONS 108
APPENDIX 111

A THE COEFFICIENTS OF HEN RY’S LAW FOR NITROGEN, HYDROGEN,
ARGON, AND HELIUM DISSOLVED IN WATER 112

vi

BIBLIOGRAPHY 119

vii

2.1
2.2

3.1
4.1
5.1

5.2

5.3

5.4

5.5

5.6

A.1
A.2

LIST OF TABLES

Values of shape factor for bubble formation in conical cavities
Calculation results of compressibility factors using the generalized com-
pressibility chart .............................

Time required for a pure gas bubble to dissolve in water .......
Experimental data for the pressure difference for bubble formation . .

The values of solubility and diffusivity of several gases dissolved in
water at 30°C ...............................
Comparison of the experimental data of AP" (MPa) with the predicted
data from the revised nucleation equation (5.43) ............
Predicted data of nucleation thresholds from Eq. (5.42) taking into
account the departure from Henry’s law .................
The estimated values of diffusivity of several gases dissolved in water
at 25°C ..................................
The calculated values of 111m, and q for nucleation in gas-supersaturated
water at 25°C ...............................
The values of solubility and diffusivity of 02 dissolved in water . . . .

The solubility of nitrogen in water at different pressures ........
The solubility of hydrogen in water at different pressures .......

viii

92

93

93

96
98

113
115

2.1
2.2

2.3
2.4
2.5
2.6
2.7
2.8
2.9

2.10
2.11
2.12
2.13

3.1
3.2
3.3
3.4
3.5
3.6
3.7

3.8

4.1
4.2

4.3

5.1
5.2

5.3

5.4

LIST OF FIGURES

System model of bubble formation in a liquid ............. 15
The curve of Gibbs function in the neighborhood of the equilibrium

radius .............. - ..................... 23
Gas concentration profiles in the liquid and in the bubble ....... 29
A vapor bubble formed at a smooth liquid-solid surface ........ 30
Definition of geometrical parameters in a conical cavity ........ 31
The effect of the liquid pressure on the nucleation rate ........ 33
The effect of the liquid temperature on the nucleation rate ...... 34
The effect of the supersaturation pressure on the nucleation rate . . . 35
Measurement and Henry’s-law prediction of the saturation concentra-

tion of 02 in water ............................ 36
Variation of H with pressure for 02 dissolved in water ......... 37

Effect of the departure from Henry’s law on nucleation rate ((15 = 0.001) 39
Effect of the departure from Henry’s law on nucleation rate ((15 = 0.005) 40

Compressibility of nitrogen ........................ 41
Radius-time relation for dissolving bubbles ............... 50
Radius-time relation for growing bubbles ................ 52
Illustration of moving grids during bubble dissolution ......... 57
Dimensionless radius vs. time curve for bubble dissolution due to dif-
fusion (numerical result) ......................... 58
Dimensionless radius squared vs. time curve for dissolving bubbles due
to diffusion (numerical result) ...................... 59
Concentration distribution around a dissolving bubble as a function of
time .................................... 59
Dimensionless radius-squared vs. time curve for growing bubbles due
to diffusion (numerical result) ...................... 60
Radius vs. time curves for a dissolving bubble ............. 61
The experimental apparatus of Finkelstein & Tamir .......... 66
Schematic illustration of the experimental apparatus used for oxygen-
supersaturation studies .......................... 72

Dependence of supersaturation pressure on capillary tube diameter.
The line and data points are an algebraic fit to many experimental

data, measured at 22°C (from Brereton et al., 1998) .......... 74
The growth of a bubble in a gas-liquid solution ............ 85
Experimental data for AP" and the predicted data from the nucleation
equation revised to include the energy of diffusive dissipation ..... 94
Saturation concentration for gas dissolved in water as a function of
pressure at 25°C ............................. 96
Dependence of AP” on temperature for 02 dissolved in water, calcu-
lated from Eq. (5.43) ........................... 99

ix

5.5

5.6

5.7

5.8

A.l
A2
A3
A.4
A5
A6
A.7
A.8

Predicted and measured dependence of 02 supersaturation pressure on

capillary tube diameter using Eq. (5.56) ................ 103
Predicted and measured dependence of 02 supersaturation pressure on

capillary tube diameter using Eq. (5.58) ................ 105
Predicted and measured dependence of 02 and He supersaturation

pressure on capillary tube diameter using Eq. (5.58) .......... 106
Predicted and measured superheat nucleation thresholds of pure water

in capillary tubes ............................. 107
Saturation concentration of N2 in water ................. 112
Variation of H with pressure for N2 dissolved in water ........ 113
Saturation concentration of H2 in water ................. 114
Variation of H with pressure for H2 dissolved in water ........ 114
Saturation concentration of He in water ................ 115
Variation of H with pressure for He dissolved in water ........ 116
Saturation concentration of Ar in water ................. 117
Variation of H with pressure for Ar dissolved in water ........ 117

CHAPTER 1
INTRODUCTION

Studies of nucleation have been carried out for many years and have led to some well-
established, classical theories which describe rates of homogeneous nucleation, or,
with some modifications for surface effects, rates of heterogeneous nucleation. They
have been used to describe formation of embryo bubbles of vapor in the parent liquid,
and of micro-bubbles of dissolved gases in dissimilar liquids. As a practical matter,
these theories are typically devised to provide simplistic macrosc0pic descriptions of
more complex microscopic phenomena. They therefore require calibrations through
the use of adjustable coefficients if they are to be used for predictive purposes.

In the work described in this study, our main interest is in providing a predic-
tive theory for nucleation thresholds of liquids supersaturated with different gases.
A second interest is in predicting effects on heterogeneous nucleation thresholds of
the confinement of gas-supersaturated liquids to small volumes, since it is well known
that such confinement can significantly increase the supersaturation threshold of nu-
cleation.

In this chapter, we first present some applications which motivate study of nucle-
ation, followed by a literature review on nucleation research. The particular problems
associated with the classical nucleation theory are pointed out, and the objectives of

this study are given.

1.1 Applications of Nucleation Theory

There are many industrial and natural processes in which bubble formation and

growth play an important role, 6.9.

o In the case of boat propellers or hydraulic machines, cavitation (the forma-

tion of bubbles) is a problem that engineers try to avoid. The high rotational

speed of propellers leads to low pressures in liquid flow which, together with
micro-bubbles of dissolved air or various particles floating around, may trigger
cavitation. The resulting bubbles produce noise, vibrations, and erosion of the

propellers.

Recent medical studies have shown that high concentrations of dissolved oxygen
in the blood system can benefit patients with lung or heart diseases. However,
formation of oxygen bubbles in the blood may cause serious problems and im-
proved understanding of bubble nucleation thresholds is needed to develop safe

medical oxygenation procedures.

Bubble production in the effervescence of carbonated beverages, such as sparkling

wine, beer, and soft drinks.
Bubble nucleation in boiling and condensation.

Divers may succumb to the ‘bends’ if bubbles of nitrogen form in their blood

while surfacing. Pressure reduction promotes the formation of these bubbles.

Understanding bubble nucleation can help to avoid contact vapor explosions.
Under certain conditions, a liquid held at an ambient pressure of 1 atm may be
superheated to close to 90% of its critical temperature before nucleation occurs.
Once boiling is initiated in a liquid superheated to this degree, the liquid will
boil explosively. Such explosive boiling presents a safety hazard in industry
and may be involved in the dangerous explosive phenomenon known as contact
vapor explosion, observed in many metallurgical processes and in paper smelt

processing (Blander and Katz, 1973).

In liquid waste treatment, dissolved air is sometimes used to produce bubbles

that selectively carry impurities to the surface.

In the petro—chemical industry, the foaming of liquid plastics is achieved through

the nucleation and growth of bubbles via diffusion processes.

o Diffusional nucleation and growth play an important role in transformations in
solid metals, e.g. the formation of alloys, the decarburization of steel, and the
development of oxidation resistant coating, etc. Understanding the mechanism
of nucleation is helpful in the manipulation of a wide range of metallurgical

microstructures.

0 Vacuum degassing processes are used to remove bubbles formed in molten steel

or glass.
0 Bubble production is important in electrolytic processes.
0 Cavitation is useful in ultrasonic cleaning devices.

a The eruption of Lake Nyos in Cameroon in 1986, which resulted in the loss
of more than one thousand lives, was caused by the sudden release of carbon
dioxide gas dissolved in the deeper regions of the lake. This sudden gas release
is very similar to the explosive degassing of magma encountered in volcanoes

(Jones, 1999).

1 .2 Literature Review

The theory of nucleation and its applications have been studied extensively by many
researchers. A literature review is now presented in four sections:

(1) classical nucleation theory,

(2) nucleation in gas-liquid solutions,

(3) nucleation in confined volumes,

(4) studies of bubble dynamics.

1.2.1 Classical Nucleation Theory

The initial ideas of classical nucleation theory date back to Vohner and Weber’s paper,

“Nucleus Formation in Supersaturated Systems,” which was published in 1926. In

that paper, the authors proposed that the probability of having a critical nucleus in
a supersaturated vapor is proportional to exp(—AW,-./ (kT)) according to fluctuation
theory, and hence the nucleation rate should be proportional to this term, where W...
is the reversible work required to form a critical-size nucleus from the supersaturated

vapor, and k is the Boltzmann constant.

Becker and D6ring (1935), and Reiss (1952) rederived Volmer and Weber’s nu-
cleation theory on a kinetic basis. This resulted in corrections for the rate at which
supercritical embryos return to subcritical sizes and for the fact that the finite rate

of reaction decreases the population of critical embryos.

Frenkel (1939) developed the statistical mechanics approach to nucleation theory
which focuses attention on the partition functions of nuclei and how the concentration

of nuclei is distributed with nucleus size.

Fisher (1948) derived an expression for the work required for reversible formation
of a bubble of radius r in a liquid under negative pressure. He also considered the effect
of surface irregularities on nucleation. He examined heterogeneous bubble nucleation
at a smooth rigid interface, a surface having a conical cavity, and a surface having a
conical cavity with a rounded apex. Based on these results, he developed an expression

for the maximum reversible work required to grow a bubble of critical size, as

an = 167r03¢/{3(AP)2},

where 43 is a factor describing the bubble shape, a is the surface tension, and AP is

the pressure difference across the bubble surface.

One assumption of the classical nucleation theory is that some macroscopic pa-
rameters, such as surface tension, can be applied to nuclei of only tens or hundreds
of molecules. Kirwood (1949) and Buff (1950, 1955) developed a theory describing

the change in surface free energy with curvature. LaMer and Pound (1949) found

that experimental agreement with classical nucleation implies that the use of the
macroscopic surface tension is valid even for nuclei at the size of 10‘8m.

Kantrowitz (1951), Probstein (1951), Wakeshima (1954), and Frisch (1957) devel-
oped theories for the lag time for redistribution of embryo sizes after an instantaneous
change of supersaturation or undercooling of the parent phase.

Cole (1974) reviewed theories and experiments related to boiling nucleation. He
predicted superheat limits for several pure liquids from homogeneous nucleation the-
ory, and found good agreement with experimental observations. The heterogeneous
nucleation from plane surfaces, spherical projections and cavities was studied and the
expressions of shape factors for those geometries were obtained. Also Cole reviewed
experimental evidence for nucleation from a pre-existing gas or vapor phase in surface
cavities. One of his conclusions was that nucleation in most boiling systems occurs
from a pre—existing gas or vapor phase. The stability of nucleation cavities under
conditions of superheat was also discussed.

Blander and Katz (1975) reviewed the theoretical and experimental aspects of
homogeneous and heterogeneous bubble nucleation. The superheat limits of many
pure substances, include some organic liquids, were calculated based on classical
nucleation theory, and the predictions were in good agreement with experimental
measurements. The superheat limits for some liquids may reach 80% to 90% of the

critical temperature.

1.2.2 Nucleation in Gas-Liquid Solutions

The classical nucleation theory originally focused on nucleation in pure superheated
liquids, condensation in supersaturated vapors, or crystal growth from a melt. Since
the 19703, nucleation in gas-supersaturated liquids has been studied by many re-
searchers, both theoretically and experimentally.

Ward and co—workers (1970) investigated the role of a dissolved gas in the ho—

mogeneous nucleation of bubbles from liquids. They presented a thermodynamic
analysis of homogeneous nucleation in weak gas-liquid solutions. They also derived
the equations for the critical bubble radius for nucleation in weak gas-liquid solutions,
considering a constant-volume, closed system of uniform and constant temperature.
It was shown that for nucleation to occur in a multicomponent solution a nucleus
must be created of at least a certain radius (i. e. the critical radius). A generalized
Kelvin equation was derived, which relates the partial pressure of the solvent vapor
inside a nucleus to the properties of the surrounding solution. To verify the equation
of critical radius in Ward’s (1970) paper, Tucker and Ward (1975) carried out an ex-
perimental study of bubbles near the critical state in gas-liquid solutions. Bubbles at
various sizes were introduced into supersaturated solutions and their growth or decay
behaviors were observed. Those at, or close to, critical size were observed to be stable
for long periods of time, while larger bubbles were observed to grow and smaller ones
to shrink, on a timescale of hours. This experiment confirmed the correctness of the
equation of critical radius for nucleation in gas-liquid solutions developed by Ward et
al. (1970). Forest and Ward (1977) conducted experiments to measure the pressure
at which homogeneous nucleation of bubbles occurs in solutions of nitrogen in ethyl
ether. The presence of the dissolved nitrogen in the ethyl ether was observed to raise

the pressure at which nucleation occurs.

Blander and Katz (1975) discussed hydrodynamic and diffusion constraints on nu-
cleation rate. A modification of classical nucleation theory was proposed for diffusion-
controlled nucleation, e.g. nucleation in a mixture with one volatile component. A
pre—exponential coefficient (SD, which accounts for the diffusion of a volatile component

to the interface, was added to the nucleation rate equation as

JD = J/(1+5o)

where J D is the nucleation rate taking into account diffusion. However this theory
does not appear to have been tested against experimental results, and will be discussed

in detail in chapter 5.

Hemmingsen (1974) conducted experiments to study nucleation / cavitation of gas-
supersaturated solutions in glass tubes (0.01—0.25 cm inside diameter). He observed
that water equilibrated with gas at high pressures is able to sustain very high supersat-
uration without any occurrence of gas bubbles during decompression to atmospheric
pressure. The maximum supersaturation tensions without cavitation, which repre-
sent the lowest possible limits of spontaneous bubble formation, were measured for

02, Ar, N2, and He.

Finkelstein and Tamir (1985) carried out experiments on bubble formation in
water supersaturated with a number of different gases. The pressure difference for
bubble formation, AP” —- an expression for the degree of supersaturation which a
solution can tolerate — was measured for difl'erent gases. While an empirical equa-
tion was found to correlate AP” with critical properties of the gas, the theoretical
explanation of this equation was questionable and the equation is inconsistent with

other experimental results. It will be discussed in detail in chapter 4.

Wilt ( 1985) studied mechanisms of nucleation in supersaturated COg~water so-
lutions, such as carbonated beverages. He examined several possible mechanisms:
homogeneous nucleation, and heterogeneous nucleation at smooth planar interfaces,
and at interfaces containing conical or spherical cavities or projections. He concluded
that the predominant mode of nucleation was heterogeneous and that it occurred in
conical cavities, presumed to be present in some form on almost all surfaces, but only
for contact angles in the range of 94 — 130°. Moreover, his analysis did not take into
account the possible influence of pre—existing gases trapped in the cavities, which can

provide nuclei for the formation of gas—vapor bubbles at the onset of nucleation.

Carr et al. (1995) carried out experimental studies on bubble nucleation rates

for the H20/C'02 / Pyrex system over the supersaturation range 1 — 7, using the
pressure release bubble nucleation technique. The dependence of nucleation rate J
on supersaturation ratio w was reported as a plot of an vs. (wP)‘2, but did not
yield a linear relation as might be expected from Wilt’s work. Carr explained the
disagreement in this way: there exists a distribution of active nucleation sites of
different sizes or geometries on the Pyrex surface; each site type displays its own rate
kinetics, and the overall bubble nucleation rate is the sum of the rates from each site
type. A two—site model was proposed and fitted the experimental data well. Again,
Carr et al. did not consider the possible effect of pre—existing gases trapped in the

cavities.

Rubin and Noyes (1987) measured supersaturation thresholds for homogeneous
nucleation of H2, N2, 02, CO, N O, 002 dissolved in water, using a new experimen-
tal approach. Instead of saturating water with gas at high pressure then decompress-
ing, as Finkelstein and Tamir had done, chemical reactions were used to generate
highly supersaturated solutions. Thus the cumbersome compressors and steel vessels
otherwise used to generate high pressures were not required in their experiments.
Supersaturation thresholds for nucleation could then be measured at atmospheric

pressure and room temperature.

Bowers et al. (1999) reported temperature dependences of supersaturation thresh—
olds for carbon dioxide, hydrogen, and oxygen in water at 1 atm pressure. The thresh-
olds were measured by generating supersaturated solutions of the gases chemically.
As temperatures increased, the supersaturation limits decreased for 002 and 02, but
increased for H2, and remained at almost the same value for N2. This result was of
considerable interest, since according to the classical nucleation theory, all the super-
saturation limits should decrease as temperature increases. Further study is needed

to explain this phenomenon.

Since the nucleation rate is extremely sensitive to surface tension, it is important to

obtain surface tension data of different gases dissolved in water at different pressures
and concentrations. Unfortunately, the metastable nature of the gas—supersaturated
solution makes this kind of experiment very diflicult. Few experimental data are avail-
able. Massoudi and King (1974) measured the surface tension of various gas-water
solutions at pressures up to 85 atms, using the capillary-rise method. Measurements
were reported for water with He, H2, 02, N2, Ar, CO, 002, N20, CH4, CgH4, C2H6,
C3H8, and n-C4H10. Their results showed that pressure has a significant effect on
surface tension for C02, N20, CH4, 02H4, CQHG, C3H3, and n-C4H10 dissolved in
water, but has little effect for He, H2, N2, or Ar solutions with water. Lubetkin and
Akhtar (1996) measured surface tension and contact angles of carbon dioxide-water
solutions at pressures in the range 1-11 bars. The surface tension of COg-water solu-
tions changes from 0.072 N / m to 0.057 N / m as the pressure of 0'02 increases from 1
to 11 bars. The advancing contact angle for aqueous 002 solutions on 316 stainless
steel changes from 77° at P = 2 bars to 44° at P = 10 bars. The effect of the changes

of surface tension and contact angle with pressure on nucleation rate was also studied.

Jones et al. (1999) reviewed bubble nucleation from cavities. They argued that
the nucleation observed in most instances, which is often at supersaturation ratios
of 5 or less, is invariably associated with the existence of gas cavities in walls of
the container. Issues concerning the formation of these gas filled cavities, and their

stability was examined.

As an alternative approach to classical nucleation theory, Talanquer and Oxtoby
(1995) applied a density functional method to predict the nucleation rates of bub-
bles in superheated, stretched, or supersaturated binary fluid mixtures. The density
functional method assumes continuous changes rather than abrupt changes of fluid
properties at the interface of nuclei, and thus avoids the macroscopic assumptions of
classical nucleation theory. Although the density functional method is not yet able

to make quantitative comparisons with experimental results, it shows some potential

for making useful predictions in nucleation studies.

1.2.3 Nucleation in Confined Volumes

When carrying out studies of nucleation in gas—liquid solutions, several researchers
have drawn attention to the effect of volumetric confinement on raising nucleation
thresholds. Hemmingsen (1974) conducted experiments to study nucleation / cavitation
of gas-supersaturated solutions in glass tubes (0.01~0.25 cm inside diameter). He ob-
served that water equilibrated with gas at high pressures is able to sustain very high
supersaturation without any occurrence of gas bubbles during decompression to atmo—
sphere pressure. The maximum supersaturation tensions without cavitation, which
represent the lowest possible limit of spontaneous bubble formation, were measured

for 02, Ar, N2, and He.

Brereton et al. (1998) studied nucleation in small capillary tubes, of inside di-
ameters of less than 100 am. It was observed in these experiments that confinement
within small capillary tubes can increase the supersaturation threshold for different
gases dissolved in water. A dual-dependence model was proposed to describe nucle-
ation of gas—supersaturated solution within confined volumes. Although this model
could. be calibrated to agree with experimental data, some of its assumptions required
more careful consideration, e.g. the assumption that the changes in the radii of cur-
vature of nuclei in crevices are neglected during decompression is questionable. Also
calculations of the critical radii of nucleation in gas-liquid solutions did not take into
account the gas species present. These shortcomings will be discussed in detail in
chapter 4 and chapter 5. Some measurements of superheat thresholds of pure water

in small capillary tubes were also reported.

10

1.2.4 Studies of Bubble Dynamics

Bubble dynamics has been studied extensively by many researchers. The case of
stationary spherical bubbles surrounded by an infinite, constant temperature liquid
is of particular interest in the later stages of nucleation. Epstein and Plesset (1950)
found an approximate analytical solution of the growth/ collapse rate for a gas bubble
in a gas-liquid solution. The equations governing spherically symmetric growth of
a bubble in an infinite medium were formulated by Scriven (1959). Bankoff ( 1966)
reviewed several cases of diffusion—controlled bubble growth: (1) asymptotic growth of
a vapor bubble in an initially uniformly superheated one-component liquid; (2) bubble
growth under nonuniform initial conditions; and (3) bubble growth in two-component
liquids. The role of bubble formation and growth in boiling heat transfer was also
studied. Bretherton (1960) studied the motion of long bubbles in a tube filled with
liquid at small Reynolds numbers, and Plesset and Chapman (1971) studied the
collapse of an initially spherical vapor cavity in the neighborhood of a solid boundary.
Ward (1984) studied the conditions for stability of bubble nuclei in solid surfaces
contacting a liquid-gas solution. Worden (1998) investigated the mass transfer of
micro-bubbles by developing a dynamic model of a single micro-bubble immersed in
an infinite stagnant liquid which was solved numerically. The model was used to

predict mass transfer coefficients in fluids with micro-bubbles.

The general conclusion from these studies is that, after formation of bubbles larger
than their critical radii, their subsequent growth in uniform media without interaction
with other bubbles is well understood. Theoretical and computational results are in
good agreement with experiment, which indicates that the continuum-level aspects of
bubble growth are quite reliable for predictive purposes. It is the micro—scale aspects

of nucleation where significant difficulties lie.

11

1.3 Objectives

Although the subject of nucleation in supersaturated and superheated liquids has been
studied extensively, understanding of the thresholds necessary for nucleation remains
incomplete and theoretical predictions can not always be reconciled with experimental
results, especially those involving different gas species dissolved in water.

In classical nucleation theories, one typically focuses on clusters of nuclei in a
liquid and considers when they are sufficiently numerous and energetic to combine
to create micro—bubbles which are large enough to avoid collapse, and grow to a
finite size. Some particular problems with these classical theories, which limit their

usefulness in practical applications, are:

1. they typically associate the conditions required for bubble formation only with
the reversible work required to create bubble surfaces of stable size, without
regard for mass transfer to the bubble surface and other irreversible effects, e.g.

diffusive dissipation, viscous dissipation, etc.

2. a nucleation model calibrated for one dissolved-gas species can be highly inac—

curate if applied to a different gas-supersaturated solution.

3. they do not predict any effect of volumetric confinement on nucleation thresh-

olds of gas-supersaturated liquid, when experiments indicate there is one.

The purposes of this study are to examine carefully the bases for nucleation the-
ories and to extend them to provide predictive models of nucleation of different
dissolved-gas species in liquids, and models of effects of confined volumes on nu-
cleation thresholds.

In this dissertation, we first present the classical nucleation theory as originally
developed for pure liquid-vapor mixtures, and as extended for gases dissolved in
dissimilar liquids. The reasons for its failure to predict the strong dependence of the

particular gas species on nucleation thresholds of gas-supersaturated liquids will be

12

discussed. Modifications are then made to the classical nucleation theory so it can
predict nucleation thresholds that are in good agreement with experimental data.

Finally the prediction of nucleation thresholds in small capillary tubes is described.

13

CHAPTER 2
THE THEORY OF N UCLEATION

The theory of nucleation has been developed from the early ideas of Volmer and
Weber (1926), Becker and D6ring (1935), and Frenkel (1955). From the classical
nucleation theory, the homogeneous bubble formation rate in a pure liquid can be

expressed as (see, for example, Carey, 1992)

J _—_ Nzgexp{—an/(kT)} (2.1)

where Wm is the maximum reversible work required to form a bubble nucleus, N1
is the number of liquid molecules per unit volume, a is the surface tension, m is the
mass of one molecule, k is the Boltzmann constant, and T is the absolute temperature.
Usually the maximum work occurs when the bubble nucleus is in a state of equilibrium
with the liquid (Fisher, 1948).

In this dissertation, we will first examine the bases of the classical nucleation the-
ory and its shortcomings. Then the classical nucleation theory is modified to describe
heterogeneous nucleation in gas-liquid solutions and nucleation in small capillary
tubes, based on the work of Ward, Carey, Wilt, and Brereton et al. Our discussion
begins with an analysis of the state of equilibrium between liquids and gases, and the

stability of bubbles.

2.1 The Equilibrium State

2.1.1 A Pure Liquid in Equilibrium with its Vapor

From considerations of equilibrium, it is possible to determine the radius of a gas
bubble in a liquid as a function of liquid pressure, temperature and the interfacial

surface tension. For a pure liquid in equilibrium with its vapor, contained inside a

14

spherical bubble (see Figure 2.1), the temperature and the chemical potential in the

two phases must be equal, (Gyftopoulos, 1991)

u:n:T an

#1 2 ”v (23)

where the subscripts l and 1) denote liquid and vapor respectively. Due to the cur-

 

 

vapor

liquid

 

 

L—__ ___

 

Figure 2.1: System model of bubble formation in a liquid

vature of the interface, the pressures in the vapor bubble and the liquid are related
through the Young-Laplace equation, as a statement of static mechanical equilibrium,

as

2
a=e+i no

8
where re is the equilibrium bubble radius, and o is the surface tension. After inte-

grating the Gibbs-Duhem property relation

du = —sdT + vdP (2.5)

15

at constant temperature from P 2 PM to an arbitrary pressure P, we obtain

P
a -— amt : / vdP (2.6)
PsatlTl)

For the vapor phase, substituting the ideal gas law (1) 2 RT/ P) into Eq. (2.6) yields

the following relation for the chemical potential of the vapor:

 

Pv
“1‘va Tu) I ,usatmcll) + R7“! 111 (P804710) (27)

Since the liquid is virtually incompressible, v is taken to be constant. Evaluation of

the integral in Eq. (2.6) for P : P, then yields
”KB, Tl) : ,usatJCFl) + vllB — Psat(:ll)l (28)

Substituting Eqs. (2.7)and (2.8) into Eq. (2.3), and noting that amt,v(T) = #sat,z(T),

allows the vapor pressure inside the bubble at equilibrium to be expressed as

(2.9)

 

vzlpz - Psat(Tl)l}

P,, = Psat(T,) exp{ RT;

Substituting Eq. (2.9) into Eq. (2.4) leads to an equation for the bubble radius (re)
at equilibrium:

2o
Psat(7l)eXP{vllH _ Psatawl/(RTlll — Pl

 

(2.10)

re:

2.1.2 Equilibrium in Gas-Liquid Solutions

For the case of an arbitrary gas, rather than the liquid’s own vapor, dissolved in
a 1iquid, Ward et al. (1970) derived equations for the equilibrium state in weak

gas-liquid solutions and the corresponding equilibrium radii of dissolved-gas/ vapor

16

bubbles. The weak-solution assumption (gas concentration in the liquid C’ << 1) is
almost always true for gases dissolved in liquid, even at very high pressures (e.g.
10 MPa). The conditions for equilibrium described by Eqs. (2.2) and (2.4) remain
valid, but in addition the chemical potentials of each component in the two phases

must be equal (Gyftopoulos, 1991)

[11,1 2 [11,” (2.11)

#21 = H2,v (2-12)

where subscripts 1 and 2 refer to the solvent and solute respectively. In the liquid

phase, the chemical potential for the solvent is (Gyftopoulos, 1991),

#1,: = #01(B,T)+RTln(y1,z)

 

1
= P T 1
M01( 1, )+RTn1+C’
z ugl(H,T)—RTC' (2.13)

where #01 is the chemical potential of pure substance 1 and y” is the mole fraction
of component 1 in the liquid phase. C’ is the concentration of component 2 in the
liquid phase, which is defined by the quotient of mole number of the solute over that

of the solvent. If the pure solvent is assumed to be incompressible, then
#1,! : “01(PsataT) + v1,l(H _ Psat) — RTC’ (214)
For the solute, the chemical potential can be expressed as (Landau, 1958)

n.2,, : 1MP), T) + RT ln(C') (2.15)

17

where a is a function that depends only on B and T. In the case of liquid 1 saturated
with gas 2 at (B,T) across a flat surface, with the saturation concentration as CO,

the chemical potential of component 2 is
am 2 ¢(H,T) + RT ln(C0) (2.16)

Solving for 2/2 from Eqs. (2.15) and (2.16), and substituting back into Eq. (2.15), gives

the chemical potential of component 2 in the liquid phase
#2,: = #02001, T) + RT 1n(C’/Co) (2.17)

In the gas/ vapor phase, the chemical potentials of the vapor and the gas in bubbles

are

H1,v = H01(P1,mT)
Z 1101(1)”, T) ‘l‘ RT1D(P1,v/Pv)
1
= P,a,T RTl P, P3,, RTl (___)
#01( t )+ Ill / 0+ n1+C”
#24; == #02(P2,v,T)
2 ”02(1)”, T) + RT111(P2’v/R,)

 

: 110201,?) + RT ln(Pv/Pz) + RTln (150”) (2.18)

where C” is the concentration of the gas in the gas—vapor bubble. After introducing
activity coefficients V1 and V2 to account for the non-ideal effects of the gas and

solution, the two above equations become

 

“L, : ”01(1),,“ T) + RT 111(Pv/P3at) + RT 1n (1 :10”) (2.19)

18

 

II
My ___ ”02(1),,7") + RT1n(P,,/P,) + RT ln (16+: ’23) (2.20)

Substituting Eq. (2.14) and Eq. (2.19) into Eq. (2.11),

Pvl V1 l : Psat(T) exp {MAB ‘— Psat) "‘ RTC' }

 

1 + C” RT
E pm, ,7 (2.21)

Usually 1) is approximately equal to unity. Substituting Eq. (2.17) and Eq. (2.20) into
Eq. (2.12), one finds

[ C’IV’ ] = C’ (2.22)

l + CH ’60
Combining Eq. (2.21) and Eq. (2.22) with Eq. (2.4), the expression for the equilibrium

radius of a spherical bubble in a gas-liquid solution can be obtained as

 

P,“ T PC’
77 t( 0+ I Pl}

— 2.2
V1 V200 ( 3)

7‘, =20/{

If the gas-vapor mixture inside the bubble behaves as an ideal gas mixture and the
solution is ideal, then V1 2 V2 2 1. If V1 2 V2 2 1 and C’ = 0, Eq. (2.23) is reduced
to Eq. (2.10), i.e. the equilibrium radius of vapor bubble in pure liquid. Note that
the equilibrium radii for dissolved gases are always smaller than those of pure vapor
bubbles. Therefore nucleation of gas-liquid solutions takes place more readily than in

pure liquids.

2.2 The Stability of a Bubble in a Liquid

Having established expressions for the equilibrium size of bubbles in liquids, we now
consider the question of stability, and when bubbles at their equilibrium size will be

stable or unstable.

19

2.2.1 The Stability of a Vapor Bubble in a Pure Liquid

If an embryonic bubble has been formed through the random collision of clusters of
vapor molecules in a pure liquid, we would like to determine whether such a bubble
is stable or unstable, and if unstable, whether it will grow or collapse. The system
model is shown in Figure (2.1). The liquid is kept in a closed system, at constant
temperature and pressure, as is typically the case in most practical applications.
In that case, the system reaches stable equilibrium when its Gibbs function is a

minimum, or a maximum for unstable equilibrium.

For convenience, we first deduce the Helmholtz free energy F of the system after a
bubble of radius 7‘ is formed and, from F, determine the Gibbs function. By definition
F = U —TS, where U is the system’s internal energy and S its entropy. F is extensive,
or additive over the subsystems in a composite system, since U and S are additive.
Therefore the Helmholtz free energy for the whole system is the sum of contributions

of the liquid, the vapor and the interface.

F=E+R+E QM)

where

F1 = Nz(uz*-Tz81)
R==Mwwowg an)

F,- 2 04m"2

and N), N, are the number of moles of the liquid and the vapor. Combining these

equations with the equilibrium assumption that T, Z T, = T, we obtain

F : N1(u, — Tsl) + N,,(uv -— Tsv) + o 47TT2 (2.26)

20

By definition, the Gibbs free energy G is given by

G _—_ U—TS+PV

= F + PV (2.27)

Using Eq. (2.26), we can rewrite the Gibbs free energy in the form of (Carey, 1999)

G : F + P1(N1'U( + vav)

: N.,g.. + Mg, + 4m2a, — va,,(P,, — 19,) (2.28)

where the specific Gibbs free energies for the liquid and the vapor, g, and gv, are

defined as

g; = U1 — T81+ PM

91) : uv ‘- T811 + vav (2.29)

The Gibbs free energy of the system before bubble formation is given by

Go 2 (NU + N091 (2.30)

Here we assume that the formation of a tiny bubble does not cause any change of
the properties of the liquid. The change in Gibbs free energy associated with bubble

formation is therefore

AG 2 G—Co

2 N,,(g,, - 91) + 47rr20 — va,,(P,, — B) (2.31)

For a pure liquid, u = g and at equilibrium, u) = 11.9, and P, — P, = 2o/re. Also,

21

37W”, it follows that at equilibrium, Eq. (2.31) becomes

since vav = 3

A0,. = -7rr§o (2.32)

If we perform a Taylor-series expansion of AG(r) in Eq. (2.31) around the equilibrium

point, we find that

AGO) : (Nvle + ANv)l(gv — gl)le + A(gv — gl)l

+47r7'2o — gnr3[(Pv — Pz)|,.3 + A(Pv — P,)] (2.33)

where

Af = 2;; J" — r.) + 0er

and the subscript e refers to properties at the equilibrium point. The symbol ‘0’
means ‘of the order of’. Substituting (9,, — gz)|e = 0 and (P, —- P))|e : 20/1}, into

Eq.(2.33) and retaining only the first order terms, we find that

AG(7‘) : 47ror2(1 — $7.1) + O(AT) (2.34)

A plot of AG versus 7‘ is shown in Figure (2.2); by the nature of the local expan-
sion, it is only accurate near the equilibrium point. From the graph, it can be seen
that the equilibrium point at re is an unstable equilibrium. If a bubble nucleus is
created by fluctuations of initial size less than the equilibrium/ critical radius, it will
collapse spontaneously. A nucleus of initial size greater than the equilibrium radius
will grow spontaneously. It is noteworthy that this result is based on equilibrium be-
ing reached when the Gibbs function is minimized or maximized, which is equivalent

to the assumption that nucleation takes place in a closed system at constant T and

H.

22

 

 

3 A
0
<1
0 I I 1
0 0.5 1 1 5
rlr.

Figure 2.2: The curve of Gibbs function in the neighborhood of the equilibrium radius

2.2.2 The Stability of a Bubble in a Gas-Liquid Solution

In the following section, the corresponding stability of a gas-vapor bubble in a gas-
liquid solution is studied. We consider a closed system of uniform and constant
temperature, in which gas-phase bubble nuclei are surrounded by liquid. The pressure
of the liquid is kept constant and effects of gravity are ignored here. In both phases,
there are 17. components. As in the previous section, it is convenient to deduce first
the system Helmholtz function, from which the desired Gibbs free energy G can be
found. The Helmholtz function F can be expressed in terms of the surface area of the
nucleus A, the number of molecules of each component in the liquid phase Nu and

vapor phase NW, the pressure of both phases Pl, Pv, temperature, entropy, etc. as
F = Z -Ni,l(ui,l — T8231) + Z Ni,v(U-i,v — TSi,v) + 014 (2-35)

Where subscript i denotes the i-th component. By definition, the Gibbs free energy

is given by

G : U—TS+PV

23

= F + PV (2.36)
Using Eq. (2.35), we can rewrite the Gibbs free energy in the form

G = F + P; (N12); + Nvuv)

Z Z Name: + 2 New“, + 0A — (PU — 190% (237)
where u,,landu,,v are defined by

[1:31 = uiJ — T8131 + szu

Him : ui,v _ Tsim 'l' vaim (238)
The Gibbs free energy of the system before bubble formation is given by

G0 = ZUVW’ + Ni,1)/li,[ (2.39)

O

l

The change in Gibbs free energy associated with bubble formation is therefore

AG : G—Go

= Z Nada... — m) + a A — (P. — PW. (2.40)

In these derivations, it is assumed that the formation of a tiny bubble will not change
the properties of the liquid, 6. g. 131,11”, etc. At equilibrium, at, = #1312 and P.” — B =

2o/re. Since vav = gnr3, it follows that, at equilibrium, Eq. (2.40) becomes

AG, 2 gargo (2.41)

24

If we carry out a Taylor-series expansion of AG(r) in Eq. (2.40) around the equilibrium

point, we find that

AG(1~,N,~) = 21{(Ni.vle + AN,,.,)[(11,-,,, - Hi,l)le + A01”, — Mull}

+47r7‘2 o — —7rr 3,,[(P— Pl)|e + A(P,,— P1)] (2.42)

where

3_f(
8r

 

M:—

 

 

e

-- Ni,v,8)} + 0(AT2, ANE, AT ANz)

. ’ZloN.

the subscript e refers to properties at the equilibrium point, and AN, 2 Ni,” — NW8.

Substituting (11,,” — Hi,l)|e = 0 and (PU - Pl)|e = 2o/re into Eq.(2.42) and retaining

only the leading and first order terms, we find that

2
AG(7‘, Ni) 2 47ro'r2(1 — 37'1)+ Z {(Nwle + AN,,v)A(1i,-,v — u,,1)}

4
— §7TT3A(PU — H)

: 47(07‘2 (1 — ’37:.) + 0(A7‘, AN,) (2.43)

e

The leading term of AG(r, N) for a bubble formed in a gas—liquid solution is not
dependent on N,, and it is identical to that in a pure liquid, Eq. (2.34). But the

expressions of re are different for nucleation of pure liquids and of gas-liquid solutions.

A plot of AG versus r is shown in Figure 2.2 which, through the nature of this
local expansion, is only accurate near the equilibrium point. From the graph, it can
be seen that the equilibrium point at re is again an unstable equilibrium. If a nucleus
is created by fluctuations of initial size less than the equilibrium/ critical radius, it
will collapse spontaneously. A nucleus of initial size greater than the equilibrium

radius will grow spontaneously. It is worth emphasizing that this result has been

25

deduced under the assumption of a closed system at constant T and Pl. Ward et al.
(1970) obtained the same result for a constant volume, closed system at uniform and
constant temperature. Thus embryo bubbles of both vapors and dissolved gases which
are dissimilar in composition to the host liquid can only grow to finite sizes if they
can be formed by random molecular collisions, at sizes greater than the equilibrium
radius re. Again, because equilibrium radii for dissolved gases are always smaller
than those for vapors, dissolved gases will, with all other factors being equal, always

nucleate more readily than vapor bubbles.

2.2.3 The Critical Radius and Gas Concentration Gradients

It was shown in the previous section that there exists a critical radius for bubble
formation in a gas-liquid solution, from the viewpoint of minimization of the Gibbs
free energy. Once a bubble is formed through random fluctuations, it tends either to
grow or to collapse depending on whether the radius of the bubble is larger or smaller
than the critical radius. Now we consider the same problem from another viewpoint:
the gradient of gas concentration across the bubble surface.

Assume a gas-vapor bubble of radius r is formed in a uniform gas-liquid solution.
Its subsequent growth or dissolution depends on whether the net mass transfer of gas
molecules is into or out from the bubble surface. From Fick’s law of mass transfer, the
concentration gradient across the bubble determines the direction of mass transfer in
continua. Therefore, the gas concentration at the bubble surface can be derived and
compared with the gas concentration in the liquid. From the Young-Laplace equation

of mechanical equilibrium, the pressure inside the bubble is

2
P. = B + 70 (2.44)

The gas concentration at the bubble surface is the saturation pressure corresponding

26

to the partial pressure of the gas inside the bubble,
CIR : Csat(Xga,P,,) (2.45)

where X gas is the mole fraction of gas in the bubble. For nucleation of gas-supersaturated
liquids, it can be shown from Eq. (2.21) that X gas 2 1. The gas concentration in the

liquid far away from the bubble is
Coo : sat(Pssat) (246)

where P3,“, is the saturation pressure corresponding to the initial gas concentration
of the solution. Thus, the bubble will grow or dissolve according to the relative

magnitudes of C I R and Coo. The bubble will grow if C | R < Coo, or

2
X90, (P, + 7") < Pm. (2.47)

We can see that as r approaches zero, the relation (2.47) will not be satisfied and the

bubble will always dissolve. At large r, the bubble will grow.

If Henry’s law is satisfied, from Eq. (2.23) we obtained

20
7'_ z ”PsatCI—l) + Pssat — [)1 (248)
At equilibrium,
2o
Xgas (Pl + _) it: Xgas (”PsatCPI) + Pssat)

% Pasat (2.49)

Therefore at the critical size re, there is no gas concentration gradient across the

bubble surface. Profiles of gas concentrations for bubbles smaller than and greater

27

than the critical radius are shown in Figure 2.3.

In summary, if a bubble is created by fluctuations of initial size less than the
equilibrium/ critical radius, it will dissolve spontaneously. A bubble of size greater
than the critical radius will grow spontaneously. For nucleation processes, extra
energy is needed to overcome the concentration gradient for a bubble to grow from
sub-critical size to critical size, since these growing processes are in the opposite

direction to gradient-driven Fickian diffusion.

2.3 The Nucleation Rate Equation

From Figure 2.2, the maximum reversible work required to form a spherical bubble

nucleus is reached at the equilibrium point,

4
an 2 Ewarg (2.50)

Wm can be regarded as the activation energy barrier for bubble nucleation that
may be reached by thermal and other fluctuations in the system. According to kinetic
theory, the probability of such an occurrence is proportional to the factor exp(— 9513;“ .
More careful considerations by Volmer and Weber (1926), Becker and Doring (1935),
Reiss (1952), and Barnard (1953), et al., gave the equation of homogeneous nucleation

rate in a liquid as,

J = N,\/%exp{—Wm/(kr)} (2.51)

where N, is the number of liquid molecules per unit volume, m is the mass of one
molecule, k is the Boltzmann constant, and T is the absolute temperature. The

homogeneous nucleation rate J is in units of 1/ (1113s).

Substituting Eq. (2.50) into Eq. (2.51), the homogeneous nucleation rate in a pure

28

VR
R<re , bubble dissolving.
mass transfer from the bubble
gas to ihe surrounding liquid
diffusion

C
A
].

 

 

R>fe , bubble growing.
mass transfer from the
surrounding liquid Into the bubble

  

gas
diffusion

C’i‘
lL————:

 

 

 

R r

 

Figure 2.3: Gas concentration profiles in the liquid and in the bubble

29

liquid or a weak gas-liquid solution can be obtained as

J 2 N,fi%exp{—§7rarg/(kT)} (2.52)

where re can be calculated from Eq. (2.23) for gas-liquid solutions, or Eq. (2.10) for

pure liquid.

 

 

Liquid
Wmm
e
///// /// /////
Solid

Figure 2.4: A vapor bubble formed at a smooth liquid-solid surface

For heterogeneous nucleation, a shape factor d), which depends on the contact
angle and the geometry of the surface, is often introduced to describe the maximum
reversible work required to form a non-spherical bubble at the surface. The shape
factor is defined as the ratio of the work required to form a non—spherical bubble to

the work to form a spherical bubble of the critical size. It always satisfies
0<¢gl
Therefore, the heterogeneous nucleation rate is expressed as
J = Nl2/3\/%exp{—§qbrrar3/(kT)} (2.53)

. . . 2 3 . .
where J 18 1n umts of l/mgs, and N,/ represents the number of liquid molecules

immediately adjacent to the solid surface per unit of surface area. For a heterogeneous

30

 

Figure 2.5: Definition of geometrical parameters in a conical cavity

nucleation process, only molecules near the solid surface can participate in embryo

bubble formation.

For bubble formation on a smooth flat surface as shown in Figure 2.4, if the bubble
shape is idealized as being a portion of a sphere, then the shape factor qb takes the

form
_ 2 +3cos€ — c0536

4 (2.54)

 

<13

where 6 is the contact angle measured in the liquid.

For bubble formation in a conical cavity as shown in Figure 2.4, the shape factor

(15 takes the form (Wilt, 1985)

(25(6’, 5) = {2 — 2sz’n(0 - B) + 6036 0032(6 - (3)/Sinfi}/4 (2.55)

Table 2.1 lists the values of shape factor at selected 6 and [3.

31

Table 2.1: Values of shape factor for bubble formation in conical cavities

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9 ¢(@fi = 5°) ¢(@fi = 10°)
60 0.562E+00 0.414E+00
62 0.480E+00 0.362E+00
64 0.405E+00 0.314E+00
66 0.337E+00 0.269E+00
68 0.276E+00 0.227E+00
70 0.222E+00 0.190E+00
72 0.175E+00 0.157E+00
74 0.135E+00 0.127E+00
76 0.101E+00 0.101E+00
78 0.728E—01 0.784E—01
80 0.504E—01 0.594E—01
82 0.330E—01 0.436E—01
84 0.201E—01 0.308E—01
86 0.111E—01 0.207E—01
88 0.5211302 0.131E—01
90 0.1901302 0.760E—02
92 0.411E—03 0.389E—02
94 0.152E—04 0.164E—02

 

2.4 Effects of P, T and Concentration on Nucleation Rate

The expression for the equilibrium/ critical radius of a spherical bubble in a gas—liquid

solution, Eq. (2.23), is restated here as

Re=2U/{
1/1

where n is approximately equal to unity, and V1 2 V2

ideal solution.

According to Henry’s law (which states that the concentration of a gas in a liquid

is proportional to the pressure), 51—9, 2: Pssat, therefore Eq. (2.56) simplifies to

Re : 20'/{
V1

”PsatCFI) + BC, _-

 

Co

Pi

11200

”Paddy-l) + Pssat

32

__H}
V2

 

1 for an ideal gas and an

 

log(JIJo)

 

 

PI’PIO

Figure 2.6: The effect of the liquid pressure on the nucleation rate

As the pressure of the liquid increases, the equilibrium / critical radius will increase.
This leads to the decrease of nucleation rate according to Eq. (2.52) or Eq. (2.53).
Figure (2.6) shows the variation of the nucleation rate with the pressure of liquid.
J0 in the figure is the heterogeneous nucleation rate for an 02-water solution at
H = 1 atm, T) = 20°C, and shape factor q5 = 0.01. The gas concentration in the
liquid (0’) is the saturation concentration at Pam : 10 MPa. It can be seen from
Figure (2.6) that the nucleation rate will decrease roughly 10 orders of magnitude, if
the pressure of the liquid increases by 1 order.

Because most of the temperature-dependent quantities appear in the exponential
term of the nucleation rate equation (2.52), a slight change in T; can have a signif-
icant effect on the rate of nucleation. As the temperature of the liquid increases,
Psat(T1) increases, 0 decreases, re decreases, and the exponential term in Eq. (2.52)
increases sharply. Although a in the pre-exponential term decreases a little, overall

the nucleation rate will increase exponentially with temperature. Figure (2.7) shows

33

45 —
4o —
35 -
30 -
25 ~
20 .
15 -
1o4
5 u
0 TV 1 l l l
270 290 310 330 350 370

W")

log(JIJo)

 

 

Figure 2.7: The effect of the liquid temperature on the nucleation rate

the variation of the nucleation rate with the temperature of liquid.

As the concentration of gas in the liquid C" increases, (i.e. the supersaturation
pressure increases), re decreases, so the nucleation rate J will increase exponentially.
Figure (2.8) shows the variation of the nucleation rate with the supersaturation pres-

sure. Presumably, this is the initial nucleation rate, because prolonged nucleation

will reduce C".

34

451
405
35)
soi

log(JIJo)

 

 

o . v T .

o 5 10 15 2o
Pant (MPa)

Figure 2.8: The effect of the supersaturation pressure on the nucleation rate

2.5 Discussions on the Applicability of Henry’s Law and

the Ideal-Gas Law at High Pressures

In this section, the applicability of Henry’s Law and the Ideal-Gas Law are dis-
cussed together with improvements on them which account for departures from ideal

gas/ solution behavior at high pressures.

2.5.1 The Applicability of Henry ’3 Law

Henry’s law states that there exists a linear relation between the partial pressure of

a gas P9 and its mole fraction mg when dissolved in a dilute solutions,
P9 = :29 H (2.58)

where H is the Henry’s law coefficient, which usually can be regarded as a constant

and be evaluated from

H : lim0 Pg/arg (2.59)

35

or more precisely,

H = 11510 fg/zg (2.60)

where fg is the fugacity of the gas in the solution. For a gas dissolved in water, the
vapor pressure of the solvent is negligible and the total pressure can be treated as the
partial pressure of the gas, without any appreciable error.

The reliability of Henry’s law at high pressures or dissolved gas concentrations

can be checked from experimental measurements of P9 and concentration. Figure 2.9

0.005 r
0.004 2
0.003 r

0.002 *

Concentration

0.001 4

 

0 l _ T T l
0 5 10 15 20
Pressure (MPa)

 

 

 

b— Henry’s Law + Experimental data

 

 

Figure 2.9: Measurement and Henry’s-law prediction of the saturation concentration
of 02 in water

shows the saturation concentration (as a mole fraction) of 02 in water at 25°C, as
a function of pressure. The experimental data were obtained from Brereton et al.
(1998) and are converted to molar units. The Henry’s constant for 02 in water at
25°C and 1 atm is 1.26 x 10‘3mol/ (d1113at1n) (Bowers, 1996). From Figure 2.9, when
the pressure is less than about 6 MPa, the error of Henry’s law is less than 5%. To
provide improved levels of accuracy, corrections should be made to Henry’s constant

before applying it at higher pressures.

36

A theoretical expression of the pressure dependence of the Henry’s law coefficient

is given by the Krichevsky—Kasarnovsky equation (Krichevsky, 1935),

ln(H/H0) = —(P — POW/(RT) (2.61)

where H is defined by P/atg, l7 is the partial molar volume of the dissolved gas in
water, and subscript 0 represents a reference point. Normally, V can be treated as
a constant at different pressures. It follows that there is a linear relation between

log(H) and P as can be seen from Figure 2.10. A least-squares fitted line relating H

 

 

 

 

5i
4.5
f
31' 41
2 ‘--3¢—¢
3.5—fl"
3 #1 1 1 1
0 5 10 15 20

Pressure (MPa)

Figure 2.10: Variation of H with pressure for 02 dissolved in water

of 02 dissolved in water at 25°C to P takes the form:

log(H) = 3.6022 + 1.1305 x 10-2 P (2.62)

where P is in MPa and log is 10—based. Once H is calculated for a given P, the

37

saturation concentration is readily found from
2:9 2 P/ H.

The same linear relation between log(H) and P exists for some other low solubility
gases dissolved in water, e.g. N2, H2, He, Ar, etc. More results are given in Appendix

A.

According to this discussion of Henry’s law and ideal solution behavior, it appears
that, for pressures higher than 7 or 8 MPa, models for dissolved gas concentration

that are more general than Henry’s law are required.

2.5.2 Eflects 0f the Departure from Henry’s Law on Nucleation Rate

Having shown how the dissolved gas concentration at high pressures can be described
by a generalized form of Henry’s law (Eq. 2.61), we now consider the effect of depar-
tures from Henry’s law on nucleation rates. For simplicity, the classical heterogeneous

nucleation rate equation (2.53)

2
J : N12/3‘/§%exp{—4i::fe } (2.63)

is used to calculate the nucleation rate of 02 dissolved in water. The critical radius

 

1",. is affected by the concentration of gas in the liquid C' as

 

(2.64)

Pm T P ’
7,8220/{17 .t(1)+ [C —B}
V1 V200

where the concentration of gas in the liquid 0’ is evaluated from

c' = P/H

38

Figures 2.11 and 2.12 show the heterogeneous nucleation rates at different shape
factors ((15 = 0.001 and 0.005) using the generalized Henry’s law. The nucleation rates
corresponding to Henry’s law are also shown in these figures for comparison. Some

conclusions can be drawn from those figures:

0 Use of the generalized form of Henry’s law always lowers the nucleation rate.
The effect is more significant at high pressures. The reason for this is that the
generalized form of Henry’s law gives lower gas concentration at the same pres-
sure, compared with Henry’s law. Lower concentration leads to larger critical

radii according to Eq. (2.64), and thus smaller nucleation rates.

0 For smaller shape factors (which are more appropriate to heterogeneous nucle-
ation region), the effect of the departure from Henry’s law on nucleation rate is

less significant, but can still be important.

201

log(J)
3

 

 

supersaturation pressure (MPa)

 

+ Henry's law + Generalized Henry's law

 

 

 

Figure 2.11: Effect of the departure from Henry’s law on nucleation rate ((0 = 0.001)

39

40 1
20 1

-20 ~

log(J)

~401

 

‘50 i) 10 20 30
-80 7

 

-100 -
supersaturation pressure (MPa)

 

 

 

+ Henry's law —x— Generalized Henry's law

 

Figure 2.12: Effect of the departure from Henry’s law on nucleation rate (<15 2: 0.005)

2.5.3 The Applicability of the Ideal-Gas Law

Having considered departures from ideal solution behavior in the previous section,
we now consider the applicability of ideal-gas behavior at high pressures (e. g. > 20
MPa) is discussed in this section. Figure 2.13 (Van Wylen and Sonntag, 1976) shows

the compressibility of nitrogen, where the compressibility factor Z is defined as

_Pv

z___
RT

(2.65)

Its deviation from unity indicates the departure of a gas from ideal behavior. It can

seen from Figure 2.13 that

e at relatively low pressures and relatively high temperatures, the compressibility
factor approaches unity, i.e. nitrogen behaves as an ideal gas. For other ranges
of relatively high pressures or relatively low temperatures, the behavior of the

gas may deviate appreciably from ideal.

40

o the deviation of the compressibility factor from unity is less than 3% for nitrogen

at a temperature of 300K and pressures of up to 10 MPa.

1.6 '-

1.4 F

1.2

l
E

F
\

1.0 —L _

/
N 200 K /
0.3 .. Set
“fated "Po:- 1

I
l
0.4 ’-' I state
I
I

0.2 1-

Saturated liqnid

0 ing—1 1
0.1 1.0

 

3r

2 4
P(MPa)

Figure 2.13: Compressibility of nitrogen

From the generalized compressibility chart (e.g. Gyftopoulos, 1991, p. 355), we
can obtain the compressibility factors for any gas at various temperatures and pres-
sures, provided the critical properties are known. Table 2.2 shows some relevant values
of the compressibility factor obtained using the generalized compressibility chart. It
can be seen from Table 2.2 and the generalized compressibility chart that at 300K,
the compressibility factor is in the range of 0.95—1.05 for N2 at pressures up to about
18 MP3, for 02 up to about 35 MPa, and for Ar up to about 34 MPa. Therefore
corrections for compressibility rarely amount to more than 5% for the supersatura-
tion pressures considered in this study. A 5% deviation in predicting supersaturation

nucleation thresholds is tolerable in this study.

41

Table 2.2: Calculation results of compressibility factors using the generalized com-
pressibility chart

 

 

 

 

 

 

Gas T, (K) PC(MPa) T(K) Tchr/Tc P(MPa) 103:13/10c 2:};
N2 126.2 3.39 300 2.38 18.6 5.5 1.05
02 154.6 5.05 300 1.94 35.4 7.0 1.05
Ar 150.8 4.87 300 1.99 34.1 7.0 1.05

 

 

 

 

 

 

 

 

42

CHAPTER 3
BUBBLE DYNAMICS

The classical nucleation theory relates nucleation rates to the work required to form
bubbles of critical size. For nucleation in dilute gas—liquid solutions, the diffusion of
gas molecules from the host liquid to the bubble surface undoubtably affects mass
transfer at bubble surfaces and so should play some role in nucleation, though it
is typically disregarded in nucleation theories. Since an important characteristic of
diffusion is the diffusion coefficient of a dissolved gas in a liquid, which takes different
values for each different dissolved gas, it is useful to study the dynamics of gas bubbles
in gas-liquid solutions to see the extent to which diffusion—coefficient values affect
bubble growth rates, and to see if they might provide a way to extend classical

nucleation theories to apply to different dissolved-gas species.

3.1 Statement of the Problem

In order to understand effects of diffusion characteristics on nucleation, we consider
an existing, spherical bubble in an infinite body of uniform liquid. The bubble may
contain a single gas species or a mixture of several species which diffuse independently
and differ in both solubility and diffusivity. Initially the concentration of gas in the
liquid is assumed to be uniform. Since we are interested in how nucleation of gas-
supersaturated liquids depends on the particular gas species, it is particularly useful to
study the extent to which bubble growth and collapse rates depend on the gas-liquid

diffusion coefficient.

3.2 Governing Equations

In this section, we first present the equations which describe the growth and collapse

of bubbles containing a single gas species. While the treatment of pure gas bubbles is

43

really a convenient first approximation to many practical situations, it also provides
a basis for extending this analysis to bubbles with multiple gas species. In our single-

species analysis, it is assumed that:

1. the system is isothermal;

2. the gas inside each bubble behaves ideally, and is well mixed;

3. the properties of the liquid are independent of dissolved gas concentration;
4. the center of the bubble does not move;

5. the pressure far away from the bubble is kept constant;

6. viscous and inertial effects are negligible.

The governing equation for gas diffusion in the liquid is

60 ac _ D (620 2g

—+"(")a - a? a»

fit ) for TB]? (3.1)

where C is the gas concentration in the liquid, r the radial coordinate, and R the
radius of the bubble. The gas concentration C’ is assumed to be a function of both
7' and t. D is the diffusion coefficient, which depends on both the gas and liquid
species involved and v(r) is the velocity of the liquid; it can be related to R, v and r
from the continuity equation applied to the (incompressible) liquid surrounding the

bubble, which can be written as

12(7") 2 I v(R) (3.2)

 

where R(t) is bubble radius and v(R) is the outward velocity of bubble surface. The
boundary conditions on concentration, for growth of a single bubble in a large liquid

medium, are

liIn C(r, t) : Coo; (3.3)

r—nc

44

C(R,t) 2 0,, t > 0 (3.4)

where C, is the saturation concentration at the bubble surface, that can be calculated

from Henry’s Law as,

C, : Pgas/H (3.5)

Here, H is the thermodynamic property which scales gas pressure to dissolved liq-
uid concentration, and is a constant for weak solutions. The subscript oo denotes

prOperties at locations far from the bubble. The initial condition is
C(r, 0) 2 C00, 1‘ > R; (3.6)

The pressure distribution in the liquid surrounding the bubble, which can change
on account of bubble growth, can be found by applying the momentum equation for
incompressible Newtonian fluids in the radial direction. The equation, in a spherical

coordinate system, can be arranged as

2 4
_8v3(t)£ 212%??? = —l%[: (3.7)
r p r

8tr2

For diffusional processes, the radial velocity at the bubble surface ’01; is usually very
small (of the order of 10‘6 m/s), so 6%) is also negligible and the pressure within the
liquid scarcely changes between the bubble surface and the far field. The pressure

inside the bubble can then be expressed as
Pgas : P00 + 2o/R (3.8)

The change of mass within the bubble satisfies Fick’s Law of mass transfer at the bub-

ble surface, which, after substitution for the mass of gas from the ideal gas equation,

45

can be written as

dm__d Poo+2o/R 4 3 _ 0C 2
dt _ dt iM< RgT ) (37TH )] ‘ Dar iRWTR’ (3'9)

where m is the mass of gas inside the bubble, C has units of kg / m3, M is the molecular
weight of gas inside the bubble and R9 is the universal gas constant. On simplification
of Eq. (3.9), we obtain

dR 8C

Ei,’ : Dakar/M (3.10)

where dR/dt can be interpreted as the bubble growth rate.

3.3 Analytical Solution

For diffusion processes between a micro bubble and a liquid, v(r), which is usually of
the order of 10‘6 m / s, can often be neglected. This assumption is reasonable, and has
been verified by numerical solutions of the complete diffusion equation for bubbles of

this size. If the convection term is neglected, the diffusion equation simplifies to

BC 620 2 0C
a — D (a: $5) ‘3'“)
with initial condition
C(7',0) 2 C00, 7‘ > R; (3.12)
and boundary conditions
r1390 C(r,t) 2 C00; (3.13)
C(R,t) : C3, t > 0 (3.14)

46

where C, is the saturation concentration at the bubble surface, and can be calculated

from Henry’s Law as as
20
C, = Pym/H = (P00 + E) /H (3.15)

C, is assumed to be constant, which is reasonable if the bubble radius is not too small

(above 1am) , so any surface tension effects can be neglected.

Epstein and Plesset (Epstein et al. 1950) solved this equation by introducing a

dependent variable,

a = r(C — C3) (3.16)
which allows the diffusion equation to be transformed to

Bu 08221

with initial and boundary conditions,

u(r,t = 0) 2 7'6, 1“ > R;

u(R,t) = 0, (3.18)
where 6 : C00 — C3. If one makes the second variable change,
5 : r — R (3.19)

and also assumes the bubble boundary moves sufficiently slowly that R is effectively

a constant, Eq. (3.17) becomes analogous to a familiar problem in heat conduction.

0'11. 82 ”a

47

The analytical solution of Eq. (3.20) can be found using the separation of variables

method, in the following form, (Carslaw, 1945)

 

u(r,t)= 2W/Ooo (-R+€) ){exp [—gzl—Bi’li — exp [—(é—IfiiJ—]}d€, (3.21)

Here the quantity of interest is the concentration gradient at r = R.

(g?) (R : 6{1+ R/(ert)’/2} (3-22)
(96%) [R = 6{1/R + 1/(7rDt)’/2} (3.23)

From considerations of the gas diffusion through the bubble surface, the mass flow

rate is

dm _ 2 (9C
d7 —47r RD(—8—T)(R (3.24)

Combining Eq. (3.23) and Eq. (3.24), the velocity of the bubble surface can be ex-

pressedas
(1R Dd l
— —— .25
d—t p —{R+ («Dal/'2} (3 )

where p is the density of the gas.

It is convenient to define f as the ratio of initial dissolved gas concentration to
the gas concentration at saturation, and define d as the ratio of the saturated gas

concentration to the gas density, so
f : Coo/Cs (3.26)
d : C's/p (3.27)

To solve for R(t), two cases are considered separately: undersaturation (0 g f <

1) and supersaturation (f > 1)-

48

3.3.1 Bubble Dissolution in an Undersaturated Solution

In order to solve (3.25), it is convenient to rewrite Eq. (3.25) in dimensionless form.

Setting

6 = R/Ro, (3.28)
x2 z (2a/Rg)t, (3.29)
a = D(Cs—Coo)/p
: Dd(l-—f), (3.30)
Eq. (3.25) becomes
de/dzi: = —a:/6 — 2) (3.31)

where 7 : (gfiglffl. In general, the constant 7 is small compared to the other term
13/ 6 so that an approximate solution can be obtained by neglecting 7 which takes the

form

62 = l — 2:2 (3.32)

or, when recast in dimensional form,

 

(R)2:1_ZD(C,—Coo)

Ea p123 t (3.33)

Figure (3.1) shows this radius-time relation for dissolving bubbles.
The time required for a bubble to dissolve completely can then be found from

Eq. (3.33) by setting R to zero, as

, _ 933
tdzssolve — 213(C;3 _ Coo) (3.34)

 

The time taken to dissolve is proportional to the initial area and density, and inversely

proportional to the diffusion coefficient and concentration difference. Table 3.1 shows

49

(RIR0)2

 

 

\/

t

Figure 3.1: Radius—time relation for dissolving bubbles

the time required for bubbles of different gases to dissolve in water. The results are
calculated based on Eq. (3.34). The gas properties are taken at a temperature of
20°C and the six kinds of gases studied are : air, hydrogen, nitrogen, oxygen, helium

and argon. It can be seen from Table 3.1 that

1. Among the six gases, it takes the least time for an H; bubble to dissolve, and

that an N2 bubble is the most difficult to dissolve.

2. Argon and oxygen have similar bubble dissolution rates. Since argon is much
more inert than oxygen to chemical reaction, argon is recommended as a good
substitute for oxygen in some experiments, in which effects of oxidation are

troublesome.

3. It takes hours for a bubble of 1 mm size to be completely dissolved in water.
Therefore it seems reasonable to wait hours to reach a saturated gas-liquid
solution with the help of stirring, during the preparation of supersaturated
solutions in the experiments. F inkelstein suggested 24 hours (F inkelstein et al.,

1985) as the waiting time for all the dissolved gas species they prepared.

50

Table 3.1: Time required for a pure gas bubble to dissolve in water

 

 

 

 

 

 

 

Gas T(sec.) T(sec.) T(hours) T(hours)
for R0 2 1pm for R0 : 25am for R0 2 1mm for R0 = 1cm
Air 0.011 6.64 2.95 295
H2 0.0057 3.53 1.57 157
He 0.0080 5.02 2.23 223
N2 0.013 7.88 3.50 350
Ar 0.0067 4.17 1.86 186
02 0.0067 4.17 1.86 186

 

 

 

 

 

 

 

3.3.2 Bubble Growth in a Supersaturated Solution

For a supersaturated solution, the positive constant a in (3.30) is redefined as
a : D(C00 — Cs)/p = Dd(f — 1). (3.35)

The constant 7 is also redefined, as

Coo _ Cs 1/2
7' Z (77) (3'36)

The dimensionless variables 6 and :17 keep the same form and the approximate solution

can be obtained by neglecting 7' as
62 : 1 + 2:2. (3.37)

In dimensional form, the solution is recast as

(3) =1+ (”€2.90”) . (3....

 

After long times, R‘2 is proportional to t, D, AC, and inversely proportional to p.

51

(R/R0)2

/ .

 

 

W

t

Figure 3.2: Radius-time relation for growing bubbles
The bubble growth rate is readily derived from Eq. (3.38) as

(112 _ 0(000 - C.)

”(z—t _ pH (3.39)

 

If diffusion is the limiting factor, the bubble growth rate is proportional to the con-
centration difference (AC), diffusion coefficient (D), and inversely proportional to
the size of the bubble and the density of the gas. Figure (3.2) shows the radius—time
relation for growing bubbles.

It is useful to restate the assumptions made to reach these quasi-stationary ana-

lytical solutions for bubble growth and dissolution rates.

1. In forming analytical solutions, it is assumed that the boundary of the bubble
is quasi-stationary. The gas diffusion equation is not identical to the heat can-
duction equation when the bubble boundary is not stationary. However, since
the bubble boundary moves so slowly (of the order of 10’6 m/s) at typical dif-
fusion rates, it is reasonable to assume that 9% z 0. This assumption has been
verified by numerical results which are discussed later. Also from Eq. (3.25), it
appears that this assumption is more accurate for larger bubbles. So the ana-
lytical solution is expected to be more accurate for bubble growth than bubble

dissolution.

52

2. It is assumed that the convection term in the diffusion equation can be ne—

glected. Since ‘17?- z 10“6 s’1 and 6(1) = d7???- < 92% for r > R, the convection

term v(r)%rcZ is usually negligibly small, except in the case of extremely large

concentration gradients.

3. It is assumed that the gas concentration at the bubble surface is constant
and that any effect of surface tension on the saturation concentration there
is ignored. This assumption is thought to be reasonable if the bubble radius

R > 10‘6m.

4. The final assumption is that the pressure is constant across the liquid field. From
the radial momentum equation, it can be estimated that ~66; z —p-§§§; m 0

and that deviations from a constant liquid pressure field are negligibly small.

3.4 Numerical Approach

The analytical solutions in the previous section are based on many assumptions and
simplifications. In order to assess the accuracy of them, the finite difference method

is adopted here to solve the complete gas diffusion equation:

 

— + ”(Ti '5? m:

8C BC _ D 02C 20C
0t 87‘ —

> for rZR (3.40)

where v(r) can be found from the (incompressible) continuity equation for the sur—

rounding liquid,

 

t 2
‘1)(7') : 127(2) ’U(R). (3.41)
with boundary conditions:
din; C(r,t) 2 C00; (3.42)
C(R,t) = 0,, t> O (3.43)

53

Here C3 is the saturation concentration at the bubble surface, which can be calculated

from Henry’s Law as

2
C, = Pgas/H = (P00 + 80) /H (3.44)
The initial condition is
C(7‘,0) : Coo, r > R; (3.45)

Initially there is a step change in gas concentration from C3 to C00 at the bubble
surface. To overcome the numerical difficulty caused by the discontinuity, the step

function is approximated by a decaying exponential term

C1730): Coo + (C(R0,0) - Coo) exPl_Kd(T - 130)] 7" 2 R0 (3-46)

where the value of K d can be adjusted to closely approximate a step function and still
preserve numerical stability. Due to the small magnitudes of D (10’9 m2/s), C and
0(7), it is better to solve these equations in dimensionless form to reduce round-off

error and improve accuracy in numerical calculations.

3.4.1 Dimensionless Form

The dimensionless variables of this problem are chosen as:

‘r' : r/RO
D

,’: —, .47

t tiff, (3 )
C, : C/CSQ
R’ = R/RO

54

where 030 is the saturation gas concentration under pressure Poo. The dimensionless

form of the diffusion equation is then

 

ac" + (d_R’ R”) 60' _ 620' 2 60'

_ ___ __ ,> , .
0t' 44 w? a" are 9 aw for 7‘ — R (348)

and the dimensionless form of the velocity of bubble boundary is

 

 

 

 

 

 

412' [33% 3,1297")
——,- z 40 (3.49)
dt [POo + 3133..
3.4.2 Discretization
For uniform grids, the diffusion equation is discretized into the following form:
07’“ — C?’ R2 073,, — 0n 1 C." + C."- — 203 v — C."
i z _ _ 1. z— : D 2+1 2—1 2 1+1 2—1 .
At + MR) 7",?" 2Ar { (Ar)2 + riAr (3 50)

620
0153

 

For non-uniform grids, the second derivative is approximated as (Ferziger, 1996)

 

(gr—g)” : i110? — ri_1)+ 031(7):“ " Ti) — C}’(ri+1 — 7””) (3.51)

1 0-5 (“+1 — 7"z'--1)(7“i+1 - Ti)(7‘i - “Pi—1)

For non-uniform grids, the diffusion equation is discretized into the following form,

 

CY‘H—Cn R20." —C."
l 1 lil x—l
tn+l_tn + U(R)7?- ri+1—ri-l
__ D (73,1(Ta—TH1)+C,”_1(Ti+i—Ti)—CI’(7‘1+1—"i—1) ZCI’Ztl—Cf’.) (3 52)
— 0.5 (7‘1+1—7'i—1)(1‘i+1—7’1)(7‘i—7‘i—1) 7‘1‘("'i+l-Ti—1) '

3.4.3 Stability Criterion

In order to simplify considerations of stability criteria, we consider only the stability

of the equations when discretized on a uniform grid, and disregard the convection

55

term. Rearranging Eq. (3.50), we find that

33-33 (Ag—,3.) (__Dm M).3_.(_D A De)

 

 

(Ar)? riAr (Ar)2 rAr
(3.53)
For numerical stability, the following three terms must be positive:
D
1— ———At > 0 (3.54)
2W )2
DAt At
D— 3.
(Ar)2 + riAr >0 ( 55’
D At
——At — D— 0 3.56
(Ar)2 rAr > ( )
The stability criterion is therefore
M > ZD (3 57)
At ' '
or, in its corresponding dimensionless form
(4702
2. ."
At’ > (3 08)

3.4.4 Moving Grids

One difficulty which arises in solving the gas diffusion equation numerically is the
problem of how to deal with the moving bubble boundary as the bubble grows. In
the approach chosen here, moving grids are used. For the case of bubble dissolution,
one may consider adding one grid point for each time step as the bubble surface
recedes. But owing to the slow movement of the bubble boundary and the small
time step, it will lead to very non-uniform grids and the small grid ratio will cause
instability. To overcome this difficulty, the following approach is adopted: initially,

the grid points are uniformly spaced with a spacing of Ar. At each time step, the

56

 

 

r0 r1 r2 r3
[___ _ I ,__ I l t>0
l'o f1 f2 I3
lnse a new
point here

Figure 3.3: Illustration of moving grids during bubble dissolution

first point is moved to the new bubble boundary. As the bubble shrinks, the space
between the first and second points becomes larger and larger. Once it reaches 1.5A7‘,
the first grid is split into two grids with equal spacing. A similar approach is adopted
for bubble growth, as the spacing between the first two grid points becomes smaller
and smaller. Once it reaches 0.5Ar, the first grid point is merged with the second

grid point.

3.4 . 5 Numerical Results

A typical spacing of grid points of Ar = 0.5pm and a time step size of At = 10‘5 sec-
onds were used to find numerical solutions to the discretized bubble growth / dissolution
equations. Figure (3.4) shows the radius change with time during a microbubble dis-

solution process. It can be seen that

1. Initially, the rate of bubble shrinkage % is very large, owing to the large gas

concentration gradient across the bubble surface at initiation of the calculation.

2. The rate of bubble shrinkage is also very large when the bubble is very small.
The reason for this is that, for a small bubble, the pressure inside the bubble is
very large on account of the surface tension. High pressure leads to a high gas
concentration at the bubble surface. The large concentration gradient drives

the bubble to dissolve faster.

57

1.2 -

0.8 r

0.2 “

 

t.

Figure 3.4: Dimensionless radius vs. time curve for bubble dissolution due to diffusion
(numerical result)

3. Except for the initial period, the rate of shrinkage increases as the bubble dis-

solves.

From Figure (3.5), it can be seen clearly that the relation between R’2 and t’ is
approximately linear, in agreement with the approximate analytical result. Figure
(3.6) shows gas concentration profiles in the liquid around the bubble. As time passes,

more gas diffuses into the liquid.

Figure (3.7) shows the change of R’2 with t’ for growing bubbles. It can be seen
that the relation between R’ 2 and t’ is almost linear, which again agrees with the
analytical result. In the early stage of bubble growth, the growth rate is controlled
by inertia and surface tension. But these factors become unimportant as the bubble
develops. Then diffusion dominates the growth of the bubble. R oc \/t is a good
approximation for bubble growth, if the initial size of the bubble (R0) is small enough

to be neglected.

58

1.2 _

0.8 ~
m 0.6 4
0.4 ~
0.2 4

 

 

Figure 3.5: Dimensionless radius squared vs. time curve for dissolving bubbles due
to diffusion (numerical result)

 

 

 

0 2 4 6 8 10

 

 

___—t'zo -——t'=10 ------ t’=13.5

 

 

Figure 3.6: Concentration distribution around a dissolving bubble as a function of
time

59

 

 

 

0 4 " '” 1 I fl

0 2000 4000 6000
t0

Figure 3.7: Dimensionless radius-squared vs. time curve for growing bubbles due to
diffusion (numerical result)

3.4.6 Comparison of Analytical and Numerical Results

Figure (3.8) shows the analytical and numerical results for radius as a function of
time for a dissolving bubble. It can be seen that the effects of the moving boundary
and of surface tension offset each other. The analytical result agrees very well with
the numerical result which accounts for the moving boundary, the convection term,

and surface tension effects, all of which are neglected in the analytical solution.

60

RI

 

 

 

1 i
0.8 ~
0.6 ~
0.4 ~
0.2 —

o . T I

0 5 10 15

t’
—— analytical solution numerical solution

 

 

 

Figure 3.8: Radius vs. time curves for a dissolving bubble

61

CHAPTER 4
TARGET DATA FOR TESTING THEORIES OF
NUCLEATION IN GAS-SUPERSATURATED SOLUTIONS

In this chapter, some experiments on nucleation of gas-supersaturated solutions are
discussed. These experimental results will be used as target data against which the
more general models of nucleation, developed as a part of this study, are tested in the
following chapter. The significance of these results to predictive theories of nucleation
is also discussed, together with theoretical interpretations of these results and prob-
lems associated with their interpretations are also pointed out. The experiments of
interest are those which provide target data for cases in which classical nucleation the-
ory is unreliable, and which provide a challenge for the more general model developed
in this study. They are experiments involving nucleation of different dissolved-gas
species under otherwise identical conditions, nucleation in volumes the size of which
has been systematically reduced, and nucleation in large systems in which surface

geometry is considered important.

4.1 Nucleation of Supersaturated C02—Water Solutions in

Large-Scale Systems

This section focuses on heterogeneous nucleation of gas-liquid solutions in large-scale
systems, where surface geometries play a role in the nucleation processes. Wilt (1985)
studied mechanisms of nucleation in supersaturated COg—water solutions, such as
carbonated beverages. He examined several possible mechanisms: homogeneous nu-
cleation, and heterogeneous nucleation at smooth planar interfaces, and at interfaces
containing conical or spherical cavities or projections. He concluded that the predom-
inant mode of nucleation was heterogeneous and that it occurred in conical cavities

(Figure 2.5, page 31), presumed to be present in some form on almost all surfaces,

62

but only for contact angles (6) in the range of 94 — 130°, over which shape factors
were computed to be small. It is important to note that, in these studies, no special
precautions were taken to pre—compress or pre—heat either the solutions or their con-
tainer, so there were probably many active nucleation sites, already partly filled with
gas, present at surfaces. Consequently, one would expect nucleation to be observed
at pressures very close to the dissolved-gas saturation pressure. A unique aspect of
this study was that Wilt did not approximate the work done to grow bubbles by their
spherical equivalent, but calculated the mechanical work done by bubbles, initially in
conical cavities, which subsequently grew first within the cavity and, second, beyond
it in a shape determined by local surface-contact conditions. His study would be
equally applicable to growth from cavities on container surfaces as it would to growth

from cavities within impurities suspended in the liquid.

Based 011 his nucleation theory analyses, in which effects of bubble shape on the
work of bubble expansion were considered, Wilt proposed a model for heterogeneous

nucleation as a variation on classical nucleation theory which took the form:

 

 

 

.47“: an M} (4.1)

J = N2/3f30(6’fl)\/7rmf?((0 [5) exp { 3kT

where N is the number of molecules per unit volume for homogeneous nucleation, or
N 2/ 3 is the number per surface area for heterogeneous nucleation, o is the surface
tension, m is the mass of one molecule, 9 is the liquid—solid contact angle of the
bubble with the cavity surface, 6 is the half apex angle of the cavity, and fa and
f 3c are functions of 6 and 6 related to the mechanical work done in growing bubbles

within and beyond surface cavities, which take the form:
f1c(9, [3) : {2 — 2sin(0 — 6) + c030 c082(6 — fi)/sinfi}/4

f3c(6,/3) : l1” sin(I9 — fill/2

63

When interpreting thresholds for nucleation, it is useful to discuss them in terms of
the ‘supersaturation ratio,’ w, defined as the ratio of dissolved gas concentration in
the liquid to the saturation concentration across a flat surface at the same liquid

pressure, minus one (so that w = 0 for saturated solutions), i.e.

C

— 1
Csat

 

w:

Based on the observation that a COrwater solution nucleates at a supersatura-
tion ratio of about 5 at atmospheric pressure and room temperature, Wilt proposed
6 = 94°,B = 4.7° as one pair of values in the above equation which enabled J to
increase substantially as the supersaturation ratio exceeded 5, and also seemed to
provide a plausible description of surface features in the relevant experiments. Wilt
also predicted that homogeneous nucleation and heterogeneous nucleation at either a
smooth planar surface, or a surface with conical or spherical projections rather than

conical cavities, will not occur at a supersaturation ratio of about 5.

4.2 Experiments on Formation of Gas-Vapor Bubbles in

Supersaturated Solutions of Cases in Water

Finkelstein and Tamir conducted an extensive series of experiments on bubble forma—
tion in water supersaturated with a gas, for several different gas species (Finkelstein &
Tamir, 1985). In this study the dissolved-gas species and the initial supersaturation
pressure were systematically varied while other experimental parameters were un-
changed. The experimental data from this study therefore provide a good test of the
predictive capabilities of nucleation models intended to apply to different dissolved

gases.

64

4.2.1 Experimental Setup and Results

The experimental apparatus used by Finkelstein & Tamir is shown in Figure 4.1.
Their experiments were carried out as follows. A supersaturated solution of gas in
water was first prepared in a small glass beaker at a high pressure Pssat, after which
the pressure of the solution was slowly and steadily reduced. During this period of
pressure reduction, the solution was observed carefully for visual signs of bubble for-
mation and nucleation. At the moment at which bubbles were first observed, the
pressure was recorded, and noted as PI. The difference between the initial super-
saturation pressure Pm“ and the pressure H, was called the pressure difference for
bubble formation and was designated as AP”. After carrying out these experiments
using nitrogen, helium, neon and argon at different initial saturation pressures, the

following results were reported:

1. For different gases, the observed values of AP" are very different. Experimen-
tally measured values of APn for different gases in water are shown in Table 4.1.
The following correlation was found to describe the behavior of AP” for different

gases,

APR/PC = 1.3(T/TC)” (4.2)

where subscript c denotes critical properties of the gas (although all the exper-

iments were conducted at the same temperature!)

2. For a given gas, the pressure difference for bubble formation (APn) appears to

be independent of the initial supersaturation pressure (Pssat).

65

 

 

 

 

 

 

 

 

 

 

1. Wgucylhda

2. WW(WW4)
3. slamuoolplpo

4. Gas llllor, 0.15 x 10"m
5. Addlllonal tiller, 5 X 10"m
0. Normal comm lot 138 IlN/m’ (AIIINCO)
7. Mn rolom valve
8. m gauge, 34.5 DIN/n13
0. We gauge, 201 mm»2
10. Promo col
11. We windows
12. Hanna or cooling locket
13. We
14. Glen balm

Figure 4.1: The experimental apparatus of F inkelstein & Tamir

66

 

 

Table 4.1: Experimental data for the pressure difference for bubble formation

 

 

 

 

 

Gas APn (MPa) at T =303K
Helium 42.0 :t 1.4
Neon 32.4 d: 1.4
Nitrogen 13.8 d: 1.4
Argon 13.1 :l: 1.4

 

 

 

 

4.2.2 Problems with F inkelstein and Tamir’s Equation on Nucleation

of Liquids Supersaturated with Gases

It is important to point out a limitation of the Finkelstein & Tamir correlation —— the
equation is a best fit of experimental data for experiments conducted at a single tem-
perature, in which the variable T/TC is included to introduce a supposed dependence
on the critical temperature (TC) of the gas, and is not necessarily applicable at arbi-
trary temperatures at all. For a specific gas, as temperature increases, AP" should
decrease, because nucleation always takes place more readily at higher temperature
according to the classical nucleation theory, e. 9. Eq. (2.53). The incorrect inclusion
of T in Eq. (4.2) leads to an intuitively incorrect result, even though it happens to
fit the data at the single temperature at which they were taken. Therefore Eq. (4.2)

is not as general as it might appear and seems to be misleading.

4.2.3 Inconsistencies between F inkelstein and Tamir’s Results and

Classical Nucleation Theory

One of the most significant findings of F inkelstein & Tamir’s experimental study
was that, for solutions of different gases dissolved in water, the bubble nucleation
characteristics are very different. The pressure differences for bubble formation (APn)
range from 13.1 MPa for argon to 42.0 MPa for helium. Bowers’ (1996) experimental
data also suggested a strong dependence of supersaturation thresholds on dissolved

gas species. However, when a nucleation rate equation of the kind derived from

67

classical nucleation theory is applied to gases like argon and helium, it is found that
the predicted value of AP" which corresponds to nucleation occuring (J increasing
rapidly with small increases in AP”) is almost the same for all gases. It varies by
less than a few percent, whereas the experimental measurements of APn can vary,
from one gas to another, by an order of magnitude. Thus nucleation—theory results,
with a weak dependence of AP” on the dissolved gas species, are contradictory to

Finkelstein & Tamir’s experimental results.

Before concluding the discussion of these target data, it is important to con-
sider whether the observed discrepancy with nucleation theory is a consequence of
an inadequate distinction between dissolved—gas species in the theory, or whether it
might arise through high-pressure/high-concentration effects such as departures from
Henry’s Law or changes in critical radii on account of the pressure of the dissolved gas.

From classical nucleation theory, the heterogeneous nucleation rate can be expressed

as (Eq. (2.53), page 30)

. / 3o -47ro q$(6’ fi)r2
2/3 ’ e
J — N —’ exp{ 3kT (4.3)

where the shape factor, (Z), may take the form given by Wilt (page 63) for conical

 

cavities,

¢3(6,[3) : {2 — 23in(0 — B) + c056 6082(6 — fi)/sinfi}/4

and the other symbols take the same meanings given previously. The equilibrium
radius of bubble formation (in a weak gas-liquid solution can be calculated from Ward’s

(1970) equation (Eq. (2.23), page 19),

 

PC'
re : 2o/{nPSm(T1) + C — H} (4.4)
/0

68

where the symbols also take their previous meanings. The coefficient 7) is

 

UK}?! - Psat) "’ RTC,
”29“) RT

and the concentration C’ is, in this case, the saturation concentration under the
supersaturation pressure Pssat. The pressure difference for bubble formation APn is

therefore Pssat — Pl. Using Henry’s Law, Eq. (4.4) is simplified to
re : 2o/ {nPsat(T1) + A3,} (4.5)

and, since 77 z 1 and Psat(T1) << APn ( Psat=2338Pa at 20°C), the above equation is

simplified further by neglecting 77pm to yield:

 

am

If we consider that Henry’s law may be inaccurate at high concentrations, a modified

expression of Henry’s law can be used.

CzWHW) 80

where H is a function of P, instead of a constant. More discussions of the applicability
of Henry’s law can be found in Section 2.5. Thus the critical radius can be more

accurately expressed as

 

20
re z HEP} (4.8)
H(Psslat)APn

For reasons of simplicity, we still assume Henry’s law is applicable in the following

P Hszl

"mama, when

derivations, but keep in mind that APn should be replaced by A

Henry’s law is no longer accurate. After substituting Eq. (4.6) into Eq. (4.3), the

69

nucleation rate for a gas-liquid solution is expressed as

 

. 3o —167ro3 (15(9 [3)
: 2/3 _ ,
J N exp { 3kTAP3

nm

For different gas species dissolved in water, the aqueous gas solution surface ten-
sion 0 is almost constant for all gases (Finkelstein & Tamir, 1985). The experimental
results of Massoudi and King (1974) showed that the surface tension changes less
than 8% for He, H2, 02, N2 or Ar dissolved in water, when the pressure changes
from 1 atm to 80 atms. Although the supersaturation pressures Pam in Finkelstein
& Tamir’s experiment are very high (13.1—42.0 MPa), the liquid pressure B at which
nucleation occurs is much lower. Therefore it is reasonable to assume that the surface
tension essentially depends only on the properties of the solvent for dilute solutions at
H. Since the experiments were carried out in the same container, surface conditions
such as the contact angle 0 and the half apex angle of conical cavities on the container
surface [3 should not change much when different gases were tested. Therefore, the
calculation results for values of AP" from Eq. (4.9) at which J begins to increase
rapidly for all the gases are expected to be almost the same. This result is in contra-
diction to the experimental result shown in Table 4.1, and indicates that the classical
nucleation rate equation fails to account for effects of different gas species. In the
next chapter, we will modify the classical nucleation theory to interpret Finkelstein
& Tamir’s data, by considering the diffusion of dissolved gas molecules towards the
bubble surface. Now, from Eq. (4.9), we can see that for a given gas, it is APn, not
the initial supersaturation pressure (Pam), that affects the nucleation rate. The pres-
sure difference for bubble formation APn is independent of the initial supersaturation
pressure (Pam). This conclusion agrees with Finkelstein & Tamir’s experimental re-
sult, and so some other feature of the nucleation—rate equation should account for the

dependence of gas species on nucleation rate.

70

Based on these analyses, it follows that F inkelstein & Tamir’s experimental results
provide useful target data for refined nucleation models. In particular, these data

provide a good test for prediction of AP" for different gas species.

4.3 Experiments on Nucleation of Gas-Supersaturated

Solutions in Small Capillary Tubes

To gain a better understanding of the mechanism of nucleation in small capillary
tubes, in which it has been observed that nucleation thresholds can be raised con-
siderably, a series of experiments which explored nucleation thresholds as functions
of dissolved-gas concentration and capillary-tube diameter were conducted by Spears

and co—workers (Brereton et al., 1998).

4.3.] Experimental Setup and Procedures

The setup used in capillary tube nucleation experiments is shown in Figure 4.2. In
these experiments, water, in which gas was dissolved at an initial supersaturation pres-
sure Pssat, (ranging from 3 to 15 MPa), was pre—compressed to even higher pressures
(Pmp, a: 100 MPa). The gas-supersaturated water solution was then decompressed
during flow along a silica capillary tube, from its pre—compression pressure at one end
of the tube to almost atmospheric pressure at discharge at the tube’s far end. The
diameters of the capillary tubes in these experiments varied from 10 to 100 um. The
tube lengths were chosen to be proportional to the square of their diameters so that
the cavitation number Ca 2 AP/ (pV2 / 2) remained constant from one experiment to
another. Also, the maximum Reynolds numbers in any experiment were also kept be-
low 2300, so that laminar flow always prevailed. These experiments were carried out
with the outer surface of the capillary tube either exposed to surroundings at room

temperature, or, if effects of temperature were to be studied, immersed in a water

71

Pure Water
Waste

Flushin Circuit
«— —-—. i
Discharge of Ca illa
Gas-enriched . p ry
Water

 

 

__ E

fifrr—Efl Pressure

Vessel

 

 

 

 

 

_ _ Gas-enriched H
_ Water

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Gas
Cylinder

 

Gas
lntensifier

 

 

 

 

 

Hydraulic
Cylinder

Figure 4.2: Schematic illustration of the experimental apparatus used for oxygen-
supersaturation studies

72

bath. In order to detect the presence of micro bubbles, fluorescein was added to the
liquid prior to gas supersaturation and pre-compression. The effluent from the capil-
lary tube was then illuminated with an argon-ion laser, which induced yellow-green

fluorescence in bubble-free liquid and blue reflections from rising bubbles.

4.3.2 Experimental Results for Nucleation of Gas-Supersaturated

Solutions in Capillary Tubes

A series of experiments (Brereton et al., 1998) on oxygen-supersaturated water were
carried out following the approach described above. In these experiments, the gas
supersaturation pressure (or concentration) was varied systematically to determine
its lowest value at which nucleation was observed in water discharged from capillary
tubes of different diameters, as determined by detection of bubbles in the effluent.
These experiments showed that pre—compression plays a important role in rais-
ing the nucleation thresholds. Without pre-compression, nucleation was observed for
supersaturation at nearly all supersaturation pressures considered. However, when
the oxygen-supersaturated solution was pre—compressed to approximately 100 MPa,
solutions of oxygen at relatively high concentrations could be decompressed to atmo—
spheric pressure within capillary tubes, without any evidence of bubble formation.
This phenomenon can be explained in the following way. It is already known that
there is very likely to be some amount of gas trapped in surface cavities and that
the entrapped gas can provide nuclei for the formation of gas-vapor bubbles at the
onset of nucleation, thereby facilitating the initial nucleation process. At high pre-
compression pressures, the solubility of gas in the liquid is also higher according to
Henry’s Law. Therefore pre—compression facilitates the dissolution of trapped gas
bubbles and drives liquid (rather than gas) into any cavities present at surfaces or on
contaminant particles. Consequently it is more difficult to initiate nucleation as the

pre—compression pressure is increased. It is also thought that pre—compression can

73

drive contaminants in the liquid into surface cavities, and eliminate some negative
effects of contaminants on nucleation. Although this mechanism provides an expla—
nation of this phenomenon, it has not been proved that this is what happens at the

microscale.

 

 

 

18a
16‘
-14“
“12‘
$10—
58"
n. 5*
4-
24

0 v T l T fl

0 20 40 60 80 100

d (u m)

Figure 4.3: Dependence of supersaturation pressure on capillary tube diameter. The
line and data points are an algebraic fit to many experimental data, measured at
22°C (from Brereton et al., 1998)

Figure 4.3 shows experimental results for the maximum achievable value of P3801,
(or, equivalently, dissolved-gas concentration) for which no nucleation is observed, or
the nucleation threshold, as a function of tube diameter d. In the area above the data
line as shown in the graph, nucleation is always observed. These experiments were
all carried out in quartz glass capillaries at a delivery pressure of 100 MPa at 22°C.
It is clear from the figure that smaller capillary tube diameters correspond to higher
nucleation thresholds of supersaturation pressure or dissolved gas concentration, so

that this elevation of the nucleation threshold appears to be a confinement effect.

74

4.3.3 Interpretations of Experimental Data Describing Nucleation in

Capillary Tubes

Using the results of these experiments on nucleation of gas-liquid solutions in small
capillary tubes by Spears and co-workers, Brereton et al. (1998) developed a dual

dependence model to interpret the experimental data.

Inside capillary tubes, the preferred mode of nucleation is heterogeneous, due to
the existence of surface cavities (which trap gas bubbles, and make the ‘energy barrier’
for bubble nucleation smaller). A cavity is ‘active’ when its radius exceeds the critical
radius for bubble formation. Experimental data suggest that the active number of

cavity sites per unit area Nactive (m’2) may be modeled as

 

TCG’U q]
Native 2 const ( ' ) (4.10)

Te

where raw is the average size of conical cavities at surfaces, and re is the critical radius
for bubble formation. q] takes the approximate value of 0.16 for boiling of water at

superheated copper surfaces (Lorenz et al., 1974.)

Brereton et al. developed a dual dependence model, which predicts that nucle-
ation rate in capillary tubes depends on sufficient numbers of both molecules of the

vaporizing species and of active cavities, and is expressed as:

30 —47ro r2¢>
J : .tANa .,.,eNs,,,.‘/—— ——£— 4.11
cons C. 1 7m eXp{ 3kT } ( l

where N13,". is the number of surface molecules of the dissolved gas species per unit
area, (i) is a shape factor for the bubble, varying between 1 for spheres and 0 for

‘pancakes’, and J has unit of s‘“lin‘2.

Although Eq. (4.11) could be calibrated to agree with experimental data, some of

the assumptions in the original paper (Brereton et al., 1998) required more careful

75

consideration, e.g. the assumption that the changes in the radii of curvature of nuclei
in crevices are neglected during decompression is questionable. Also more detailed
calculations of the critical radii of nucleation in gas-liquid solutions could have been
made. More importantly, the effect of dissolved gas species on nucleation thresholds

was not considered.

The target data of Brereton et al. (1998) therefore provide a test for the capa—
bilities of nucleation models in predicting effects of varying surface area (and also
varying surface area to volume ratio), effects of dissolved-gas concentration, and ef-
fects of precompression pressures on nucleation thresholds. In the next chapter, a
nucleation model with several refinements intended to allow prediction of these ef-

fects will be developed and tested.

4.4 Some Shortcomings of the Classical Nucleation Theory

In classical nucleation theories, one typically focuses on clusters of nuclei in a liquid
and considers when they are sufficiently numerous and energetic to combine to create
micro-bubbles which are large enough to avoid collapse, and grow to a finite size.
Some particular problems with these classical theories, which limit their usefulness in

practical applications, are:

1. they typically associate the conditions required for bubble formation only with
the reversible work required to create bubble surfaces of stable size, without
regard for mass transfer to the bubble surface and other irreversible effects, e.g.
diffusive dissipation, viscous dissipation, etc. It is possible that bubble forma-
tion is also strongly dependent on the gas species involved, possibly through
(species dependent) diffusive effects adjacent to the bubble surface, or through

other species dependent effects;

76

2. a nucleation model calibrated for one dissolved-gas species can be highly inac-
curate if applied to a different gas-supersaturated solution and the reason for

this shortcoming will be explored in Chapter 5;
3. they do not predict any effect of volumetric confinement on nucleation thresh-

olds of gas-supersaturated liquid, when experiments indicate there is one.

Modifications to the classical nucleation theory, which account for these shortcom-
ings and are tested against the target data described in this chapter, are presented in

Chapter 5.

77

CHAPTER 5
MODIFICATIONS TO THE CLASSICAL NUCLEATION
THEORY

Several problems associated with the classical nucleation theory were pointed out in
the previous chapter, together with target data sets with which an improved theory
should provide better agreement. In this chapter, modifications to the classical theory
of nucleation are described and tested. They take into account effects of diffusion of
gas molecules, volumetric confinement, and departures from ideal gas and solution

behavior.

5.1 Prediction of Nucleation Thresholds of Liquids

Supersaturated with Different Gases

In Section 4.2, F inkelstein and Tamir’s experiments on bubble formation in water
supersaturated with a number of different gases were described and discrepancies be-
tween the experimental results and predictions from the classical theory of nucleation
were pointed out. The classical nucleation theory relates nucleation rate only with
work required to create bubble surfaces of stable size without regard for mass transfer
to the bubble surface and other surface effects, and this is possibly the reason it fails
to predict accurately the nucleation thresholds of liquid supersaturated with different
gases. From the classical nucleation theory, the predicted value of AP", an expression
for the degree of supersaturation which a solution can tolerate, is almost the same for
all the gases. This nucleation-theory result, with a weak dependence of AB, on the
dissolved gas species, is in contradiction to F inkelstein & Tamir’s experimental data
of (APn) ranging from 13.1 MPa for argon to 42.0 MPa for helium. In this section,
the classical nucleation rate equation is modified to account for species-dependent dif—

fusive effects in a way that allows it to reproduce Finkelstein & Tamir’s experimental

78

results for the dependence of AB, on species.

Becker (1938) appears to be the first to consider effects of diffusion on nucleation
and proposed the following type of expression for the rate of nucleation in a condensed

system, e.g. a liquid-solid phase transformation,

.12 K exp{—————Wrw +q}

M, (5.1)

where Wm, is the maximum reversible free energy necessary for nucleus formation, q
is the energy of activation for diffusion across the phase boundary, K is a constant

that accounts for the free collisions between molecules, and k is Boltzmann’s constant.

Blander and Katz (1971) measured the nucleation temperatures for n-pentane
and hexadecane mixtures, in which the solute, n-pentane, is relatively volatile. They
proposed a modification to the nucleation rate equation to account for effects of

diffusion as

JDZJ/(1+(SD) (5.2)
_ 20
— D\/27rka(C — C3...)

 

5:)

where J is the nucleation rate for pure solvent at the same pressure and temperature,
J D is the nucleation rate for the solution with the volatile solute, (SD accounts for the
diffusion of the volatile solute to the interface, m is the mass of one molecule of the
solute, C is the initial concentration of the volatile solute, and Csat is the equilibrium
concentration at a vapor pressure equal to the ambient pressure. While this pre-
exponential modification appeared to reconcile some of their observations with the
theory, a calculation of nucleation-rate dependence on AP" using Eq. (5.2) shows that
it fails to explain the large difference of AP", for different gases dissolved in water.
As discussed earlier, the nucleation rate, J or JD, is insensitive to pre—exponential

factors. Similar values of AR, would be obtained, even if (SD were varied by several

79

orders of magnitude. So this pre—exponential modification for diffusion effects does

not seem adequate.

The classical nucleation theory relates the nucleation rate to the reversible work
required to create a bubble of critical size, without regard for any irreversible effects
due to diffusion or internal friction. We suspect that the dissipation of energy due to
viscous and diffusive effects may not be negligible during nucleation process, therefore,
the equations for these dissipative terms will be derived and their order of magnitude

will be estimated.

5.1.1 Dissipation of Energy through Difiusion and Viscous Efiects

In this subsection, the formulas of dissipation of energy through diffusion and viscous
effects during bubble growth will be derived. Their significance to nucleation of gas-
supersaturated liquids will be discussed afterward.

During processes such as bubble growth, the entropy of a fluid system increases
as a result of the irreversible process of thermal conduction, internal friction and

diffusion. The rate of increase of entropy of a binary solution in a finite volume can

be expressed as (Landau and Lifshitz, 1987)

- 2
% fps dV = f ——’”(gradT) dV

 

T2
77 av,- ka 2 '01), 2 fg , 2
+ fZT (317k + 393i 3611:8321) dV+ T(dZ’UV) dV
-2
1
+ ffidv (5.3)

where s is the entropy, v is the velocity, K is the thermal conductivity, 17 is the dynamic
viscosity, C is the second viscosity coefficient, a = [Dp/(Bu/aC)p,T], u is the chemical
potential of the mixture fluid, D is the diffusivity, C is the concentration defined by

mass fraction of the solute, p is the density of the liquid, and i is the diffusion flux,

80

i.e. the amount of the solute transported by diffusion through unit area in unit time.

The first term on the right side of Eq. (5.3) is the rate of increase of entropy owing
to thermal conduction, the second and third terms are due to internal friction, and
the fourth corresponds to diffusion. These terms are always positive. The general

form of diffusion flux is

i : —pD {gradC + (kT/T)gradT + (kp/P)grad P} (5.4)

For incompressible flow under constant pressure and temperature, Eq. (5.3) sim-

 

plifies to
d 01),- 0v__k_ 2 01212
— d _ —j[— ——. r
dtiips V 2T (Bxk+ 0x.- 36"“ 6:1: —) dV (5")
+ f (ngradC)2 dV
aT

Further, if the flow is spherically symmetrical, and the only component of velocity is

the radial one 1),, then in a spherical coordinate system

W = i%(2(%i’)2+4(%)2) W
+ pr-—(—%C) 2(g—C)PT W (5-6)

For weak solutions (C << I), the chemical potentials (in joules per molecule) for

 

the solute and solvent are (Landau and Lifshitz, 1987, p. 234)

p2 : kBTlnC + f2(P, T) (5.7)

M : kBTln(1 —- C) +- f1(P, T) (5.8)
The potential of the mixture (in joules per unit mass) is defined by (Landau, 1987,

81

p. 228)
p.dC : p2 dng + #1 atm (5.9)

where M; is Boltzmann constant, and n1,n2 are the numbers of solvent and solute

molecules in unit mass of solution. The numbers n1, n2 satisfy the relation

nlml + ngmg : 1

where ml (or m2) is the mass of one molecule of solvent (or solute). By definition,

C = ngmg. From Eq. (5.9), it follows that

H = #2/m2 — [Ll/ml (5-10)

Combining Eqs. (5.7) —— (5.10), the change of [.L relative to concentration can be

obtained as

 

 

(6“) ~ kBT — RJ (5.11)

CC PT 0mg AIC
where Ru is the universal gas constant, and M is the molecular weight of the so—

lute. Substituting Eq. (5.11) back into Eq. (5.6), the rate of entropy increase in an

isothermal, spherically symmetrical, weak solution with spatially uniform pressure is

if st — f3 2 8”” 2+4.(3’1)2 dV
dt ”‘ ‘ T 07‘ 7‘ ‘

JFWD—‘M (%)2R“W

 

(5.12)

The two items in Eq. (5.12) represent the rate of viscous dissipation of energy as

\Ilmm =77/f (2(80’)+ +4(3—:‘)2) (1th (5.13)

 

82

and the rate of dissipation of energy due to diffusive effects as

 

Twig— )2 .
WWW-on: ff ’0 0111)}?qu (5.14)

5.1.2 The Significance of Difiusion and Viscous Effects t0 Nucleation

Having derived expressions for the dissipation of energy due to viscous and diffusive
effects, the order of magnitude of each dissipation term will be estimated, and its
significance to nucleation of gas-supersaturated liquids discussed. We consider an
initially uniform gas-water solution in which a bubble of critical size (Re) forms as a
result of density / thermal fluctuations. The classical nucleation theory indicates that

the reversible work required to form such a bubble is, Eq. (2.50)

4 .
WM, 2 gnaR: (5.15)

and the homogeneous nucleation rate is determined by the reversible work through

the following relation, Eq. (2.51)

30 W.
J:N — — m’ 7.1
(WM m) <0 6>

However, since the nucleation process may not necessarily be reversible, the dissipa—

 

tion of energy due to viscous and diffusive effects may not necessarily be neglected, as
in the classical nucleation theory. Therefore it is useful to estimate the order of mag-
nitude of dissipative energy terms, so we know whether viscous and diffusive effects

are significant to nucleation.

Since the path of a real, irreversible nucleation process can not be specified

uniquely, a simple representative path is assumed: the bubble grows from nearly

83

zero size to the critical size at a constant growth rate, i. e.

(112

__ : ’Uc :: const. or R = vet
dt

where R is the radius of the bubble. From the continuity equation, the velocity of
liquid is
———vc, 7‘ > R (5.17)

The dissipation of energy due to viscous effects is then obtained as

61), 2 2),. 2
11mm, _ ffn{2( 07') ”(7) }dth (5.18)
Re/vc 00 61),. 2 v? 2 2
——/0 [R n{2(6r> +4(—r—) }47r7' drdt

Re /vc
: / 16777rR v3 dt
0

 

 

: 8n7rch:

Comparing this viscous dissipation term with the reversible work (Eq. 5.15) required

to form a bubble of critical size, the bubble growth rate should satisfy

10

l 71 x 10"3
6 1 x 10‘3
~ 12 m/s

for the two terms to be of the same order and viscous dissipation to be significant.

This velocity seems implausibly large for bubble growth in physical systems.

Next we will estimate the order of magnitude of the dissipation of energy due
to diffusive effects. Figure 5.1 shows the growth of a bubble in a gas-liquid solution.

Applying the conservation of mass principle, the mass flow rate of gas into the growing

84

bubble is

dm d
"(F : a<pgasVX908) (520)

where pgas is the density of the mixture inside bubble, and X903 is the mass fraction
of gas inside the bubble. For nucleation of gas-supersaturated liquids, gas is the

dominant component inside the bubble, i.e. X903 z 1, which can be deduced from

Eq. (2.21) and Eq. (2.22).

Liquid

905 difiUSiO“ bubble growth

Bubble

Figure 5.1: The growth of a bubble in a gas-liquid solution

The density of gas mixture inside a steady—growing bubble can be estimated as

_ P903
”9‘” ‘ (Ru/M) T
P + 20/R

(Rm/1W ) T
20

RT(&/M)

22

(5.21)

where mechanical equilibrium is assumed and the Young-Laplace equation (2.4) is
used to relate Pgas to P of the liquid. Ru is the universal gas constant, and Al is
the molecular weight of the gas. The pressure of the liquid P is negligible compared
with the pressure inside bubbles of size smaller than the critical radius. Substituting

Eq. (5.21) into Eq. (5.20), one finds

:12 _ 1(37I'UIl'IngRvC
(it — 3RuT

 

85

To calculate the dissipation of energy due to diffusion, the concentration profile of
dissolved gas in the liquid is required. We may imagine that the bubble acts as a sink
which draws gas molecules from the liquid. Assuming a quasi-steady process, i.e. at
each instant of bubble growth, there is a steady flux of gas molecules flowing towards

the bubble surface,
6C _ dm/dt
(9r _ )0D47r7'2

 

(5.23)

where p is the density of the liquid, D is the diffusivity of the gas in the liquid,
and diffusion follows Fick’s law: dm/dt : pDAOC/a'r. Substituting Eq. (5.20) into
Eq. (5.23) yields

E _ EOIWXgas ’UCR
01‘ _ 3 RquD 1‘2
Ach

7.2

(5.24)

 

 

where A denotes the constant
_ 4 all! X gas

A ’ 5 RquD

Eq. (5.24) is subject to the boundary condition

C(r—> 00) :000

Therefore, the concentration profile can be found by integrating Eq. (5.24),

_ EOMXgasch
3 RquDr
_ Ach

’I"

(5.25)

 

C(r) 2 Coo

2 C00 (5.26)

 

Once the concentration profile is known, the dissipation of energy by diffusion can be

86

calculated as

 

 

 

 

TpD( (8_C_)2Ru
‘sz'f/usion — /]{ CM dV dt (5.27)
Re/vc
= [0 [004 WTpDRu( LC)? 7" 2dr (1t
_ Re/vc WTpDRflA 113122
’ /o [004 7‘ (Cm—Ach/r)drdt

 

_ Re/vc TpDE, Coo
— [0 47r 1W .Avaln(C,oo 30.24116) dt

 

_ TpDRu Coo R3
— 47f M Aln(°o — Ave) 2
8 C: 2 .—
_ gwanas lIl (C——ooA—’UC) Re (028)

Comparing this dissipation term with the reversible work required to form a bubble
of critical size, for the two terms to be of the same order, the bubble growth rate

should satisfy

Coo
2Xgas 111m N 1
Coo
c N 0.39—+— 5.29
v A ( )

In the above derivation, the estimation X gas ~ 1 was used. For 02 dissolved in water

at 30°C, the value of A is

4 0’1”me

3 Rump

4 0.071 x 32.0 x 1.0
33314.0 x 303 x 996 x 2.8 x 10-9

: 0.43 (s/m) (5.30)

A:

 

The saturation concentration of ()2 dissolved in water at 10 MPa corresponds to a
mole fraction of about 0.002 or a. mass fraction of 0.0035. If 000 takes this value, then

for the dissipation of energy by diffusion to be of the same order as the reversible

87

work to form a bubble of critical size, it requires

vc ~ 0.003 (m/s)

At this bubble growth rate, it would take about 3.3 micro seconds to form a bubble
of size at 10‘8 m, which seems plausible. In contrast, a bubble growth rate of 12 m/s,

required for viscous dissipation to be significant, seems highly improbable.

It is noteworthy that if we assume diffusion occurs only in a limited range, say

R g r g nR, the integration of Eq. (5.27) from 'r = R to r : nR yields

 

8 Cm _ A C
In which case,
0.0053
c N -—-—- . 2
v 1.65 — g,- (5 3 l

and vc is still of the same order regardless of whether 71 is set to 10, or 100, or infinity.

In real nucleation processes, bubble formation and growth may not necessarily
follow spherical growth at a constant rate. However, the largest contributions to each
integral in Eq. (5.12) take place as R approaches Re, and so the dissipation terms in
real bubble growth are still likely to be of the same order as these estimates. If we
instead assumed that the bubble grew at a constant £31,2-, the same order of magnitude
for (Pm-”1,3,0” would be obtained. Therefore, for nucleation of gas-supersaturated liq-

uids, it seems reasonable to add the irreversible dissipation of energy through diffusion

to the reversible work in a nucleation rate equation like Eq. (2.52) or Eq. (2.53).

88

5.1.3 A New Nucleation Model for Liquids Supersaturated with

Different Gases

Based on the findings of the previous section, it is proposed that the energy required
for nucleation to occur in a weak gas-liquid solution includes not only the reversible
work of formation of a gas / vapor nucleus of critical size, but also the energy of diffusive
dissipation. Therefore one term exp{—q/(kT)} is added to the classical nucleation

rate equation for a weak gas-liquid solution, and the modified nucleation rate is

 

3 w...
J ‘1 NW3 fiexp{— (7TH) $015)} (533)
2
2 NW3 %exp{(—4’;Z;— ’31,) 6,09 3)} (5.34)

where q is the dissipation of energy due to diffusion which always inhibits nucleation.

If Henry’s law is assumed, substituting Eq. (4.6) into the above equation yields

3 16 3 0
J z N2/3‘/ EO}? exp { (— ggflfim — g?) 42(6), 5)} (5.35)

No model for q appears to have been published for nucleation in weak gas—liquid

 

solutions, so a preliminary expression for q is developed here.

While the dissipation of energy due to diffusion in nucleation processes is clearly a
quantity that depends on microscopic effects, some insight into its likely dependence
on measurable properties can be gained by considering the equation of diffusive dis-

sipation we derived in the previous section, which is restated here

 

 

T D(.—
\Ildiffusion: /f p C AI) RudV dt (5.36)
so,
T919027.) R. ,
(Irv/f C 11 1 (1V dt (.37)

89

where p is the density of the liquid, D is the gas diffusion coefficient in the liquid, C
is the gas concentration in the liquid, R, is the universal gas constant, and M is the

molecular weight of the gas.

The \Ildz- ”1,3,0" term is too complicated to use in a nucleation rate equation, so it
is useful to approximate it by simplifying its functional dependence. The functional

dependence of q on these variables and other coefficients is estimated as follows.

1. The larger the diffusion coefficient of a dissolved gas in a liquid, the more rapidly
a uniform dissolved-vapor concentration will be restored after any perturbation
and the more unlikely it will be that sufficient free collisions between vapor
molecules can create a nucleus of critical size. Therefore, with J proportional

to e“q, it follows that q oc D.

2. The larger the deviation of dissolved gas concentration from its equilibrium
saturated concentration (AC 2 C —Csat), the more metastable the solution, and
nucleation is more likely to occur. The increase of AC should lead to a smaller
q, and a larger nucleation rate. Therefore q should be inversely proportional
to AC, suggesting that q oc D/AC or q or D/C. For solutions approaching
supersaturation nucleation limits, the following approximations can be made:

C >> Csat and C m AC

3. The inversely proportionality of q to C or AC can also be verified in this way.
The larger the gas concentration, the more gas molecules exist in a unit volume
and the more likely that sufficient gas and vapor molecules collide together
to create a nucleus of critical size. From an energy balance viewpoint, it needs
more work to bring sparsely distributed gas molecules together to form a bubble.

Thus a smaller C should lead to a larger q and a smaller nucleation rate.
4. A combination of din‘iensional analysis and the above reasoning suggests that

90

the energy of activation for diffusion may take the form

q = f {D (3%) kT} (5.38)
343(5)}

where f and F are unknown functions. A simple linear model is proposed. q
could take the form:
1

q = constant (D) (KC) kT (5.40)

where the constant has a unit of s/m2. If Henry’s law is assumed, it can be

simplified as

 

q : constant (D) (aAlP ) kT (5.41)

In Eq. (5.40), q is proportional to DT/AC or DT/ C. The same term DT/C

appears in the equation of \Ildiffusion, (5.36).

The Bunsen coefficient a is adopted here to represent the solubility. It is defined as
the volume of gas reduced to 20°C and 1 atm, which is absorbed by unit volume of
solvent at a stated temperature under a partial pressure of gas of 1 atm. Substituting

Eq. (5.40) into Eq. (5.34) gives the revised nucleation rate equation,

 

30 47m 7'2 1
: 2/3 _ e _ , __ F,
J N 7rm exp {( 3kT constant (D) (AC)) ¢(6,fi)} (o 42)

If Henry’s law is assumed, the above equation simplifies to

/ 3o 167ro3 1
J : N2/3 ir—m exp { (—W — constant (D) (GAP )) ¢>(6,/3)} (5.43)

 

91

5.1.4 Calculation Results of the New Nucleation Model for Water

Supersaturated with Difierent Gases at a Constant Temperature

Eq. (5.43) can be solved by setting J to a constant value (e. g. 1) and calibrating
the constant term in q, and the shape factor. The values of solubility and diffusivity
of different gases dissolved in water can be found from Wise and Houghton (1966),
Janssen (1987), “CRC Handbook of Chemistry and Physics,” or “Solubility Data

Series.” Some of them are listed in Table 5.1.

Table 5.1: The values of solubility and diffusivity of several gases dissolved in water
at 30°C

 

 

 

 

 

Gas Solubility a Diffusivity
Bunsen absorption coef. D (x10'9 m2/s)
Helium 0.0087 8.0
Nitrogen 0.0129 3.5
Argon 0.0273 2.7
Oxygen 0.0271 2.8

 

 

 

 

 

Table 5.2 shows the predicted values of the pressure difference for bubble formation
(APn) from the revised nucleation equation (5.43), as well as the experimental data
of Finkelstein & Tamir at 30°C and of Hemmingsen at 25°C. Since Finkelstein and
Hemmingsen claimed that AB, is weakly dependent on temperature in the range of
293 to 303K, we can compare AP" data at 30°C with those at 25°C.

In Eq. (5.43), Henry’s law was used to estimate the dissolved gas concentration at
different pressures. As discussed in Section 2.5, departures from Henry’s law should
be considered for pressures higher than 7 or 8 MPa. The supersaturation thresholds in
this study are in the range of 13 to 42 MPa, therefore the generalized form of Henry’s
law (Eq. 2.61) is needed to calculate saturation concentration in the nucleation equa-
tion (Eq. 5.42). We suspect that the large discrepancy between experimental data
and calculation data of AP" for nitrogen is due to the use of Henry’s law.

Table 5.3 and Figure 5.2 show the supersaturation nucleation thresholds for dif-

92

Table 5.2: Comparison of the experimental data of AR: (MPa) with the predicted
data from the revised nucleation equation (5.43)

 

 

 

 

 

 

 

Gas Experimental Result of AB, Calculation Result of AP”
(MPa) (MPa)
Helium 42.0 :1: 1.4 44.02
Neon 32.4 i 1.4 28.45
Nitrogen 13.8 :1: 1.4 18.66
Argon 13.1 :1: 1.4 13.44
Oxygen 13.5 - 14.0 at 25°C from Hemmingsen 13.56

 

 

 

 

* If unspecified, the experimental data are from Finkelstein at 30°C.

 

ferent gases dissolved in water at 25°C calculated from Eq. (5.42) with (b = 0.003 and

J : 1, taking into account departures from Henry’s law. The reason we choose 25°C

Table 5.3: Predicted data of nucleation thresholds from Eq. ( 5.42) taking into account
the departure from Henry’s law

 

 

 

 

 

 

Gas Experimental Result of AP" Calculation Result of APn
(MPa) (MPa)
Helium 42.0 :1: 1.4 40.60
Nitrogen 13.8 :t 1.4 14.75
Argon 13.1 :t 1.4 10.50
Oxygen 13.5 - 14.0 at 25°C from Hemmingsen 14.00

 

 

 

 

* If unspecified, the experimental data are from Finkelstein at 30°C.

instead of 30°C is that we only have solubility data (for different gas species dissolved
in water under a wide range of pressures) at 25°C. The diffusivity data at 25°C are
estimated using a model supplied by Spalding (1963), together with experimental
diffusivity data at 20 and 30°C from Wise and Houghton (1966).

“2) (“2)
_ Z — — 5.44
(IVSC ) 20° C ( pT / pT 200 C ( l

93

 

 

A E‘
G — .

l 50 x’ .
E 4::

c _ .
O. 40 ’I’ i
<1 /

“a 304

3 x’

N 20 ‘ z’

> I

a a"

3;- 10 . ’,’HIH

5

a 0 ’ I T l 1 I

 

O 10 20 30 40 50
Experimental data of A P" (MPa)

 

A Helium I Nitrogen 2: Argon
0 Oxygen —---y=x

 

 

 

Figure 5.2: Experimental data for AP“ and the predicted data from the nucleation
equation revised to include the energy of diffusive dissipation

94

where the Schmidt number is defined as

NS. = — (5.45)

Table 5.4 lists the estimated diffusivity data of gases dissolved in water at 25°C.

Table 5.4: The estimated values of diffusivity of several gases dissolved in water at
25°C

 

 

 

 

 

Gas Diffusivity D (x10‘9 m2/s)
Helium 7.05
Nitrogen 2.97

Argon 2.38
Oxygen 2.63

 

 

 

 

Comparing Table 5.3 with Table 5.2, it is found that the values of nucleation
thresholds for nitrogen and oxygen are in much better agreement when the departure
from Henry’s law is taken into account. This can be explained in this way: the
departure from Henry’s law for nitrogen dissolved in water is much less than that
for oxygen, as shown in Figure 5.3. As we discussed in Section 2.5, departures from
Henry’s law always lower the nucleation rate, and therefore require a higher AP", for
nucleation to occur. Taking into account the departures from Henry’s law leads to
a greater increase in AP" for oxygen than for nitrogen, thus reducing the nucleation
threshold discrepancy between oxygen and nitrogen.

As we discussed in Section 2.5, accounting for the departure from Henry’s law
always leads to higher nucleation thresholds. However, this trend is obscured by
the calibration of the two constants in the nucleation rate equations. Due to different
calibration constants used in the calculations, we can not compare the absolute values

in the two tables, Table 5.2 and Table 5.3.

While the use of the generalized form of Henry’s law, Eq. (5.42) produces more

accurate results, the simpler model Eq. (5.43), which assumes ideal solution behavior,

95

 

0.004 --

 

 

g 0.003 ~
E
5 0.002 ‘
0
5
0 0.001 -
O T T l 1
0 5 10 15 20

Pressure (MPa)

 

+ Oxygen + Nitrogen

 

 

 

Figure 5.3: Saturation concentration for gas dissolved in water as a function of pres-
sure at 25°C

requires much less solubility data, and can still produce useful information on the
species dependence of nucleation thresholds.

Table 5.5 lists the values of the reversible work required to form a bubble of
critical size, and the energy of diffusive dissipation at nucleation limits at 25°C from
Eq. (5.42). It can be seen that wm, and q are of the same order for 02, N2, and Ar, but
not for He, for which q dominates. Those data again confirm the idea that the energy
of diffusive dissipation should not be neglected for nucleation of gas-supersaturated

liquids.

Table 5.5: The calculated values of wm, and q for nucleation in gas-supersaturated
water at 25°C

 

02 H8 N2 AT
wm, / [CT 16020 1061 8493 15648
q / kT 8299 23258 15826 8671

 

 

 

 

 

 

 

 

 

Table 5.2 indicates that, after calibration, the revised nucleation equation predicts

96

the values of AP” from Finkelstein & Tamir’s experimental data quite accurately, in-
troducing a strong species dependence to AP” through energy of diffusive dissipation.
Therefore the addition of a diffusive dissipation term to the nucleation rate equation
appears to be an important and physically realistic improvement to nucleation mod-

eling.

5.1.5 Temperature Dependence of Nucleation Thresholds of Gas-

S'upersaturated Liquids

Several experiments have been conducted to study the temperature dependence of su-
persaturation thresholds for gas dissolved in water, but the results are not conclusive.
Hemmingsen (1975) concluded that “over a wide range of temperatures, from 3.5 to
40°C, there was only a relatively small change in the cavitation properties (of N2 and
Ar dissolved in water). However, with decreasing temperatures over a narrow range
below 35°C, there was a significant and substantial increase in the supersaturation
required for the onset of any cavitation.” Finkelstein and Tamir (1985) observed that
“AP,, is completely independent of temperature in the range of 293 to 303K (for all the
tested gases, N2, He, Ne, and Ar).” In his technical notes, Bowers (1999) reported
the temperature variation of supersaturation thresholds for oxygen, nitrogen, hydro-
gen, etc. in water at 1 atm pressure. The measurements were made by generating
solutions of gases chemically. Bowers reported the bubble nucleation limit for oxygen
decreased from 0.15 to 0.10 mol/dm3 over the range of 283 to 298K. This corresponds
to a decrease of more than 4 MPa of supersaturation pressure. Bowers’s result appears
to contradict Finkelstein’s and Hemmingsen’s observations. One possible explanation
is that the discrepancy may be due to the different gases studied and the different
methods they adOpted to measure the supersaturation thresholds. Finkelstein and
Hemmingsen used the method of saturating water with gas at high pressures then de—

compressing, while Bowers used chemical reactions to generate highly supersaturated

97

 

solutions. The results may not be comparable.

Table 5.6 shows the values of solubility and diffusivity of oxygen dissolved in water
at different temperatures. Data at 20 and 30°C are taken from Wise and Houghton
(1966). The diffusivity at 25°C is estimated from Spalding (1963). Solubility at 25°C
is obtained from Fog (1990).

Table 5.6: The values of solubility and diffusivity of 02 dissolved in water

 

Oxygen
T(°C) Solubility a Diffusivity
Bunsen coef. D (x10’9 m2/s)

 

 

 

 

20 0.0323 2.3 .H
25 0.0295 2.63
30 0.0271 2.8

 

 

 

 

 

According to Eq. (5.43), as temperature increases from 20 to 30°C, the calculation
result of AP" for 02 dissolved in water increases from 13.3 to 13.6 MPa as shown
in Figure 5.4, a relatively small change. This result agrees with Finkelstein and
Hemmingsen’s experimental observations, but contradicts those of Bowers.

In Eq. (5.43), as temperature T increases, D increases and C or AC decreases.
This leads to a decrease of wrw, but an increase in q. Thus it would be possible that J
may decrease as temperature increases for some gases over some temperature ranges,
as indicated by Bowers’ experimental result that “the (supersaturation nucleation)
limit for hydrogen increases from 0.03 mol/dm3 at 290K to 0.08 mol/dm3 at 308K.”

This phenomenon can not be explained from the classical nucleation theory.

98

 

 

14‘ ‘_———t———i
A12‘
E105
'3: 6~
4.
2_
0 w T I I
0 10 20 3O 40
T(°C)

Figure 5.4: Dependence of AP" on temperature for 02 dissolved in water, calculated
from Eq. (5.43)

5.2 Prediction of Nucleation Thresholds of Gas-Liquid

Solutions in Small Capillary Tubes

There have been some attempts to understand the effects of confinement of gas-
supersaturated solutions within small volumes and their ability to raise nucleation
thresholds. Some experiments on nucleation in small capillary tubes (diameter less
than 100nm) were reported by Brereton, et al. (1998), and were described in Sec-

tion 4.3.

5.2.1 Dual Dependence Model to Account for Effects of Volumetric

Confinement on Nucleation

Brereton et al. developed a dual dependence model to predict nucleation rate, i.e.
the heterogeneous nucleation in capillary tubes depended on sufficient numbers of

both molecules of the vaporizing species and active cavities at the surface at which

99

nucleation took place. The equation for nucleation rate (s‘lm’2) is expressed as:

/ 3o —47ro 72¢
: 3 ‘t A Nae ive Nsur _ ____e_ 4
J cons z 1 7rm exp { 3kT } (5 6)

More details of the dual dependence model can be found in Section 4.3.

Although Eq. (5.46) could be calibrated to agree with experimental data, some of
assumptions in the original paper (Brereton et al. 1998) seem questionable after closer
inspection. For example, one assumption concerning the initial state of embryonic
nuclei compressed within surface cavities, and another assumption that the changes
in the radii of curvature of nuclei in crevices are neglected during decompression was
questionable. Also more accurate estimations of the critical radii of nucleation in
gas-liquid solutions could have been made. In addition, the effect of the dissolved gas
species on the observed threshold for nucleation (or, maximum dissolved-gas concen-

tration above which nucleation always took place) was not considered.

Based on the model of nucleation of gas-supersaturated liquids proposed in Sec-
tion 5.1 and Brereton’s dual dependence model, we propose an improved nucleation

rate equation to describe nucleation of gas-supersaturated liquids in small capillary

/ 3o —47ro 73¢ qu I,
J — COIlSt A Nactive leur % exp {—3k_f—— — 76?} (0.47)

where Nactive and N13,), are numbers of active sites and surface molecules per unit

tubes:

area, and q accounts for the dependence of nucleation of gas-supersaturated liquid on

diffusion, and takes the form (see page 91)

q = constant (D) (23%) ktT (5.48)

The critical radius (re) for bubble formation in a gas-liquid solution can be calculated

from Ward’s equation, Eq. (2.23). This model (Eq. 5.47) does not make any assump—

100

tion about the initial state of embryonic nuclei compressed within surface cavities, or
that the changes in the radii of curvature of nuclei in crevices are neglected during

decompression.

The total number of surface molecules n13“, is determined from the average spacing
s between molecules and the capillary-tube surface area. The ratio of these molecules

around the perimeter to those in the bulk liquid in the capillary is then

nlsur 43

 

 

 

= — 5.49
m d ( )
For oxygen dissolved in water,
nlsur 0o 4300
—’ ‘ = ‘ 5.50
NI,O2 d ( )
The concentration of oxygen in water is defined as
C = —"£- (5.51)
"1,1120
and can be estimated from Henry’s Law as
Pssat
C = ” 5.52
H < >

where H is Henry’s constant for the two species and P380, is the pressure under which

the saturated oxygen-water solution was prepared. Thus,

43 Pssa
”13211302 2 "(£20 d02 7t (5-53)

 

The average spacing between oxygen molecules can be expressed as

 

85120
- z 5.54
802 (Pssat/H)1/3 (O )

101

Substituting Eq. (5.54) into Eq. (5.53) yields the expression for the number of surface

molecules of the dissolved gas per unit area,

 

N lsur,02 : nlsur,02 /A
71.1,}120 43HQO Pssat 2/3

The nucleation rate J(s’1m’2) for 02-water solution in capillary tubes is

/ 30 —47ro r305
J : const A Nactive leur.02 g7; exP {—3-k—T—_ - %}

rm.) ‘11 /3o —47ro r34) qu
—— COHStlA ( T‘e ) leur,02 $exp{—3—k-T—— — Ff} (5.56)

where Mama, can be calculated from Eq. (5.55), Native can be estimated from

 

Eq. (4.10), and the critical radius (re) for bubble formation in a gas-liquid solution

can be calculated from Ward’s equation, Eq. (2.23).

Since the conditions for nucleation are quite insensitive to the rate at which bub—
bles are formed, similar values of P330, would be obtained even if J were varied by
several orders of magnitude, between, say, 1 and 1000. Eq. (5.56) can be solved by
setting J to a constant value and calibrating the shape factor and constl. Figure 5.5
shows the predicted supersaturation pressure thresholds of 02 dissolved in water from
Eq. (5.56), predicted over a range of different capillary tube diameters, as well as the
experimental data from Brereton, et al. (1998). It can be seen from the figure that
Eq. (5.56) can provide a reasonable agreement with the experimental data when suit-
able calibration values have been chosen for the two constants ((15 = 8.76 x 10‘5 and
J = 1). Thus Eq. (5.56) can correctly interpret the effect of volumetric confinement
on nucleation thresholds for gas-liquid solutions through the use of a nucleation rate
equation in which the pre—exponential scaling is jointly dependent on both the num-

ber of active surface cavities and the number of molecules of dissolved gas adjacent

102

to the surface.

Punt (MPa)
8 K3

 

 

 

0 20 4O 60 80 1 00
(NM M)

 

 

 

+calculation I experiment

 

Figure 5.5: Predicted and measured dependence of 02 supersaturation pressure on
capillary tube diameter using Eq. (5.56)

From the classical heterogeneous nucleation model, no volumetric confinement
effects are predicted. Thus according to the classical model, the supersaturation
thresholds for nucleation in different size of capillary tubes are almost the same, i.e.

a horizontal line would be obtained in Figure 5.5.

5.2.2 A Preliminary Model to Account for Efiects of Volumetric

Confinement on Nucleation

Although the dual dependence model can reproduce experimental data of nucleation
thresholds of oxygen dissolved in water in small capillary tubes with reasonable ac-

curacy, some questions remain:

103

o The model introduces the effect of volumetric confinement on nucleation thresh-
olds through pre—exponential factors in the nucleation rate equation (5.56). As
we mentioned before, the nucleation rate is quite insensitive to pre-exponential
factors. Therefore the predicted P330) vs. (1 curve from Eq. (5.56) usually has
less curvature than the experimental curve, as shown in Figure 5.5. Also the

calibrated shape factor is much smaller than expected.

Those questions suggest one might instead introduce a term inside the exponential
part, which depends on the diameter of the capillary tube. The following assumption
was proposed: the shape factor for nucleation is inversely proportional to the diameter
of capillary tube, i.e.

(brxa or qbz— (5.57)

where (to is a function of contact angle, and is independent on tube diameter. do is
usually in the order of 10‘8 or 10‘7 meters.

Some reasons which lead to this assumption include:

a As capillary tube diameter decreases to the order of one micrometer, it is pos-
sible that the average cavity size of the tube surface is reduced, which inhibits
nucleation and corresponds to large shape factors (this hypothesis is currently

under experimental investigation).

0 The maximum value of shape factor is unity, which corresponds to the occur-
rence of homogeneous nucleation. Experimental data show that as capillary
tube diameter tends to zero, the nucleation threshold approaches homogeneous

nucleation limits.

With these assumptions concerning shape factor, the heterogeneous nucleation

rate equation becomes

30 47m R2 (I (150
: N2/3,/__ - e _ _. — 5’8
J nmeXp{( 3kT kT) d} ( O)

104

 

Eq. (5.58) was used to calculate nucleation thresholds of 02 dissolved in water
in small capillary tubes. The results, shown in Figure 5.6, fit the experimental data

from Spears very well, indicating a dependence of nucleation threshold on volumetric

   

 

 

confinement.
20 -
15 r
E
E 10 -
l
5 l
0 . . r ,
0 20 40 60 80

d(u m)

 

 

 

+calculation In experiment

 

Figure 5.6: Predicted and measured dependence of 02 supersaturation pressure on
capillary tube diameter using Eq. (5.58)

Once calibration constants are obtained for oxygen dissolved in water in small
capillary tubes, the nucleation thresholds for other gas species can be readily calcu-
lated from Eq. (5.58) without further calibrations. The gas species dependence is
introduced through the activation energy for diffusion term, q, as discussed in the
previous section. Figure 5.7 shows the predicted nucleation thresholds of 02 and He
dissolved water in small capillary tubes with do = 1.2 x 10‘7 and J = 1, as well as
the experimental data from Spears. It can be seen from the figure that the shape

factor assumption (Eq. 5.57) together with an activation energy for diffusion term

105

can produce good agreement with experimental data.

100 -
§
3' 10 '1 I
O.
1 I I I I I I

 

 

0 20 4o 60 80 100 120
dlu m)

 

 

 

+He-cal. I He-exp. -o—02-cal. A OZ-exp.

 

Figure 5.7: Predicted and measured dependence of 02 and He supersaturation pres
sure on capillary tube diameter using Eq. (5.58)

As another test, we attempted to predict the superheat thresholds for nucleation
of pure water in small capillary tubes. The heterogeneous nucleation rate equation

for pure water in capillary tubes is

30 47ro R2 d
: 2/3 _ _ e _9
J N VnmeXp{( 3kT )d}

The experiment was described in detail in Brereton et al. (1998) and essentially in—

 

(5.59)

volved immersing small capillary tubes filled with pro-pressurized water in oil baths.
Figure 5.8 shows the comparison of predicted superheat nucleation thresholds and ex—
perimental data. The reasonable agreement provides some support for the assumption

that the shape factor for nucleation in small capillary tubes is inversely proportional

106

to tube diameter.

 

 

 

G l

3' 260 ~

2

a u

a

E

3 180 ~ I ¢ _

C I

3.

g 140 3

O

3

z 100 1 I I I 1
O 40 80 120 160 200

Tube diameter ([1. m)

 

+cal. I exp.

 

 

 

Figure 5.8: Predicted and measured superheat nucleation thresholds of pure water in
capillary tubes

The effect of volumetric confinement is really one of reduced surface area. It is
a result of heterogeneous nucleation depending on both the number of active surface
nucleation sites (cavities) and the number of molecules of the dissolved-gas species
at the surface, as well as on bubble shape factors and other surface-dependent ef-
fects. As surface area is reduced, the nucleation rate in a dual dependence model is
also reduced, possibly until homogeneous nucleation becomes more likely. This ex-
planation of raising the nucleation threshold by volumetric confinement is essentially
through surface area reduction, consistent with the nucleation model developed in

this dissertation.

107

CHAPTER 6
CONCLUDING REMARKS AND RECOMMENDATIONS

In this dissertation, we first presented the classical nucleation theory as originally
developed for pure liquid-vapor mixtures, and as extended for gases dissolved in
dissimilar liquids. The reason for its failure to predict the strong dependence of su-
persaturation nucleation thresholds on gas species was explained primarily as one
of neglecting diffusive dissipation. Finally nucleation in small capillary tubes was
addressed. Some modifications were made in this study to the classical nucleation
theory, by considering the irreversible effects of diffusion of gas molecules and volumet—
ric confinement, so it could reproduce the experimental data of nucleation thresholds
of gas-liquid solutions correctly.

New contributions of this study to nucleation theory are listed below:

0 The equations for dissipation of energy due to viscous and diffusive effects dur—
ing bubble growing processes were derived. The order of magnitude of these
dissipative terms was estimated. The significance of the energy of dissipation
due to diffusion to nucleation of gas-supersaturated liquids was pointed out.

(Section 5.1)

o A new model including diffusive dissipation was proposed to predict nucleation
thresholds of water supersaturated with different gas species, which is superior
to an earlier correlation equation proposed by Tamir & Finkelstein for the same
purpose. The model predictions are in good agreement with experimental data.

(Section 5.1)

o Ward’s equation of critical radius and Brereton’s dual dependence model have
been incorporated into nucleation theory to predict nucleation thresholds of

gas-supersaturated liquids in small capillary tubes. Calculation results indicate

108

that the new model can predict the effect of volumetric confinement on rais—
ing nucleation thresholds well, after an appropriate choice of two calibration

constants. (Section 5.2)

o The limitations of Henry’s law at high pressures were recognized. The effects of
the departure from Henry’s law on nucleation were also discussed. An improved
dissolved-gas concentration model was used in calculations of supersaturation

thresholds. (Sections 2.5, 5.1, and 5.2)

I A new model, which assumes the shape factor for nucleation in small capillary
tubes to be inversely proportional to the tube diameter, was proposed. It was
applied to predict the supersaturation nucleation thresholds of water supersatu-
rated with different gas species, as well as the superheat nucleation limits of pure
water in small capillary tubes. The predicted results were in good agreement

with experimental data. (Section 5.2.2)

Due to the limited time frame of this research, many questions and uncertainties

still remain, and need further study. Some of them are:

0 Some experiments are recommended to study the physical chemistry of surfaces
of small capillary tubes. Experimental data on how contact angle, average cav-
ity size, shape factor, and other parameters vary with tube diameter and com-
position would benefit the modeling of nucleation in confined volumes. Also
some special treatments to surfaces or surface coating, which changes the con-
tact angle and other properties, and therefore affect nucleation, are worthy of

investigation .

o For experiments measuring nucleation thresholds of liquids flowing through
small capillary tubes, the temperature of effluent should be measured by a

thermo—couple. The liquid temperature may rise by well over 20°C due to fric—

109

tion between the tube surface and the liquid, particularly when tube diameters

are of micron dimensions.

As an alternative approach to classical nucleation theory, the density functional
method assumes continuous changes rather than abrupt changes of fluid prop—
erties at the interface of nuclei, and thus avoids the macroscopic assumptions
of classical nucleation theory. Although the density functional method is not
yet able to make quantitative comparisons with experimental results, it shows
some potential for making useful predictions in nucleation study. Further study

should be conducted on this topic.

In recent years, the lattice Boltzmann method (LBM) has been applied to simu-
lations of single-phase and multiphase fluid flows (Chen, 1998). The fundamen-
tal idea of the LBM is to construct simplified kinetic models that incorporate
the essential physics of microscopic processes so that the macroscopic average
properties obey the desired macroscopic equations. It is possible that the LBM
may be applied to simulate the nucleation process in liquids, and improve un-

derstanding of non-equilibrium phase-change problems.

110

APPENDIX

111

APPENDIX A
THE COEFFICIENTS OF HENRY’S LAW FOR
NITROGEN, HYDROGEN, ARGON, AND HELIUM
DISSOLVED IN WATER

In Section 2.5, the applicability of Henry’s Law was discussed, together with im-
provements on it which accounts for departures from ideal solution behavior at high
pressures. Here more values of the coefficients of Henry’s Law for Nitrogen, Hydrogen,
Argon, and Helium dissolved in water, are listed.

Figure A.2 shows the variation of log(H) with pressure for N2 dissolved in water

at 25°C. The line which best fits these data for H of N2 dissolved in water at 25°C is

log(H) = 3.9684 + 3.0061 x 10’3 P (A.1)
where P is again in MPa.

1.E-02 -
8.E-03 *
6.E-03 r

4.E-03 ~

Concentration

2.E-O3 —

 

 

0.E+OO 1 . . .
o 50 100 150

Pressure (MPa)

Figure A.1: Saturation concentration of N2 in water

112

Figure A] shows the saturation concentration (in mole fraction) of N2 in water at
25°C, as a function of pressure. The experimental data were obtained from Krichevsky

(1935), as shown in Table A1

 

 

5 -
4.5 ~
9 /
‘5 4 _
2
3.5 -
3 T I I
0 50 100 150
Pressure (MPa)

Figure A.2: Variation of H with pressure for N2 dissolved in water

Table Al: The solubility of nitrogen in water at different pressures

 

 

 

 

 

 

 

 

 

Pressure Saturation Concentration
(MPa) (mole fraction)

2.5 2.800x10’4

5 5.420x 10-4

10 1.015 x10:3

20 1.812x 10-3

30 2.455x10‘3

50 3.558x10‘3

80 4.909x10‘3
100 5.720x10'3

 

 

 

 

Figure A.3 shows the saturation concentration (in mole fraction) of H2 in water at
25°C, as a function of pressure. The experimental data were obtained from Krichevsky

(1935), as shown in Table A2. Figure A.4 shows the varition of log(H) with pressure

113

 

 

 

2.0E-02 -

1.5E-02 A

   
 
 

1.0E-02 3

Concentration

5.0E-03 ~

 

 

ODE-+00

l T I

0 50 100 150
Pressure (MPa)

Figure A.3: Saturation concentration of H2 in water

 

 

 

5 1
4.5]
E
‘5 41
2 n- - ; ¢ ; ‘4.
3.5 3
3 T T I
0 50 100 150

Pressure (MPa)

Figure A.4: Variation of H with pressure for H2 dissolved in water

114

for H2 dissolved in water at 25°C. The best-fit line for H of H2 dissolved in water at

25°C is

log(H) : 3.7598 + 8.6111 x 10“4 P (A2)

where P is again in MPa.

 

Table A2: The solubility of hydrogen in water at different pressures

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Pressure Saturation Concentration ‘
(MPa) (mole fraction) _.
2.5 4.306x 10‘4 ‘
5 8.635x10‘4 .
10 1.709><10’3
20 3.351x10-3
40 6.390x10“3
60 9.252x10—3
80 1.191x10-2
100 1.425x 10-2
5.0E-03 —
4.0E-03 ~
:
8
g 3.0E-03 3
I:
8
c 2.0E-03 _
O
U
1.0E-03 ~
1
0.0E+OO I T I I l I
0 20 40 60 80 100

Pressure (MPa)

Figure A.5: Saturation concentration of He in water

Figure A.5 shows the saturation concentration (in mole fraction) of He in water at

25°C, as a function of pressure. The experimental data were obtained from “Solubility

115

Data Series, Volume 1, Helium and Neon _ Gas Solubilities.” Figure A.6 shows the

1D
0

log(H)
A

 

 

3 i i
0 50 1 00

Pressure (MPa)

Figure A.6: Variation of H with pressure for He dissolved in water

varition of log(H) with pressure for He dissolved in water at 25°C. The best-fit line

for H of He dissolved in water at 25°C is

log(H) = 4.2120 + 8.0044 x 10’4 P (A.3)

where P is again in MPa.

Figure A.7 shows the saturation concentration (in mole fraction) of Ar in water
at 25°C, as a function of pressure. The experimental data were obtained from Spears.
Figure A.8 shows the varition of log(H) with pressure for Ar dissolved in water at
25°C. The best-fit line for H of Ar dissolved in water at 25°C is

log(H) = 3.6706 + 2.7759 x 10-3 P (A.4)

where P is again in MPa.

116

1.2E-02 3
1.0E-02 3
8.0E-03 3
6.0E-03 3
4.0E-03 3
ZOE-03 3

0.0E+00 i r i i
0 20 40 60 80

Pressure (MPa)

Concentration

 

 

Figure A.7: Saturation concentration of Ar in water

 

 

 

5 I
4.5 3
EE
‘5 4 3
2 A 3 f : :4:
3.5 l
3 r i . ,
0 20 40 60 80

Pressure (MPa)

Figure A.8: Variation of H with pressure for Ar dissolved in water

117

 

 

BIBLIOGRAPHY

118

 

BIBLIOGRAPHY

[1] Adamson, A., 1990, “Physical Chemistry of Surfaces,” p. 364, fifth edition, John
Wiley & Sons, Inc.

[2] Bankoff, S. G., 1966, “Diffusion-Controlled Bubble Growth,” Adv. Chem. Eng,
6, p. 1.

[3] Becker, R., and Diiring, W., 1935, “The Kinetic Treatment of Nuclear Formation
in Supersaturated Vapors,” Ann. Phys., 24, pp. 719-752.

[4] Bird, R. B., Stewart, W. E., and Lightfoot, E. N., 1960, “Transport Phenomena,”
John Wiley & Sons, Inc.

[5] Blander, M., Hengstenberg, D., and Katz, J. L., 1971, “Bubble nucleation in
n—Pentane, n—Hexane, n-Pentane + Hexadecane mixtures, and Water,” J. Phys.
Chem., 75, p. 3613.

[6] Blander, M., and Katz, J. L., 1973, “The Role of Bubble Nucleation in Explosive
Boiling,” Fourteenth National Heat Transfer Conference, Atlanta, GA.

[7] Blander, M., and Katz, J. L., 1975, “Bubble Nucleation in Liquids,” AIChE J .,
21, p. 833.

[8] Bowers, P. G., Hofstetter, C., Letter, C. R., and Toomey, R., 1995, “Super-
saturation Limit for Homogeneous Nucleation of Oxygen Bubbles in Water at
Elevated Pressure: Superhenry’s Law,” J. Phys. Chem., 99, p. 9632.

[9] Bowers, P. G., Bar—Eli, K., and Noyes, R. M., 1996, “Unstable Supersaturated
Solutions of Cases in Liquids and Nucleation Theory,” J. Chem. Soc., Faraday
Trans, 92, p. 2843.

[10] Bowers, P. G., Hofstetter, C., Ngo, H. L., and Toomey, R. T., 1999, “Tempera-
ture Dependence of Bubble Nucleation Limits for Aqueous Solutions of Carbon
Dioxide, Hydrogen, and Oxygen,” J. Colloid Interf. Sci., 215, pp. 441-442.

[11] Brereton, G. J., Crilly, R. J., and Spears, J. R., 1998, “Nucleation in Small
Capillary Tubes,” Chemical Physics, 230, p. 253.

[12] Bretherton, F. P., 1960, “The Motion of Long Bubbles in Tubes,” J. Fluid Mech.,
10, p. 166.

[13] Buff, F. P., and Kirkwood, J. G., 1950, “Remarks on Surface Tension of Small
Droplets,” J. Chem. Phys., 18, p. 991.

[14] Buff, F. P., 1955, “Spherical Interface. II. l\«Iolecular Theory,” J. Chem. Phys,
23, p. 419.

119

[15] Burmeister, L. C., 1993, “Convective heat transfer,” 2nd edition, p. 113, John
Wiley & Sons, Inc.

[16] Cable, M., and Frade, J. R., 1987, “Diffusion Controlled Growth of Multi-
Component Gas Bubbles,” J. Materials Sci., 22, p. 919.

[17] Cahn, J ., and Hulliard, J ., 1958, “Free Energy of a Nonuniform System. I. Inter-
facial Free Energy,” J. Chem. Phys, 28, p. 258.

[18] Cahn, J., and Hulliard, J., 1959, “Free Energy of a Nonuniform System. III.
Nucleation in a Two-Component Incompressible Fluid,” J. Chem. Phys, 31,
p. 688.

[19] Carey, V. P., 1992, “Liquid-Vapor Phase Change Phenomena,” Hemisphere Pub-
lishing Co.

[20] Carey, V. P., 1999, “Statistical Thermodynamics and Microscale Thermo-
physics,” Cambridge University Press.

[21] Carr, M. W., Hillman, A. R., and Lubetkin, S. D., 1995, “Nucleation Rate
Dispersion in Bubble Evolution Kinetics,” J. Colloid Interf. Sci., 169, pp. 135-
142.

[22] Carslaw, H. S., 1945, “Introduction to the Mathematical Theory of the Conduc-
tion of Heat in Solids,” p. 158, Dover Publications.

[23] Chen, R, and Chen, S. H., 1985, “Diffusion of Slightly Soluble Gases in Liquids
Measurement and Correlation with Implications on Liquid Structures,” Chem.
Eng. Sci., 40, p. 1735.

[24] Chen, S., and Doolen, G. D., 1998, “Lattice Boltzmann Method for Fluid Flows,”
Annu. Rev. Fluid Mech, pp. 329-364.

[25] Clever, H. L., 1979, “Solubility Data Series, Volume 1, Helium and Neon — Gas
Solubilities,” Pergamon Press.

[26] Cole, R., 1974, “Boiling Nucleation,” Adv. Heat Transfer, 10, p. 85.

[27] Defay, R., Prigogine, I., Bellemans, A., and Everett, D. H., 1966, “Surface Ten-
sion and Adsorption,” John Wiley 81. Sons, Inc.

[28] Dergarabedian, P., 1953, “The Rate of Growth of Vapor Bubbles in Superheated
Water,” J. Appl. Mechanics, p. 953.

[29] Epstein, P. S., and Plesset, M. S., 1950, “On the Stability of Gas Bubbles in
Liquid-Gas Solutions,” J. Chem. Phys, 18, p. 1505.

[30] Ferziger, J. H., and Peric, M., 1996, “Computational Methods for Fluid Dynam-
ics,” Springer—Verlag.

120

[31] Finkelstein, Y., and Tamir, A., 1985, “Formation of Gas Bubbles in Supersatu-
raed Solutions of Gases in Water,” AIChE Journal, 31, p. 1409.

[32] Fisher, J. C., 1948,“The Fracture of Liquids,” J. Appl. Phys, 19. p. 1062.

[33] Fogg, P. and Gerrard, W., 1990, “Solubility of Gases in Liquids: a Critical
Evaluation of Gas/ Liquid Systems in Theory and Practice,” John Wiley & Sons
Ltd.

[34] Forest, T. W., and Ward, C. A., 1977, “Effect of a Dissolved Gas on the Homo—
geneous Nucleation Pressure of 3 Liquid,” J. Chem. Phys, 66, p. 2322.

[35] Forest, T. W., and Ward, C. A., 1978, “Homogeneous Nucleation of Bubbles in
Solutions at Pressures above the Vapor Pressure of the Pure Liquid,” J. Chem.
Phys, 69, p. 2221.

[36] Frenkel, J ., 1939, “A General Theory of Heterophase Fluctuations and Pretran-
sition Phenomena,” J. Chem. Phys, 7, p. 538.

[37] Frenkel, J., 1946, “Kinetic Theory of Liquids,” Chap. VII, Dover Publications,
New York.

[38] Frisch, J. F., 1957, “Time Lag in Nucleation,” J. Chem. Phys, 27, p. 90.

[39] Geankoplis, C. J., 1983, “Transport Process: Momentum, Heat, and Mass,”
Allyn and Bacon, Inc., p. 391.

[40] Gyftopoulos, E. P., and Beretta, G. P., 1991, “Thermodynamics Foundations
and Applications,” Macmillan Publishing Co.

[41] Hemmingsen, E. A., 1970, “Supersaturation of Gases in Water: Absence of Cav-
itation on Decompression from High Pressures,” Science, 167, pp. 1493-1494.

[42] Hemmingsen, E. A., 1975, “Cavitation in Gas-Supersaturated Solutions,” J.
Appl. Phys, 46, p. 213.

[43] Hemmingsen, E. A., 1977, “Spontaneous Formation of Bubbles in Gas—
Supersaturated Water,” Nature, 267, pp. 141-142.

[44] Hirth, J. P., and Pound, G. M., 1963, “Condensation and Evaporation, Nucle—
ation and Growth Kinetics,” Progr. Mater. Sci., 11, pp. 1-167.

[45] Holman, J. P., 1990, “Heat Transfer,” 7th edition, PP 301, PP 349, McGraw-Hill.

[46] Huntington, H. B., and Seitz F ., 1942, “Mechanism for Self-Diffusion in Metallic
Copper,” Physical Review, 61, p. 315.

[47] Huntington, H. B., and Seitz F., 1942, “Self-Consistent Treatment of the Vacancy
Mechanism for Metallic Diffusion,” Physical Review, 61, p. 325.

121

[48] Jenssen, L., and Warmoeskerken, M., 1987, “Transport Phenomena Data Com-
panion,” Edward Arnold Ltd.

[49] Jones, S. F., Evans, G. M., and Galvin, K. P., 1999, “Bubble Nucleation from
Gas Cavities .. a Review,” Adv. Colloid Interf. Sci., 80, p. 27.

[50] Jost, W., 1960, “Diffusion in Solids, Liquids, Gases,” third printing with adden-
dum, Academic Press.

[51] Kantrowitz, A., 1951, “Nucleation in Very Rapid Vapor Expansions,” J. Chem.
Phys, 19, p. 1097.

[52] Katz, J. L., and Blander, M., 1973, “Condensation and Boiling: Corrections to
Homogeneous Nucleation Theory for Nonideal Gases,” J. Colloid Interf. Sci., 42,
p. 496.

[53] Kirwood, J. G., and Buff, F. P., 1949, “The Statistical Mechanical Theory of
Surface Tension,” J. Chem. Phys, 17, p. 338.

[54] Krichevsky, I. R., and Kasarnovsky, J. S., 1935, “Thermodynamical Calculations
of Solubilities of Nitrogen and Hydrogen in Water at High Pressures,” J. Am.
Chem. Soc., 57, p. 2168.

[55] Krieger, I. M., Mulholland, G. W., and Dickey, C. S., 1967, “Diffusion Coeffi-
cients for Cases in Liquids from the Rates of Solution of Small Gas Bubbles,” J.
Phys. Chem., 71, p. 1123.

[56] Landau, L. D., and Lifshitz, E. M., 1958, “Statistical Physics,” Addison-Wesley
Pub. Co., p. 275.

[57] Landau, L. D., and Lifshitz, E. M., 1987, “Fluid Mechanics,” 2nd edition, Perg-
amon Press.

[58] LaMer, V., and Pound, G., 1949, “Surface Tension of Small Droplets from Volmer
and F lood’s Nucleation Data,” J. Chem. Phys, 17, p. 1337.

[59] Liebermann, L., 1957, “Air Bubbles in Water,” J. Appl. Phys, 28, p. 205.

[60] Lorenz, J. J ., Mikic, B. B., and Rohsenow, W. M., 1974, Proc. Fifth Int. Heat
Transfer Conf., 4, p. 35

[61] Lothe, J ., and Pound, G. M., 1962, “Reconsiderations of Nucleation Theory,” J.
Chem. Phys, 36, p. 2080.

[62] Lubetkin, S. D., 1988, “The Nucleation of Bubbles in Supersaturaed Solutions,”
J. Colloid Interf. Sci., 26, p. 611.

[63] Lubetkin, S. D., 1995, “The Fundamentals of Bubble Evolution,” Chem. Soc.
Rev., 24, p. 243.

122

[64] Lubetkin, S. D., and Akhtar, M., 1996, “The Variation of Surface Tension and
Contact Angle under Applied Pressure of Dissolved Gases, and the Effects of
these Changes on the Rate of Bubble Nucleation,” J. Colloid Interf. Sci., 180,
p. 43.

[65] Massoudi, R., and King, A. D., 1974, “Effect of Pressure on the Surface Tension
of Water. Adsorption of Low Molecular Weight Gases on Water at 25 °C,” J.
Phys. Chem., 78, p. 2262.

[66] Merte, H., 1973, “Condensation Heat Transfer,” Adv. Heat Transfer, 9, p. 181.

[67] Oliver, J. F ., 1977, “Resistance to Spreading of Liquids by Sharp Edges,” J.
Colloid Interf. Sci., 59, p. 568.

[68] Oriani, R. A., 1962, “Emendations to Nucleation Theory and the Homogeneous
Nucleation of Water from the Vapor,” J. Chem. Phys, 58, p. 2082.

[69] Ozisik, N., 1980, “Heat Conduction,” John Wiley & Sons, Inc.

[70] Probstein, R. F., 1951, “Time Lag in the Self-Nucleation of a Supersaturated
Vapor,” J. Chem. Phys, 19, p. 619.

[71] Reid, R., Prausnitz, J ., and Sherwood, T., 1977, “The Properties of Gases and
Liquids,” third edition, McGraw-Hill Book Co.

[72] Reiss, H., 1952, “The Statistical Mechanical Theory of Irreversible Condensation.
I,” J. Chem. Phys, 20, p. 1216.

[73] Rubin, M., and Noyes, R., 1987, “Measurements of Critical Supersaturation for
Homogeneous Nucleation of Bubbles,” J. Phys. Chem., 91, p. 4193.

[74] Rubin, M., and Noyes, R., 1992, “Thresholds for Nucleation of Bubbles of N2 in
Various Solvents,” J. Phys. Chem., 96, p. 993.

[75] Scriven, L. E., 1959, “On the Dynamics of Phase Growth,” Chem. Eng. Sci., 10,
p. 1.

[76] Sherwood, T. K., Pigford, R. L., and Wilke, C. R., 1975, “Mass Transfer,”
McGraw-Hill, Inc.

[77] Skripov, V. P., 1974, “Metastable Liquids,” John Wiley & Sons, Inc.

[78] Smoluchowski, R., Mayer, J. E., and Weyl, W. A., 1951, “Phase Transformation
in Solids”, Symposium held at Cornell University, Aug. 23—26, 1948. Sponsored
by the committee on solids division of physical sciences, the national research
council. John Wiley & Sons Inc.

[79] Spalding, D. B., 1963, “Convective Mass Tiansfer,” h‘chraw-Hill, pp. 139-143.

123

[80] Talanquer, V., and Oxtoby, D., 1995, “Nucleation of Bubbles in Binary Fluids,”
J. Chem. Phys, 102, p. 2156.

[81] Tucker, A., and Ward, C. A., 1975, “Critical State of Bubbles in Liquid-Gas
Solution,” J. Appl. Phys, 46, p. 4801.

[82] Turnbull, D., and Fisher, J. C., 1949, “Rate of Nucleation in Condensed Sys-
tems,” J. Chem. Phys, 17, p. 71.

[83] Volmer, M., and Weber, A., 1926, “Nucleus Formation in Supersaturated Sys-
tems,” Z. Phys. Chem., 119, p. 277.

[84] Wakeshima, H., 1954, “Time Lag in the Self-Nucleation,” J. Chem. Phys, 22,
p. 1614.

[85] Ward, C. A., Balakrishnan, A., and Hooper, F. C., 1970, “On the Thermody-
namics of Nucleation in Weak Gas-Liquid Solutions,” Trans. ASME, 92, p. 695.

[86] Ward, C. A., Tikuisis, P., and Venter, R. D., 1982, “Stability of Bubbles in a
Closed Volume of Liquid-Gas Solution,” J. Appl. Phys, 53, p. 6076.

[87] Ward, C. A., Johnson, W. R., Venter, R. D., and Ho, S., 1983, “Heterogeneous
Bubble Nucleation and Conditions for Growth in a Liquid-Gas System of Con-
stant Mass and Volume,” J. Appl. Phys, 54, p. 1833.

[88] Ward, C. A., and Levart, E., 1984, “Conditions for Stability of Bubble Nuclei in
Solid Surfaces Contacting a Liquid-Gas Solution,” J. Appl. Phys, 56, p. 491.

[89] Wilt, P. M., 1985, “Nucleation Rates and Bubble Stability in Water-Carbon
Dioxide Solutions,” J. Colloid Interf. Sci., 112, p. 530.

[90] Wise, D. L., and Houghton, G., 1966, “The Diffusion Coefficients of Ten Soluble
Gases in Water at 10-60°C,” Chem. Eng. Sci., 21, pp. 999-1010.

[91] Worden, R. M., and Bredwell, M. D., 1998, “Mass-Transfer Properties of Mi-
crobubbles. 2. Analysis Using a Dynamic Model,” Biotechnol. Prog., 14, p. 39.

124