Investigation of Phase Equilibrium In Magnesium Alloys through Diffusion Couples
AN INVESTIGATION OF THE STABLE EQUILIBRIUM … · An Investigation of the Stable Equilibrium State...
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AN INVESTIGATION OF THE STABLE EQUILIBRIUM STATE OF A BUBBLE IN A A thesis submined in conformity with the requirements for the degree of Master of Applied Science and Engineering Graduate Depariment of Mechanical and Industriai Engineering University of Toronto O Copyright AIi Keshavarz 1998
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Transcript of AN INVESTIGATION OF THE STABLE EQUILIBRIUM … · An Investigation of the Stable Equilibrium State...
AN INVESTIGATION OF THE STABLE EQUILIBRIUM STATE OF A BUBBLE IN A
A thesis submined in conformity with the requirements for the degree of Master of Applied Science and Engineering
Graduate Depariment of Mechanical and Industriai Engineering University of Toronto
O Copyright AIi Keshavarz 1998
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To Aida and Tannaz
ABSTRACT
An Investigation of the Stable Equilibrium State of a Bubble in a Finite Volume of a Water-Nitmgen Solution
Ali Keshavarz
A thesis submitted for the degree of Master of Applied Science and Engineering, Graduate Department of Mechanical and Industrial Engineering, University of Toronto,
1998
A theory has been previously developed for predictiag the size of a bubble immersed
in a closed volume of a Liquid-gas solution. An experimental apparatus has been designed
and constructed to investigate the theory. The apparatus permits the measurement and con-
trol of the temperature and the pressure in the system containing a water and nitrogen
solution. The theory was found to accurately predict the radius of the bubble in the stable
equilibrium. A method was also presented to rneasure the total number of moles of the sys-
tem components, i.e. water and nitrogen. The results obtained from this method were corn-
pared with those obtaùied with earlier methods, e.g. of the Van Slyke apparatus. It was
found that this method yields more accurate measurements than previous ones.
The Laplace equation has been traditionally used to predict the equilibrium size of a
bubble immersed in a solution by neglecting the changes in the thermodynamic properties
of the solution surrounding the bubble. If the changes in the properties of the solution sur-
rounding the bubble are not taken into account, the Laplace equation has been found to
lead to disagreement with the observations.
ACKNOWLEDGMENTS
1 h t extend my sincere gratinide to my supervisor, Dr. Charles A. Wmd, for suggest-
h g this study and for his continual support throughout it's duration. The oppominity of
investigating a hindamental thermodyuamic issue h m both theoretical and expenmental
perspectives with Dr. Ward has k e n a wholly rewarding experience.
1 am grateful to Dainis Stanga for many Uuminatiog technical discwion. m d y dur-
h g the construction of the experimental apparatus. 1 also thank Barbara Ward for her gen-
erous support and advice during the course of the present work.
My th& go to ail my friends for their support and encouragement. In particular, I
would iike to thank my lab mates and fnends: Gang Fang, Michael Sasges, Aibeno Nieto,
Payam Rahimi, Payam Tangestanim and Boja. Popovic. 1 am also indebted to the faculty
and staff in the department of Mechanical and Industrial Engineering at the University of
Toronto.
Many thanks go to the Naniral Sciences and Engineering Research Council of Canada.
the University of Toronto and to LACEC Energy System Inc. for the financial support
throughout the duration of this work.
TABLE OF CONTENTS
1.1 Background .... ...... ... ... ...... . .... .. ....UtlUtl...... . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation of Present Snidy ....................................................................... ........... -5
1.3 Scope of the Thesis ............... ............ ......... .... .... ........................... . .... .. ............... .6
2.1 htroduc tion . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . ..., . . . . . . . . . .. .. .. . . . ........-. .. . . . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . -. . -8
2.2 Availability and Irreversibility . . . ... . ..... . . . ... ....... . . ... ... ... ... ..*.... .. . .. . .-. . . .. . . ..... . . ... . . .. . . . -9
2.3 Concept of Thermodynamic Potential for a Homogeneous System ..................... 1 1
2.3.1 Helmholtz Potential .... .................... .......... ,,,... ..................... . . . 1 1
2.3.2 Gibbs Potential ................... .....,........... .... .... . ..... ....-... ........... ...... .-.. ...... ....... 12
2.4 Concept of Themodynarnic Potential, B, for a Heterogeneous System and Its Der- ivation ...... . ...... ...... ................. .................. . ................... ......... . . . 12
2.5 Necessary Conditions For Equilibrium . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6
2.6 Concept of Equilibrium Radius ....................... .. ................ . . . . . . . .............. 18
2.7 Henry's Law ............................................................................................. 18
2.8 Equilibrium State of a Gas Bubble in a Finite Volume of a Liquid-Gas Solution and Its Expressions ........................... .. ................... .................. ........ . .... .......... 18
2.9 Expression for the Themodynamic Potential, B, Near Equilibrium .................... 23
2.10 Possibility of Equilibrium States for a Single Bubble in a Closed Volume of a Liq- uid .,. ... . .-. . .. . ... ... ... ... . .. . ....... .. ..-. ...-....... .-.....-......... .*......... ...-.. .............. ..-. ... .. . .. .. ... ..26
2.1 1 Theoretical Rediction of the Radius of a Single Bubble in a Finite Volume of a
Liquid-Gas Solution. Due to a Change in the Liquid Pressure by the Themodynam- ic Potential cuves .................................... ., ..,.,,... . 28
2.12 Figures ............................................................................................................. 3 0
CEWPTER 3: EXPERIMENTAL INVESTlGATION ..a.ew.e....).e.H.. .......................... 35
3 . 1 Introduction .......................................................................................................... 35
.............................................................. 3.2 Experimental Design ...................... .. 3 6
3.2.1 Water Reparation Unit ........................................................................... 3 6
3 .2.2 Bubble Preparation Unit ............................................................................ -37 3.2.3 Liquid Pressure Control Unit ...................................................................... 38
3.2.4 Data Acquisition Unit ......................... ... ............................................... 38
3.2.4.1 Temperature .............................................................................. 39
3.2.4.2 Pressure ................................................................................... 39
3.2.6 Experimentai Vesse1 ................................................................................ 41
3.3 Cleaning hocedure and Contamination Monitoring Method .............................. 42
3.4 Experimental Procedure ....................................................................................... 43
3.5 Method Used to Measure Total Gas and Liquid Contents of a Closed Volume of a Liquid-Gas Solution ............................................................................................ 45
3.6 Experimental Results ........................................................................................... 48
3 .6.1 First Experiment .......................... ... .................................................... -49 3-62 Second Expriment .................................................................................... -50
3 .6.3 Third Expriment ............................ ... ................................................. -51
3.7 Error Andysis ....................................................................................................... 52
3.8 Uncertainty Due to Temperature and Pressure Ructuations ................................ 53 3-9 Tables .................................................................................................................... 57
3.10 Figures .......................................................................................................... 65
C-R 4: SUMMARY AND CONCLUSIONS .................................................... 75
Appendix B: Capillarp Method for Surface Tension Masurement ............. ........ .... .80
vii
LIST OF TABLES
CHAPTER 3: EXPERIMENTAL INVESTIGATION ....m...m..................................... 3 5
TABLE 3.1. Surface tension measurements of water during the experiment as an indication for possible system contamination .................................................................................... 57
TABLE 3.2. Stable equilibrium bubble sizes for experiment number 1 ......................... 58
...... TABLE 3.3. Total number of moles of water and nitrogen in experirnent number 1 58
TABLE 3.4. Predicted and measured equiïbrium bubble radü for experiment number 1 W
TABLE 3.5. Stable equilibrium bubble sizes for expriment number 2 .......................... 59
TABLE 3.6. Total number of moles of water and nitrogen in expriment number 2 ...... 59
TABLE 3.7. Predicted and measured equilibrium bubble radii for experiment number 2 59
TABLE 3.8. Stable equiiibrium bubble sizes for experiment number 3 .......................... 60
TABLE 3.9. Total number of moles of water and nitrogen in experiment number 3 ...... 60
.......... TABLE 3.10. Predicted and rneasured equilibrium bubble radii for experiment 3 60
TABLE 3.1 1 . Variation in R, due to temperature fluctuation for experimeat number 1 . 6 1
TABLE 3.12. Variation in R, due to pressure fluctuation for experimeot number 1 ....... 61
TABLE 3.13. Variation in R, due to temperanire fluctuation for experiment nurnber 2 -62
....... TABLE 3.14. Variation in R, due to pressure fluctuation for experiment number 2 62
TABLE 3.15. Variation in R, due to temperature fluctuation for experiment number 3 -63
TABLE 3.16. Variation in R, due to pressure fluctuation for experiment number 3 ....... 63
TABLE 3.17. A cornparison between the predicted and the measured equilibrium bubble radii .................................................................................................................................. .64
CHAPTER 4: SUMMARY AND CONCLUSIONS .. .......................... ...o............. 7s
AppendSr A: Pressure Transducer Calibration ..,U.......~.......~w..~~H~.~~. ...*.. 77
TABLE A . 1 . The pressure transducer calibration measurements ..................................... 78
Appendix C: Propertïes of Nitmgen .............. ..,.........~.~..m...e~*..*a~.~~~~m.*.... 82
TABLE C . 1 . Solubility of the nitrogen in the water and Henry's constant at different tem- peratures ...................... .......................... ........................................................................... -82
Appendix D: Pmperties of Water .,..,.,, .....,.,..,...**c..*..*.......e.....*.....m.... ..... 83 TABLE D . 1 . Saturation pressure and surface tension of water ...................................... 83
LIST OF FIGURES
FIGURE 2.1. A schematic of the system under consideration. Note that the piston is free .................................................................. to move. ............................ .,.. -30
FIGURE 2.2. Plot of equilibrium radius versus bubble radius for a single bubble imuiersed in a water-nitrogen solution at 298.35 K and 154.5 kPa Total water and nitmgen gas contents are 1.984 mol and 35.233 pmol, respectively. Possible equilibrium states labelled and are 0.12 1 mm and 0.979 mm, respectively. corresponds to an unstable equilibrium and to a stable one. -3 1
FIGURE 2.3. A conceptual plot of the themodynamic potential versus bubble radius. The potential is a maximum at the unstable equilibrium and a minimum at the stable equïiibrium. The s m d e r equilibrium bubble radius is of the unstable and the larger bubble radius is of the stable equilibnum. ............................. 32
FIGURE 2.4. Variation of the equilibrium radius with the radius of a single bubble immersed in a water-nitrogen solution at 298.35 K and 154.5 kPa. Water content is 1.984 mol. Total nitrogen gas content for profiles labeled A, B and C are 35.233,35.056 and 35.000 p o l , respectively. There are two possible equilibrium states for case A while non is possible for case C. Case B shows the minimum amount that could possibly give rise to an equilibrium state. 33
FIGURE 2.5. A theoretical prediction of the size of a single bubble immersed in a water- nitrogen solution at 298.35 K. due to change in liquid pressure form 154 kPa, profde A to 154.5 kPa, profile B. Total water and nitmgen gas contents are 1.984 mol and 35.233 p o l , respectively. ..................... ... ...................... 34
FIGURE 3.1. Schematic diagram of the experimental apparatus. ..................... .. ...... 65
FIGURE 3.2. Schematic diagram of the water preparation unit. .................................. 66
FIGURE 3.3. Schematic diagram of the bubble preparation unit. ............................... 67
FIGURE 3.4. Schematic diagram of the iiquid pressure control unit. ........................ 6 8
FIGURE 3.5. Schematic diagram of the data acquisition unit. ................................... 69
FIGURE 3.6. (A) Bubble image as caphued by the processing unit and (B) the same image with the edge hding filter. Bubble diameter is 223 pixels which equais
FIGURE 3.7. Schematic diagram of the experimental vesse1 and its accessories ........ 71
FIGURE 3.8. Photopph of the experimentai vesse1 .................................................. 72
FIGURE 3 .9 . Photograph of the experimental apparatus ......................................... 73
FIGURE 3.10. Cornparison between the predicted and the measured stable equilibrium .................................................................................................. bubble radii -74
CHAP'X'ER 4: SUMMARY AND CONCLUSIONS .................................................... 75
Appendix A: Pressure Transducer Cdibration ,,., ................................ 77 FIGURE A . 1 . The pressure Transducer calibration cume ........................................... 79
AppendiP B: Capülary Method for Surface Tension Measurement ........................ 080 Appendnr C: Properties of Nitrogen ....*m..,,,.. ...~.e~....~...e.~..~~....~.m.~...~ 82 Appendix D: Properties of Water m . ~ . - - ~ ~ e - ~ ~ m m ~ - o - ~ ~ ~ ~ œ - - m œ - e - ~ - - ~ œ ~ - - m - - m - ~ ~ - ~ - - - - e m - - - - 8 3
AL' surface area of the liquid-gas surface phase
A" surface area of the solid-liquid surface phase
AO surface area of the reservoir surface phase
B thermodynamic potential for a heterogeneous system
Bo themodynamic potential of the reference state
C number of the components in the system
C, a constant
C2 a constant
C, a constant
C4 a constant
C5 a constant
Cb a constant
O, a property at state one
D, a property at state two
F Helmholtz potentiai
F' Helmholtz potential for the gas phase
fiG Helmhottz potential for the liquid-gas surface phase
G Gibbs potential
Gibbs potential for the liquid phase
g acceleration of the gravity
h capillary rise
H enthalpy
I irreversibility
KH Henry's constant
N, total number of mole of the solvent
nmber of mole of the solvent in the gas phase
N I number of mole of the solvent in the liquid phase
N l LG nurnber of mole of the solvent in the liquid-gas surface phase
N, total number of mole of the solute
N , ~ number of mole of the solute in the gas phase
N~~ number of mole of the solute in the liquid phase
L N, number of mole of the solute in the solvent saturated with the solute across a
fiat surface
LG NZ number of mole of the solute in the liquid-gas surface phase
Nf> number of mole of cornponent "i" in the reservoir bulle phase
N: number of mole of component "i" in the reservoir surface phase
Af number of mole of cornponent "i" in the gas phase
N: nurnber of mole of component "i" in the liquid phase
LG Ni number of mole of cornponent ''Y in the Liquid-gas surface phase
Ni number of mole of component '3"
N, number of mole of component "i'
P pressure
P , partial pressure of the solvent in the gas phase
G P, partiai pressure of the solute in the gas phase
pb pressure of the reservoir bulk phase
xiii
pG pressure inside the gas phase
L P pressure inside the liquid phase
L P , pressure inside the liquid phase at state one
L P, pressure inside the liquid phase at state two
P , saturation pressure of the solvent
p number of the phase in the system
R radius of the bubble
R universai gas constant
R, equilibrium radius of the bubble
R, equilibrium radius of the bubble at state one 1
R equüibrium radius of the bubble at state two
( Rc)mc t z~~red rneasured equilibrium radius of the bubble
( Rc)predicred P redicted equilibrium radius of the bubble
Ri,, iimiting equilibrium radius of the bubble
S entropy
sb entropy of the bulk phase
s0 entropy of the surface phase
T temperature
Ib temperature of the bulk phase
T' temperature of the gas phase
e temperature of the iiquid phase
TG temperature of the liquid-gas surface phase
TFL temperahlre of the solid-liquid surface phase
TO temperature of the surface phase
xiv
U intemal energy
d total energy of the bulk phase
LIG total energy of the gas phase
uL total energy of the liquid phase
uLG total energy of the iiquid-gas surface phase
#L total energy of the solid-liquid surface phase
LJa total energy of the surface phase
V volume
V velocity
vb total volume of the reservoir bulk phase
V" total volume of the gas phase
vL totai volume of the liquid phase
v variance
v, saturation volume of the pure solvent
WC, work crosshg the control volume
W,,, reversible work
x l mole fraction of the solvent in the gas phase
xzG mole fraction of the solute in the gas phase
Z elevation
y surface tension of the liquid
surface tension of the liquid-gas surface phase
surface tension of the resenoir surface phase
AA changes in area A
AD changes in property D
AP changes in pressure P
AS changes in entropy S
AU changes in internai energy U
A V ~ changes in volume of the reservoir bulk phase vb
AV' changes in volume of the gas phase vG
hvL changes in volume of the liquid phase vL
Aû changes in property D
E uncertainty
&cxpcriment U~certahty for experiment nurnber one
E~~~~~~~~~~~ uncertainty for experiment number two
E u p c r i m e n t ~ uncertainty for expairnent number three
E~ uncertainty of variable "i"
uncertainty of the predicted equiiibrium radius E(RC)prcdicied
uncertainty of the measured equilibnum radius
- py chemicai potential of solvent in the gas phase
Cr: chemical potential of solvent in the liquid phase
chemical potentiai of the solvent in the Liquid-gas surface phase
OG pi chemical potentiai of pure solvent in the gas phase
OL p chernical potential of pure solvent in the liquid phase
L pi, chernicai potential of the solvent in the liquid phase reference state
pf chernical potential of solute in the gas phase
chemical potential of solute in the iiquid phase
pgG chemical potential of the solute in the iiquid-gas surface phase
pi.ZG chemical potentiai of pure soiute in the gas phase
p2L chemical potential of pure solute in the liquid phase
chernical potential of the solute in the liquid phase reference state 2o
pi chemicai potential of component 'Y
b pi chemical potential of component "5" in the buik phase
G pi chemical potentiai of component "i" in the gas phase
chemical potentid of component "i" in the liquid phase
LG pi chernical potential of component "i" in the îiquid-gas surface phase
p y chemical potential of component "i" in the soiid-liquid surface phase
p: chemical potential of component "i" in the surface phase
p density of water
Superscripts
O refen to a property of a pure substance
b refers a property to a b u k phase
G refen a property to a gas phase
L refers a property to a liquid phase
LG refers a property to a iiquid-gas surface phase
SL refen a property to a solid-liquid surface phase
a refen a property to a surface phase
subscripts
O refers to a reference state
1 refers fo solvent
2 refers to solute
2s refers to saturated solute
c refers to equilibrium state
i refen to compownt '5"
j refers to component "j"
lim refers to limiting value
H refers to henry's
= refers to sanuated properties
CHAPTER 1: INTRODUCTION
1.1 Background The behavior of gas bubbles has been studied extensively since 1870, when for the
first iime Hoppe suggested that the sudden death of the compressed air worken was due to
the formation of gas bubbles in their blood and tissues. in 1878. Paul Bert confirmed the
existence of gas bubbles within the animal tissues following a rapid decompression. He
then stated that if enough gas bubbles are present, they could kill or paralyze the animals.
These hdings have formed the bais for studies of decompression sichess. ' The theoretical predictions of Rofessor J. Willard Gibbs of Yale University were a
nvning point in the studies of the gas bubbles. In 1875, his derivation of the Gibbs phare
rule was ranked among the highly significant contributions to physical science. He consid-
ered a system without auy chernical reactions, and then he wrote his phase d e for such a
system as
where p is the number of the phase in the system; v is the variance and C is the number
of the components in the system. He explained the term variance as the number of inten-
sive properties that were required to completely specify the state of the system.I3 For
instance, lets consider a two phase mixture of water and nitrogen. For this system, C, the
number of the components equals 2, the number of the phases p is 2, therefore the vari-
ance v is also 2. This means that a total of 2 intensive properties is required to completely
fuc the state of the system.
Moreover, in 1878, by considering an equiiibrium size gas bubble irnmened in a
Liquid with an W t e extent, Gibbs predicted that the bubble is in an unstable equilibrium.
1
He also predicted that, any bubble smaller than eqdibrium size mut dissolve and larger
than that must gow? The expression for the equilibrizun radius given by Ward et al. in
1970, did in fact simpMy the investigation on the gas bubble. ' ' They coosidered a closed
system containing a gas vapor nucleus surrounded by a liquid at a constant temperature.
Furthemore, they added another constraint by considering a weak solution of the gas dis-
solved in the liquid. The beauty of this work was that they expressed the equilibnum
radius of a sphencal nucleus purely in terms of properties of the solution. if we assume
that the gas vapor mixture inside the bubble behaves as an ideal gas mixture, the expres-
sion for the equilibriurn bubble radius may then be written as
L wbere q = Exp[$(P - PJ - $1 ; Rc is the equifibrium radius of a spherical gas N:
bubble; P , and v, are the saturation pressure and volume of the pure liquid, respectively:
L L y is the surface tension of the liquid; pL is the pressure inside the liquid; Nz and Nz, are
the number of mole of the solute in the liquid phase and in the liquid saturated with the gas
across a flat surface. respectively; N: is the number of mole of the solvent in the Liquid
phase; T is the temperature in degree of Kelvin and R is the universal gas constant. The
concept of the equilibriurn radius will be discussed and its equation will be derived in
chapter 2.
In 1975, Tucker and Ward examined both the stability and the predicted value of
the equîiibrium radius of a bubble by making two bubbles present simultaneously in a liq-
uid, one larger than the equüibrium site and the other smallere3 They observed that the
former grows while the latter dissolves and hence they concluded that the equilibrium
state is unstable and the value of the equiïibrum radius Lies between the two initial radii.
Furthemore, by considering a closed volume of a liquid, it was predicted by Ward
et al. in 1982, that depending on the amount of the gas present in the Liquid, there may be
none, one, or OHO equilibrium bubble sizes available to the system? This was contrary to
the case of an infinite extent of the liquid where there was only one possible equilibnum
state for the system. In the case where two equilibriurn bubble sizes are possible, the state
corresponding to the smaller size, as it was mentioned earlier, was an unstable equilib-
num, whereas the larger was a stable equilibrium. The latter is the state which was not
considered by Gibbs. In 1982, dso, the term thermodynarnic potentiai, B. was defined and
introduced for the first tirne, by Ward et al? They predicted that, the nature of the equilib-
num state can be determined where the thermodynarnic potential is an extremum. Follow-
ing that they discussed the possible equilibrium States for a bubble in a closed volume of a
Liquid.
In 1984, an expression for the thermodynamic potential B was denved for a heter-
ogeneous system by Ward and Levart. They stated that the necessary condition for the sys-
tern to be in the state of equilibrium is that the differential of the potential B should
vanishg They considered an arbitrary large but closed volume of a liquid-gas solution
maintained at constant temperature and pressure. Pockets of a gaseous phase. i.e., bubble
nuclei were predicted to be in stable equilibrium in the roughness of the wails of the con-
tainer. The experimentai observation confirmed the existence of the stable equilibrium
state, provided that the gas concentration in the îiquid phase is slightly greater than the
equilibrium value. In the present study, in chapter 2, the thermodynamic potential B for
the system under investigation, will be derived in a way simüar to the work described
above.
It is worthwhile to mention that, in general, gas bubbles cm be formed in a liquid-
gas solution in two dinerent ways. Le., homogeneous and heterogeneous formations. In
the former case, based on the homogenous nucleation theory, if there exist enough thermo-
dynamic fluctuations, then a gas bubble may be formed in the Liquid whereas in the latter
case, the gas bubble is injected into a liquid? There has been a great deal of attention
given to the heterogeneous nucleation in recent attempts to study the gas bubble related
problems.
There have also been a number of investigations involving the size of a gas bubble
to determine the surface tension of surfactant solutions. The concept of the stability of the
equilibrium size gas bubble was not considered by anyone of thern.14*1618 For instance. in
1977, the pulsating bubble technique for evaluating the puimonary surfactant was devel-
oped by Enhorning in order to measure the surface tension of different surfactant solu-
tions.14 He considered a Liquid-gas, two phase system for which the pressure across the
Liquid-gas interface was recorded by his apparatus and the maximal and minimal bubble
sizes were obtained with a microscope. The system was kept at a constant temperame of
37°C and the surface tension for different samples were then determined directly by the
Laplace equation. explained in the next section. He considered pure water as one of his
experimentai sample. He then looked at the same bubble within the water at a constant
temperature, measured the maximal and the minimal bubble radii, recorded their corre-
sponding pressures and caiculated the surface tensions by the Laplace equation. For a
period of five minutes, the recorded surface tensions varied between 72k1.2 to 73k1.5
dydcm for the maximal and -6 to 75S.5 dynkm for the minimal bubble size. The
4
error comparing to the tabulated surface tension of the water at 37OC. ir.. 70.096 dydcm,
is very high.15 The question which these results raise is that of whether those maximal and
minimal bubble radü were in fact equilibrium radii.
In 1994, Chang and Franses studied the factors affecthg the dynamic surface ten-
sion measurernents with the pdsating bubble surfactometer. They clearly stated that the
Laplace equation was strictly valid for a srnail bubble and a static interface. They aiso
added bat, tbis equation had been commonly assumed to be valid under dynamic condi-
tions.16 Chang et al. in 1996 used the same equation to measure the surface tension for dif-
ferent preparations of dipalmitoylphosphatidylcholine (DPPC)-hexadecanol dispersed in
s a h e at 37"C, with the pulsating bubble system, under dynamic ~onditions.'~
1.2 Motivation of Present Study With the advent of the computer aided experiments, there has been provided more
confidence in dealing with such a delicate problem, i.e., the equilibrium of a gas bubble.
Recently as a result of an investigation on the contact angle hysteresis in a systern that
contained a Line of contact between solid, liquid and vapor phases and was subjected to
gravity, Ward et al. showed experimentallY6 and proved theoretically7 that the contact
angle can also be a function of pressure. Foliowing that a senes of experiments was
designed and carried out to observe the response of the contact angle to the pressure, but
this time, in a nonequiïibriurn pulsating bubble system.
Moreover, a two component, water-nitrogen, two phase, liquid-gas, systern was
considered. The pressure inside the liquid was changed sinusoidaily while the pressure
inside the gas was held constant. The size of the gas bubble and its corresponding liquid
pressure were recorded venus thne by two cornputers. The Laplace equation was used to
check the surface tension of the water while the actual bubble radius and its corresponding
mN Liquid pressure were recorded. A fluctuation of (k30)- was found in the surface tension
rn
of the water during one cycle. For the case of a spherical gas bubble the Laplace equation
is: pG - pL = 3 where P' and pL are the pressures inside the gas and the iiquid R '
phase, respectively; y is the surface tension of the water and finally, R is the radius of the
bubble. The expriment was nin at a constant temperature.
Furthemore, the surface tension of the water was measured by capiliary rise
technique8, once at the beginning and once more at the end of the expriment. No changes
were found which confirmed that there was no impurîty or surfactant present in the water
during the experiment. By having assumed a constant surface tension for the water, growth
and dissolution of the bubble were also predicted and found to be conaary to the achiai
behavior of the gas bubble. This contradiction between the Laplace equation and our
observation of the gas bubble became the major motivation for the present work.
1.3 Scope of the Thesis In this study, a two component, water-nitrogen, two phase, liquid-gas systern in
which gas had k e n dissolved into a finite volume of a stirred iiquid, was considered. The
necessary relations among the involved parameters, namely temperature, pressures, bubble
radius, sizes of the system and the thermodynamic potentiai were derived from the equi-
librium conditions. Terms and calculations are defined and explained in chapter 2. As it
was shown by Ward et al?, the equilibnum state was then determined where the thermo-
dynamic potentiai was an extremum. Maximum thermodynamic potentiai was interpreted
as an unstable equilibrium state and the minimum one as a stable equilibnum state. The
former case is when the bubble size is smaller and the latter case is when the bubble size is
larger. The subject of this work is the stable equilibrium state.
Moreover, the thermodynamic potential curves were used to predict the change in
bubble sue when subjected to different liquid pressures. Agreement was achieved between
the observation and the theory. Also. the tendency of the bubble to grow or to dissolve to
the next equilibrium size, as a result of a change in the liquid pressure c m be predicted
accurately. by using the thermodynamic potential curves. Finally, it was concluded that the
Laplace equation c m be used to predict the size of a bubble provided that, the change in
the properties of the iiquid surroundkg the bubble has already k e n taken into account.
CHAPTER 2: THEORY ON THE STABILITY OF A GAS BUBBLE IN A FINITE VOLUME OF A LIQUID
2.1 Introduction
According to the fmt postdate of thermodynamics, for a macroscopic "simple"
fluid system, entropy and intemal energy are the thermodynamic properties.'O If this sys-
tem is in a state of equilibrium, it c m then be characterized completely at the macroscopic
level by its independent variables, i-e., S,VJV, 4,. . . ,N,. Thus the intemal energy can be
cl = U(S.V,N, fi2?-.-.Nr) (2.1)
where U is the intemal energy, S is the entropy, V is the volume and N, is the number of
mole of species 'Y' inside the fluid. These variables are calied the extensive properties of
the system.
Moreover, if the Eq. (2.1) is Merentiated one may write
Furthemore. the intensive properties in thermodynamic are defined as the differ-
entials of the extensive properties and cm be expressed as follows:
Temperature:
Chernical Potentiai of component "i":
and if these definitions are used in Eq. (2.2), one can then write
dU = TdS - PdV + pidNi
2.2 Availability and Irreversibility In the analysis of the complex thermoàynamic systems, the concepts of the avail-
ability and the irreversibility are nomaiiy used. l 3 According to Ref. 13, the irreversibility
can be written as
= wrev - WC" (2.7)
where I is the irreversibility, W,,, is the reversible work and WC, is the work crossing the
control surface during the process. Also, for a system having a fixed m a s and maintained
at a constant temperature, the expression for the reversible work is
where a subscript 1 or 2: refers a property to state one or two, V is the velocity and Z is
the elevation.
Moreover, the expression of the reversible work for a steady-state, steady-flow pro-
cess maintained at a constant temperature may be written as
where H is the enthalpy and it may also be expressed in terms of U, P and V
H = U + P V (2.10)
The reversible work for a steady-state, steady-flow process is a maximum when the final
state is in equilibrium with the surroundings with superscript O. This maximum reversible
work is designated as availability and may be expressed as foiiows
where is the availability of the system.
Furthemore, in Eq. (2.8) the quantity ( U - TS) is cded the Helmholtz potential
which is a rhemodynamic property of a substance and may be show to be
F = U - T S (2.12)
where F is the Helmholtz potential. h the case where the changes in the kinetic and
potential energies are insignificant, by considering Eqr. (2.8) and (2.12) one may then
write
Also, in Eq. (2.1 1) the quantity (H - TS) is cailed the Gibbs potential which is a thermo-
dpamic property of a substance and may be shown to be
G = H - T S (2.14)
where G is the Gibbs potential. By neglecting the changes in the kinetic and the potential
energies and using Eqs. (2.1 1) and (2.14), one fin&
w = Gl-Go (2.15)
The concepts of the Helmholtz and the Gibbs potentials will be discussed in the following
section,
2.3 Concept of Thermodynamic Potentid for a Homogeneous System
2.3.1 Helmholtz Potential
For a system contained in a rigid vesse1 with diathermal waiis, that has a constant
mass and is surrounded by a reservoir, the thermodynamic potential is the Helmholtz
potential or Helmholtz free energy. For a homogeneous system, it is by definition, the par-
tial Legendre transform of the intemal energy LI, that replaces the entropy S by the tem-
perature T as an independent ~ariable.'~ By considering Eqs. (2.1) and (2.3)- Helmholtz
potential cm be written as
F = U - T S (2.17)
Thew relations will be used in the derivation of the therrnodynamic potential B.
2.3.2 Gibbs Potential
When a system is in interaction with both a pressure and a temperature reservoir,
i.e., a process maintained at a constant pressure and temperature, Gibbs potential or Gibbs
free energy is admVably suited. It is the Legendre transform of the intemal energy LI. that
simultaneously replaces the entropy S by the temperahue T and the volume V by the
pressure P as independent variables. l0 Then, by considering Eqs. (2.1 ), (2.3) and (2.4).
Gibbs potential may be written as
G = U - T S + P V (2.19)
Based on the Euler relation, for a multi-component system, we have
where pi and Ni are respectively, the chernical potentiai and the nurnber of moles of spe-
cies i . B y inserting Eq. (2.20) in Eq. (2.19). we find
These equations will be used in the following sections.
2.4 Concept of Thermodynamic Potential, B, for a Heterogeneous System and Its Derivation
We shall consider a single sphericai gas bubble trapped inside a liquid phase, con-
12
tained in a piston-cylinder arrangement, within a temperature-pressure reservoir. This is
illustrated in Fig. (2.1). Also, suppose that the reservoir consists of a bulk phase b and a
sudace phase o. For such a system, we can not use the Helmholtz potential because the
volume of the system is ailowed to change.
In Gibbs potential although both the temperature and the pressure are controlled
but, it is valid only for a homogeneous system, i.e. a system in a single phase. Therefore,
the thermodynamic potential, B for a heterogeneous system was found to be suitable for
the present work. The thermodynamic potential, B is the one which was introduced by
Ward et al. for a gas bubble in 1982~ and generaüzed by Ward and Levart in 1984.' The
constraints for such a system are as foilows:
The total energy is constant:
where U' is the total energy of phase j; CI is a constant: a superscript L or G refers a
property to the Liquid or gas phases; a superscript LG or SL refers a property to the liq-
uid-gas or solid-liquid surface phases and a superscript b or refers a property to the
bulk or surface phases.
r The total volume of the system and the reservoir is constant:
where V' is the total volume of phase j and C2 is a constant.
The total area is constant:
where A' is the surface area of phase j and C:, is a constant.
For the reservoir:
N~~ = C4
and
= C5 (2.26)
where i = 1,2,. . . ?r ; N, is the total number of mole of species r ; C4 and Cg are con-
stants.
For the liquid-gas phase inside the cylinder:
where Co is a constant.
The Euler relation for the buk phase is
for the surface phase the above relation may be written as
where T , P , p and y are the intensive properties. temperature, pressure, chernical poten-
tial and surface tension, respectively.
Now suppose the bubble is held at radius, R which is not an equilibrium radius, by
means of a constraint, until the system has corne to equilibrium state. If we release that
constraint, the system may undergo a spontaneous transition process, and hally, the equi-
Librium is restored. For such a system, AD may be defined as the changes in property D, in
transition from one equilibnum state to another. It may be written as
hD = D 2 - D ,
For intemal eaergy, as there is no work done on the system, one may write
If we use Eqs. (2.28) and (2.29) in Eq. (2.3 1) and by considering Eqs. (2.25) and (2.26).
one fin&
note that
According to the second postulate of thermodynamics, for an isolated system in stable
equilibrium, total entropy, either increases or remains constant:
by combining Eqs.(2.32), (2.33) and (2.34) we may have
Eqs. (2.23) and (2.24) may be used in Eq.(2.35)
The liquid phase has the same pressure as the reservoir and the Merences between
and have been conçidered negligible, we may then organize the Eq. (2.36) as
L L 6(uL- T S ~ + P v ) +A(C I ' -TS~ )
~ ( u ~ ~ - T S ~ ~ ) + A( L P ~ - 1-9~) + p L ~ v G - Y S L ~ S L 5 O
if we negiect the adsorption on the solid surface, by using Eq. (2.29)
After combining Eqs. (2.37) and (2.38), one fin&
A ( U ~ - TSL + PV) + A@ - TS') + A( wLG - T S ~ ' ) + p L d v G s O
Eqs. (2.1 7) and (2.19) may be used
A ( G ~ + F' + FLG) + P~AV' s O (2.40)
This is where we de fine thermodynamic potentid, B for the system under consideration
L G B = G ~ + F ~ + F L ' + P v by using Eqs. (2.40) and (2.4 1) together, one may get
ABSO (2.42)
which means that, for any spontaneous changes in the system frorn one stable equilibrium
state to another, B is reduced.
2.5 Necessary Conditions For Equilibrium in approaching the equilibrium state, while spontan&us processes continue, the
differentiai of B tends to zero, where at complete equilibrium state, one may write
Moreover, by using Eq. (2.41) and Euler relations, Eqs. (2.28) aad (2.29), we may rewrite
the Eq. (2.43) as
Now if we dflerentiate Eq. (2.27)
then, after combining Eqs. (2.44) and (2.45). one h d s
i = 1 i = 1
Furthemore, for a sphencal bubble of radius, R, we may wnte
and
3 d~~~ = d ( 4 ~ R - ) = ~ x R (2.48)
Bnally, by considering Eqs. (2.46), (2.47) and (2.48), we are lead to the following equilib-
num conditions:
where, i = 1,2 ,..., r and
where, R, is the equilibrium radius.
2.6 Concept of Equilibrium Radius When a bubble of gas is in contact with a liquid-gas solution, there may exist one
or two radii at which bubble neither grows nor dissolves. This radius is c d e d the equilib-
rium radius and it is denoted by the symbol &.' ' It depends only on the thermodynamic
properties of the iiquid-gas solution and thus is itself a property of the system. It cm be
either positive or negative values. As it was stated in the opening of the previous chapter,
in the case of two equilibnum radii, the srnaller one is referred to as an unstable equilib-
rium radius and the larger one as a stable equilibrium radius. The latter is the subject of
this work. The expression for the eqdibrium radius will be derived in this chapter.
2.7 Henry's Law In eariy nuieteen century, Henry determined the quantity of different gases
absorbed in water at different temperatures.l2 He found that the mole ratio of the gas dis-
solved in the water was proportional to the pressure at which the gas had k e n dissoived in
the water. The statement may be expressed as
where pL is the pressure inside the iiquid; KH is the Henry's constant which is basicaily a
L function of the temperature only; is the number of mole of the gas in the liquid satu-
rated by the same gas across a Bat surface and N: is the number of moie of the liquid.
2.8 Equüibrium State of a Gas Bubble in a Finite Volume of a Liquid-Gas Solution and Its Expressions
In the present work a two phase, liquid-gas, two component, water-nitrogen, sys-
tem was studied. Subscrîpt " 1" refers a property to the solvent component which is water
and subscript "2" refen a property to the solute component which is nitrogen gas in our
investigation. There are three major assumptions to be made at this point which are as fol-
Iows:
s The pure solvent is incompressible.
The tiquid-gas solution is a weak solution.
Gas phase is assumed to form an ideal gas mixture.
The chemical potentials for a weak solution where there is only one dissolved gas,
may be expressed as1 '
and
where the superscript "0" indicates a property of the pure substance in the phase indicated
by the second superscript; v , and P , are the saturated volume and pressure of the solvent
at temperature T, respectively; R is the universal gas constant; T is the temperature in
L L degree of Kelvin; N I and N2 are the number of moles of the solvent and solute in liquid
L phase, respectively and Nb is the number of mole of the solute in the solvent saturated
by the same solute across a flat surface.
Also, according to an ideal gas mixture assumption, the chemical potentials for the
gas phase may be Wntten as' l
and
w here
and
1 L
are the mole fractions of the solvent and the solute in the gas phase, respectively.
When Eqs. (2.52) and (2.54) are used in Eq. (2.49), one may get
G p , = v',
G where P I is the partial pressure of the solvent* in the gas phase and
Moreover. if this the, Eqs. (2.53) and (2.55) are used in Eq. (2.49), one fin&
G where P , is the partial pressure of the solute, in the gas phase.
Furthemore, for an ideal gas mixture. form Dalton's Law, the total pressure id3
Then Eq. (2.64) may be combined with Eq. (2.50) in order to get the expression for the
equilibrium radius
By considering Henry's Law, Eq. (2.5 1). we may simpliQ the above equation
As we may notice, the equilibrium radius is expressed in terms of the properties of the iiq-
uid. only.' ' Thus it is a property of the liquid phase itseif. In other words, at equilibnum, it
is required for the bubble to have a radius equal to the equilibrium radius R, .
Since the system is closed to mass transport, we may write
and
of solvent and solu where N l and N2 are the totai number of m [te in the system,
G respectively. Ideal Gas Law rnay be used to express NI and NZG. thus the Eqs. (2.67)
and (2.68) may be rewritten as
and
where vG is the totai volume of the gas phase, Le., the bubble. If we use the Eqs. (2.59)
and (2.62) in Eqs. (2.69) and (2.70), respectiveiy, we may get
and
When Eqs. (2.71), (2.72) and (2.5 1) are combined. one may find the equation for N~~
G -1 N: = N ~ ( N , R T - ~ P , V ' ) [ ~ + N , K T - ~ P , V ] (2.73)
Then, the Eqs. (2.7 1) and (2.73) can be used to express the ratio N ~ / N ,
i V 2 k V i L = (N~RT)/[V'K~ + N$T- q~,~G] (2.74)
Therefore, by considering the Eqs. (2.66) and (2.74) we may simpli@ the expression for
the equilibrium radius as foilows
For the case of a spherical bubble
and it is worthwhile to note that q is always near unity for ail the conditions we consider.
2.9 Expression for the Thermodynamic Potential, B, Near Equili brium
In Section (2.4), the thennodynamic potential, was defined in ternis of the Gibbs
and Helmholtz potentials and a cross term, in general form. We write the equation once
again in this section but this t h e in more specified form. for the system under consider-
ation. Then by having Eq. (2.41), one may write
B = G ~ ( T , P ~ , N , L , ~ 2 L ) + F ~ ( T , v ~ , N ~ =,NZG) + (2.77)
L G F L C ( ~ ~ L G ~ , L C , ~ , L G ) + P v
or by considering Eqs. (2.17) and (2.19) and Euler relations Eqs. (2.28) and (2.29) we get
Near the equilibrium, after applying Eqs. (2.49) and (2.50), the above expression may be
rewritten as
where from conservation of rnass
and
L G N2 = N2 + N2 + N2 LG (2.8 1 )
For simplicity, at this point, we may choose the potential of a system filled with two com-
ponents but in a single phase, i.e., liquid phase as a reference state and write its potential
L L Bo = N ~ P , ~ + N ~ P & , (2.82)
L L where plo and ph are the chernical potentials of components one and two in liquid phase
at the reference state, respectively and Bo is the thermodynamic potential of the reference
system.
After combining Eqs. (2.79) and (2.82), one fin& that B - Bo near the equilibrium state is
given by
We fkst rewrite the Eq. (2.52) for the chemical potential of the component one, while there
exist two phases in the system
and then the chemicai potential of the same component, but this time where only one
phase, i.e. liquid, is present rnay be written as
L L OL N2 plo(T,P 4 , N 2 ) = pl (TS,) + v_(pL -Pm) - RT- (2.85)
NI
sidarly, by ushg Eq. (2.53), the expressions for the component two, may be written as
and
From there we may write
and if Eq. (2.74) and Henry's Eq. (2.5 1) are used in the above Eqs. (2.88) and (2.89), one
fin&
L L G -1 PI -Pl0 = A T V ~ ( N , / N , ) ( ~ P , - P ~ ) [ v ~ K ~ + N ~ R T - ~ P , V ]
and
After combining Eqs. (2.83), (2.90) and (2.9 1) one 6nds the fuial expression for B - Bo
where vG and A LG are volume and area of a spherical bubble of radius R , respectively
and q equals to unity as it was explained before.
2.10 Possibility of Equiübrium States for a Single Bubble in a Closed Volume of a Liquid
To investigate the existence of possible equilibrium states for a system we may
plot the b c t i o n for the equilibrium radius versus the bubble radius. i.e., Eq. (2.75). The
intersections between the line, x = y, and the plot, indicate the existence of the equilib-
rium states and the values of their correspunding equilibrium radii. An example is shown
in Fig. (2.2), where, R,, represents an unstable equiübrium radius and R,, represents a
stable one. We also, may plot B - Bo versus the bubble radius to s p e c e the stable and the
unstable equilibrium states. As it was stated earlier in ihis chapter and also shown in Fig.
(2.3). at equilibrium the differential of B must vanish. In other words, B is an extremum
at the state of equilibrium. It is a maxirnum at the unstable equilibnum where the bubble
radius is smaller and a minimum at the stable equilibrium where the bubble radius is
larger.
Moreover. in Eq. (2.75), if we hold ail the variables, i.e., the temperature T , the
liquid pressure pL and the total number of mole of the liquid N, . constant and then plot
the equilibrium radius R, versus the bubble radius R , for different values of the total gas
content N2, we might get a plot similar to Fig. (2.4). It is shom clearly that, dependhg on
the conditions under which the system was prepared and also, is king operated we rnay
have none, one or two equilibnum states. In the case where NO equilibrium states are
present, Fig. (2.4) plot "A". the one comsponding to the bigger radius is a stable equilib-
rium and the one corresponding to the smaller radius is an unstable equilibrium. Plot "B".
in the same figure, represents the condition where only one equilibrium state is possible.
We may designate that equilibrium radius by RI,, as it is corresponding to the lower lirnit
of the gas content for a possible equilibrium state during an expefiment.
Furthemore. we rnay observe similar effects on the possibility of the equilibrium
L states, by altering the other variables, i.e., temperature T, iiquid pressure P and the total
number of mole of the liquid NI . Thus, it is crucial to control the parameters precisely and
measure them accurately.
By considering all of those mentioned above, for the system under investigation,
one may express Rl,, through the foilowing function:
The possibüity of an equilibrium radius could be specified by usiag Eq. (2.93).
2.11 Theoretical Prediction of the Radius of a Single Bubble in a Finite Volume of a Liquid-Gas Solution, Due to a Change in the Liquid Pressure by the Thermodynarnic Potential curves
T h e d y n a m i c potential curves are able to predict the behavior of a gar bubble,
provided that the size of the system, its temperature and pressure are known. Once these
variables are given, we may use the theory and trace the bubble between the equilibrium
States accurately. This cm be achieved by assuming that the bubble which was initially in
equilibrium state on a known constant pressure curve, moves to a new one due to change
in liquid properties as a result of change in its pressure. Then the bubble either grows or
dissolves to the equilibrium state correspondkg to the new pressure. To clariS the idea,
we may consider the following closed system for which
and
The system is in a state of stable equiiibrium under these conditions. The equilibnum
radius cm be calculated fiom Eq. (2.75) as
R,, = 1 .lZ6mm .
We may plot Eq. (2.92) for the aforementioned system and get Fig. (2.5). According to this
figure, point (1) represents the k t stable equilibrium state.
Moreover, at this point we may increase the pressure instantaneously to
P; = 154.5kPo ,
and plot Eq. (2.92) for this new pressure. According to the theory, there will be a shift from
the initial constant pressure cuve to the new one due to the change in the properties of the
liquid. This is shown as point (O) in Fig. (2.5) where the bubble is no more at equilibriurn.
Furthemore, the bubble dissolves to the correspondhg stable equilibrium size
which may be calculated from Eq. (2.75) as
Rc2 = 0.979mm .
This new stable equilibrium state is shown as point (2) in Fig. (2.5). It should be noted that
this has been conbned in the present work throughout a number of experiments for
which the details are given in the foilowing chapters.
2.12 Figures
FIGURE 2.1. A schematic of the system under
consideration. Note that the piston is fiee to move.
FIGURE 2.2. Plot of equilibrium radius versus bubble radius for a single
bubble irnmesed in a water-nitrogen solution at 298.35 K and 154.5 kPa.
Total water and nitrogen gas contents are 1.984 mol and 35.233 pmol.
respectively. Possible equilibrium States labeiled R, and R are 0.12 1 mm I c2
and 0.979 mm, respectively. R,! corresponds to an unstable equilibriurn and
R to a stable one. =.r
Unstable Equilibnum
FIGURE 2.3. A conceptual plot of the thermodynamic potential venus bubble
radius. The potential is a maximum ai the unstable equilibnurn and a minimum at
the stable equilibrium. The srnalier equilibrium bubble radius is of the unstable
and the larger bubble radius is of the stable equilibrium.
FIGURE 2.4. Variation of the equilibrium radius with the radius of a single
bubble immersed in a water-nitrogen solution at 298.35 K and 154.5 kPa.
Water content is 1.984 mol. Total niaogen gas content for profiles labeled A,
B and C are 35.233, 35.056 and 35.000 v o l , respectively. There are two
possible equilibrium States for case A while non is possible for case C. Case B
shows the minimum amount that could possibly give rise to an equilibrium
state.
FIGURE 2.5. A theoretical prediction of the size of a single bubble immersed in
a water-nitrogen solution at 298.35 K, due to change in Liquid pressure form
154 kPa, profile A to 154.5 kPa, pronle B. Total water and nitmgen gas contents
are 1.984 mol and 35.233 pmol. respectiveiy.
CHAPTER 3: EXPERIMENTAL INVESTIGATION
3.1 Introduction The objective of this expriment was to examine the predicted stable equilibrium
size of a single bubble immersed in a h i t e volume of a liquid-gas solution. Aiso. an
apparent contradiction to Laplace equation was investigated during the nonequilibrium
transient fkom one stable equilibrium to another. It was shown that a change in the liquid
pressure while the temperature is constant, might affect the other iiquid properties, e-g..
critical radius. The Laplace equation is not able to predict the direction of the change in
the bubble size because the liquid pressure is the only parameter which is changed in that
equation during the process. Aithough it is quite valid at equilibrium. but care should be
taken to use this equation during nonequiübrium stages.
Moreover. by reviewing the past w o r k ~ ~ - ~ ~ * " that were related to the present
study, it was found bat, the system is a very sensitive function of its variables, Le., pres-
sure, temperature and size of the system. The experimental apparatus should be quite
capable of controlling the parameters precisely and measuring them accurately. Therefore,
all the measurhg tools were chosen carefully and calibrated according to the expenmentd
requirements. The errors associated with each one of them, were considered separately in
the results. Specid consideration was aven to the measurement of the size of the system.
i.e., the total number of moles of the liquid and the gas components.
Furthemore, water was chosen as the experimentd Liquid, as it is by nature. a
major part of many physical and biologicd system for which a number of bubble expen-
ments have already k e n carried out and results exist for comparisons. Also, nitrogen is a
well behaved gas with a very low solubility in water which allows to maximire the change
in the liquid pressure. Therefore, water and nitrogen were the ideal components for this
series of the experiments. It is worthwhile to mention that, both of this substances are of
special interest to decompression investigations.
In this chapter, k t , an experimental apparatus used in this investigation will be
described. Dinerent subsystems in the apparatus and their design considerations will be
discussed individuaily. Then the appropriate methods used to control certain thermody-
namic properties such as temperature and pressure will be explained. The overall expen-
mental procedures will be outlined and ha l ly a powerfid technique will be presented to
measure the size of the system to a high degree of accuracy.
3.2 Experimental Design The basic goveming parameters involved in an experimental examination of a bub-
ble stability are temperature, liquid pressure, volume of the liquid surrounding the bubble
and the total gas content of the system. The designed apparatus must be able to connol and
measure those parameten, in order to meet the experimental requirements. The conceptual
design expressed in the form of a schematic diagram is displayed in Fig. (3.1). The appara-
tus consists of five distinguishable units, an experimental vesse1 and a temperature enclo-
sure. Each will be described in the foilowing sections.
3.2.1 Water Preparation Unit
Figure (3.2) displays a schematic diagram of the water preparation unit. As it can
be seen in this figure, it consists of a boiler, a special filter, a gassing-ciegassing chamber, a
vacuum pump and a nitmgen gas supply. Tap water was passed through a demineralizer
(coniingm D 2 a ) where its purity increased to approximately 1 MR -cm at the exit. This
filter is designed to deliver denaùieraiized water while continuously monitoring purity with
a glas-platinum electrode. The purity was measured as resistivity in MR -cm . The water
was then distilled by an automatic 3 liter still boiler (corninga AG-3ADA) heated by an
immersion heater (vycor@). Also, to increase the Level of the purity of the disùlled water, a
nanopure bioresearch deionization system (~ranstead~) was used. The hlter was designed
to produce type4 regent grade water equai to or exceeding standards established by
ASTM, CAP and NCCLS with bacterial endotoxin levels below 0.01 . The water at the ml
tilter outlet was 18.2MR -cm pure as it was shown by the system indicator.
Moreover, the nItered water was transferred to a clean chamber for degassing and
gassing processes. The chamber was comected to a vacuum pump (Duo seal@, model
1402, made by Welch Scientific Co.). There was dso a second co~ec t ion to the chmber
to provide the supply of the nitrogen gas. The temperature was controlled by means of a
water bath (~aake@) swounding the entire chamber. The chamber was also equipped with
a pressure gauge and a themorneter. First the water was degassed by the vacuum pump to
-30inHg, while stirring it, for a sufficient length of time. It was then gassed by a supply
of the pure nitrogen under the required pressure. After this stage, water was ready ro be
transferred to the experimental vessel.
3.2.2 Bubble Preparation Unit
Figure (3.3) exhibits a schematic diagram of this unit. A compact infusion pump
(Harvard aratus tus@) which could control the Bow rate, was used. A syringe (30cc B-D@)
was mounted on the pump. It codd be operated by the pump or even rnanually. The sys-
tem was flushed with the ultra high pure nitrogen supplied by the ( ~ a t h e s o n ~ CO.), prior
to the operation. A single slug of the same nitrogen was then produced and directed to the
experimental vesse1 through a special method which will be explained later in this chapter.
3.23 Liquid Pressure Control Unit
A srnail diameter piston was used to directly change the liquid pressure in the
closed system of the Liquid-gas solution. This method had the advantages of not aecting
the total gas content of the system and also avoiding contamination. Fig. (3.4) depicts the
principle of the liquid pressure control unit. A crank shaft was moving a hydraulic ram in
and out of a cylinder containing oil. The rnovement was achieved manuaily, in our case.
but a continuous pulsating motion was aiso possible. The motion of the hydraulic ram pro-
duced a srnail movement of a large diameter piston and consequently that of the smaii
diameter piston afnxed to it. Due to the fact that the water can be considered incompress-
ible in this case, this srnail movement caused a significant change in the liquid pressure.
Proper fitting and sealing material were used to ensure no leak from the experimental ves-
sel. The working procedures wiil be explained later in this chapter. It should be noted that
a similar design was used by Enhoming, in 1977, when he introduced the pulsating bubble
technique for the h t time. l4
3.2.4 Data Acquisition Unit
hessure and Temperature were the two major governing parameters in the present
experiment. It was shown by the previous investigators 3-5v9*1 l that the validity of the
results was highly related to how accurate the pressure and the temperature of the system
were measured. Therefore, a data acquisition software ( ~ a b ~ i e w ~ , version 3.1) was used
to acquire the measurements of the thermodynamic properties, i.e. the temperature and the
pressure. Also, the differential inputs were configured by a board (Phoenix contactB type
UMK-SE). The program use& was custom made according to the experimental require-
ments and tested pnor to the operation. The componding temperature and pressure sen-
sors used in conjunction with the above mentioned software, will be described in the
foiiowing subsections. Fig. (3.5) displays a schernatic diagram of the data acquisition unit.
Teflon insulated thermocouple wires (K-type, OmegaB) were used for the tempera-
hue measurements. lg The temperature reference point was provided by an ice point refer-
ence cell (bye@ Instrument). A stainless steel thermocouple weii was used to
accommodate the thennocouple wires into the experimental vessel. The overall instrumen-
tation error due to temperature measurement was investigated by dipping the themocou-
ple into the ice point reference cell. No significant deviation from zero was found. The
temperatures were recorded continuously versus time by the data acquisition system. they
were averaged over a desired length of time during which the system was considered in a
stable equilibrium state. The average values of the temperatures were then used in the
equations. The standard deviations were calculated and the correspondinp errors will be
discussed later in this chapter.
3.2.4.2 Pressure
A flush diaphragrn differential pressure transducer (statharnB, mode1
PM260T-15-350), was used to monitor the pressure inside the liquid phase. Its working
range was f 15 psig which roughly equals to 2 ami. absolute with an innnitesimal resolu-
tion. As the temperature kept constant within the working range of the transducer during
each experiment, thermal sensitivity shift and thermal zero shift were found insignificant
for the condition of the experiment. A 5 volt supply (GFC ~ammond? was used to power
the transducer. The pressure transducer was calibrated in the range needed for the experi-
mental conditions, pnor to the actual nui, very carefully. A linear relation with the comla-
tion factor of one was found between the voltage and differential pressure. The caiibration
curve is available in appendix A Thexfore, the instrumentation error for the condition of
our experiments was found negligible. Moreover, similar to the case of the temperature,
the acquisition system was recording the pressures venus t h e and their average values
were used in the equations. Also, their associated errors and the procedures WU be
detailed later in this chapter.
3.2.5 Image Processing Unit
A solid state camera (~ohu@, mode1 4815-5000/0000) dong with a macro lens
anon on@ FD 1- k4) and an extension tube (canon@ FD 50) were used to capture the
images of the bubble. A variable intensity light source (Fiber ~ite@) and a ground glass
diffuser plate were produchg a concentrated beam of light, for better resolution. An image
processing software (NM image" version 1.61) was used to process the captured images
in the computer and to facilitate the bubble size measurement. The software was calibrated
according to the instructions given in the manuai, from pixel to millimeter by a known
Iength, i.e. bubble injection tube diameter, before the experiment and since then the serting
of the camera was Ieft ~ n c h a n ~ e d . ~ ~
The error due to the calibration wiil be discussed later in this chapter and other
than that, the instrumentation error was considered negligible for the condition of this
experiment. Fig. (3.6) shows a captured image of a singie bubble caught between two
glass fibers. Because of the limitation imposed by the windows, a s m d portion from the
top of the bubble could not be seen by the camera. The bubble diameter was directly mea-
sured by the image measuring tool, by using the edges of the image, according to the
instruction rnanua.Lt0
The effect of the gravity on the bubble was investigated by cdculating the Bond
number for the experiment. The Bond number is the ratio of the gravitational force and the
surface tension force. From Table 3- 17 the maximum measured bubble radius was 1 -32
mm. Also, by considering the surface tension of the water at 2YC to be 72.0 mN/m, one
may calculate a bond number of 0.2. Since this number was calculated for extreme cases
and still is smaller than unity, we s h d neglect the effect of gravity in the present work.
The experimentd procedures are detailed later in this chapter.
3.2.6 Experimental Vesse1
The body of the experimental vessel was made in one piece from stainless steel
type 3 16. Four large stainless steel M e s with g l a s windows were used to provide a
good visibili ty. The glass thickness was chosen according to the pressure requirements
with a high safety factor to ensure a safe ~peration.~' The vessel was aligned properly with
the pressure control unit rarn and held firm on a bdanced stand, by means of four clamps.
Fig. (3 -7) depicts a schematic diagram of the experimental vessel. The inlet for the water
and the slug of the nitrogen was provided through a three way valve connected to the bot-
tom of the vessel by means of a one eighth of an inch diameter tubing and appropriate fit-
tings. The method of operation of this speciai valve wili be explained later in this chapter.
A type 3 16 stainless steel themocouple well was used for the temperature measurement.
The liquid pressure control unit piston was inserted in the vessel through a proper fitting
and a special O-ring. Care was taken to ensure that the piston was fkee to rnove while hav-
ing an air tight seal between the closed volume and the smu11ding.
Moreover, a pressure tramducer, as described in Sec. (3.2.4.2). was rnounted on
top of the vessel dong with a ball valve to isolate the flush diaphragm pressure transducer
during the stage when the vessel was exposed to a very high pressure to dissolve excess
gas pockets. The bubble was caught between a glas fiber arrangements designed specially
for this purpose. As show in Fig. (3.7). to ease the excess bubble drainage from the ves-
sel. if any. a quarter inch O. D. chamfered fitting and an isolating valve were mounted on
top of the vessel right over the iniet port. The experimental vessel was also connected to a
bubble drain valve as weli as a high pressure nitrogen gas supply valve by a quarter inch
O. D. tubing, above the isolathg valve. The tube was long enough to avoid gas difision
during the pressunting period.
Furthermore. a srnail stir bar with a stir magnet were used to maintain a uniform
gas concentration around the bubble in the liquid-gas solution. Furthermore. a temperature
air bath was installed to control the vessel temperature during the experiment. Fig. (3.8)
displays a photograph of the experimental vessel assembly with a scale beside it to be tter
understand its actud sue. The worhg procedures wiii be explained later in this chapter,
under a difkrent section.
3.3 Cleaning Procedure and Contamination Monitoring Method
The cleaning procedure was based on the Ref. 22. AU the stainless steel compo-
nents and the glassware were first Rnsed with puriiied, dried and distiUed acetone (Cale-
don?. They were then drained and rinsed with the pure water before introducing a
solution of a detergent (~lconox@ laboratory) and demineralized-deionized distilied
water. The items were allowed to rernain in the solution for 24 hours before being rinsed
with demineralized-deionized distiued water at least ten times to remove the detergent
particles. Fmally, they were soaked in a solution of Chromic and Sulfunc acid (made by
~isher@ Chernical) for a sumient time before king drained. The cornponents then were
rinsed with demineralized-deionized distilled water at least ten times to remove any traces
of the acid. The O-ring seals and the gaskets were cleaned only by the detergent solution
and then rinsed with demineralized-deionized distilled water at least ten times. At this time
the items were considered clean and ready to assemble.
Moreover, due to change in surface tension of the water as a result of the presence
of any impurity or surfactant, this parameter was used to indicate the contamination of the
system, if any. Based on this method, a water sample was taken fiom the system in differ-
ent locations. i.e. before, in the middle and after the experiment. The surface tension of the
water was then measured by the capillary technique8 which is described in Appendix B.
No sigmficant deviation fiom the handbook values'5 were found. Thus, the system was
considered free of surfactant. The results of those tests are given in Table (3.1). Each
recorded surface tension is actudy an average of three separate rneasurements, therefore
its corresponding standard deviation is also shown.
3.4 Experimentai Procedure Conceptually, the experimental procedure must investigate the stable equilibrium
state as predicted in the previous chapter for a single bubble immersed in a finite volume
of a liquid gas solution. The governing themodynamic properties, i. e. pressures and tem-
peratures were considered known and the sizes of the system, i. e. number of moles of the
water and the nitmgen were calculated fiom a mathematical mode1 based on a technique
which will be described later.
A sufficient quantity of the demheralized-deionized distilled water was first trans-
ferred to the precleaned water preparation chamber. It was then degassed by means of a
vacuum pump until no further gas bubbles leaving the surface of the Liquid, were observed.
After isolating the vacuum pump, pure nitmgen was introduced into the chamber under
the required pressure. These components are shown in Fig. (3.2). To speed up both the pro-
cesses a clean st i r bar and a stirrer unit were used to stir the Liquid continuously during the
operation. Stirring also, ensured a UILifom desirable liquid-gas solution. The prepared
water was then ûmsferred to the experimental vessel which was kept under a pressure
slightly lower than that of the water preparation chamber, very slowly to avoid the produc-
tion of any gas pocket inside the vessel. Aiso, all the connections to the ceil were deaerated
by allowing the water to overtlow from their fittings More tightening them. The transfer-
ring process was continued until the water was dripping from the drain when at this time.
the drain valve and the supply were shut, respectively.
Moreover, to dissolve all unwanted gas bubbles, if any, the pressure inside the ves-
sel was increased to lûûpsig by means of a supply of the nitrogen gas from the top as
shown in Fig. (3.71, for about 3 hours. The pressure was then released to the working pres-
sure very slowly to avoid any bubble evolution due to decompression. At this time. the
vesse1 was comected to the bubbie preparation unit by the three way valve. The unit was
already flushed with the nitrogen gas and kept at the same pressure as the inside of the ves-
sel. A slug of the nitmgen was then pushed into the vessel with a very slow rate by the
infusion pump and caught between the fhed glass fiben installed on the vessel ceihg. A
constant pressure was rnaintained in the vessel during this process with the drain valve
partially open to ailow the bubble king pushed into the vessel and also drain any extra
44
ones produced during the operation, if any. FinaUy the tubing was flushed with the water
and the vessel was isolated to run the experiment. The data acquisition and image process-
ing units dong with their accessories and correspondhg cornputers were started. Light
was focused on the bubble while the camera was iive and the image was seen on the corn-
puter monitor. Each tirne when a single image of a bubble was saved the atmosphenc pres-
sure and room temperature were also recorded accordingly.
Furthemore, the pressure inside the liquid was controiled by the in and out move-
ment of the hydraulic piston. The advantages of this hydraulic system were to fine tune the
liquid pressure precisely as well as to smoothen the motion. The stroke of the piston was
changeable and the limits were designed according to the experimental requirements. The
temperature of the vessel was aiso controllable by means of an air bath as mentioned ear-
lier in this chapter. Due to a good heat transfer characteristic of the stainless steel type 3 16.
the air bath was successful in maintainhg the desired temperature during the course of the
experiment. Fig. (3.9) shows a photograph of the whole apparatus.
3.5 Method Used to Measure Total Gas and Liquid Contents of a Closed Volume of a Liquid-Gas Solution
Since the value of R, is very sensitive to the total gas content of the system, to
determine its value accurately, one must measure the gas content of the system precisely.
A number of different rnethods have aiready been used by investigators to measure the
total gas content of a iiquid-gas solution. Van Slyke gas analyzer has been the most corn-
mon apparatus for this purpose. 3, 26
Using this method, the Van Slyke apparatus must be connected directly to the
experimental vessel so that the liquid could be transferred into the Van Slyke apparatus
without exposure to air. This extra connection would violate our interest of having least
connection to the experimental vessel. However, Ward et al. in 1986. used the Van Slyke
method and found a standard deviation of 0.012 in four repeated measurements for the
ratio of the initial gas concentration to the saturated c~ncentration.~~ They also, used
another method by taking the initial gas concentration as a fitting parameter and improved
the error to 0.00 1.
The technique of measuring the liquid and gas contents in the present work. was
achlally introduced by Ward et al., in 1982, but the equations were not investigated.?
According to this method, to determine the total number of moles of the water and the
nitrogen present in the system, two of the observed equüibrium measurements were used.
To make it more clear, one may rewrite the expression for the equilibrium radius, Eq.
(2.75) as
where, at equiiibrium
L L If two observed equilibrium measurements, i.e. (RCI,PI ) and (Rc2Q2) are used in the
above equation to form two independent equations, one finds
The total number of mole of water, NI and the total gas content, N2, are the solutions to
this system of equatiom.
To discuss the accuracy of this methoà, we can consider for instance, experiment
number two which was conducted at a constant temperature of 37.1"C. B y using two sets
of stable equilibrium measurements, based on the procedures discussed in experimental
result section, the total nurnber of moles of the water and the nitrogen content were calcu-
lated as 2.01651 mol and 35.9164 p o l , respectively. Moreover, the nitrogen solubility at
37.1°C and 101.325 kPa is 10.21 jmol of the nitrogen per mol of the wateg4 and from
there by knowing the number of mole of the water, i.e. 2.0165 1 mol, the nitrogen satura-
tion concentration is obtained as 20.5886 p o l . The ratio of the total niuogen content to
the saturated content can also be found as 1.744479955. The uacertainty of the measure-
ment of the radius of a bubble in tbis experiment was found to be M.O22mm, as discussed
in the error analysis section. If one adds this value to the initial radii in the measurement
sets while pressures left unchanged, the total number of moles of the water and the nitro-
gen can then be calculated as 2.08440mol and 37.1364pno1, respectively. These values
give a nitrogen sanirated concentration of 2 1.28 17pmol. Furthermore, the ratio of the total
nitrogen content to the saturated content can also be found as 1.744992 176. By comparing
the two ratios, i.e. More and after applying the measurement uncenainty, one finds a dif-
ference of 0.0005 which implies an accuracy of up to four significant numbers.
This is a very simple method for measuring both the gas and the iiquid content
accurately without any extra apparatus or fitting parameters, only by using two stable
equilibrium measurements. Moreover, by improving the temperature and pressure mea-
surements, one can even increase the accuracy of this method with a signincant amount.
The experimental resuits and their corresponding error analysis are given in the foilowhg
47
sections.
3.6 Experimentai Resuits As explained earlier, the experimental method involved the inaoducing of a single
bubble into a finite volume of a water-nitmgen solution held at a constant temperature.
The solution was stirred continuously while the goveming thermodynamic properties were
recorded versus time by a computer. h g e s of the bubble were also capnired by a second
computer and the pressure was adjusted and fine tuned manually by a hydraulic pressure
control unit. Prior to the actual experiment, a senes of preliminary theoretical calculations
specified the possible working range for the operation, i.e., liquid pressure and gas content
by considering the initial fixed constraints such as temperature and the approximate vol-
ume of the experimental vessel.
Three series of experiments were conducted. In each experiment, four distinguish-
able observed stable equilibrium measurements were recorded out of which two were used
to calculate the total number of moles of water and nitrogen and the other two for compar-
ison with the theoretical predicted values. In the first experiment it was aimed to hold the
vessel temperature at 25OC and the solution was saturated with nitrogen at approximately
1.5 atm. absolute. In the second experiment, temperature was increased to 37OC while the
airn was to saturate the solution under the same saturation pressure, i.e., 1.5 atm. absolute.
Third experiment was conducted at the same temperature as the f h t one, Le., 2S°C, but
this time the nitrogen saturation pressure was increased to approximately 1.8 atm. abso-
lute. The aforementioned conditions were chosen in this way to serve a better cornparison
while having some common experimental basis, i.e. 25 and 37°C. Once again, it should be
noted that, the actual number of moles of water and nitmgen, i.e. NI and N2 were calcu-
lated h m the set of Eqs. (3.2) and (3.3).
3.6.1 First Experiment
As it was outlined in the previous section, based on the preliminary calculations,
the a h was for a total gas content of approximately, 36 p o l and a temperature of W C .
By considering the values of nitrogen solubility in water at different t e rnp~ature~~, given
in Appendur C, and the properties of watep, given in Appendix D, a pressure of 1.5 atm.
absolute was chosen for the gassing process. Water was king saturated at that pressure for
a sufficient length of time until a pressure of 1.5 atm. absolute was maintained in the
chamber with the supply shut. The water-nitrogen solution was then transferred to the
experimental vesse1 as it was described in Sec. (3.4).
Moreover, a bubble was introduced and measurements were started. The pressure
was adjusted in the range established by the preLiminary caiculations and then held con-
stant dong with the temperature with a minimum fluctuation. The bubble diameters were
measured by a computer every ten minutes and the corresponding atmosphenc pressures
were recorded manuaily at the same tirne. Aiso, as it was mentioned in Sec. (3.4), the tem-
peratures and the pressures were recorded continuously versus rime, with the rate of one
data per minute, in the other computer. The criterion for a state of stable equilibrium was,
a constant bubble diameter at a constant pressure and temperature.
Furthemore, a total of four stable equilibrium measurements was collected and
Listed in Table (3.2). The stable equilibrium Liquid pressures are given in the second col-
umn and their corresponding equilibrium bubble îadii, in the third column. It must be
noted that the values for the pressures were averaged over a length of one equilibrium state
observation t h e . The h t and the last measurement sets were used to calculate the size of
the system b r n Eqs. (3.2) and (3.3). Table (3.3) shows the calculated total number of
moles of the water NI, and the nitmgen N2, present in the system, respectively.
Once the size of the system was specified, the predicted values of bubble radii for
the other two pressures were calculated by using Eq. (3.1) and Appendix D. These values
are displayed in Table 3-4. The liquid pressures are given in the first column and the corre-
sponding measured and predicted bubble radii in the second and third, respectively-The
overaii average temperature for the experiment was calculated as 252°C. The errors asso-
ciated with the temperature and the pressure averaging wiil be discussed later in this chap
ter.
3.6.2 Second Experiment
In this experiment, it was aimed for the same total gas content as that of the first
experiment, i.e. 36 p o l , but the working temperature was increased to 37°C. Under the
similar procedure, water was saturated with nitrogen at 1.5 atm. absolute and then tram-
ferred to the experimental vessel. Once again, preliminary calculations were made pnor to
the actual experiment, to spec* the possible range for the working pressure, Le. the pres-
sure range through which the stable equilibrium size bubble could be achieved. A bubble
was then introduced and the measurements were started, as described in the previous sec-
tion.
Moreover, four sets of distinguishable equiiibrium measurements were coiiected
and listed in Table (3.5). in a similar order as that of the fint experiment. The first and the
third measurements were used to calculate the size of the system, similar to the first exper-
iment. Table (3.6) displays the calculated total number of moles of the water N1, and the
nitrogen N2, present in the system. respectively. By considering these calculated values,
liquid pressures and Appendix D, once again, two predicted bubble radii were found from
Eq. (3.1). They are shown in Table (3.3, in a similar order as that of the f h t experiment.
The overall average temperature for the experiment was found to be 37.1°C. The errors
associated to the temperature and the pressure averaging will be discussed later in this
chapter.
3.63 T k d Experiment
It was decided to nin this experiment at the same temperature as that of the fint
experiment, i.e. 25OC, but with slightly increase in total gas content. Therefore, water was
saturated witû nitrogen at approximately, 1.8 atm. absolute. The preliminary theoretical
calculations were carried out to establish a proper working range for the liquid pressure.
The water was then tramferred, a bubble was injected and measurements started based on
the same procedures outlined in the previous sections.
Furthemore, the measurements of the distinguishable stable equilibriums are
shown in Table (3.8). The stable equilibrium liquid pressures are given in the second col-
umn and their corresponding equilibrium size bubble radü, in the thkd one. The size of the
system was found by considering the first and the third stable equilibrium measurement
sets and using Eq. (3.2). Those results are displayed in Table (3.9). The f m t column repre-
sents the calcuiated total number of moles of the water NI, while the second column
shows that of the nitrogen N2, present in the system. Fially, a cornparison of the mea-
sured and predicted equilibriurn bubble radii, are exhibited in Table (3.1 O), in the similar
order to that of the last two experiments. The overall average temperature for this experi-
ment was found to be the same as experiment number one, Le. 252°C. The erroa associ-
ated with the temperature and the pressure averaging will be discussed later in this chapter.
3.7 Error Analysis Generally speaking, here is a total of three main sources of error, involved with
the bubble experiments, i.e. contamination error, instrumentation error and measurement
error. In Sec. (3.3), a method for contamination monitoring during the experiments was
described. Surface tension of the water was used as an index for the level of contamina-
tion. This property was detemiùied in dinerent locations at dinerent t h e , by using the
capillary technique. explained in Appendix B. By considering the surface tension mea-
surements shom in Table 3- 1, it was concluded that, the system did not have any signifi-
cant contamination or impurities. Also, as outlined in Secs. (3.2.4) and (3.2.5). under the
data acquisition and image processing, respectively. no significant instrumentation errors
for the conditions of these experiments were found.
Moreover, the only considerable source of error for these series of experiments
was found to be the measurement error. As outlined in Sec. (3.2.5), the image processing
unit was calibrated according to the instruction manual by a known length, i.e. the bubble
injection tube diameter. An average value with a total uncertainty of M.022mm was used
for the reference length. This uncertainty is illustrated as an error bar for the measured
equilibrium radü in Fig. (3.10).
Furthemore, the temperatures and the pressures were measured versus t h e by the
cornputer, as explained before in this chapter. They both were kept constant with a mini-
mum fluctuation during one stable equilibrium state, and therefore averaged over the
length of the tune of that equilibriurn state. These averaged values were then used in Eq.
(3.1) to calculate the predicted values of the equilibrium radü. The temperature and pres-
sure errors associated with the predicted equilibrium radius wdl be discussed in the fol-
lowing section. An overail uncertainty was calculated according to Ref. 25. and was then
52
used to construct the error bar for the predicted equilibrium radii in Fig. (3.10).
3.8 Uncertainty Due to Temperature and Pressure Fluctuations In experiment number one, it was decided to hold the temperature at 2S°C. Due to
a slight fluctuation, the average temperature, i.e. 25.2OC with a standard deviation of
0.12OC, was used. The variation in the value of the predicted equilibrium bubble radii due
to this temperature fluctuation was investigated for both the liquid pressures by using Eq.
(3.1 ) and Appendix D and given in Table (3.1 1). Two averaged Liquid pressures are shown
in the first column, temperatures and their corresponding predicted equilibrium radii are
depicted in the second and third column, respectively, and the percent deviations are given
in the last column. The overall average percent deviation due to temperature fluctuation
i.e. 7.19%. was then used to find the total uncertainty for the experiment number one.
Moreover, the variation in the value of the predicted equilibrium bubble radii due
to fluctuation in pressure was studied. As it was shown in Table (3.4), the two average
pressures in this experiment were 153.996 and 154.510 kPa, respectively, with the stan-
dard deviations of 0.090 and 0.032 kPa, respectively. In Table 3- 12, the effect of these
pressure fluctuations on the predicted equilibrium bubble radii, are displayed. It should be
noted that. the temperature was considered constant, i.e. 25.2OC, during this analysis. The
temperature is shown in the k t column, the liquid pressures and their corresponding pre-
dicted equilibrium size bubble radii are depicted in the second and third column, respec-
tively, and the last column belongs to the percent deviations. The overall average percent
deviation due to pressure fluctuation i.e. L .5 1 %, was then used to find the total uncertainty
for the experiment number one.
Also, from Ref. 25, the total uncertainty due to both independent variables, i.e. the
temperature and the pressure can be written as2'
and from there, the total uncertainty for experiment number one is
( E ) ~ ~ ~ ~ ~ ~ ~ ~ ~ = M.0735mm. (3 -6)
Furthemore, in experiment nurnber two, the overaii average temperature was
found to be 37. 1°C with a standard deviation of 0.08"C. By the siiniiar procedure as that
of experiment number one, the variation in the values of the predicted equilibrium bubble
radii due to this tempera- fluctuation, for both the iiquid pressures, is illustrated in
Table (3.13). The overall average percent deviation due to the temperature fluctuation Le.
3.69%, was then used to h d the totai uncertainty for this experiment. The variation in the
value of the predicted equilibrium bubble radii due to fluctuations in the pressure was then
calculated, same as in the previous experiment, and shown in Table (3.14). It should be
noted that, the two average pressures in this experiment were 179.851 and 180.300 kPa
with the standard deviations of 0.090 and 0.032 kPa, respectively. The overail average per-
cent deviation due to pressure fluctuation i.e. 0.08%, was then used to find the total uncer-
tainty for the experiment number two. Also, by considering Eq. (3.4), the uncertauity can
be written as
('1 ~ x ~ e r i r n c n r 2
For the case of the third experiment, the overaii average tempera
(3.8)
.Rire was
25.=. 1 I0C. The variation in the value of the predicted equilibrium bubble radii due to
this temperature fluctuation for the two üquid pressures are given in Table (3.15). The
averaii average percent deviation due to temperature fluctuation was found to be 3.75%.
Also, by considering the two average pressures in this experiment. Le., 187.998H.008 and
188.995+O+0007 kPa, respectively, s d a r to that of the previous experiments, Table (3.16)
was made. From there, an overd average percent deviation of 0.09%, was found due to
the pressure fluctuation and then used to find the total uncertainty for the experiment num-
ber three. According to Eq. (3.4), the uncertainty is
and after simplifying, the total uncertainty for experiment number three can be written as
( E ) ~ ~ ~ ~ ~ ~ ~ ~ < ~ = M*0375mm (3.10)
At this point the overaii uncertainty of the predicted equilibriurn size bubble radii is taken
as an average of the Eqs. (3.6), (3.8) and (3.10) as foiiows
= f 0,0735 + 0.0369 + 0.0375 '( RC )prrdicwd 3 (3.1 1)
and after simprning, the overail uncenaiaty of the predicted equiiibnum size bubble radii
during the experimeots, can be written
Finally, Table 3- 17 displays a cornparison between the predicted stable equilibrium
bubble radü and their corresponding measwed ones. It should be noted that the values
were given with their uncertainties as descnbed before. To better understand the cornpari-
son between the predictions and the measurements, (RJprcdicled was plotted versus
( Rc)mensnred in Fig. (3.10) with their correspondhg error bars. A 45 degree iine was also
drawn to better judge the accuracy of the prediction of the equilibrium bubble radii by the
theory outlined in chapter 2. From this plot, it is inferred that the theoretical thennody-
namic method of predicting the stable equilibrïum bubble radius, is in good agreement
with the experimental obsexvations.
3.9 Tables
TABLE 3.1 Surface tension measurements of water during the experiment as an indication
II before 1 71.72W.11 1 72.00
for possible system conramination
II Water Gassing I @2S°C I 025°C
~abuiated l5 Surface Tension
mN/m
Water Sampling Location
l
Measured Surface Tension
mN/m
II at the Beginning of I @23.S°C I @23S°C
after
Chamber fiom Vesse1
1 Experiment 1 1
7 1.8=.06
72.08M.08
Experiment fiom Vesse1
at the End of
72.00
72.24
72.05W.06
@24"C
72.16
@ 24°C
TABLE 3.2 Stable equilibrium bubble sizes for experiment number 1
No.
I 2 3
TABLE 3.3 Total number of moles of water and nitrogen in experiment number 1
Liquid Pressure PL kPa
153.485
153.996
154.5 10
Number of Mole of Water mol
TABLE 3.4 Predicted and measured equilibrium bubble radii for experiment number 1
Measured Eqfibrium Bub ble Radius
~%)m~asurrd* mm 1 -24
1.130
1.010
Number of Mole of Nitmgen
m o l
Liquid Pressure PL kPa
Meas ured Equilibrium Bubble Radius
(Wrneas~fed* mm
Predicted Equilibrium Bubble Radius
(&)oredicted* inrn
TABLE 3. Stable equilibrium bubble sizes for experiment number 2
Liquid Pressure PL kPa
Measured Equilibrium Bubble Radius
@c)muisured*
TABLE 3.6 Total number of moles of water and nitrogen in experiment number 2 I 1 11
Number of Mole of Water mol
Number of Mole of l Nitrogen
umol
TABLE 3.7 Predicted and measured equilibrium bubble radii for experiment number 2
Liquid Pressure Measured Predicted Equilibrium Bubble Radius Equilibrium Bubble Radius 1 (%)oredicteci? mm
TABLE 3.8 Stable equilibrium bubble sizes for experiment number 3 i f
No.
1
TABLE 3.9 Total number of moles of water and nitrogen in experiment number 3
Number of Mole of Water mol
TABLE 3.10 Predicted and measured equilibrium bubble radii for experiment 3
Liquid Pressure PL kPa
187.801
Number of Mole of Ni trogen
umol
Liquid Pressure l?=
Measured Equilibrium Bubble Radius
(%),ad. mm
1,360
kPa (Khnea-i* Illm
Measured Equiiibnum Bubble Radius
Predicted Equilibrium Bubble Radius
TABLE 3.12 V'ation in R, due to pressure fluctuation for experiment number 1
TABLE 3.1 1 V'ation in R, due to temperature fluctuation for expriment number 1
% deviation
+5.24
Experimentd Temperature, T
O c
25.2
Predicted Eqdibri~m Bubble Radius
(Rc)prrdicteci* mm
1.1 85
Liquid Pressure PL kPa
153.996
Experimental ~emperam, T
OC
25.2+0.12
Liquid Pressure P' kfa
153.906
Predic ted Equilibrium Bubble Radius
(%)predictedv mm 1.147
deviahon
+1.86
TABLE 3.13 Variation in R, due to temperature fluctuation for experiment number 2 l
TABLE 3.14 Variation in R, due to pressure fluctuation for experiment number 2 Ir I t I
Liquid Pressure 1
I
II Experimentai 1 Liquid Pressure 1 Predicted 1
Experimental P icPa
179.85 1
Predic ted
Temperature. T "c 37.1
Tempera-, T OC
P= kpa
179.848
Equilibn~m Bubble Radius Wp&ia& mm
% deviation
37.1+0.08 1 1.358
Equilibnum Bubble Radius (%&redicted. mm
1.317
t3.11
% deviacion
M.00
TABLE 3.15 Variation in R, due to temperature fluctuation for experiment number 3
TABLE 3.16 Variation in R, due to pressure fluctuation for experiment nurnber 3
96 deviaûon
L
Predicted Eqfibnum Bubble Radius
(Wprsdi&. mm
Liquid Pressure PL kPa
Expe rime n ta1
Temperature, T O c
Experimentd Temperature, T
OC
187.998
Predicted Equilibrium Bubble Radius
(%)prediccedy
Liquid Pressure
P= kPa
25.2M.11 1 1.37 1 1 +2.93
% deviation
25.2
25.2 1
25.2
25.2
25.2
187.998
188.006
188.988
188.995
189.002
1.332
1.33 1
1.171
1.170
1.169
0.00
-0.08
M.09
0.00
4.09
TABLE 3.17 A cornparison between the predicted and the measured equilibrium bubblt radii
NO. L
1
2
3
4
5
6
Predic ted Equilibrium Bubble Radius
(%)Predicted~ mm 0.979M.049
1.1 26,+0.M9
1.17W.049
I.24M.049
1.3 17f0.049
1 .33m.049
Measured Equilibrium Bubble Radius
~ ) r n ~ ~ ~ mm
1 .O 1 M . 0 2 2
1.130.022
1 -2W.022
1,245W.022
1.32W.022
1.3 10.022
3.10 Figures
Data image Acquisition Unit Cornputer
I
I l Processing Unit
- - - - - - - ., 7
Tempe- 1 I -sure rature 1 wdow sensor Seosor 1
I 1 , I
Bubble and I G l a s Fibers
Diffuser 1
Enclosure ' - - - t - - - 1 Conml Unit
Vacuume
Filter
1 Bubble (
Pure Nitrogen Cylinder
FIGURE 3.1.Schematic diagram of the experimental
apparatus.
Tap Water
Demineraiizer
Boiier
Deionizer
To Pressure Gauge ,
+ To Thennometer
- - - - - - I
I Water I Preparation I F o Nitrogen supply
vacuum 1 - , -P
To Experimentaî vesse1
FIGURE 3.2.Schematic diagram of the water preparaûon unit.
To Drain
FIGURE 3.3. Schemaùc diagram of the bubble preparation
unit.
Piston and Cyiinder Arrangement
FIGURE 3.4. Schematic diagram of the liquid pressure control unit.
Ice Point Reference Machine
Acquisition I-1 Differential input Board
Power
q- Pressure Transducer
FIGURE 3.5. Schematic diagram of the data acquisition
unit,
RGURE 3.6. (A) Bubble image as captured by the processing unit and (B) the
same image with the edge hding filter. Bubble diameter is 223 pixels which
equals to 2.74mm.
Pressure Gauee 1
Drain
Isolating Valve
Pressure Conuol
Bar Magnet
Water Bubble Preparation Preparation Unit Unit
Three way Valve
FIGURE 3.7.Schematic diagram of the experimental vesse1 and its
accessones.
FIGURE 3.8. Photograph of the experimental vesse1
FIGURE 3.9. Photograph of the experimental apparatus
Measureâ Bubble Radidmm
FIGURE 3.10. Cornparison between the predicted and the
measured stable equüibriurn bubble radii.
CHAPTER 4: SUMMARY AND CONCLUSIONS We have investigated a heterogeneous system consisting of a bubble h e n e d in a
finite volume of a water-nitrogen solution. Both the thermodynamic potential and the pos-
sible equilibrium States for such a system were studied. The thermodynamic potentia1,b ,
was dehed for a heterogeneous system and the postdates of thermodynamics were used
to infer the condition under which the potential B was an extremum. Sirice the differential
of B vanishes at an equilibrium state, the necessary conditions for equilibrium couid be
derived. An expression for the radius of a bubble when the system was at equilibririm, Le.
equilibrium radius was then found in t e m of the liquid properties, i.e. temperature, pres-
sure and total number of moles of water and nitrogen. The potential B , was also expressed
as a function of these liquid properties and the radius of the bubble.
Moreover, it was found that depending on the conditions of the system. there may
be none, one or two equilibrium state sizes available for a bubble irnmersed in a water-
nitrogen solution. In the case of two equilibrium bubble sizes, the smaîler size is an unsta-
ble equilibrium state and the larger one is a stable equilibrium state. The latter was the
subject of the present work.
Furthemore, by considering the goveming parameten involved in this system, i.e.
mainly the temperature. the liquid pressure, the number of moles of the water and the
nitrogen. an experimental apparatus was designed and constructed. The radius of a bubble
under dinerent equilibrium conditions. Le. different temperature and pressure, was mea-
sured. AU the possible sources of error were investigated. The theoretically predicted bub-
ble radius, i.e. the one caiculated fiom the expression of the equilibriurn radius, was
compared with the one measured during the experiment. They were found to be in agree-
ment with each other at ail equilibrium conditions considered.
Based on the experimental observations, when a system is in a state of stable equi-
Librium and the liquid pressure is then changed, as a result of that, the properties of the liq-
uid smunding the bubble are also changed, e.g. the equiiibrium radius. If these changes
in the Liquid properties are considered, one can then use the Laplace equation to calculate
the equilibrium size of the bubble. However, if the changes in the properties of the liquid
are neglected, the Laplace equation may lead to contradiction.
Finally, the method used CO measure the gas content of the system was described.
Two stable equilibriurn measurements were considered in each expriment. By using the
expression for the equilibrium radius, a system of two independent equations was
obtained. The solutions to this set of equations gave the total number of moles of water
and nitrogen present in the system. The level of accuracy of this method was examined by
analyzing the error associated with the toial gas content measurement in one of the experi-
ment and then comparing to the previous methods used by a number of investigators. It
was found that, the method used in the present work is more accurate than those of the pre-
vious investigations.
Appendix A: Pressure 'Ikansducer Calibration The pressure transducer used in this work was a differential pressure msducer
(stathama, mode1 PM26ûTClt15-350). It was calibrated prior to the experiment by means
of known pressures and their comsponding voltage readings. The flush diaphragm of the
transducer was exposed to different pressures within its working range. These pressures
were then read by using a U-type manometer in miUrneten of water while the corre-
sponding voltages were shown by a multimeter in millivolts. A total of fourteen measure-
ment points was coilected. Table (A. 1) displays the recordeci measurements and Fig. (A. 1 )
shows a plot of those. A correlation factor of one was found that ensum a sufficient accu-
racy for the pressure transducer.
TABLE A. 1. The pressure transducer calibration measurements
Voltage / mv Differential Pressure / Pa
Appendix B: Capillary Method for Surface Tension
Surface tension can be measured by observing the position of the meniscus in a
capillW8 In the case of a glas capiiiary, wened by water, the meniscus rises and exhibits
an upward concave kee surface. If the same capillary is not wetted by a iiquid, e-g. mer-
cury, the meniscus is depressed and the surface is convex. In both cases, the pressure on
the concave side is higher by
where y is the surface tension of the liquid and R denotes the radius of curvature of the
surface of meniscus, assurned sphencal in shape and equal to the capiiiary inner radius.
In the fimt case. when the water is wening the glas capiiiary, the pressure differ-
ence is bdanced by the hydrostatic pressure. By considering that the water nses to a height
of h , then one may write
AP = pgh (B -2)
where p is the density of the water and g is the acceleration due to gravity. By combining
Eqs. (B. 1) and (B .2), one fmds
and after simpliSing
A very accurate cathetometer (~~ t ikon@) was used to rneasure h . The density of the
water, p , was found from handbookU by howing the room temperature. The imer diam- 7
eter of the capillary was measured to be 0.41 mm, and g was taken as 9.8 1 rn/sb. Three
same I.D. capiiiary were used in each measurement and the average value of h was used
in calculation of surface tension. Finaily this measured surface tension was compared with
the tabulated one at the same temperature.15 These values are ail shown in Table (3.1).
Appendix C: Properties of Nitrogen Nitrogen is a normally gas element with a m e l ~ g point of -2 10.OO°C and a boiling
point of -195.7g°C. It is slighty soluble in water while showing no chernical reaction. Its
solubility in water and corresponding Henry's constant are given in table below?
p.mol nitrogen /mol water I Pa
TABLE C. 1. Solubility of the nitrogen in the water and Henry's constant at Merent temperatures
Henry's Constant, KH Temperature Solubility
Appendix D: Properties of Water The properties of the water used in the present work were taken from Refs. 15 and
23. Table below displays those properties.
TABLE D. 1. Saturation pressure and surface tension of water
Temperature OC
1
20.0
25.0
25.2
30.0
35.0
37.0
37.1
Saturation Pressure, P, Pa
2339
3 169
3212
4246
5628
5979
6365
Surface Tension, y mN/m
I
72.7800
72.0050
7 1.9740
7 1 .23ûû
70.4200
70.0960
70.0798
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