EXPERIMENTAL SYSTEM EFFECTS ON INTERFACIAL SHAPE
AND INCLUDED VOLUME IN
BUBBLE GROWTH STUDIES
a Thesis Submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE
in the Department of Mechanical, Industrial and Nuclear Engineering
of the College of Engineering
2012
by
Gabriel Benjamin Wickizer
B.S. (Mechanical), University of Cincinnati, Cincinnati, USA, 2006
Committee Chair: Professor Raj M. Manglik
Abstract
Measurements in experimental studies of adiabatic single bubble growth
dynamics bear the combined effects of both the testing parameters and the test system
features. The present study investigates the impact of specific experimental methods and
system features, namely gas flow path, system volume, orifice construction, and
visualization surface, on the measurement of adiabatic single-bubble growth dynamics at
the tip of submerged capillary orifices. The present work jointly focuses on
characterization of bubble volume and shape during nucleation and growth. Photos of
bubble growth from a 1.75 mm capillary tube orifice were taken for glycerin, water, and
75 wt% aqueous glycerin for system volumes from 0.2 – 301.5 mL over a range of flow
rates from 0.01 – 1.6 mL/s, photographed through both planar and curved surfaces.
Interfacial aspect ratio and included volume from each system modification were
analyzed to determine the effect of system volume and to understand the impact of flow
metering on the constant gas flow boundary values in water and aqueous glycerin as well
as the influence of curvature in the visualization surface and the effects of liquid viscosity
in the presence of these system features. It was found that interfacial aspect ratio
decreases with increasing system volume and with decreasing viscosity over the full
range of flow rates considered. Additionally, interfacial aspect ratio decreases when a
cylindrical visualization surface is used, owing in part to horizontal magnification.
Furthermore, it is observed that bubble shape must be treated distinctly from bubble
volume when surface curvature is present or system volume is minimized.
Acknowledgements
To my advisor, Professor Raj M. Manglik: thank you. I would have done none of
this without you. I am grateful for the whole experience.
Dr. Jog, you have consistently been an inspiration and offered refinement at its
time. Dr. Iroh, your voice from the material perspective was enlightening to the formation
of the early experiments. Larry, it‘s all about computers; we both know that. Dr. K., Dr.
Kukreti, Dr. Lim, Dr. Son, Dr. Burrows, Dr. Ghia, and Dr. Stalcup, your educational,
professional and friendly input were the best of impromptu: timely and refreshing.
Special thanks to Steve Turek for helping me with the balancing table, Aravind
Subramani for your time, suggestions and friendship, not to mention your scholarly
contribution, Advait Athavale for your support in surface tension experiments, and
Deepak Sagar for the many inspiring conversations, data for comparison, unique ideas
and interesting perspective.
For those in my life who have been forced to listen to descriptions, justifications,
and hoi polloi from academia, I am blessed by listening ears. Wes, your unique input
helped shore up the underbelly of this process tremendously. Ted, I was aided by your
human resourcefulness and friendship. Les, your couch made this possible. Dan, you
helped inspired me to think about this in the first place.
Labmates! Omar, Deepak, Rohit, Rupesh, Prashant, Mohammad Ryad, Anirudh,
Vishaul, Utkarsh, Vignesh, Abinash, Amrutha: I am honored to share precious days.
Paul, Paul, Kallie, Ariel, Rachel, and Ingrid, you remind me each moment how
good a thing I have going in my family.
Dad, I have learned that an M.D. and an M.S. do not matter much in fatherhood
and sonship. I confess that it took a long time to sink in. Your love and passion get me
started and keep me going, from the first pizza napkin ever to reach absolute zero until
now. Mama, your ears are anointed, your wisdom is incomparable, and your love is
sure—this work reflects that.
Mariana, my dear, this is for you. Thanks for helping me learn to earth-bend.
Father, thank you for leading me to it. It was not where I thought it would be. Glory to
your name!
Table of Contents
Abstract
Acknowledgements
Table of Contents
i
List of Tables
iv
List of Figures
v
Nomenclature
x
1
Introduction
1
1.1
Motivation
1
1.2
Scope of Present Work
4
2
3
Literature Review
6
2.1
Important variables
6
2.2
Liquid Wetting and Orifice Construction
7
2.3
Flow Boundary Condition
11
2.4
Complex Rheology
16
2.5
Photographic Technique
18
Experimental Design and Method
i
21
3.1
Introduction
21
3.2
Fluid Selection
21
3.3
Fluid Property Measurement
24
3.3.1 Surface Tension
24
3.3.2 Fluid Rheology
25
3.3.3 Solution Preparation
25
Experimental System
25
3.3.1 Liquid Pool
25
3.3.2 Gas Flow Path
26
3.3.3 Gas Flow Rate
27
Experimental Procedure
28
3.4.1 Bubble Generation
28
3.4.2 Photographic Technique
28
3.4
3.4
4
Results and Discussion
31
4.1
Low Volume, Constant Flow
31
4.2
Effect of Liquid Viscosity
37
4.3
System Effects
42
4.3.1 Flow Metering
42
ii
5
4.3.2 System Volume
51
4.3.3 Orifice Construction
73
4.3.4 Visualization Surface
77
Conclusion
90
References
95
Appendix
100
iii
List of Tables
Table
Page
Table
Page
3.2.1
Thermophysical properties of test liquids
22
4.3.4.1
Comparison of areas measured from images taken through different
79
visualization surfaces
4.3.4.2
Common visual methods employed by researchers for bubble study
85
from 1941 – 2011
A.1
Temporal volumetric bubble development data
100
A.2
Temporal bubble aspect ratio development data
102
A.3
Comparative bubble aspect ratio development data
104
iv
List of Figures
Figure
2.3.1
Page
Schema of gas flow devices used in bubble growth
12
experiments from 1941 – Present
3.2.1
Schematic of experimental setup
23
4.1.1
Bubble departure volume for aqueous glycerin solutions at
34
Nc = 0.1, d0 = 1.75 mm
4.1.2
Temporal volumetric bubble development in glycerin for
35
Nc = 0.1, d0 = 1.75 mm
4.1.3
Bubble departure volume for aqueous glycerin solutions at Nc = 0.1,
36
d0 = 0.32 mm
4.2.4
Bubble necking behavior in a low-volume, constant flow air-water
37
system with porous restrictor, capillary orifice and plane visualization
4.1.5
Bubble necking behavior in a low-volume, constant flow air-glycerin
38
system with porous restrictor, capillary orifice and plane visualization
4.2.1
Bubble departure volume for researchers using capillary orifices in
41
aqueous glycerin at d0 = 2 mm
4.2.2
Bubble departure volume in aqueous glycerin for researchers using
42
capillary orifices, effect of orifice diameter
4.2.3
Bubble departure volume in aqueous glycerin solutions for
researchers using capillary tubing with varying orifice construction,
effect of liquid viscosity and orifice length
v
43
4.3.1.1
Temporal volumetric bubble development in glycerin, Nc = 0.1,
47
d0 = 1.75 mm
4.3.1.2
Temporal volumetric bubble development in water, Nc = 0.1
48
d0 = 1.75 mm
4.3.1.3
Interfacial shape in a low-volume, constant flow air-glycerin
49
system with porous restrictor, capillary orifice and plane
visualization surface
4.3.1.4
Bubble aspect ratio development in air-water interfaces, Nc = 0.1,
50
d0 = 1.75 mm
4.3.1.5
Interfacial shape in a low-volume, constant flow air-water
51
system with porous restrictor, capillary orifice and plane
visualization surface
4.3.1.6
Bubble aspect ratio development in water, Nc = 0.1, d0 = 1.75 mm
52
4.3.2.1
Bubble departure volume in glycerin for different system volumes at
58
d0 = 1.75 mm
4.3.2.2
Bubble departure volume in water for different system volumes at
59
d0 = 1.75 mm
4.3.2.3
Bubble interval versus gas flow rate in water and glycerin at Nc ~ 100, 60
d0 = 1.75 mm
4.3.2.4
Temporal volumetric bubble development in glycerin at Nc = 138.9,
61
d0 = 1.75 mm
4.3.2.5
Temporal volumetric bubble development in water, Nc = 110.4,
d0 = 1.75 mm
vi
62
4.3.2.6
Temporal volumetric bubble development for water and glycerin at
63
Nc ~ 100, d0 = 1.75 mm
4.3.2.7
Temporal volumetric bubble development in water at different system 64
volumes at d0 = 1.75 mm, ReL = 150
4.3.2.8
Temporal volumetric bubble development in water and
65
glycerin, the effect of system volume and viscosity at d0 = 1.75,
Q = 1 mL/s
4.3.2.9
Temporal volumetric bubble development in glycerin at
66
d0 = 1.75 mm, ReL ~ 1
4.3.2.10
Interfacial shape in an air-glycerin system with Nc = 138.9,
67
capillary orifice, and plane visualization surface
4.3.2.11
Temporal bubble aspect ratio development in glycerin at Nc = 138.9
68
d0 = 1.75 mm
4.3.2.12
Interfacial shape in an air-water system with Nc = 110.4,
69
capillary orifice, and plane visualization surface
4.3.2.13
Temporal bubble aspect ratio development in water and
70
glycerin at Nc ~ 100, d0 = 1.75 mm
4.3.2.14
Interfacial shape in air-glycerin and air-water systems with
71
Nc ~ 100, capillary orifice, and plane visualization surface
4.3.2.15
Temporal bubble aspect ratio development in water, Nc = 110.4,
72
d0 = 1.75 mm
4.3.2.16
Temporal volumetric bubble development in glycerin for researchers
with capillary orifices, effect of system volume, d0 = 2 mm, ReL ~ 0.1
vii
73
4.3.2.17
Bubble departure volume in water and glycerin for different system
74
volumes at d0 = 1.75 mm
4.3.2.18
Bubble necking behavior in a constant flow air-glycerin system with
75
added volume, capillary orifice and plane visualization
4.3.2.19
Bubble necking behavior in a constant flow air-water system with
76
added volume, capillary orifice and plane visualization
4.3.3.1
Temporal volumetric bubble development in water, capillary
79
orifice versus plate orifice construction at Q = 1.6 mL/s
4.3.3.2
Interfacial shape in air-water systems with plate orifice and
80
cylindrical visualization surface (a,b) versus capillary orifice
and plane visualization surface (c)
4.3.4.1
(top to bottom) Steel spheres of 3.96, 4.12, 9.52 mm diameter
83
on an identical base in (left to right) open air, water bath in
plane-faced Perspex, water bath in cylindrical-faced borosilicate
4.3.4.2
Temporal aspect ratio development in glycerin visualized through
84
different surfaces at d0 = 1.75 mm, Vb = 0.27 mL
4.3.4.3
Temporal bubble aspect ratio development in water visualized through 85
different surfaces at d0 = 1.75 mm, Vb = 0.05 mL
4.3.4.4
Interfacial shape in air-water systems with capillary orifice of
86
d0 = 1.75 mm, effect of visualization surface
4.3.3.5
Interfacial shape in air-glycerin systems with capillary orifice of
d0 = 1.75 mm, effect of visualization surface
viii
87
4.3.4.6
Visualization surfaces used for bubble studies depending on image
88
data from the 1940s to the present
A.1
Snell's-Law magnification of a silhouette: a. Refraction through a
a curved pane b. Traces of 1 Critical, 2 Supercritical, and 3 Subcritical
light rays c. Dependence of image magnification on distance
of image magnification on distance
ix
110
Nomenclature
A
surface area [cm2]
A‘
surface area per unit volume [cm-1]
a
semimajor elliptical axis
b
semiminor elliptical axis
BI
bubble interval [ms]
c
semiminor ellipsoidal axis in depth normal
d
diameter [mm]
H
height [mm]
L
capillary orifice length
N
number of elements
P
pressure [Pa]
Q
volumetric gas flow rate [mL/s]
R
tank radius [mm], linear regression error where specifically noted (4.1)
r
radius [mm]
ReL
Liquid-Displacement Reynolds Number (4Qg/d0l)
V
Volume [mL]
W
width [mm]
Greek Alphabets

aspect ratio H/W

discrete element
x

gradient operator

viscosity [cPs]

refractive index

density [Kg/m3]

gas-liquid interfacial tension [mN/m2]

mathematical summary of elements

elapsed time for a given period [ms]
Subscripts
0
value taken at the orifice tip
b
bubble
d
departure
e
ellipsoid
g
gas phase
i
internal to the orifice
l
liquid phase
m
optically magnified value
max
maximum value on the range
O
orifice
s
system
sph
spherical
RMS
root-mean-squared value
xi
Chapter 1
Introduction
1.1 Motivation
Fluid particles have claimed a space in the nexus joining science and industry
from before the industrial era [1-4]. The continuous, closed gas-liquid interface—of
which the bubble is an important instance—ranks among the most important members of
the fluid particle class, playing a central role in a panoply of processes in the industry of
our common era and the natural human environment. Consequently, many researchers
have invested time on the subject throughout the past 150 years and several have
thoroughly reviewed these efforts, lending a firm foundation to the growing discourse on
what, how, when, and where the bubble comes to be [2,3].
The formation of gas bubbles in liquid media characteristically signal the
occurrence of boiling heat transfer, foaming, chemical reactions, and pressure-driven
diffusion. Early in the twentieth century, the attraction of fine suspended solids to the
gas-liquid interface was studied in conjunction with electrostatic properties as the bubble
became a charged surface [5]. This work would have direct applications in the refinement
and pumping of petrochemicals, exemplified by the 1950 ―Conference on the Formation
and Properties of Gas Bubbles‖ held by the Institution of Chemical Engineers [6]. At the
same time, the development of plated columns and the need for more highly effective
distillation brought emphasis to the dynamics of bubble growth and its relationship to the
geometry with which bubbles are formed [5]. Later researchers would consider the
bubble for rotary and stagnant sparges as well as bubble reactors [7]. As the knowledge
1
of single bubble mechanics and the physics involved in their nucleation, growth,
departure, rise, and oscillatory behavior grew, the bubble claimed attention from other
fields as the principal analogy to other phenomena, such as the emergent economic
terminology referring to the near-instantaneous growth and collapse of industrial sectors,
e.g. the ―Housing Bubble.‖ The gas-liquid interface has also offered a tangible system for
the study of extra-mechanistic explorations into chaotic behavior, under-water acoustics,
detonation systems, bio-computational devices, and even work on the Grand Unified
Theory [8-10]. The expanding field of single-bubble growth and its new and notable
applications roots itself in the systematic study of the interaction of a continuous gas
phase in contact with a continuous liquid phase in the presence of a driving force. To this
elemental system we return, focusing on the system characteristics and fluid properties
that have influenced this field of study in past and recent years.
Fluid mechanics considers the bubble through analytical and computational
means, founded on the observer‘s experience with a controlled system. This method
relies on the Navier-Stokes equations and their ability to relate phenomena occurring on
surfaces and bodies. Green‘s Theorem provides a method by which a surface integral
may be transformed into a volumetric integral, allowing physical forces acting on a plane
and those inherent to a body in its environment to be compared directly, each being a
mathematical bijection of the other. Single bubbles have been considered fundamentally
as both a surface of contact between two continuous phases [11] and a volume of gas
surrounded by liquid [6]. The simplifying assumption of axisymmetry enables the direct
comparison between these two interpretations of the bubble, linked by the radius of
curvature. This assumption has been reported to be adequate under quite a wide range of
2
liquid properties and system characteristics [12]. Researchers concern themselves either
with understanding the total volume developed during the bubble growth cycle, the gasliquid contact area present at the time of departure, or both simultaneously.
In much of the literature, researchers assume that the bubble is a sphere (implying
axisymmetry) with some degree of justification [2]. While this simplifies the analysis
considerably, such understanding of the accurate profile of the bubble and its total surface
area is then restricted to the classic expression for the surface area of a sphere of radius r:
Vsph = (4/3)r3
(1.1.1)
Asph = 4r2
(1.1.2)
Asph‘ = 4r2/(4/3)r3 = 1/3r-1
(1.1.3)
where Asph‘ is the ratio of surface area Asph to volume Vsph. Commentary and data
from the literature clearly show that where axisymmetry holds, the bubble geometry is
often aspherical [13]. Researchers such as McCann and Prince [14] have considered the
geometric state of bubbles at departure to define regimes of growth or state diagrams. It is
evident that the bubble may take on different values of the ratio A‘ depending on which
regime is under consideration. A simple secondary case would be the prolate (or
conversely oblate) ellipsoid. It is a seemingly tractable problem, yet it involves an
elliptical integral of the second kind and has no analytical closed-form solution. On the
other hand, if taken to its limit, an upper bound for the surface area of a prolate or oblate
ellipsoid (where a, b and c are axes and a = b) may be stated as in expression
Ae = 2ab
(1.1.4)
Ve = (4/3)abc
(1.1.5)
Ae‘ = 2ab / (4/3)abc = 3/2c-1
(1.1.6)
3
where Ae‘ is the ratio of ellipsoidal surface area Ae to ellipsoidal enclosed volume Ve. If
the researcher were to consider a bubble that could be described as a prolate ellipsoid, the
possible range of estimation for the surface area, regardless of experimental uncertainty,
would be almost an order of magnitude, as Ae‘/As‘ is nearly equal to five. The
fundamental importance of phase contact area to future research in the growing field of
bubble growth applications demands increased analytical accuracy, improved modeling,
and refined experimental methods that will enable the scientific community to develop a
clearer understanding of the shape of gas-liquid interfaces.
1.2
Scope of Present Work
Even under adiabatic conditions without the occurrence of a chemical reaction or
the introduction of oscillatory energy fields, bubbles may be both induced (e.g.,
cavitation, air-jet entrainment) and produced. It is to the latter that this work refers,
dealing exclusively with the entrance of a gas phase into a liquid pool through an orifice
under the influence of a gas pressure gradient, whether the orifice back-pressure is static
or dynamic. The present study investigates adiabatic, single bubble growth at constant,
low rates of gas flow (0.002 – 1.6 mL per second). Bubble growth occurred in isothermal,
infinite static pools of water and aqueous solutions of glycerin and hydroxyethyl cellulose.
The primary experimental system used was augmented to ensure a constant instantaneous
gas flow rate throughout the entire range of consideration, with very nearly zero pressure
capacitance (i.e., the absence of a gas chamber effect). The fluids chosen represent two
pure reference liquids of different viscosity and a comparator liquid exhibiting NonNewtonian (Power-Law) rheology but having the same static viscosity as the aq. Glycerin
solution. This study is completed with a 1.75 mm orifice, but the work is extended to an
4
orifice of 0.32 mm to examine the effect of orifice diameter and supplemented with
photos that offer temporal resolution of the growth dynamics to tenths of a millisecond.
The experimental system and its results are compared broadly with the systems
and results of other researchers to provide insight into first and second-order system
effects on the experimental results and to propose revisions to future experimental
methods that will lead to more harmonious presentation and enhanced understanding. The
relationship among system volume, interfacial included volume, and bubble aspect ratio
are explored with the help of hi-speed digital photography and edge-tracing analysis.
Additional effort has been undertaken to document the difference between the two
classic visualization surface geometries used in bubble growth studies, i.e. pools with
rectangular and circular cross-section. The underlying optics of the common silhouette
photography method in bubble growth study is explored in a cylindrical vessel where a
rectangular vessel may be considered one instance of a generalized cylinder and the
impact of surface curvature on the temporal bubble volume and bubble profile
development are presented for a single case in both water and glycerin.
5
Chapter 2
Literature Review
2.1
Important variables
Tate‘s 1864 work on droplet mass precedes a majority of the literature on bubble
growth (and other fluid particles as well) and is occupied with the relationship between
mass and orifice diameter at a constant flow rate [4]. He also documents the effect of
different liquid systems at constant density, showing a definite effect from the liquid
interfacial forces. He controlled not only for flow rate but also for the secondary effect of
surface wetting, using surfaces of knife edge thickness all the way to infinite flat plates
while documenting the effect on the resultant droplet‘s weight. He supplemented his
discussion with an account of the general droplet geometry observed by the naked eye.
Several years later, Coppock‘s and Meiklejohn‘s unique study reports bubble volume
independent of the flow rate, as their experimental aim was to study mass transfer in
bubble columns [15]. They used a photographic method to study bubbles rising in a
cylindrical column and correlated the bubble volume through a proportional relationship
with liquid surface energy, density, and orifice diameter, modified by the gravitational
constant. They assume that the resultant bubble shape is spherical and do not consider
contact area in relationship to mass transfer. Their results are cited by following studies
because of their conclusion that viscosity has little effect on bubble size in the flow range
considered, yet they emphasize the range wherein the gas flow rate itself does not show
an appreciable effect, e.g., the constant-volume regime. They used glass capillaries for
6
their experimentation and assume a spherical shape, using this and the bubble count to
determine equivalent volume by collecting the bubbles in a burette. Both studies favored
a static treatment of the interface, but later work adheres to their focus on interfacial
included volume and profile.
In the opposite and most recent extreme, some researchers have found single
bubble growth to be a unique phenomenological problem pertaining to chaotic or
aperiodic behavior, and have conducted experiments with only minor modifications from
their predecessors [8-10]. Cieslinski [8] used a cylindrical apparatus equipped with a
hydrophone to examine the onset of chaotic behavior and the pressure oscillations
transmitted to the liquid by the growing bubble from a glass orifice, in combination with
a more classic photographic technique to determine the frequency of bubble growth. His
apparatus was equipped with a large chamber volume to ensure constant-pressure and he
varied the flow to achieve different frequencies, finding that a single flow rate could
result in multiple frequency values. His remark on the shape of the departing bubble is
that it results from deterministic forces and that the shock of break-off tends to elongate
the bubble; he does not report the volume of the bubble during growth. Although this
would seem to contradict the primary importance of volume and profile, it simply isolates
the temporal element that binds both key variables, illustrating the transition from the
early static approach to the now dominant dynamic treatment of bubble growth and
development.
2.2
Liquid Wetting and Orifice Construction
Early in the recent century, Eversole et al conducted a study of bubble growth by
using photography to record the silhouette of air-alcohol interfaces obscuring light from a
7
mercury vapor lamp [5]. They reported photos taken at regular time intervals and
measures of the gas flow rate with a careful emphasis on the periodic nature of bubble
growth as well as the interaction of gas, liquid and solid phases at the point of
introduction of gas into the pool. They worked with nitrogen gas and alcohols in a square
glass tank and corrected the flow rate for vapor pressure owing to the volatile nature of
the liquids observed. They focused on relationships among bubble pressure, gas flow rate,
liquid surface tension, hydrostatic head, and the shape and volume developed during the
bubble growth cycle. They did not consider viscous effects other than the viscosity of
nitrogen and their system used glass capillary orifices and alcohols, all of which are
characterized by very low surface energy leading to excellent wetting behavior. Siemes
and Kauffman report a lesser volume at departure for bubbles in aqueous glycerin than
for water at low to moderate flow rates and a viscosity of less than 100 cPs [21]. These
researchers, like Eversole et al, use glass capillaries with a metal disc attached at the tip.
They also rely on capillary length to develop a continuous, smooth pressure gradient in
the air supply to the orifice tip.
Datta et al report some of the earliest parametric results dealing with viscous
effects and other system effects, such as orifice orientation [6]. They compared the
volume of bubbles growing from vertical upward-facing capillaries to those in a
horizontal alignment to bear out the effects of interfacial attraction between the orifice
material and the gas-liquid interface. Their finding exemplifies a place where the bubble
mechanics are system-dependent in a way that has not been thoroughly investigated by
many of the subsequent researchers, although Kumar and Kuloor do consider the effect of
the orifice axial angle in their 1970 review [3]. Additionally, Datta et al report a negative
8
correlation of interfacial included volume with increasing liquid viscosity over a range of
viscosities 1 to 100 times the value for water. This has been explained later by many
researchers as pertaining to the range of viscosities and flow rates considered, though
none has undertaken to determine a critical value of viscous resistance required to
significantly impact single bubble growth. Datta et al used aqueous glycerin solutions to
vary the value of system viscosity, which in consequence decreases the gas-liquid
interfacial tension. The glass capillaries employed also decreased the solid-liquid
interfacial tension. Though they see little influence of viscosity on departure size, they
report that higher viscosity produces a higher degree of geometric regularity in the
departing bubbles. Additionally, Datta et al discuss the relationship of bubble shape and
orifice size in the static volume regime, where bubbles are spherical at detachment in
orifices of a diameter from 0 – 0.4 mm, ellipsoidal from 0.4 – 4.0 mm, and develop into
spherical caps at higher diameters.
Leibson et al, 1956 [22,23] used a long cylindrical steel column modified with flat
viewing ports for photographs to research the nozzle behavior in bubble growth as
compared to air jets flowing into ambient air. They augment the orifice tip with a disc
and use square-edge and round-edge orifices to determine the relationship between the
orifice end condition and the vena cava of flow, considering a range of flow rates from
the single bubble regime to the formation of fully-developed turbulent jets. They use an
orifice Reynolds‘ number, define the regimes of flow, and report a critical pressure ratio
for different orifices. The discussion makes evident that the flow interacted with the walls
of the column at higher Reynolds Numbers.
9
In their much-cited work from 1960 [24,25], Davidson and Schueler used a
cylindrical tank fitted with three different types of orifices above a gas chamber, fed
alternately by a compressed-gas cylinder or a compressor pump. They designed their
orifices based on the flow boundary condition desired, using a brass fitting with a plate
orifice insert above the chamber to create bubbles at constant gas pressure. To achieve
the opposite extreme for ―very small flow rates‖ in the range of 0 – 120 mL per minute,
they use long capillaries fed by an air compressor and small plate orifice inserts with a
brass fitting atop a 45 L drum, claiming that both give a constant rate of gas flow. There
is appreciable variance in their results in this range. For constant, large flow rates, a plate
orifice is used in conjunction with a sintered fitting to ensure a high pressure drop across
the orifice. In all cases, a Perspex ring is attached to the orifice to dispel recirculation in
the pool from the forming bubble.
Ramakrishnan et al raised the assertion that, for a given liquid, bubble departure
volumes at a sufficient flow rate will not depend on the orifice diameter [20]. It is
interesting to note that they used a glass packing in their orifice fitting to maintain a
smooth pressure gradient in the air supply to the orifice. Both in this study and their 1970
review article [3], Kumar and Kuloor urge the need for researchers to consider the head
of the column when calculating the gas flow rate to maintain consistency in the measure
of this key value for future bubble growth research. Kumar and Kuloor also depict a
different anemometer for the measurement of gas flow that ensures bubble volume is
taken at ambient pressure. They take great care to saturate the gas phase with liquid vapor
before introducing it to the pool.
10
Jamialahmadi‘s relatively recent study uses plate orifices with a glass packing to
ensure a constant-flow condition [17]. Like the previous work on mass transfer in bubble
growth, these researchers take the concentration of dissolved gas into account and
perform a systematic study of the effect of liquid properties on bubble growth. Their data
shows that, below about 240 mL per minute, bubbles generated show weak dependence
on the viscous properties of the liquid pool.
2.3
Flow Boundary Condition
The literature records experiments conducted at a constant-flow or a constant-
pressure boundary condition. The definition behind this claim is unclear for some
researchers, but Davidson and Schuler worked to exclusively achieve one or the other
condition in high and low flow rate ranges by augmenting the gas flow path and the
experimental geometry (they do not study flow rates lower than 1 mL/s) [24,25]. Park et
al [26] demonstrate the impact on bubble growth when it occurs at flow conditions
intermediate to both these extremes, and provide a point of reference for other works that
do not directly report the regime of gas flow during experimental operation. Figure 2.3.1
demonstrates schema of typical flow metering used in bubble experimentation from 1941
to the present.
11
P1
P0
P0
P0
P0
P1
Q0 P
1
P0
P1
P1
Q0,1
Q0,2
Q0
Q0
P1
IV. Mass Flow
Control (MFC)
P2
V. MFC with Flow
Straightener
Q0
I. Capillary
Tube
II. Gas-Fed
Chamber
III. Syringe
Pump
Figure 2.3.1 - Schema of gas flow devices used in bubble growth experiments from 1941
– Present
Kupferberg and Jameson conduct a photographic study at intermediate and
constant-pressure flow conditions using water alcohol mixtures [11]. They use the
sintered-fitting technique developed by Davidson and Schueler for constant flow at high
flowrates and find that the agreement in their results is inhibited by their own disclusion
of the disc that Davidson and Schueler had attached to the orifice tip. Both researchers
use cylindrical tanks, yet Kupferberg and Jameson choose a diameter only 70% the value
of Davidson‘s and Schueler‘s work, making recirculation more apparent.
McCann and Prince present bubbling regimes that result from intermediate and
constant-pressure conditions using a cylindrical tank with plate orifices and a variablevolume orifice chamber [14]. They describe phenomena such as ―double bubble pairing,‖
showing depictions of paired bubbles during growth that have complex geometry. They
describe six different bubble regimes that develop and make the observation that
detachment from the orifice is the result of a pressure difference between the bubble and
the chamber. They urge the need for researchers to take account of phenomena such as
12
pairing and coalescence as limitations to the sphericity assumption often used to model
growth. Kulkarni and Joshi restate this in a comment about the limitation of
Ramakrishnan et al‘s model [2].
A study by Hayes et al, 1959 directly examined the effect of a gas chamber
mounted prior to the orifice [16]. They consider the systems where orifice and chamber
radii are equivalent and increase the chamber radius to a limiting size such that further
increase no longer affects bubble growth. Tadaki and Maeda studied the effects of
chamber pressure capacitance on bubble growth in different fluids using photographic
techniques [27]. They recount the gas flow boundary condition as directly dependent on
the chamber volume. Their definition of the gas chamber is unique in that they consider
the entire volume from the metering device all the way to the orifice in their measure,
varying this entire volume to discover its effects. Tadaki and Maeda also introduce the
bubble diameter as a mean statistical quantity, according to a logarithmic distribution.
The vast majority of experimental literature on single bubble growth represents the
bubble without treatment of volumetric variance even in cases where the bubble shape is
reported to be poorly approximated by a sphere.
Some researchers, such as Khurana and Kumar, 1968 and Park et al, 1976, have
endeavored to deal with the gas chambers by mapping the chamber pressure fluctuations
and bubble dynamics at a range of conditions until no further change is seen with an
increase in chamber volume [26,28]. Others have much less rigorously treated the flow
characteristics in their systems, focusing on the achievement of consistent average
volumetric air flow and steady, periodic bubble formation. Owing to this, understanding
of flow conditions intermediate to constant-pressure and constant instantaneous
13
volumetric flow are not well harmonized across the same variety of liquids in which
bubble growth has been studied, as some researchers remark [29]. The literature is quite
clear that periodic single bubble growth achieved by different gas flow sources shows
great variation in the bubble profile [29]. Park et al study the effect of different chamber
volumes by inserting a transducer in the chamber just below the orifice and filling this
cavity with different volumes of the test liquid. They use pressure-time traces and
thermodynamic analysis to investigate the transition of bubble behavior at the constantflow condition to the behavior at the constant-pressure condition along with supporting
expressions [26]. Satayanarayan et al. [30] study exclusively the constant pressure regime
by increasing the volume of the gas chamber to ensure freedom from the effect of
pulsation in the chamber.
Fainerman et al [31] study the pressure developed within a bubble in various
solutions to understand dynamic surface tension by the maximum bubble pressure
method. They use a device that has a large chamber volume to damp pressure oscillations
in the chamber, ensuring a constant-pressure condition, along with glass capillaries in a
downward-facing vertical orientation. Their work clearly demonstrates, under constantpressure conditions, the presence of viscous interaction during Stoke‘s Flow bubble
growth. This range of velocities is the same that has been reported by workers using the
alternative flow extreme to show very little viscous effect [1,6,15,25].
Subramani et al, 2007 [19,32], refine the understanding of the effects of liquid
properties in a careful study of pure liquids at a wide range of viscosity and surface
tension, but similar specific gravity. This photographic study took place in a cylindrical
beaker and employed a syringe pump as a constant volumetric displacement source
14
driving air through a network of tubes to bubble from stainless steel capillaries. Although
the study does not characterize the gas flow boundary condition, it does report the
temporal and spatial evolution of the growing single bubble with direct photographic
comparison, allowing one to easily see that it takes place at an intermediate flow
condition. The system volume is reported to include four 60 mL syringes, totaling more
than 250 mL. This is near the center of the intermediate chamber region as reported by
Tadaki and Maeda, 1963, and the bubbles formed do experience a waiting time, as
observed for other constant-pressure and intermediate results in the literature [26,28].
Subramani et al take the primary thermophysical factors affecting bubble growth to be
liquid surface tension, viscosity and density, but also point out that the viscous effects
deliver the majority of their influence during necking and departure. Care is taken to
demonstrate how both the profile and the volume of a bubble at departure depend on the
liquid properties and the gas flow rate, as well as the orifice diameter. This study was
conducted with a non-wetting orifice of negligible thickness and sees viscous retardation
play a role even at low flow rates in contrast to the research conducted with glass
capillaries and very low gas chamber volumes.
Teresaka and Tsuge deal specifically with the non-spherical shape of bubble
profiles with a model that is applicable only for the constant flow condition, i.e. where
the instantaneous and average volumetric gas flow rates are equal [13]. They report both
shape and volume based on a video-camera study of bubble profile and formulate a nonspherical linear element approximation solved computationally. They use an orifice
packing in some experimental cases to straighten the flow as well as capillary tubes in
other cases, but with no measure of the instantaneous volumetric flow rate. They design a
15
constant flow experiment with hydrogen and water or aq. glycerin based on the proposal
by Takahashi and Miyahara, 1976 that the constant flow condition can be claimed if the
orifice geometry has L/di4 > 1012 m-3, where L is the orifice length and di the orifice
diameter. In addition to flow rate, Teresaka and Tsuge measure the pressure difference
across the orifice. Capillary flows are compared to the packed orifice flows to find that
the condition for constant flow is that the pressure drop across the orifice P is such that
P/(4/di) > 1. As with the studies on intermediate flow condition, pressure-time traces
refine the understanding of an individual growth cycle. They compare the effect of
different orifice geometries, yet do not report the impact of these geometries on the
pressure difference. No direct attention is given to the mechanistic impacts of secondorder orifice-liquid interactions in the experimental literature.
2.4
Complex Rheology
Miyahara et al used a tank fitted with a pressure chamber and a bronze capillary
nozzle with glass bead packing as well as stainless steel capillaries to study the effects of
high viscosity for shear-independent and shear-dependent systems in the form of aqueous
glycerin and carboxymethylcellulose (CMC or Cellulose gum), respectively [33]. They
used videography to find that aqueous CMC shows no tendency to reach a point
independent of chamber volume as did the other liquids for the same range of volumes
[33]. They fed the gas chamber with great care to avoid shell vibration by capillary
introduction of the air to the axis of the volume, above the liquid depth, where it passed
through the orifice and bubbled into a tank with square cross section. They report higher
viscosity values for glycerin than do any of their comparators for equivalent liquids.
16
Viscosity in general is a strong function of temperature, yet especially in the case of
glycerin [34]. Miyahara et al control the temperature using a circulating water bath, but
do not directly report it. This study‘s finding was that viscosity affects the necking
behavior during bubble growth and the subsequent rise of the bubble. Furthermore, the
bubble is pictured as a simple sphere in the first stage of growth, whereas the neck is
considered to consist of tangential arcs defining the collapsing portion of the bubble
volume during departure. Viscosity‘s effect is related to the solid-liquid contact angle,
which is included as a term modifying the surface tension force in their proposed model.
They make the additional observation that viscous resistance will develop a higher initial
velocity at nucleation than would occur in an inviscid liquid. Rabiger found in his 1984
study that single bubbles tend to elongate in aqueous polyacrylamide (PAA) as compared
to viscous pure liquid pools. He employs a gas chamber to observe bubble formation at a
submerged orifice under high rates of flow (2,100 mL per minute) [35].
In 1959, Acharya et al studied aqueous solutions of glycerin, PAA, polyethylene
oxide (PEO), and CMC, finding that the presence of complex rheological behavior does
not significantly alter the volume or rising behavior of the bubble; however, they study
relatively low concentrations of these rheological enhancers and make no comment on
the deformation of the bubble profile in viscoelastic liquids [36]. Coppock and
Meiklejohn [15] found only that methyl cellulose additives in water alter the behavior in
the same way that surfactants do, only to a lesser extent owing to a smaller reduction in
the system interfacial tension. In a series of studies by Jiang and other colleagues [37-39],
aqueous polyacrylamide and aqueous glycerin were investigated photographically with
the help of a laser. Jiang et al mention the shape of the bubble as an ―inverse tear-drop,‖
17
which is the more classic observation in pure liquids. Jiang et al find that, in the
intermediate flow regime (gas chambers of 30 – 270 mL) at low rates of flow (0.1-0.6
mL/s) that low concentrations of PAA may not appreciably deform the bubble profile.
2.5
Photographic Technique
A recent study by de Witt et al [5] points to the importance of imaging technique
by conducting a study to correct the optical distortion arising from light-wave refraction
through cylindrical faces. Though precautions have been taken in many of the singlebubble experimental methods presented here [22-25,37,45], several of these studies rely
on photographic methods without stating action taken to minimize uncertainty brought
about through optical distortion, analogue-to-digital conversion, image enhancement, or
lighting [1,6,8-11,14,15,27,32,37,46-50]. Many of these studies present photos taken
through cylindrical faces. The dispersing of light rays by curvature in transparent media
has been well known since the birth of optical physics. As bubble research depends
heavily on photographic and optical techniques, some efforts to minimize distortion are
worth mention here. One of the earliest studies to note the possibility of optical
interference was Garner and Suckling, 1958 [41][41]. These researchers machined an
optical correction of the same material as the pipe material, then further corrected for the
impact of the difference in index of refraction between water and acrylic resin. This is in
contrast to a previous study released by Garner and Hammerton [51] where no effort was
made. Although many researchers used exclusively flat-faced geometries for viewing,
recent studies have begun to take note of optical uncertainty when using photography to
study bubbles. Some studies rely on optical verification and take no further action, some
18
match the refractive index of the test liquid to that of the tank material, and others simply
use flat-faced geometry [5].
Quigley et al [12] relied on a square apparatus and orifice plates above a gas
chamber to photograph bubble growth and study mass transfer at the interface. They
claim that the pictures justify the sphericity approximation used, although their
photographs show irregular prolate ellipsoids with unsteady surfaces in some cases. They
find both viscosity and density to have little effect at the depth considered, though cubic
feet are the operative scale and the maximum viscosity considered was only about 420
times the value for water.
In a pair of studies in 2002 and 2003 [43,44], Li et al use photographic methods
with a cylindrical tank fitted with plate orifices of different sizes above a gas chamber to
investigate wake development in the bubble column and its effects on the subsequent
formation of bubbles. Birefrigence photographs show visual record of the residual
stresses in a cross-section of the pool, observed to relate to the elongation of the lower
portion of the bubble, creating the ―inverse teardrop‖ shape. Li et al conclude that single
bubble growth in viscous Newtonian and Non-Newtonian liquids may be represented by
two-dimensional models and make no mention of system augmentation for optical
distortion.
Van Krevelin and Hoftijzer‘s 1950 study deals with bubble columns and reports
the gas-liquid contact area developed during growth as the independent variable for the
study of mass transfer [52]. They conducted their photographic experiment in a cylinder
with no amelioration of the optic effects brought about by the curvature of the surface,
while holding the assumption of sphericity as the basis for their calculation of contact
19
area. Krevelin and Hoftijzer also treat the impact of viscosity and contact angle of the test
liquids, where viscosity tends to lower the onset of a constant frequency regime (similar
to Ramakrishnan et al‘s observation on surface tension) and the wetting behavior of the
orifice is actually modified by a parafin wax coating to improve the validity of the results.
Experimental approaches to the study of single bubble growth in infinite, stagnant
pools show a considerable variation in design, leading to differences in the impact of
secondary effects on the results. Owing to the scope of the studies conducted, their
individual conclusions must be re-applied with care to results generated in similar, but
not the same, experimental systems. Researchers have mentioned the tendency of
viscoelastic fluid to develop elongated bubble profiles [33,35]. Kulkarni and Joshi [2]
urged the evolution of a common approach to the study of bubble size; yet, in the
tradition of McCann and Prince, there is also a need for a revised taxonomy of bubble
shape [14]. de Witt et al remind us that we must correct for unintended system effects in
the visual results [5]. The interrelationship of viscosity, liquid wetting behavior and the
flow boundary condition demand that emergent research calculate and report the flow
boundary condition [5]. To enable the furtherance of the field, some of these effects
deserve detailed treatment; the present study will concern itself with the change in
expression of deterministic forces during bubble growth owing to system effects from the
flow metering device, the system volume, the orifice construction, and the visualization
surface employed in the study of volumetric and spatial development in bubble growth.
With the help of such treatment, future studies will be better equipped to make general
connections among data over an ever wider range of parameters.
20
Chapter 3
Experimental Design and Method
3.1
Introduction
To study adiabatic single bubble growth in ―infinite‖ pools of pure and complex
liquids at steady state conditions with constant low rates of gas flow, an experimental
system has been designed and is depicted in its entire assembly in Figure 3.2.1. All
experiments take place at near-standard-temperature-and-pressure (STP) conditions, with
an ambient pressure of 745 ± 10 mmHg and a conditioned ambient temperature of 23 ± 2
o
C. An explanation of the development of this system and the experimental method
follows.
3.2
Fluid Selection
Filtered, dry air provided by a compression system is selected for the gas phase in
all cases, yet the system is verified in a single experiment with Argon from a cylinder to
establish the expected behavior using clean inert gas. Steam, condensed as de-ionized
water, purchased from Kroger® was used as a pure fluid and solvent for experimentation.
Solutions of Glycerin (Tedia™ 99.5% purity, Sigma Aldrich Chemicals, 97% purity)
were created to investigate the effects of Newtonian and Non-Newtonian rheology on
single bubble growth under the chosen conditions. The hydrostatic head at the orifice tip
is controlled by first choosing a minimum column height of 101.6 mm to ensure that the
liquid column was not a dominant influence on the bubble behavior [2,14] and then
21
augmenting the column height of solutions with lower density to achieve agreement in
the hydrostatic head at the orifice tip within 3%. Liquid column height is measured by
using a leveling table (not depicted in the schematic) and superimposed rule in 1/8 inch
increments. A tabulated comparison of the thermophysical properties of test liquids is
included as Table 3.2.1.
Table 3.2.1 – Thermophysical properties of test liquids
Liquid

Concentration


3
[wt% Glycerin] [Kg/m ] [mN/m] [Cps]
Water
0
998
72.4
1
aq. Glycerin
75
1184
66.6
24
Glycerin [34]
100
1258
63.0
773
22
9
8
11
14
12
13
7
6
10
5
1
4
3
2
Figure 3.2.1 - Schematic of experimental setup
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Air compressor
Dessicator
Filter-Drier
Pressure Regulator
Mass Flow Controller
Sintered Flow Straightener
Square-Edged Syringe Tip
Perspex Tank
Glass Funnel
Soap-Film Flow Meter
Low-Heat High-Intensity Lamps (2)
Diffuse white reflector
High Speed CCD Camera
NAC Hi-D Cam II Image Capture System
23
3.3 Liquid Property Measurement
3.3.1 Surface Tension
Equilibrium and dynamic gas-liquid interfacial tension (surface tension) is
measured by the maximum bubble pressure method using a Sensadyne QC6000 system.
Both glass and stainless steel capillaries are used and the most consistent results are
reported as the average of a group of three readings, where a reading consists of the
average of more than five bubble cycles, following a stabilization period. The
temperature probe is calibrated daily using the lab spirit-thermometer after a stabilization
period of about two minutes. The measured voltage signal in the Sensadyne is generated
by a differential pressure transducer that is calibrated for each individual reading using
de-ionized water and ethanol as reference liquids.
Equilibrium surface tension must be measured in the Stoke‘s Flow regime, where
ReO << 1. This may also be achieved by increasing the bubble interval beyond a value
where the value of liquid surface tension depends on the bubble interval. This is ensured
by a dynamic surface tension experiment where the bubble interval was successively
increased and the change in surface tension determined over several intervals. It was
found that the change in the measured value is on the order of experimental precision at a
bubble interval of about 10 seconds. All equilibrium values are reported based on values
taken at intervals of 30 seconds or greater.
24
3.3.2 Fluid Rheology
Liquid rheology is measured using an AR2000 rheometer (TI Instruments) with
concentric conical spindle, held at a constant temperature of 23 oC by a circulating water
bath. The working range of this device can be determined using Newtonian liquids and
observing the range of values at which the rheological properties are not overly skewed
by surface effects. Viscosity values are verified by taking an average of the static values
measured across the experimental range that were treated as a population mean,
accounting to some degree for random error in the viscosity measure.
3.3.3 Solution Preparation
All solutions were prepared using de-ionized water. Care is taken to avoid any
moisture contamination both in storage and preparation of the solutions. Solutions of
glycerin are observed to maintain their rheological properties for a period of several
months, but were periodically checked to avoid a deviation from the expected behavior.
3.4 Experimental System
3.4.1 Liquid Pool
Single bubble growth events are observed through the walls of a 6x6.5x10 inch
Perspex tank fed by compressed air. The gas flow path terminates in an inverted funnel
feeding a soap film anemometer, as expressed in Figure 3.2.1. All surfaces of contact for
the test liquids are cleaned and then rinsed three times with distilled water and let dry
inverted to avoid particulate contamination. After liquid introduction, the inverted funnel
is immediately placed into the free surface of the liquid bath and the tank covered with
25
Parafilm to avoid any air-born contaminants from entering as well as to minimize solvent
evaporation. The tank is fixed to a level table, which is reset to level after the liquid is
introduced. The gas flow rate could then be modified at will using a control knob on the
mass flow controller.
The air is introduced through a valve, a desiccator, and a filter into a pressure
regulator maintained in all but specially noted cases at 30 psi. From the pressure
regulator, air flows through a Porter Instruments TM mass flow controller (or in some
cases a low-flow rotameter with needle valve) with a sintered flow element designed for
120 mL per minute maximum flow rate. Air from the mass flow controller passes through
an additional sintered insert acting as a flow straightener and into a Luer fitting. Squareedged stainless steel syringes of half an inch in length and varying inner diameter (0.32
mm, 1.75 mm) were attached at this fitting. Air passed through this syringe to enter the
stagnant, isothermal pool as a bubble.
3.4.2 Gas Flow Path
Before each experiment, the airline is thoroughly inspected for leaks. The orifice
is run through three times with distilled water and, for aqueous glycerin, an additional
run-through with ethanol is performed. The air system is set at the maximum flow rate for
several minutes to ensure the orifice was clean and dry before introduction of the test
liquid. It is observed that, when the internal surface of the orifice had been wet by liquid,
the bubble necking and break-off behavior changes significantly. Air mass flow control
was accomplished by a combination of porous media and a diaphragm, the resistance of
which can be modified by turning the control knob as previously described. This device
is said to reach a steady gas flow rate within 45 seconds and provides fine control over
26
the gas flow rate even at very low rates of flow. The sintered insert used to straighten the
flow at the orifice fitting is also produced by Porter Instruments and enables a nearly
complete isolation of the bubble event from the air-flow system, keeping pressure
oscillation to a minimum and ensuring an instantaneously steady entry of air into the
liquid pool.
3.4.3 Gas Flow Rate
Air flow is measured by collecting bubbles in an inverted funnel and passing them
through a soap-film anemometer composed of a 50 mL burette. The elapsed time for a
given volume of air is measured with a stop watch over several volumetric intervals. The
volumetric rate per time was calculated from the soap film itself while moving at constant
velocity. For the initial low, constant flow experiments with water and aqueous glycerin
an average of the measured volumetric flow rate is compared to the value calculated from
photos of the bubble silhouettes and a majority of the data were found to agree within 5%
in the range of flow rates considered. This comparison is then used to refine the flow rate
measurement technique and design subsequent tests. In some cases, bubble interval and
bubble photos are used to calculate the gas flow rate values directly.
27
3.5 Experimental Procedure
3.5.1 Bubble Generation
The setup as described is observed to achieve a steady periodic rate of uniform
bubbles, as many prior researchers have demonstrated using low rates of gas flow in a
variety of conditions [2,3]. A flow stability experiment is conducted in which the elapsed
time following a flow rate adjustment was measured and bubble images taken (by the
same method described later) at several subsequent times. The bubble departure volume
and flow rate were measured and these measurements compared to determine the
minimum elapsed time required to consider the flow steady. The time required for
steadiness in the present system is found to be about 40 minutes. Actually, the time may
be much closer to 30 minutes, as the change in value between data taken at 30 minutes
and that taken at 40 minutes was not appreciably different.
Images of the bubbles were taken using low-resistance, high-intensity lamps
focused on a diffuse white reflector using parabolic mirrors. The lamp light is obscured
by the bubble and the resultant silhouette photographed by a CCD camera (NAC
Instruments), in general at a rate of 2000 images per second. The photographic method is
covered in greater detail in the subsequent section. Bubble volume at departure is
calculated from these photographs and found to vary within 1% once a steady condition
was reached.
3.5.2 Photographic Method
The imaging method is refined iteratively over several preliminary experiments to
determine the most accurate protocol. As described earlier, the flow rate measurement is
28
verified against the volume calculated from the bubble image and the method adjusted
until agreement within 5% of the equivalent spherical diameter is achieved for the range
of flow rates considered. Additionally, an analysis of the change in bubble volume during
break off shows that the differential in the calculated value amounted to about 1% from
the image just prior to the image immediately following departure. The two images are
taken 0.5 milliseconds apart and it was arbitrarily taken that the post-departure image
would be considered to be the departure volume of the bubble.
To record a single bubble flow, images are continuously recorded over an interval
of several seconds, ensuring at least 11 successive bubbles. The frame rate is held
constant at 2000 frames per second except in special cases. These images are saved in a
buffer using NAC Hi-D Cam II image capture hardware and analyzed immediately to
determine the bubble interval. Values of bubble interval are calculated as an average of
the elapsed time over the complete growth cycle—nucleation, growth, and departure—of
several bubbles (between 11 and 500, depending on the flow rate and spatial resolution of
the image). In all but the case of extremely low rates of flow, the buffer memory is
sufficient to take successive images for 6 or more bubbles from which to calculate the
departure volume. These images are saved as ‗.tif‘ files to preserve full resolution and
analyzed in Matlab by a script included in Appendix C. The bubble volume was
calculated from the sum of cross-sectional area elements A along the vertical growth
axis. The equivalent spherical radius and bubble departure volume could then be
calculated respectively using equations 3.5.2.1 and 3.5.2.2.
rb = (A/)1/2
(3.5.2.1)
Vb = 4/3rb3
(3.5.2.2)
29
Verification of the image analysis in Matlab is carried out with ImageProTM, a
commercial software that employs intensity-level analysis to trace object edges. The area
calculated by these two methods agreed within acceptable experimental uncertainty. The
measurement uncertainty arising from the pixel–intensity threshold is considered and
treated in Appendix D.
30
Chapter 4
Results and Discussion
4.1
Low Volume, Constant Flow
The results presented are among the very lowest scrutinized in the literature. The majority
of the studies reviewed here report ―low flow rates‖ or even ―extremely low flow rates‖
to mean gas flow rates of about 1 mL per second [17,24,25]. At the orifice sizes
commonly chosen in the literature, bubble volume at departure is frequently on the order
of 0.1 – 1 mL per bubble regardless of the choice of fluid, at the low end of the study‘s
experimental range of flow rates. It can be seen that the emphasis of this work hinged on
an understanding of single bubble departure characteristics at even lower flow rates
where the definition of constant flow was actually instantaneous, rather that the common
approach of a constant root-mean-square (QRMS) approach. The root-mean-squared
approach is implied, though not guaranteed, by the common experimental choice of
integrating bubble volume over several periods and dividing by time to establish the flow
rate. The results for pure fluids such as water and glycerin demonstrate weak dependence
on viscosity below a value of 87 centipoises. At more greatly reduced flow rates, there is
a slight deviation in the departure volume of bubbles formed in aqueous glycerin
solutions and in water owed to surface tension, as Tate‘s Law stated quite early in the
history of the field. This subject will be given more attention in following pages. 75%
aqueous Glycerin has a viscosity of about 25 times the value for water at room
temperature and, remarkably, can be seen to have about 5% less included volume on
31
average below ReO = 1. It should be noted that the transition to Stoke‘s flow, where ReO
<< 1, may be said to begin here and is precisely where the curve levels out, approaching
its asymptotic limit (Figure 4.1.1). Above ReO = 1, the departure volume begins to grow
appreciably with increasing flow rate. For pure glycerin, this is the point where viscosity
begins to influence the final dimensions of the departing bubble. This is much earlier than
some researchers found [17]. Smooth, instantaneously constant volumetric bubble growth
over the measured range occurs in the present experimental apparatus‘ flow system
elements including a sintered flow straightener, a low-volume double-male Luer-lock
orifice fitting, and a 0.5 inch capillary orifice. This flow characteristic may be considered
in a basic, mathematical sense to be the physical realization of a system in which the
second derivative of the temporal volumetric gradient is made close to zero. The data in
Figure 4.1.2 show volumetric growth versus time with linear curve fits. A linear
regression analysis gives R2 error of less than 3% and agrees well with the theoretical
limit. An additional feature of this system is the near-zero dead time. A normalized time
parameter is used, namely g/BI, and the maximum typical for g in the present system is
1 within 0.5%. Even where the Orifice Reynolds Number is much less than one, dead
time was not appreciably increased. As Davidson and Schueler stated in their two-part
1960 work, the experimentalist must design a pressure drop to ensure constant volumetric
air flow for bubble experiments [24,25]. The literature demonstrates two fundamental
ways of introducing such a restriction while preserving constant pressure in the growing
bubble: a long capillary tube or a porous flow restrictor in-line, set close to the growing
interface [3,24,25]. The present experiment chiefly relies on the second method, although
gas flow throughout the system was facilitated using capillary tubing. For bubble growth
32
in aqueous glycerin from an orifice with an inner diameter of 0.32 mm, the data show
behavior consistent with those from a 1.75 mm inner orifice diameter (Figure 4.1.3).
From about ReO = 0.1 to 100, the effects of viscosity 1000 times greater than water
become apparent, while the effect of viscosity 25 times that of water is not appreciable
(Figures 4.1.1 and 4.2.1). Bubble departure volume in glycerin may also be seen to reach
even lower values than in water, provided the flow rate is sufficiently reduced. This
clearly demonstrates that, for an instantaneously constant flow system, bubble departure
volume ceases to depend on viscosity as flow rate is reduced without limit. This author is
not aware of any true counterexample to this behavior where the volumetric bubble
growth rate may be shown to be constant; that is, no example is known where the
experimentalist also demonstrates that the second derivative of bubble volume in time is
zero. Based their explicatory efforts for this condition and, in contrast to the other
literature, Teresaka and Tsuge hold this as a necessary and sufficient condition for
constant flow [13,27]. They show that the deviation of experimental data from their
results occurs where additional system volume called chamber volume is present and
their stated constant volumetric growth condition is not satisfied (also see Figure 4.2.3).
33
1
Nc = 0.1
d0 = 1.75 mm
Vs = 0.2 mL
Vb [mL]
Water
75% aq. Glycerin
Glycerin
0.1
0.01
0.01
0.1
1
10
100
ReL
Figure 4.1.1 - Bubble departure volume for aqueous glycerin solutions at Nc = 0.1, d0 = 1.75 mm
34
1000
0.25
Nc = 0.1
d0 = 1.75 mm
Vs = 0.2 mL
0.20
ReL = 0.04 (0.04 mL/s)
ReL = 0.11 (0.09 mL/s)
ReL = 0.12 (0.10 mL/s)
ReL = 0.04 (0.20 mL/s)
ReL = 0.04 (0.95 mL/s)
Vb [mL]
0.15
0.10
0.05
0.00
0.00
0.20
0.40
g/BI
0.60
0.80
Figure 4.1.2 - Temporal volumetric bubble development in glycerin for Nc = 0.1,
d0 = 1.75 mm
35
1.00
1
Nc = 0.1
d0 = 0.32 mm
Vs = 0.2 mL
Glycerin, (Sagar, 2012)
aq. Glycerin
Water
Vb [mL]
0.1
0.01
0.001
0.01
0.10
1.00
10.00
ReO
Figure 4.1.3 - Bubble departure volume for aqueous glycerin solutions at Nc = 0.1, d0 = 0.32 mm
36
100.00
6 mm
3
0.641
0.731
0.821
0.910
1.000
/BI
0
a. Water; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 1125.5 (1.55 mL/s); BI = 42 ms
6 mm
3
0.709
0.781
0.854
0.927
1.000
/BI
0
b. Water; d0 = 1.75 mm; Nc = 0.1 mL (0.2 mL); ReL = 522.8 (0.72 mL/s); BI = 66.21 ms
6 mm
3
0.790
0.842
0.895
0.947
1.000
/BI
0
c. Water; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 123.4 (0.17 mL/s); BI = 234 ms
6 mm
3
0.909
0.932
0.955
0.977
1.000
/BI
0
e. Water; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 21.8 (0.03 mL/s); BI = 1022 ms
Figure 4.2.4 – Bubble necking behavior in a low-volume, constant flow air-water system
with porous restrictor, capillary orifice and plane visualization
37
8 mm
4
0.647
0.736
0.824
0.912
1.000
/BI
0
a. Glycerin; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 1.12 (0.95 mL/s); BI = 250 ms
8 mm
4
0.690
0.768
0.845
0.923
1.000
/BI
0
b. Glycerin; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 0.24 (0.20 mL/s s; BI = 250 msec
8 mm
4
0.735
0.801
0.867
0.934
1.000
/BI
0
c. Glycerin; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 0.11 (0.09 mL/s; BI = 477 msec
8 mm
4
0.798
0.849
0.899
0.950
1.000
/BI
0
d. Glycerin; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 0.04 (0.95 mL/s; BI = 976 ms
Figure 4.1.5 – Bubble necking behavior in a low-volume, constant flow air-glycerin
system with porous restrictor, capillary orifice and plane visualization
38
4.2
Effect of Liquid Viscosity
Data from the present experimental system compares well with capillary fed
systems in the literature [21]. A comparison of researcher data for low and high rates of
air flow at a 2 mm nominal orifice diameter may be seen in Figure 4.2.1. The common
factor of the experimental systems included here is the direct application of air flow to a
long capillary of a bore on the order of millimeters. These experiments also saw direct
contact between the capillary and the liquid pool, as opposed to systems relying on a
plate orifice to adjust the orifice size. It should be noted that the present system relied on
a capillary flow path, but that in addition, a sintered insert was present at the syringe
fitting to isolate the orifice tip and reduce system volume as much as possible. The three
data sets compared in Figure 4.2.1 show that in aqueous glycerin solutions of a viscosity
less than 100 times the value for water, the departure volume approached a lower value
than for water as the flow rate decreased. At higher rates of flow, bubble behavior in the
low viscosity solution is indistinguishable from the pure water pool.
Figure 4.2.2 illustrates orifice dependence in several bubble experiments with
aqueous glycerin solutions of a low concentration versus pure water. The curves show
that, in a certain set of experiments, the effect of viscosity was not significant at values
lower than 100 centipoises. The experimental systems in Figure 4.2.2 are fed by capillary
tubing as well. Ramkrishnan et al and the present study both employ a different technique
much more seldom seen in the literature [20]. Steel capillary tubes are combined with a
restrictive obstruction (effectively a porous media) that is intended to straighten the
airflow and lend to constancy. This has been described in more detail in Chapter 2. It is
observed that viscosity did not affect departure volume in these experimental systems in
39
the range of viscosity, orifice diameter, and Orifice Reynolds Number form 0 – 100 cPs,
0.32 – 5.95 mm, and 0.1 – 1000, respectively. In contrast to the data shown for the
present system and comparative capillary experiments, researcher data for water and
aqueous glycerin was found in the stated range of flow rates and orifice diameters that
does not correspond entirely to the behavior described, yet still rely on capillary tubing
and do not employ chambers to enhance system volume. These experiments are shown in
Figure 4.2.3 and warrant some explanation in comparison to the previous observations.
Teresaka and Tsuge use a small, relatively short capillary tube atop an air feed line of
comparatively larger diameter [13]. It must also be noted that the value of viscosity for
the solution they employ is just above 100-fold the value of water. Jamialahmadi et al
demonstrate a slight increase in departure diameter in aq. glycerin as compared to water
when the solution has a viscosity of 86.6 cPs (Lower values approach that of water) [17].
They conducted experiments with plate orifices having an inner diameter from 0.5 – 4
mm, requiring the bore underneath the plate to be somewhat larger. Our attention turns to
the effect of system volume, orifice geometry, flow metering, and visualization systems
in the following section, and will address the effect of plate orifice geometry on single
bubble development. The most significant differentiating element in the two systems is
the lack of strict isolation in the gas flow path via either a smooth long capillary or a
porous restrictor. In both counterexamples, the value of viscosity is on the border of the
stated range and there is reason to believe that some volumetric capacitance or system
volume was present in the gas flow path. The direct impacts of this feature on the single
bubble growth cycle and bubble shape are summarized in the following section.
40
100.00
H2O Glycerin T

d Researcher
[wt.%]* [wt.%]* [OC] [cPs] [mm]
[-]
Vb [mL]
10.00
46%
46%
39%
39%
35%
35%
-
54%
54%
71%
71%
75%
75%
20
20
20
20
23
23
10
10
25
1
24
1
2
2
2
2
2
2
[52]
[52]
[21]
[21]
1.00
0.10
0.01
0.01
0.1
1
10
100
1000
10000
100000
ReL
Figure 4.2.1 - Bubble departure volume for researchers using capillary orifices in aqueous glycerin at d0 = 2 mm
*Values for other researchers estimated based on experimental conditions
41
100
H2O Glycerin T

d Researcher
[wt.%]* [wt.%]* [OC] [cPs] [mm]
[-]
31%
100%
31%
100%
46%
46%
39%
39%
35%
35%
Vb [mL]
10
1
-
79%
0%
79%
0%
54%
54%
71%
71%
75%
75%
25
25
25
25
20
20
20
20
23
23
50
1
50
1
10
10
25
1
24
1
6
6
4
4
2
2
2
2
2
2
[20]
[20]
[20]
[20]
[52]
[52]
[21]
[21]
0.1
0.01
0.1
1
10
100
ReL
1000
10000
100000
Figure 4.2.2 - Bubble departure volume in aqueous glycerin for researchers using capillary orifices, effect of orifice diameter
*Values for other researchers estimated based on experimental conditions, with 25OC assumed for [20]
42
100.00
H2O Glycerin T

d Researcher
[wt.%]* [wt.%]* [OC] [cPs] [mm]
[-]
16%
16%
16%
12%
12%
Vb [mL]
10.00
-
84%
84%
84%
88%
88%
22
22
22
20
20
10
10
25
1
24
2
2
2
2
2
[17]
[17]
[17]
[13]
[13]
1.00
0.10
0.01
1
10
100
ReL
1,000
10,000
Figure 4.2.3 - Bubble departure volume in aqueous glycerin solutions for researchers using capillary tubing with
varying orifice construction, effect of liquid viscosity and orifice length. *Values for [17] estimated based on
experimental conditions, with 20OC assumed for [13]
43
4.3 System Effects:
4.3.1 Flow Metering
When system volume is low, bubble growth occurs in the present system at an
instantaneously constant volumetric rate for glycerin. Figure 4.3.1.1 shows the
characteristic normalized behavior of single bubble growth from the time of departure of
the previous bubble to the time of departure of the bubble under consideration. It can be
seen that, in a normalized sense, the cycle behaves nearly the same over a decade of flow
rates. That is to say, the characteristic pattern of growth is relatively unchanged. This
would suggest that the balance of deterministic forces is not altered significantly by
increasing gas flow per time, and that the force balance is a scalar multiple of some
fundamental balance. A balance of this kind has recently been refined and presented
where gas momentum, pressure, and buoyancy are opposed by viscous stresses, surface
tension and liquid inertia throughout the bubble growth cycle [32]. The principal balance
that has been proposed in flow at a sufficiently low flow rate is a simple sum of gas
pressure, surface tension and buoyancy, corroborated above by Figure 4.1.2 even for the
case of liquids with slightly higher viscosity than water. Small deviations can be seen at
the lowest flow rate in Figure 4.3.1.1, which appear as dips in the curve. The first of these
dips is the point at which the onset of necking occurs, while the second, the maxima that
can be observe just before breakoff, is thought to be the excess volume required to
overcome the interfacial tension at the vena cava of the bubble neck. For water, which
has a very low viscosity, a very different effect is seen at low system volume. The single
bubble growth curves in Figure 4.3.1.2 show that gas flow rate alters the dynamic force
44
balance. Rather than confusing the issue, this helps to illuminate the role of viscosity.
Viscous damping at a low system volume dominates the influence of gas momentum,
causing the force balance to maintain a similar character over a decade of gas flow rates.
The relatively minor influence of gas momentum has a much more significant impact in
the case of water, an inviscid liquid. The key restraining force for bubble growth in water
is surface tension, which does not depend on the gas flow rate. Hence, the momentum
imparted by gas flow plays a bigger role. At the same time, a slightly higher critical
pressure is needed to achieve bubble growth in water under equivalent conditions. This
additional restraint to nucleation, coupled with a stronger capillary interaction with the
steel orifice at breakoff, depends on the rate of gas flow to determine the time required to
attain critical nucleation pressure. For the low volume system employed in the present
study, a plate including three sets of images visualizing the flow across a decade of flow
rates is included in Figure 4.3.1.3. The signature shape of departing bubbles in this
system can be seen to be very similar to an airfoil: a nearly-spherical top half joined to an
ogive of minimum tip radius. The development of profile shape is represented by plots
temporal aspect ratio development as in Figure 4.3.1.4. The nucleation phase can be seen
here as the process of rapidly forming a spherical nucleus. This newly formed nucleus is
more of an ellipsoid than a spheroid, having an aspect ratio above 1.3. From this stable
nucleate bubble, growth is paired with continued elongation. The rapid expansion phase
here is a relatively short portion of the cycle, whereas much of the growth occurs
simultaneously with translatory movement in the face of heightened liquid drag force,
leading to elongation. Shape development for air-water interfaces at 0.2 mL system
volume is shown in Figure 4.3.1.6. It can be seen that at the highest flow rate, the growth
45
is essentially two-phase, as when greater volume is present, yet for the lower gas flow
rates, an additional shift takes place at breakoff, elongating the bubble somewhat. As in
the case where greater volume is present, the final aspect ratio is constant over a decade
of gas flow rates. Instead of a spherical nucleus, the profile already has a larger vertical
axis than it does a horizontal, and the nucleus enters the growth and translation phase
here and undergoes steady growth and elongation throughout the remainder of the cycle.
46
0
0.25
0.5
0.75
1
1
1.00
Glycerin
Nc = 0.1
d0 = 1.75 mm
Vs = 0.2 mL
Vb / Vb,max
0.75
ReL = 0.24 (0.20 mL/s)
ReL = 0.12 (0.10 mL/s)
ReL = 0.04 (0.03 mL/s)
0.75
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.1.1 - Temporal volumetric bubble development in glycerin, Nc = 0.1, d0 = 1.75 mm
47
0.00000
0.25000
0.50000
0.75000
1.00000
1
1.00
Water
Nc = 0.1
d0 = 1.75 mm
Vs = 0.2 mL
Vb / Vb,max
0.75
0.75
ReL = 529 (0.72 mL/s)
ReL = 81 (0.11 mL/s)
ReL = 22 (0.03 mL/s)
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.1.2 - Temporal volumetric bubble development in water; Nc = 0.1, d0 = 1.75 mm
48
8
4
0.00
0.20
0.40
0.60
0.80
1.00
/BI
0 mm
a. Glycerin; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 0.24 (0.20 mL/s); BI = 250 ms
8
4
0.00
0.20
0.40
0.60
0.80
1.00
/BI
0 mm
b. Glycerin; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 0.12 (0.10 mL/s); BI = 430 ms
8
4
0.00
0.20
0.40
0.60
0.80
1.00
/BI
c. Glycerin; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 0.04 (0.03 mL/s); BI = 976 ms
Figure 4.3.1.3 – Interfacial shape in a low volume, constant flow air-glycerin system with porous restrictor, capillary orifice and
plane visualization surface
49
0 mm
 [Hb/W b]
0
0.25
0.5
0.75
1
1.75
1.75
1.50
1.5
1.25
1.25
1.00
1
0.75
0.75
Glycerin
Nc = 0.1
d0 = 1.75 mm
Vs = 0.2 mL
0.50
ReL = 0.24 (0.20 mL/s)
ReL = 0.12 (0.10 mL/s)
ReL = 0.04 (0.03 mL/s)
0.25
0.00
0.00
0.5
0.25
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.1.4 - Bubble aspect ratio development in glycerin, Nc = 0.1, d0 = 1.75 mm
50
8
4
0.00
0.20
0.40
0.60
0.80
1.00
/BI
a. Water; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 529 (0.72 mL/s); BI = 66 ms
0 mm
8
4
0.00
0.20
0.40
0.60
0.80
1.00
/BI
b. Water; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 81 (0.11 mL/s); BI = 324 ms
0 mm
8
4
0.00
0.70
0.78
0.85
0.93
1.00
/BI
c. Water; d0 = 1.75 mm; Nc = 0.1 (0.2 mL); ReL = 22 (0.03 mL/s); BI = 1022 ms
Figure 4.3.1.5 – Interfacial shape in a low-volume, constant flow air-water system with porous restrictor, capillary
orifice and plane visualization surface
51
0 mm
0
0.25
0.5
0.75
1
1.75
1.75
1.50
 [Hb/W b]
1.25
Water
Nc = 0.1
d0 = 1.75mm
Vs = 0.2 mL
1.5
ReL = 529 (0.72 mL/s)
ReL = 81 (0.11 mL/s)
ReL = 22 (0.03 mL/s)
1.25
1.00
1
0.75
0.75
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
g/BI
0.75
1.00
Figure 4.3.1.6 - Bubble aspect ratio development in water, Nc = 0.1, d0 = 1.75 mm
52
4.3.2 System Volume
Researchers have borne out the effects of a gas chamber on bubble growth quite
thoroughly since the middle of the 20th century; however, many researchers did not take
the volume of their experimental system‘s gas flow path into direct numeric consideration
[6,13,17,19]. Several rigorous studies provide excellent, reliable results at a condition
where the average gas flow rate has a constant root mean squared value, driven either by
volumetric displacement or compression. These experiments are characterized by the
―intermediate flow condition‖ described by Park et al in 1976 [26]. The present system
was augmented with a stainless steel cylinder of precisely 300 mL and bubble growth
experiments conducted for both glycerin and water. Figure 4.3.2.1 displays the measures
of bubble departure volume with the addition of this system volume and in the absence of
the porous insert earlier described. The volume was installed immediately downstream of
the mass flow controller. It may be observed that the results are very similar to
Subramani, 2007, who used a syringe pump and an identical orifice to that employed in
the present work (1.75 mm diameter). Because of the importance of the system volume
parameter and its lack of cohesive definition in the literature, it will be redefined here as
the volume of gas bounded by the orifice and the first significant pressure restriction in
the gas flow path. This definition reflects the present system design and is similar to the
treatment of Miyahara et al, Teresaka and Tsuge, and Park et al, to name a few
[26,29,33]. It can easily be seen that an increase in chamber volume tends to increase the
departure volume for bubble growth in an infinite liquid at a constant periodic rate and
steady gas flow (QRMS). Figure 4.3.2.2 depicts the influence of system volume on bubble
departure in pure water. For the data taken in the present study, a large change in the
53
system volume produces a change in departure volume for water of about 10% of the
total value. Here, Subramani‘s work is seen to be almost indistinguishable from the low
system volume set from the present study at the range of flow rates considered. The
increased radial viscous stresses in glycerin help explain the pronounced enhancement of
departure volume with the addition of system volume as compared to water. System
pressure is not appreciably higher, owing to the fact that surface tension is similar for
both liquids [26,29]; yet, pressure capacitance is significantly increased, resulting in a
larger volume of gas at the point that critical nucleation pressure is attained. On the other
hand, bubbles in glycerin in a system of appreciable volume show significantly higher
dependence on the rate of gas flow over a single decade. Pressure capacitance in this case
causes a lag, which is followed by very rapid volumetric growth once the critical excess
pressure has been reached. It can be seen from Figure 4.3.2.4 that the normalized lag time,
or ―dead time‖ as many researchers have called it, decreases with increasing flow rate.
This is exactly what one might expect, given that the critical pressure required for bubble
nucleation is reached at a more rapid rate for fixed system volume when the rate of gas
flow is increased. This also comments on why the departure volume at a given system
pressure ultimately reaches the same value as that reached in other systems once the gas
flow rate has been increased to a sufficient degree. Gas momentum plays a greater and
greater role as the gas flow rate increases, as Kassimsetty et al commented in 2007 [53].
It can be seen in Figure 4.3.2.5 that increased system volume enhances the impact
described for water as well, increasing the dependence of single bubble growth in water
on the rate of gas flow. Here, a dead time is also seen, even at flow rates where the postnucleation growth occurs at a fairly linear rate. The key point of differentiation between
54
the resistance of surface tension and the resistance of volumetric capacity is that it alters
the momentum before nucleation occurs, by allowing the system pressure to drop below
the system pressure in a comparable system with low volume. The lag time explains why
the departure volume for bubbles in water depends less on system volume: volume in
experimental bubble growth systems most directly impacts departure volume in liquids
where viscous capacitance is also present. The coupled gas-liquid capacitance results in
larger bubbles at lower frequencies for the same rate of gas flow. The curves in Figure
4.3.2.6 show the explicit comparison described. It is important to note that the growth
time in glycerin versus water is about 5 to 1, or 300 milliseconds and 60 milliseconds,
respectively. Figure 4.3.2.7 emphasizes the degree to which system volume interacts with
different liquids, namely water and Glycerin. At this lower set of flow rates, system
volume can be seen to dominate the growth pattern, and Figure 4.3.2.8 depicts a
comparison of five unique researchers with differing system volumes. It shows that effect
obeys the relationship described, although other system factors are present. As such, few
truly comparative results are available on a single-bubble growth cycle basis, yet the
contribution by Subramani in 2007 lent a significant advancement in that it rigorously
and systematically compared the visual (―Image data‖) data, the growth versus time, and
the departure volume versus flow rate. As such, it is an experimental study that offers a
unique comparative case for the present study, as may be observed in all the previous and
following figures contrasting the two works.
Subramani et al report data taken at a system volume intermediate to the two
chosen for the present work and demonstrates behavior intermediate to the two
behavioral extrema of the present study. This is true at higher and at lower flow rates,
55
although it is also evident that the rate of volumetric expansion is higher just after the
initial emergence of the bubble from the orifice tip. The expansion appears to be bounded
by the low-volume curve, which would indicate that the volumetric growth actually
slowed before the maximum departure volume was reached. Figure 4.3.2.10 displays
several sets of images of single bubble growth in glycerin where the system has been
augmented with about 300 mL additional volume. The images capture the onset of the
rapid expansion portion of bubble growth, which can be seen as an inflection in the
volumetric and aspect development curves plotted against time for the same conditions.
As the gas momentum increases owing to higher gas flow rate, the nucleate bubble at the
onset of rapid expansion can be seen to grow larger, then to elongate. This elongation
from heightened gas momentum occurs in the departing bubble as well. The departure
volume stays relatively constant throughout the range. A comparison of the interfacial
shape that is developed under mass-flow control with 300 mL additional system volume
is depicted in Figure 4.3.2.12 and represented by the aspect ratio versus time curves in
Figure 4.3.2.13. It is immediately clear that the effect of viscosity in the equivalent
system at the same flow rate tends to round the departing bubble profile. It can also be
seen that the rapid initial expansion has a comparatively less rapid character in glycerin.
The air-glycerin bubble grows from nucleate bud too full development on a steady
translational trajectory, with a minor inflection marking the hemispherical cap stage. The
fully developed profile in glycerin is essentially a sphere plus a neck, accounting for the
slightly greater height at departure. Together, these examples substantiate the claim that
system interactions have direct repercussion on the balance and expression of
deterministic forces in bubble growth according to the key parameters of interest, most
56
notably with respect to the impact of gas momentum and viscous stresses. This impact
also extends to resultant bubble shape, and the characteristic volumetric and aspect
development of the interface. The impact of water‘s capillary behavior can be seen in
Figures 4.3.2.14 and 4.3.2.15. The retreat of liquid into the capillary orifice at bubble
breakoff is much deeper than for glycerin, which one might expect for the liquid with the
higher surface energy; however, the entrance is also time-dependent and thus depends on
liquid viscosity, for which water is better-suited to enter the orifice after breakoff as well.
The nucleating water bubble, then, interacting with system volume, sees a very flat initial
profile, followed by a nearly instantaneous assumption to the spherical nucleus. After this
nuclear bud is formed, growth and translation continue until bubble breakoff at a
relatively constant volume and constant aspect ratio. Without the effect of viscosity, there
is no additional elongation at higher flow rates within the experimental range. It can also
be seen that overshoot occurs for higher gas flow rates during the initial expansion,
followed by settling during the growth and translation phase. Inspection of Figure
4.3.2.14 sheds some light, as the departing bubble resides much closer to the meniscus
during the time needed to achieve critical nucleation pressure, owing to the higher gas
flow rate. It is likely that this overshoot marks the onset of wake interactions for this
system. Single Bubble growth cycles at very similar flow rates are contrasted in Figure
4.3.2.16. The comparison is made to illustrate the complexity in ascribing causality to the
impact of system effects on the deterministic forces and their resultant impact on
interfacial development. Most notably, the work of Subramani and the present work, at a
comparable volume to Jiang et al, can be seen to behave differently. All quoted
57
1
Vb [mL]
0.1
0.01
Glycerin
d0 = 1.75 mm
Nc = 138.9 (301.5 mL)
Nc = 119.8 (260 mL), (Subramani, 2007)
Nc = 0.1 (0.2 mL)
0.001
0.001
0.01
0.1
ReL
1
Figure 4.3.2.1 - Bubble departure volume in glycerin for different system volumes at d0 = 1.75 mm
58
10
Vb [mL]
1
0.1
Water
d0 = 1.75 mm
Nc = 110.4 (301.5 mL)
Nc = 95.2 (260 mL), (Subramani, 2007)
Nc = 0.1 (0.2 mL)
0.01
100
1000
ReL
10000
Figure 4.3.2.2 - Bubble departure volume in water for different system volumes at
d0 = 1.75 mm
59
10000
Nc ~ 100
d0 = 1.75 mm
Vs = 301.5 mL
Glycerin, Nc = 138.9
Water, Nc = 110.4
BI [msec.]
1000
100
10
0.01
0.10
1.00
10.00
Q [mL/s]
Figure 4.3.2.3 - Bubble interval versus gas flow rate in water and glycerin at Nc ~ 100,
d0 = 1.75 mm
60
1.00
Glycerin
Nc = 138.9
d0 = 1.75mm
Vs = 301.5 mL
Vb / Vb,max
0.75
ReL = 1.28 (1.08 mL/s)
ReL = 0.75 (0.63 mL/s)
ReL = 0.13 (0.11 mL/s)
0.50
0.25
0.00
0.00
0.25
0.50
0.75
g/BI
Figure 4.3.2.4 - Temporal volumetric bubble development in glycerin, Nc = 138.9,
d0 = 1.75 mm
61
1.00
0
0.25
0.5
0.75
1
1
1.00
0.75
Water
Nc = 110.4
d0 = 1.75 mm
Vs = 301.5 mL
0.75
Vb / Vb,max
ReL = 776.94 (1.07 mL/s)
ReL = 508.98 (0.70 mL/s)
ReL = 145.22 (0.20 mlL/s)
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.2.5 - Temporal volumetric bubble development in water, Nc = 110.4,
d0 = 1.75 mm,
62
0.00000
0.25000
0.50000
0.75000
1.00000
1
1.00
Nc ~ 100
d0 = 1.75mm
Vs = 301.5 mL
Vb / Vb,max
0.75
Water, Nc = 110.4, ReL = 776.9
Water, Nc = 110.4, ReL = 588.2
Glycerin, Nc = 138.9, ReL = 1.3
Glycerin, Nc = 138.9, ReL = 1.0
0.75
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.2.6 - Temporal volumetric bubble development for water and glycerin at
Nc ~ 100, d0 = 1.75 mm
63
0
0.25
0.5
0.75
1
1
1.00
Vb / Vb,max
0.75
Water
ReL = 150 (0.2 mL/s)
d0 = 1.75mm
Nc = 110.4 (301.5 mL)
Nc = 0.1 (0.2 mL)
0.75
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.2.7 - Temporal volumetric bubble development in water at different system
volumes at d0 = 1.75 mm, ReL = 150
64
0.00000
0.25000
0.50000
0.75000
1.00000
1
1.00
d0 = 1.75 mm
Q = 1 mL/s
Vb / Vb,max
0.75
0.75
Glycerin, Nc = 138.9, ReL ~ 1
Glycerin, Nc = 0.1, ReL ~ 1
Water, Nc = 110.4, ReL ~ 1000
Water, Nc = 0.1, ReL ~ 1000
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.2.8 - Temporal volumetric bubble development in water and glycerin, the effect of system
volume and viscoisty, d0 = 1.75, Q = 1 mL/s
65
0
0.25
0.5
0.75
1
1
1.00
Glycerin
ReL ~ 1 (1 mL/s)
d0 = 1.75mm
Vb / Vb,max
0.75
Nc = 138.9 (301.5 mL)
Nc = 119.8 (260 mL), (Subramani, 2007)
Nc = 0.1 (0.2 mL)
0.75
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.2.9 - Temporal volumetric bubble development in glycerin at d0 = 1.75 mm,
ReL ~ 1
66
12
6
0.00
0.50
0.75
0.68
0.88
1.00
/BI
0 mm
a. Glycerin; d0 = 1.75 mm; Nc = 138.9 (300 mL); ReL = 1.28 (1.08 ml/s); BI = 260 ms
12
6
0.00
0.70
0.85
0.78
0.93
1.00
/BI
0 mm
b. Glycerin; d0 = 1.75 mm; Nc = 138.9 (300 mL); ReL = 0.75 (0.63 mL/s); BI = 450 ms
12
6
0.00
0.90
0.95
0.93
0.98
1.00
/BI
c. Glycerin; d0 = 1.75 mm; Nc = 138.9 (300 mL); ReL = 0.13 (0.11 mL/s); BI = 2200 ms
Figure 4.3.2.10 – Interfacial shape in an air-glycerin system with Nc 138.9, capillary orifice, and plane visualization
surface
67
0 mm
0
0.25
0.5
0.75
1
1.5
1.50
1.25
 [Hb/W b]
1.00
Glycerin
Nc = 138.9
d0 = 1.75 mm
Vs = 301.5 mL
1.25
ReL = 0.13 (0.10 mL/s)
ReL = 0.75 (0.63 mL/s)
ReL = 1.28 (1.08 mL/s)
1
0.75
0.75
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.2.11 - Temporal bubble aspect ratio development in glycerin, Nc = 138.9,
d0 = 1.75 mm
68
8
4
0.00
0.20
0.40
0.60
0.80
1.00
/BI
a. Water; d0 = 1.75 mm; Nc = 110.4 (300 mL); ReL = 777 (1.07 mL/s); BI = 54.8 ms
0 mm
8
4
0.00
0.20
0.40
0.60
0.80
1.00
/BI
b. Water; d0 = 1.75 mm; Nc = 110.4 (300 mL); ReL = 509 (0.71 mL/s; BI = 75 ms
0 mm
8
4
0.00
0.70
0.78
0.85
0.93
1.00
/BI
c. Water; d0 = 1.75 mm; Nc = 110.4 (300 mL); ReL = 145 (0.2 mL/s; BI = 243 ms
Figure 4.3.2.12 – Interfacial shape in an air-water system with Nc = 110.4, capillary orifice, and plane visualization
surface
69
0 mm
0.00
0.25
0.50
0.75
1.00
2.25
2.25
2.00
1.75
 [Hb/W b]
1.50
Nc ~ 100
d0 = 1.75mm
Vs = 301.5 mL
2
1.75
Water, Nc = 110.4, ReL = 776.9
Water, Nc = 110.4, ReL = 588.2
Glycerin, Nc = 138.9, ReL = 1.3
Glycerin, Nc = 138.9, ReL = 1.0
1.5
1.25
1.25
1.00
1
0.75
0.75
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.2.13 - Temporal bubble aspect ratio development in water and glycerin at
Nc ~ 100, d0 = 1.75 mm
70
12
6
0.00
0.50
0.75
0.68
0.88
1.00
0 mm
/BI
a. Glycerin; d0 = 1.75 mm; Nc = 138.9 (300 mL); ReL = 1.28 (1.08 mL/s; BI = 260 ms
8
4
0.00
0.20
0.40
0.60
0.80
1.00
/BI
b. Water; d0 = 1.75 mm; Nc = 110.4 (300 mL); ReL = 777 (1.07 mL/s; BI = 54.8 ms
Figure 4.3.2.14 – Interfacial shape in air-glycerin and air-water systems with Nc ~ 100, capillary orifice, and plane
visualization surface
71
0 mm
0.00
0.25
0.50
0.75
1.00
2.25
2.25
2.00
1.75
 [Hb/W b]
1.50
Water
Nc = 110.4
d0 = 1.75 mm
Vs = 301.5 mL
2
1.75
ReL = 776.94 (1.07 mL/s)
ReL = 508.98 (0.70 mL/s)
ReL = 145.22 (0.20 mlL/s)
1.5
1.25
1.25
1.00
1
0.75
0.75
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.2.15 - Temporal bubble aspect ratio development in water, Nc = 110.4,
d0 = 1.75 mm
72
1.00
Glycerin
ReL ~ 0.1
d0 = 2 mm
Vb / Vb,max
0.75
Nc = 0.1 (0.2 mL)
Nc = 119.8 (260 mL), (Subramani, 2007)
Nc = NA (270 mL), (Jiang et al., 2008)
Nc = 138.9 (301.5 mL)
0.50
0.25
0.00
0.00
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.2.16 - Temporal volumetric bubble development in glycerin for researchers with
capillary orifices, effect of system volume, d0 = 2 mm, ReL ~ 0.1
73
12 mm
6
0.840
0.880
0.920
0.960
1.000
/BI
0
a. Glycerin; d0 = 1.75 mm; Nc = 138.9 (300 mL); ReL = 1.28 (1.08 mL/s); BI = 260 msec
12 mm
6
0.872
0.905
0.936
0.968
1.000
/BI
0
b. Glycerin; d0 = 1.75 mm; Nc = 138.9 (300 mL); ReL = 0.98 (0.83 mL/s); BI = 330 ms
12 mm
6
0.958
0.969
0.979
0.990
1.000
/BI
0
c. Glycerin; d0 = 1.75 mm; Nc = 138.9 (300 mL); ReL = 0.132 (0.27 mL/s); BI = 1000 ms
12 mm
6
0.980
0.985
0.990
0.995
1.000
/BI
0
d. Glycerin; d0 = 1.75 mm; Nc = 138.9 (300 mL); ReL = 0.13 (0.11 mL/s); BI = 2200 ms
Figure 4.3.2.18 – Bubble necking behavior in a constant flow air-glycerin system with
added volume, capillary orifice and plane visualization
74
8 mm
4
0.787
0.840
0.894
0.947
1.000
/BI
0
a. Water; d0 = 1.75 mm; Nc = 110.4 (300 mL); ReL = 784 (1.08 mL/s); BI = 55 ms
8 mm
4
0.816
0.862
0.908
0.954
1.000
/BI
0
b. Water; d0 = 1.75 mm; Nc = 110.4 (300 mL); ReL = 508 (0.70 mL/s); BI = 70 ms
8 mm
4
0.854
0.890
0.927
0.963
1.000
/BI
0
c. Water; d0 = 1.75 mm; Nc = 110.4 (300 mL); ReL = 349 (0.48 mL/s); BI = 101 ms
8 mm
4
0.944
0.958
0.972
0.986
1.000
/BI
0
d. Water; d0 = 1.75 mm; Nc = 110.4 (300 mL); ReL = 153 (0.21 mL/s); BI = 243 ms
Figure 4.3.2.19 – Bubble necking behavior in a constant flow air-water system with
added volume, capillary orifice and plane visualization
75
experimental parameters are very similar, with the exception of two: placement of system
volume and origination of gas flow. In Jiang‘s experiment, the system volume is placed
far away from the orifice tip, removed by a long capillary tube. Subramani also places the
system volume at a distance from the orifice, while the present study investigates the
influence of a volume set very near to the orifice fitting. In Subramani‘s case, the flow is
driven volumetrically, which requires system air pressure to increase slightly to maintain
the same flow rate on average over time. His is also a closed system from the volume to
the orifice, while Jiang et al feed the volume through additional tubing from a constant
pressure source (gas compressor). Additionally, there is a valve for metering the flow
placed in line between the additional volume and the orifice. It is possible that in Jiang‘s
case, the main effect of the volume at such a great distance from the orifice would be
found at resonant orders, which would exist at specific frequencies. It would actually be
common practice to remove such outlying data for presentation; Jiang et al do not
mention any such effect explicitly. It can be clearly seen that, where the system volume is
included close enough to the orifice that there is no significant resistance to air flow
downstream, the effect of volume is as described in the majority of the literature and
exposed in somewhat greater detail in the present study. Further effort would be needed
to truly understand the impact of orders in bubble growth and their system dependence.
4.3.3 Orifice Construction
Teresaka and Tsuge (Figure 4.3.2.16) show a similar effect to that of system volume,
though they use a compressor, air tubing, and a plate from which exits a capillary orifice
of precise length to achieve the condition L/di4 > 1x1012, where L is orifice length and di
the inner diameter. This was proposed by Tadaki and Maeda, 1963 to assure the constant
76
flow condition [27]. Here it can be seen that the step change between inner diameter for
the larger tube and the capillary orifice allows the system to behave as if there were a
slight volumetric capacitance: dead time is observed. Figure 4.3.3.1 compares the present
work at low system volume and instantaneously constant volumetric air flow to McCann
and Prince, who use a chamber of various sizes and an orifice plate. As a consequence of
the orifice plate, the emergence of the bubble occurs very early, but something very
similar to dead time is observed as the pressure required for nucleation is attained, at
which point, a rapid expansion ensues. Interestingly, the point at which the bubble rises
to full height is marked by a significant dip in the bubble curve, which McCann and
Prince term ―Delayed Release.‖ After this point, the bubble grows slightly larger in a
linear fashion and departs. Both researchers offer good examples of the class of
experiments dealing with orifice construction including a step change in diameter near
the orifice tip, where the bubble cycle and bubble shape take on values not observed in
systems with smooth capillary orifices (Figure 4.3.3.2).
77
0
0.25
0.5
0.75
1
1
1.00
Water
ReL = 1200 (1.6 mL/s)
Vb / Vb,max
0.75
d0 = 6.35 mm, Vs = 170.0 mL,
(McCann and Prince, 1971)
d0 = 1.75 mm, Vs = 0.2 mL
0.75
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.3.1 - Temporal volumetric bubble development in water, capillary orifice versus
plate orifice construction at Q = 1.6 mL/s
78
a. Water, d0 = 6.35 mm; Vs = 2205.0 mL; ReL = 200 (0.28 mL/s (Kupferburg and Jameson, 1969)
b. Water, d0 = 6.35 mm; Vs = 170.0 mL; ReL = 1200 (1.62 mL/s), (McCann and Prince, 1971)
c. Water, d0 = 1.75 mm; ReL = 1200 (1.62 mL/s)
Figure 4.3.3.2 Interfacial shape in air-water systems with plate orifice and cylindrical visualization surface (a,b) versus capillary
orifice and plane visualization surface (c)
79
4.3.4 Visualization Surface
To understand the effect of the visual method on interfacial shape and included
volume, two tank geometries, a cylindrical column and a rectangular column, were filled
with different liquids. Steel spheres were chosen as reference objects in three
denominations: 3.957, 4.117 and 9.520 mm in diameter by Vernier screw-caliper
respectively. Bubbles having an equivalent diameter equal to the spheres would be in the
range of 0.01 – 0.07 mL, values typical of the experimental range of the present study.
These spheres were placed on a stand (the same stand used throughout) in the open air
and in both tank geometries while filled with water. Silhouettes of the spheres were
photographed using the imaging method described in Section 3.5.2. Images are displayed
in Figure 4.3.4.1 and their respective area measures are described in Table 4.3.4.1. The
scale stated in the image at bottom right is typical for all images in the figure. Horizontal
distortion becomes evident as the sphere in consideration grows in comparison to the
radius of surface curvature. The researchers who did not use plane surfaces, in general,
used cylindrical columns. From the standpoint of optical physics, these two geometries
may be related in that the plane is simply a cylindrical column where the radius is infinite.
In this way, the optics may be compared and borne out in a predictive manner, an
exercise which was carried out schematically for the present case and has been included
as Appendix A. It can be seen that, out of the bubble studies here reviewed from some 70
years of research, 46% (23 of 50) of those employing visualization as an analytical
method also used a cylindrical column of finite curvature (Table 4.3.4.2). To better
understand the present body of literature on the subject of bubble growth and improve
80
future experimental and analytical efforts in bubble growth, researchers depend on a
more holistic understanding of the impact of visual methods on their experimental results.
Although both interfacial size and interfacial shape play a central role in bubble
growth study, primary importance is historically placed on size reported either in the
form of bubble diameter or bubble volume. This is probably owed in some part to the
degree of precision possible to both parameters. At the point that images relied on hand
measurements, several methods were available to gain a cross-checked, accurate, and
relatively precise measure of the bubble volume [3]. From the volumetric measure, aided
by the assumption of spherical geometry, researchers could calculate the bubble diameter
reliably, often noting that there was a limited range for which the spherical assumption
holds.
81
Figure 4.3.4.1 – (top to bottom) Steel spheres of 3.96, 4.12, 9.52 mm diameter on an
identical base in (left to right) open air, water bath in plane-faced Perspex, water bath in
cylindrical-faced borosilicate
Table 4.3.4.1 – Comparison of areas measured from images taken through
different visualization surfaces
Visual Methods Used
by Researchers
Aggregate
[# studies]
Cylindrical
[# studies]
Plane
[# studies]
Bubble Visualizations
50
23
27
No Action
40
14
26
Validation
2
1
1
Index Matching
2
1
1
Surface Modification
7
7
0
82
0.00
0.50
0.75
1.00
2.25
2.00
2
1.75
1.50
 [Hb/W b]
0.25
2.25
Glycerin
d0 = 1.75mm
Vb = 0.27 mL
1.75
Plane Surface, Nc = 138.9 (301.5 mL)
Cylindrical Surface, Nc = 119.8 (260.0 mL),
(Subramani, 2007)
1.5
1.25
1.25
1.00
1
0.75
0.75
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.4.2 - Temporal bubble aspect ratio development in glycerin visualized through
different surfaces at d0 = 1.75 mm, Vb = 0.27 mL
83
0.00
0.25
0.50
0.75
1.00
2.25
2.25
2.00
 [Hb/W b]
1.75
Water
d0 = 1.75mm
Vb = 0.05 mL
2
Plane Surface, Nc = 110.4 (301.5 mL)
Cylindrical Surface, Nc = 95.2 (260.0 mL),
(Subramani, 2007)
1.75
1.50
1.5
1.25
1.25
1.00
1
0.75
0.75
0.50
0.5
0.25
0.25
0.00
0.00
0
0.25
0.50
0.75
1.00
g/BI
Figure 4.3.4.3 - Temporal bubble aspect ratio development in water visualized through
different surfaces at d0 = 1.75 mm, Vb = 0.05 mL
84
8
4
0.20
0.40
0.60
0.80
1.00
/BI
a. Water; Nc = 110.4 (300 mL); ReL = 500 (0.71 mL/s); BI = 75 ms; plane surface
0 mm
8
4
0.50
0.68
0.75
0.83
1.00
/BI
b. Water; Nc = 95.2 (300 mL); ReL = 500 (0.67 mL/s); BI = 85 ms; cylindrical surface (Subramani, 2007)
0 mm
8
4
0.20
0.40
0.60
0.80
1.00
/BI
0 mm
c. Water; Nc = 0.1 (300 mL); ReL = 500 (0.72 mL/s); BI = 66 ms; plane surface
Figure 4.3.4.4 – Interfacial shape in air-water systems with capillary orifice of d0 = 1.75 mm, effect of visualization surface
85
12
6
0.90
0.93
0.95
0.98
1.00
/BI
0 mm
a. Glycerin; Nc = 0.1 (0.2 mL); ReL = 0.1 (0.1 mL/s; BI = 2200 ms; plane surface
8
4
0.89
0.91
0.97
0.94
1.00
/BI
0 mm
b. Glycerin; Nc = 0.1 (260.0 mL); ReL = 0.1 (0.1 mL/s); BI = 1470 ms; cylindrical surface (Subramani, 2007)
8
4
0.20
0.40
0.80
0.60
1.00
/BI
0 mm
c. Glycerin; Nc = 0.1 (301.5 mL); ReL = 0.1 (0.1 mL/s); BI = 430 ms; plane surface
Figure 4.3.3.5 – Interfacial shape in air-glycerin systems with capillary orifice of d0 = 1.75 mm, effect of visualization
surface
86
12
Cylindrical Column
Rectangular Prism
Number of Publications
10
8
6
4
2
0
1940
1950
1960
1970
1980
1990
Decade of Publication
2000
2010
Figure 4.3.4.6 - Visualization surfaces used for bubble studies depending on image data
from the 1940s to the present
87
Table 4.3.4.2 – Common visual methods employed by researchers for bubble study from
1941 - 2011
Visual Methods Used
by Researchers
Aggregate
[# studies]
Cylindrical
[# studies]
Plane
[# studies]
Bubble Visualizations
50
23
27
No Action
40
14
26
Validation
2
1
1
Index Matching
2
1
1
Surface Modification
7
7
0
88
Recent advancements in optics and visual technology have led to faster camera
responses, x-ray visualization of liquid metal systems, in situ measurement of bubbly
flows, and many other previously impossible investigations [1]. Nevertheless,
experimental investigations of bubble growth have not often quantitatively treated the
interfacial aspect ratio. The present study deals with the temporal dynamics of bubble
aspect ratio for the first time to elucidate the impact of system effects on bubble growth.
Figures 4.3.4.2 and 4.3.4.3 show the unique signature of the cylindrical column on the
visualization of single bubbles. In both sets of images the departing bubble visualized
through a cylindrical column is closer to an oblate ellipsoid than its comparator. The
visual effect observed is superposed on the effect of system volume. System volume can
be seen from the previous sections to be the key impact on the size of departing bubbles.
But it is to the shape of departing bubbles that this work turns.
Many computational and analytical studies depend on experimental data for
validation—the same data from which bubble size was reported on the basis of a
spherical assumption. The importance of interfacial shape at departure is paramount to
the understanding of the gas-liquid interactions that take place across the interface in a
host of theoretical studies as well as practical applications. Even in the fundamental case
of a nearly perfect sphere, visualization alters shape in a manner that depends on the ratio
of the spherical radius to the radius of curvature of the surface. This cannot be mapped
out precisely in the present comparison without an exact understanding of the interaction
of the camera lens and the silhouette image, but the impact can be quantitatively borne
out by a study of the temporal aspect ratio development as measured using both surfaces.
89
Measures of interfacial aspect ratio from the present study‘s bubble images are
contrasted with measurements of images reported by Subramani, 2007 in Figures 4.3.4.2
and 4.3.4.3 [19]. The most distinct comparison is that of the departure value obtained,
which is far closer to unity for Subramani than for the present study. It cannot be
assumed from this, of course, that the visualizations measured by Subramani are more
spherical; rather, their maximum width and their maximum height are more or less equal
in certain cases. For water, it can be seen that the development of bubble height in
comparison to bubble width has a very similar pattern both for Subramani‘s images and
the present work. Coupled with Figure 4.3.4.5, it can be seen that the ultimate shape of
the bubble visualization through a cylindrical column is relatively wider, just as Figure
4.3.4.1 predicts.
For glycerin, the contrast increases, owing to the impact of system volume and
viscosity on the temporal volumetric development of the bubble. In the plane surface
visualization, there is a smooth, monotonically ascending function for the majority of the
growth cycle, as height slowly dominates width and the bubble departs at an aspect ratio
of about 1.5. Figures 4.3.4.3 and 4.3.4.5 show a much lower aspect ratio at departure for
glycerin, approaching unity. The additional differentiation effected by the joint impact of
system volume and increased viscosity causes the bubble to reach a larger size during
rapid expansion than in water at equivalent conditions. The additional radial growth
during this phase happens more rapidly than the previous and subsequent portions of the
growth cycle. As magnification depends on the ration rb/R, or the radius of the bubble
and the surface radius, this magnification accelerates as the radial expansion accelerates.
This is observed to occur at precisely the point the nucleate bubble attains spherical
90
geometry, having expanded into the pool from its initial quasi-hemispherical meniscus.
The combined impact of system volume, liquid viscosity, and visualization surface
creates a corner in the temporal aspect ratio. This is significant because it displays the
dynamic characteristic of the influence of the visualization surface. Magnification in the
horizontal plane is dependent on the curvature of the surface as well as the illuminated
body, as in lens theory for the case of an infinite hemispherical lens. For viscous liquids,
not only is the overall volume greater at departure where system volume is appreciable,
but the interfacial expansion occurs much more rapidly (Figure 4.3.2.5). The rate of
change in bubble radius is suddenly much larger during this period, and the magnification
in the horizontal plane grows from nearly zero to some finite value in a very short
period—depicted by the nearly horizontal advance as time elapses. This visual effect
decays during the subsequent slow growth and translation stage occurring just after this
initial, rapid expansion.
Figures 4.3.4.2 – 4.3.4.5 give the sense that, while interfacial included volume
depends much less strongly on the visualization surface, the interpretation of interfacial
shape stands to be better understood through attention to the visual method employed by
the experimentalist. Additional to the visualization photographs included in this
subsection, the visualizations by McCann and Prince and Kupferburg and Jameson
depicted in Figure 4.3.4.1 were taken through a cylindrical column face. In the case of
Kupferburg and Jameson, these results were used to verify later computational modeling
efforts [11,14,54]. Researchers have dealt with this visual dependency in different ways
throughout the literature, though the majority of reviewed cases do not record any
specific action taken to reduce the impact of the visualization surface. The most popular
91
method reviewed here took the form of a modification to return the visualization surface
to planar geometry (Table 4.3.4.2). Others used image validation, either contrasting
comparative measures or directly measuring a reference scale. Still others used either a
combination of validation and refractive index matching or the latter in isolation to
ensure a minimal effect from the surface-liquid combination. It is interesting to note that
these steps were taken not only by those using cylindrical columns, but also in the present
study and by other researchers using plane surfaces. In addition, Figure 4.3.4.6 gives a
clear indication that researchers continue to use cylindrical surfaces in the study of
bubbles while also relying on visualization as a component of the study. It is, therefore,
critical to the refinement of analytical and experimental methods used in the study of
bubble growth to understand the impact of surface geometry on the results obtained. The
horizontal magnification imbued by the cylindrical visualization surface alters the visual
interpretation of interfacial shape, both in its static and dynamic aspect, and comprises a
unique system impact on the expression of deterministic forces in bubble growth.
92
Chapter 5
Conclusion
The present study investigates the effects of the experimental system and method
on the data that resulted from the study of gas-liquid interfaces. Bubble literature from
1941 to the present is reviewed with an eye to the characteristics and commonalities of
the experimental system, as well as a focus on the expression of the primary measures
sought for the bubble growth problem: interfacial size and shape. These measures are
expressed severally as bubble radius, bubble departure diameter, and bubble volume in
the case of interfacial size; bubble silhouette, bubble profile, and bubble aspect ratio in
the case of interfacial shape. Where the growth of bubbles is concerned, increased
importance has historically been given to the interfacial size in experimental study and
the convention has been to assume a spherical bubble shape. Recent efforts that focus on
the necking and departure, nucleation, and coalescence phenomena involved in bubble
growth expand our view of the bubble growth process and its nuances, as well as lend
important observations to the continued pursuit to rigorously and systematically describe
the impact of liquid properties on adiabatic interfacial development in infinite, stagnant
liquid pools over the full range of gas flow rates and orifice diameters.
Many researchers have investigated bubble growth where the volumetric rate of
gas flow is held constant (Section 2). Other researchers have investigated the alternate
boundary condition, where gas pressure in the system is held constant even while bubbles
depart from the orifice into the pool. These researchers investigate the impact of gas
chamber volume and increase it to the point that it shows no further impact on the
resultant departure diameter [27]. Still other researchers state the system volume and the
93
gas flow rate, without adhering completely to operation at either boundary condition.
These experiments at mixed flow conditions provide an additional body of data that aids
future researchers in understanding the impact of system volume and orifice geometry by
allowing such an analysis as is conducted in the present study. It is found that so-called
―chamber volume‖ may or may not include all possible sources of pressure fluctuation
and volumetric capacitance, leading to bubble formation at a mixed boundary condition.
Instead, system volume can be stated as the included volume between the orifice exit and
the final upstream restriction. This definition is applied in the present study and found
consistent even where researchers have not stated the presence of a chamber [17,29]. It is
also found that increased system volume tends to decrease relative bubble height for
aqueous glycerin at low flow rates. Increased system volume also impacts the force
balance during bubble growth, enhancing the effect of liquid viscosity and resulting in
larger single bubbles over a longer growth period for the same rate of gas flow. In the
present study, a porous restrictor was inserted at the base of the orifice to create a
restriction that both minimized system volume and caused instantaneously constant gas
flow, or a linear volumetric growth rate. This effect is achieved for glycerin. This is also
achieved for water at higher flow rates, probably owing to un-damped interfacial
vibrations in the bubble wake and the liquid meniscus. This low volume, constant flow
system minimized the impact of viscosity on bubble growth at low flow rates. It was
found that, under these conditions, bubbles grown in aqueous glycerin solutions may
have smaller departure volumes than those in water at equivalent rates of flow, consistent
in principal with the Tate‘s Law Bubble. Where the value of viscosity exceeds 100 fold
the value for water this was not observed until ReO was less than 0.10. Tate‘s Law over-
94
Table 5.1 – System impacts on interfacial included volume and aspect ratio
System Characteristic
Interfacial Included Volume:
Interfacial Aspect Ratio
System Volume
increases with increasing
system volume
decreases with increasing
system volume
Visualization Surface Curvature
not significantly impacted
decreases with decreasing
radius of surface curvature
Orifice Construction
develops less monotonically
where smooth capillaries are
not used
-
Liquid Viscosity
shows increasing impact of
system volume with increasing
viscosity of the test liquid
shows increasing impact of other
characteristics with increasing
viscosity of the test liquid.
95
predicts the asymptote both for water and aq. glycerin, but is based on the assumption of
a spherical bubble, which is shown to fit less well in the present system at low volume
(Figures 4.3.1.4 – 4.3.1.7). The effects described are confirmed over a broad range of
flow rates and orifice diameters based on this and other works. It was also found that the
abrupt change in the internal diameter of the gas flow path, such as with a plate orifice,
demonstrates an effect similar to that of system volume for aqueous glycerin and that the
dynamic volumetric development in water is unique for plate orifice systems.
The experimental study of bubbles depends on visualization to validate the
assumption of a spherical bubble and to enhance the researcher‘s observations on the
nature of adiabatic bubble growth. Measures and visualizations made through the surface
of cylindrical columns are impacted by the surface itself, which has the tendency to
behave like an infinite, hemispherical lens and magnify the image in the horizontal
slightly. This produces a complex effect, as the magnification depends on the focal length
of the lens and the ratio of the object boundary‘s horizontal displacement from the center
of curvature to its radius. It is therefore concluded that the visualization surface must be
taken into account in the future and modified as needed to improve the comparability of
future results. This imperative is underscored by the current reliance of analytical and
computational bubble growth study on experimental data that was obtained from
visualization in systems using cylindrical surfaces. Furthermore, it is recommended that
future research makes a clear distinction between interfacial included volume and shape,
as they serve unique purposes in industry and cannot be assumed to relate through
sphericity in some cases. Specifically, the aspect ratio of an interface with equivalent
included volume is show in the present study to vary up to 35% from one system to
96
another, based on the system impacts described. A common emphasis on the impact of
flow metering, system volume, orifice construction and the visualization surface will
improve the robustness of future research in adiabatic bubble growth in infinite, stagnant
liquid pools. A new treatment of bubble shape as a unique, yet primary parameter will
open novel avenues and set the stage for future scientific advancement. It is to these joint
aims that this study dedicates itself, with the results described and concluded here.
97
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102
Appendix A
Experimental Data
A.1 Temporal volumetric bubble development data
Water
Water
Plane Surface
Plane Surface
d0 = 1.75 mm
d0 = 1.75 mm
Vs [mL] = 0.2
Vs [mL] = 301.5
Q [mL/s] = 0.03
Q [mL/s] = 0.11
Q [mL/s] = 0.72
Q [mL/s] = 0.2
Q [mL/s] = 0.10
BI [ms] = 1022
BI [ms] = 324
BI [ms] = 66
BI [ms] = 243
BI [ms] = 75
Q [mL/s] = 0.70
BI [ms] = 55
Vb [mL] = 0.04
Vb [mL] = 0.05
Vb [mL] = 0.06
Vb [mL] = 0.06
Vb [mL] = 0.06
Vb [mL] = 0.06
 g/BI
Vb/Vb,m ax
 g/BI
Vb/Vb,m ax
 g/BI
Vb/Vb,m ax
 g/BI
Vb/Vb,m ax
 g/BI
Vb/Vb,m ax
 g/BI
Vb/Vb,m ax
0.00
0.01
0.00
0.01
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.00
0.02
0.01
0.03
0.01
0.02
0.00
0.03
0.00
0.02
0.00
0.04
0.01
0.04
0.01
0.06
0.03
0.04
0.00
0.05
0.00
0.05
0.00
0.06
0.01
0.06
0.02
0.09
0.04
0.06
0.00
0.08
0.00
0.07
0.00
0.08
0.01
0.09
0.02
0.12
0.06
0.07
0.00
0.09
0.00
0.09
0.00
0.10
0.01
0.11
0.03
0.15
0.08
0.09
0.00
0.12
0.01
0.11
0.00
0.12
0.01
0.13
0.03
0.18
0.10
0.11
0.01
0.15
0.03
0.14
0.00
0.14
0.01
0.15
0.04
0.21
0.12
0.13
0.04
0.17
0.05
0.16
0.00
0.16
0.01
0.17
0.05
0.24
0.14
0.15
0.06
0.23
0.09
0.18
0.00
0.18
0.01
0.19
0.06
0.27
0.16
0.17
0.08
0.24
0.11
0.20
0.00
0.20
0.01
0.22
0.07
0.30
0.19
0.19
0.10
0.27
0.13
0.23
0.00
0.22
0.01
0.24
0.09
0.33
0.21
0.20
0.11
0.28
0.15
0.25
0.00
0.23
0.01
0.26
0.10
0.36
0.24
0.22
0.13
0.30
0.16
0.27
0.00
0.25
0.01
0.28
0.11
0.39
0.26
0.24
0.15
0.31
0.17
0.30
0.00
0.27
0.01
0.30
0.13
0.42
0.29
0.26
0.16
0.32
0.18
0.32
0.00
0.29
0.01
0.32
0.14
0.45
0.32
0.28
0.17
0.34
0.19
0.34
0.00
0.31
0.02
0.35
0.16
0.48
0.35
0.30
0.18
0.36
0.22
0.36
0.00
0.33
0.02
0.37
0.18
0.51
0.38
0.31
0.20
0.38
0.23
0.39
0.00
0.35
0.02
0.39
0.19
0.54
0.42
0.33
0.22
0.39
0.24
0.41
0.00
0.37
0.02
0.41
0.21
0.57
0.45
0.35
0.24
0.42
0.27
0.43
0.00
0.39
0.03
0.43
0.23
0.60
0.49
0.37
0.25
0.43
0.28
0.46
0.00
0.41
0.03
0.45
0.25
0.63
0.52
0.39
0.27
0.46
0.31
0.48
0.00
0.43
0.03
0.47
0.27
0.66
0.56
0.41
0.28
0.47
0.33
0.50
0.00
0.45
0.04
0.50
0.29
0.69
0.59
0.43
0.30
0.50
0.36
0.52
0.00
0.47
0.05
0.52
0.31
0.72
0.63
0.44
0.33
0.51
0.37
0.55
0.00
0.49
0.06
0.54
0.34
0.75
0.67
0.46
0.35
0.52
0.39
0.57
0.00
0.51
0.07
0.56
0.36
0.78
0.71
0.48
0.37
0.54
0.41
0.59
0.00
0.53
0.09
0.58
0.38
0.81
0.75
0.50
0.40
0.55
0.42
0.61
0.01
0.55
0.10
0.60
0.41
0.84
0.80
0.52
0.42
0.58
0.45
0.64
0.02
0.57
0.12
0.63
0.43
0.87
0.85
0.54
0.45
0.60
0.48
0.66
0.05
0.59
0.14
0.65
0.46
0.90
0.89
0.56
0.47
0.62
0.50
0.68
0.08
0.61
0.16
0.67
0.48
0.93
0.94
0.57
0.49
0.63
0.52
0.71
0.12
0.63
0.19
0.69
0.51
0.96
0.98
0.59
0.51
0.64
0.53
0.73
0.16
0.65
0.22
0.71
0.54
0.99
1.00
0.61
0.54
0.66
0.55
0.75
0.21
0.67
0.24
0.73
0.56
0.63
0.56
0.70
0.60
0.77
0.26
0.68
0.27
0.76
0.59
0.65
0.58
0.71
0.62
0.80
0.32
0.70
0.30
0.78
0.62
0.67
0.60
0.72
0.64
0.82
0.38
0.72
0.34
0.80
0.65
0.69
0.62
0.75
0.67
0.84
0.45
0.74
0.37
0.82
0.68
0.70
0.65
0.77
0.69
0.87
0.52
0.76
0.41
0.84
0.72
0.72
0.67
0.79
0.73
0.89
0.59
0.78
0.44
0.86
0.75
0.74
0.70
0.81
0.75
0.91
0.67
0.80
0.48
0.88
0.79
0.76
0.72
0.82
0.77
0.93
0.76
0.82
0.52
0.91
0.83
0.78
0.75
0.83
0.78
0.96
0.85
0.84
0.56
0.93
0.87
0.80
0.77
0.85
0.81
0.98
0.94
0.86
0.60
0.95
0.91
0.81
0.80
0.86
0.82
1.00
1.00
0.88
0.65
0.97
0.96
0.83
0.83
0.87
0.85
0.90
0.69
0.99
1.00
0.85
0.85
0.93
0.92
0.92
0.74
0.87
0.87
0.94
0.94
0.94
0.79
0.89
0.90
0.95
0.96
0.96
0.85
0.91
0.92
0.97
0.98
0.98
0.92
0.93
0.94
0.98
0.99
1.00
1.00
0.94
0.96
0.99
1.00
0.96
0.98
0.98
1.00
1.00
1.00
103
Glycerin
Glycerin
Plane Surface
Plane Surface
d0 = 1.75 mm
d0 = 1.75 mm
Vs [mL] = 0.2
Vs [mL] = 301.5
Q [mL/s] = 0.03
Q [mL/s] = 0.10
Q [mL/s] = 0.20
Q [mL/s] = 0.11
Q [mL/s] = 0.63
BI [ms] = 976
BI [ms] = 430
BI [ms] = 254
BI [ms] = 2200
BI [ms] = 450
Q [mL/s] = 1.08
BI [ms] = 260
Vb [mL] = 0.04
Vb [mL] = 0.05
Vb [mL] = 0.07
Vb [mL] = 0.26
Vb [mL] = 0.32
Vb [mL] = 0.31
 g/BI
Vb/Vb,m ax
 g/BI
Vb/Vb,m ax
 g/BI
Vb/Vb,m ax
 g/BI
Vb/Vb,m ax
 g/BI
Vb/Vb,m ax
 g/BI
Vb/Vb,m ax
0.01
0.00
0.01
0.00
0.02
0.01
0.35
0.00
0.02
0.00
0.04
0.00
0.02
0.00
0.02
0.00
0.04
0.01
0.76
0.00
0.05
0.00
0.08
0.00
0.04
0.00
0.04
0.01
0.06
0.02
0.76
0.00
0.07
0.00
0.16
0.00
0.05
0.00
0.05
0.01
0.08
0.03
0.77
0.00
0.10
0.00
0.20
0.01
0.06
0.00
0.06
0.01
0.11
0.05
0.77
0.00
0.12
0.00
0.25
0.01
0.07
0.00
0.08
0.02
0.13
0.07
0.78
0.00
0.15
0.00
0.29
0.01
0.08
0.00
0.09
0.03
0.15
0.09
0.78
0.00
0.17
0.00
0.35
0.01
0.10
0.01
0.10
0.04
0.17
0.11
0.79
0.00
0.20
0.00
0.39
0.02
0.11
0.02
0.11
0.05
0.19
0.14
0.79
0.00
0.23
0.00
0.43
0.03
0.12
0.02
0.13
0.06
0.21
0.16
0.80
0.00
0.25
0.00
0.47
0.04
0.13
0.03
0.14
0.07
0.24
0.18
0.80
0.00
0.28
0.00
0.52
0.07
0.15
0.04
0.15
0.09
0.26
0.20
0.81
0.00
0.30
0.00
0.56
0.11
0.16
0.05
0.17
0.10
0.28
0.23
0.81
0.00
0.32
0.00
0.60
0.16
0.17
0.07
0.18
0.11
0.30
0.25
0.82
0.00
0.35
0.00
0.64
0.22
0.18
0.08
0.19
0.12
0.32
0.27
0.82
0.00
0.37
0.00
0.69
0.29
0.19
0.09
0.20
0.14
0.35
0.29
0.83
0.00
0.40
0.00
0.73
0.38
0.21
0.10
0.22
0.15
0.37
0.32
0.83
0.00
0.42
0.00
0.77
0.48
0.22
0.11
0.23
0.16
0.39
0.34
0.84
0.00
0.45
0.00
0.81
0.58
0.23
0.12
0.24
0.18
0.41
0.36
0.84
0.00
0.47
0.00
0.86
0.69
0.24
0.13
0.25
0.19
0.43
0.39
0.85
0.00
0.50
0.00
0.90
0.80
0.26
0.14
0.27
0.20
0.45
0.41
0.85
0.00
0.52
0.00
0.94
0.91
0.27
0.15
0.28
0.22
0.48
0.44
0.86
0.00
0.55
0.01
0.98
0.99
0.28
0.17
0.29
0.23
0.50
0.46
0.86
0.00
0.57
0.01
0.29
0.18
0.31
0.24
0.52
0.49
0.87
0.00
0.59
0.01
0.30
0.19
0.32
0.26
0.54
0.51
0.87
0.00
0.62
0.01
0.32
0.20
0.33
0.27
0.56
0.54
0.88
0.00
0.64
0.02
0.33
0.21
0.34
0.28
0.58
0.56
0.88
0.01
0.67
0.03
0.35
0.23
0.36
0.30
0.61
0.59
0.89
0.01
0.69
0.06
0.36
0.25
0.37
0.31
0.63
0.62
0.89
0.01
0.72
0.09
0.38
0.26
0.38
0.32
0.65
0.64
0.90
0.01
0.74
0.13
0.39
0.27
0.40
0.34
0.67
0.67
0.90
0.01
0.77
0.19
0.40
0.28
0.41
0.35
0.69
0.70
0.91
0.01
0.79
0.26
0.41
0.29
0.42
0.37
0.71
0.72
0.91
0.02
0.81
0.33
0.43
0.31
0.43
0.38
0.74
0.75
0.92
0.03
0.84
0.42
0.44
0.32
0.45
0.39
0.76
0.78
0.92
0.04
0.86
0.51
0.45
0.33
0.46
0.41
0.78
0.81
0.93
0.06
0.89
0.61
0.46
0.34
0.47
0.42
0.80
0.83
0.93
0.09
0.91
0.71
0.47
0.34
0.49
0.43
0.82
0.86
0.94
0.13
0.94
0.82
0.49
0.35
0.50
0.43
0.84
0.89
0.94
0.17
0.96
0.91
0.50
0.35
0.51
0.44
0.87
0.91
0.95
0.23
0.99
0.99
0.51
0.36
0.52
0.46
0.89
0.94
0.95
0.30
0.52
0.36
0.54
0.48
0.91
0.96
0.96
0.37
0.54
0.38
0.55
0.50
0.93
0.98
0.96
0.45
0.55
0.41
0.56
0.51
0.95
0.99
0.97
0.53
0.56
0.43
0.58
0.53
0.97
1.00
0.97
0.62
0.57
0.44
0.59
0.55
1.00
0.99
0.98
0.71
0.58
0.46
0.60
0.56
0.98
0.80
0.60
0.48
0.61
0.58
0.99
0.89
0.61
0.49
0.63
0.59
0.99
0.97
0.62
0.50
0.64
0.61
1.00
1.00
0.63
0.52
0.65
0.63
0.65
0.53
0.66
0.64
0.66
0.55
0.68
0.66
0.67
0.56
0.69
0.68
0.68
0.58
0.70
0.69
0.69
0.59
0.72
0.71
0.71
0.60
0.73
0.72
0.72
0.62
0.74
0.74
0.73
0.63
0.75
0.76
0.74
0.65
0.77
0.77
0.76
0.66
0.78
0.79
0.77
0.68
0.79
0.81
0.78
0.70
0.81
0.82
0.79
0.71
0.82
0.84
0.80
0.73
0.83
0.86
0.82
0.74
0.84
0.87
0.83
0.76
0.86
0.89
0.84
0.78
0.87
0.91
0.85
0.79
0.88
0.92
0.86
0.81
0.90
0.94
0.88
0.83
0.91
0.95
0.89
0.85
0.92
0.97
0.91
0.88
0.93
0.98
0.92
0.89
0.95
0.99
0.93
0.91
0.96
1.00
0.95
0.93
0.97
1.00
0.96
0.95
0.98
1.00
0.97
0.96
1.00
0.97
0.98
0.98
1.00
1.00
104
A.2 Temporal bubble aspect ratio development data
Water
Water
Plane Surface
Plane Surface
d0 = 1.75 mm
d0 = 1.75 mm
Vs [mL] = 0.2
Vs [mL] = 301.5
Q [mL/s] = 0.03
Q [mL/s] = 0.11
Q [mL/s] = 0.72
Q [mL/s] = 0.2
Q [mL/s] = 0.10
BI [ms] = 1022
BI [ms] = 324
BI [ms] = 66
BI [ms] = 243
BI [ms] = 75
Q [mL/s] = 0.70
BI [ms] = 55
Vb [mL] = 0.04
Vb [mL] = 0.05
Vb [mL] = 0.06
Vb [mL] = 0.06
Vb [mL] = 0.06
Vb [mL] = 0.06
 g/BI
 [H b/Wb]
 g/BI
 [H b/Wb]
 g/BI
 [Hb/Wb]
 g/BI
 [Hb/Wb]
 g/BI
 [Hb/Wb]
 g/BI
 [Hb/Wb]
0.00
0.18
0.00
0.13
0.00
0.11
0.00
0.05
0.00
0.01
0.00
0.00
0.02
0.11
0.03
0.19
0.05
0.35
0.02
0.00
0.03
0.00
0.05
0.00
0.04
0.14
0.07
0.29
0.10
0.57
0.04
0.00
0.07
0.00
0.09
0.13
0.06
0.15
0.10
0.39
0.15
0.66
0.06
0.00
0.10
0.05
0.14
0.62
0.08
0.15
0.14
0.49
0.20
0.77
0.08
0.00
0.13
0.40
0.19
1.01
0.10
0.18
0.17
0.57
0.25
0.81
0.10
0.00
0.17
0.65
0.23
1.05
0.12
0.18
0.20
0.65
0.30
0.86
0.12
0.00
0.20
0.74
0.28
0.90
0.14
0.18
0.24
0.72
0.35
0.92
0.14
0.00
0.23
0.84
0.32
0.93
0.16
0.19
0.27
0.77
0.40
0.94
0.17
0.00
0.27
0.93
0.37
0.96
0.18
0.20
0.31
0.81
0.45
0.97
0.19
0.00
0.30
0.98
0.42
0.97
0.20
0.20
0.34
0.86
0.50
1.04
0.21
0.00
0.34
0.98
0.46
1.05
0.22
0.23
0.37
0.90
0.55
1.08
0.23
0.00
0.37
0.99
0.51
1.16
0.23
0.23
0.41
0.91
0.60
1.11
0.25
0.00
0.40
1.01
0.56
1.23
0.25
0.24
0.44
0.93
0.65
1.15
0.27
0.00
0.44
1.03
0.60
1.26
0.27
0.26
0.47
0.97
0.70
1.19
0.29
0.00
0.47
1.07
0.65
1.28
0.29
0.27
0.51
0.99
0.75
1.25
0.31
0.00
0.50
1.10
0.69
1.32
0.31
0.28
0.54
1.01
0.80
1.30
0.33
0.00
0.54
1.14
0.74
1.37
0.33
0.31
0.58
1.03
0.85
1.35
0.35
0.00
0.57
1.17
0.79
1.42
0.35
0.32
0.61
1.07
0.90
1.40
0.37
0.00
0.60
1.20
0.83
1.44
0.37
0.35
0.64
1.08
0.95
1.48
0.39
0.00
0.64
1.24
0.88
1.46
0.39
0.38
0.68
1.11
1.00
1.53
0.41
0.00
0.67
1.26
0.93
1.49
0.41
0.42
0.71
1.13
0.43
0.00
0.70
1.28
0.97
1.50
0.43
0.45
0.75
1.17
0.46
0.00
0.72
1.30
1.00
1.50
0.45
0.50
0.78
1.21
0.48
0.00
0.77
1.34
0.47
0.54
0.81
1.24
0.50
0.00
0.81
1.37
0.49
0.61
0.85
1.28
0.52
0.00
0.84
1.40
0.51
0.66
0.88
1.32
0.54
0.00
0.87
1.43
0.53
0.72
0.92
1.36
0.56
0.01
0.91
1.46
0.55
0.76
0.95
1.46
0.58
0.11
0.94
1.51
0.58
0.81
1.00
1.59
0.60
0.23
0.97
1.53
0.61
0.87
0.62
0.37
1.00
1.55
0.63
0.88
0.64
0.51
0.65
0.92
0.66
0.67
0.67
0.95
0.68
0.81
0.68
0.96
0.70
0.88
0.70
0.99
0.72
0.93
0.72
1.01
0.75
0.98
0.74
1.02
0.77
1.01
0.76
1.07
0.79
1.06
0.78
1.06
0.81
1.08
0.80
1.10
0.83
1.13
0.82
1.12
0.85
1.17
0.84
1.15
0.87
1.20
0.86
1.17
0.89
1.27
0.88
1.21
0.91
1.30
0.90
1.24
0.93
1.36
0.92
1.28
0.95
1.42
0.94
1.30
0.97
1.48
0.96
1.37
1.00
1.54
0.98
1.46
1.00
1.58
105
Glycerin
Glycerin
Plane Surface
Plane Surface
d0 = 1.75 mm
d0 = 1.75 mm
Vs [mL] = 0.2
Vs [mL] = 301.5
Q [mL/s] = 0.03
Q [mL/s] = 0.10
Q [mL/s] = 0.20
Q [mL/s] = 0.11
Q [mL/s] = 0.63
BI [ms] = 976
BI [ms] = 430
BI [ms] = 254
BI [ms] = 2200
BI [ms] = 450
BI [ms] = 260
Vb [mL] = 0.04
Vb [mL] = 0.05
Vb [mL] = 0.07
Vb [mL] = 0.26
Vb [mL] = 0.32
Vb [mL] = 0.31
Q [mL/s] = 1.08
 g/BI
 [H b/Wb]
 g/BI
 [H b/Wb]
 g/BI
 [Hb/Wb]
 g/BI
 [Hb/Wb]
 g/BI
 [Hb/Wb]
 g/BI
 [Hb/Wb]
0.00
0.19
0.00
0.14
0.00
0.08
0.00
0.14
0.00
0.27
0.00
0.07
0.02
0.00
0.05
0.08
0.08
0.31
0.35
0.05
0.02
0.15
0.04
0.00
0.04
0.04
0.09
0.34
0.16
0.66
0.76
0.20
0.04
0.12
0.08
0.00
0.06
0.13
0.14
0.60
0.24
0.82
0.76
0.20
0.07
0.09
0.12
0.03
0.08
0.23
0.19
0.73
0.32
0.95
0.76
0.20
0.09
0.06
0.19
0.07
0.10
0.35
0.24
0.82
0.40
1.06
0.77
0.22
0.11
0.05
0.23
0.12
0.12
0.46
0.28
0.91
0.48
1.15
0.77
0.22
0.14
0.03
0.27
0.16
0.14
0.56
0.33
0.97
0.56
1.25
0.78
0.22
0.16
0.02
0.32
0.27
0.17
0.67
0.38
1.03
0.64
1.34
0.78
0.22
0.18
0.00
0.36
0.36
0.19
0.74
0.42
1.07
0.72
1.42
0.79
0.23
0.20
0.00
0.40
0.46
0.21
0.78
0.47
1.14
0.80
1.50
0.79
0.23
0.23
0.00
0.44
0.54
0.23
0.82
0.52
1.19
0.88
1.57
0.80
0.23
0.25
0.00
0.48
0.61
0.25
0.86
0.57
1.22
0.96
1.66
0.80
0.23
0.27
0.02
0.51
0.69
0.26
0.86
0.59
1.24
1.00
1.69
0.81
0.25
0.30
0.02
0.55
0.74
0.27
0.89
0.61
1.28
0.81
0.25
0.32
0.02
0.59
0.79
0.29
0.91
0.66
1.31
0.82
0.25
0.34
0.03
0.63
0.85
0.31
0.93
0.71
1.36
0.82
0.26
0.36
0.05
0.67
0.90
0.33
0.95
0.75
1.41
0.82
0.26
0.39
0.06
0.71
0.95
0.35
0.97
0.80
1.45
0.83
0.28
0.41
0.08
0.75
1.01
0.37
0.98
0.85
1.50
0.83
0.28
0.43
0.09
0.78
1.06
0.39
1.00
0.90
1.55
0.84
0.29
0.45
0.11
0.82
1.13
0.41
1.03
0.94
1.60
0.84
0.29
0.47
0.14
0.86
1.19
0.43
1.04
0.99
1.64
0.85
0.31
0.50
0.17
0.90
1.26
0.44
1.05
1.00
1.66
0.85
0.31
0.52
0.21
0.94
1.32
0.46
1.05
0.86
0.32
0.54
0.26
0.98
1.38
0.48
1.08
0.86
0.32
0.56
0.29
1.00
1.43
0.50
1.09
0.86
0.34
0.56
0.32
0.52
1.09
0.87
0.35
0.59
0.39
0.54
1.12
0.87
0.37
0.61
0.48
0.56
1.13
0.88
0.38
0.63
0.55
0.58
1.17
0.88
0.40
0.65
0.63
0.60
1.17
0.88
0.43
0.68
0.67
0.62
1.19
0.89
0.46
0.70
0.71
0.64
1.20
0.89
0.49
0.72
0.75
0.66
1.22
0.90
0.54
0.74
0.80
0.68
1.24
0.90
0.58
0.76
0.84
0.70
1.26
0.91
0.66
0.79
0.88
0.72
1.27
0.91
0.70
0.81
0.92
0.75
1.31
0.92
0.72
0.83
0.97
0.77
1.33
0.92
0.76
0.85
1.02
0.79
1.35
0.93
0.79
0.88
1.07
0.81
1.37
0.93
0.81
0.90
1.12
0.83
1.40
0.94
0.82
0.92
1.18
0.85
1.43
0.94
0.85
0.94
1.24
0.87
1.45
0.94
0.87
0.97
1.30
0.89
1.47
0.95
0.90
0.99
1.39
0.91
1.48
0.95
0.92
1.00
1.44
0.93
1.51
0.95
0.93
0.95
1.54
0.96
0.98
0.97
1.58
0.96
1.02
0.99
1.60
0.97
1.05
1.00
1.62
0.97
1.10
0.98
1.15
0.98
1.19
0.99
1.25
0.99
1.29
1.00
1.34
1.00
1.40
106
A.3 Comparative bubble aspect ratio development data
Water
Glycerin
d0 = 1.75 mm
d0 = 1.75 mm
Plane Surface
Cylindrical Surface
Plane Surface
Plane Surface
Cylindrical Surface
Vs [mL] = 301.5
Vs [mL] = 260.0
Vs [mL] = 0.2
Vs [mL] = 301.5
Vs [mL] = 260.0
Q [mL/s] = 0.48
Q [mL/s] = 1.67
Q [mL/s] = 0.72
Q [mL/s] = 1.08
Q [mL/s] = 1.33
Vb [mL] = 0.05
Vb [mL] = 0.05
Vb [mL] = 0.06
Vb [mL] = 0.27
Vb [mL] = 0.27
 g/BI
 [Hb/Wb]
 g/BI
 [Hb/Wb]
 g/BI
 [H b/Wb]
 g/BI
 [H b/Wb]
 g/BI
 [Hb/Wb]
0.00
0.00
0.00
0.11
0.00
0.11
0.00
0.07
0.00
0.27
0.02
0.00
0.02
0.00
0.02
0.04
0.02
0.04
0.02
0.18
0.03
0.00
0.04
0.00
0.03
0.20
0.03
0.00
0.04
0.08
0.05
0.00
0.07
0.00
0.05
0.31
0.05
0.00
0.06
0.00
0.06
0.00
0.09
0.00
0.06
0.36
0.06
0.00
0.08
0.00
0.08
0.00
0.11
0.00
0.08
0.42
0.08
0.00
0.11
0.00
0.09
0.00
0.13
0.00
0.09
0.50
0.09
0.01
0.13
0.00
0.11
0.08
0.16
0.09
0.11
0.58
0.11
0.01
0.15
0.00
0.12
0.21
0.18
0.20
0.12
0.59
0.16
0.04
0.17
0.00
0.14
0.32
0.20
0.33
0.14
0.62
0.17
0.06
0.19
0.00
0.15
0.39
0.22
0.46
0.15
0.66
0.19
0.07
0.21
0.00
0.17
0.48
0.25
0.53
0.17
0.71
0.21
0.09
0.23
0.00
0.18
0.59
0.27
0.56
0.18
0.76
0.22
0.10
0.25
0.00
0.20
0.68
0.29
0.60
0.20
0.77
0.24
0.12
0.28
0.00
0.21
0.76
0.31
0.64
0.21
0.79
0.25
0.15
0.30
0.00
0.23
0.81
0.34
0.67
0.23
0.78
0.27
0.16
0.32
0.00
0.24
0.82
0.36
0.70
0.24
0.78
0.28
0.19
0.34
0.00
0.26
0.84
0.38
0.73
0.26
0.80
0.31
0.25
0.36
0.00
0.27
0.87
0.40
0.75
0.27
0.83
0.33
0.28
0.38
0.03
0.29
0.90
0.43
0.75
0.29
0.85
0.34
0.33
0.40
0.05
0.30
0.92
0.45
0.76
0.30
0.86
0.36
0.36
0.42
0.07
0.32
0.93
0.47
0.76
0.32
0.89
0.38
0.42
0.45
0.10
0.33
0.95
0.49
0.76
0.33
0.90
0.39
0.44
0.47
0.15
0.35
0.96
0.51
0.78
0.35
0.91
0.41
0.48
0.49
0.20
0.36
0.97
0.54
0.76
0.36
0.92
0.42
0.51
0.51
0.27
0.38
0.98
0.56
0.79
0.38
0.92
0.44
0.54
0.53
0.33
0.39
1.00
0.58
0.79
0.39
0.94
0.45
0.58
0.55
0.42
0.41
1.01
0.60
0.81
0.41
0.95
0.47
0.60
0.57
0.53
0.42
1.02
0.63
0.83
0.42
0.95
0.48
0.63
0.59
0.55
0.44
1.03
0.65
0.84
0.44
0.97
0.50
0.65
0.62
0.56
0.45
1.05
0.67
0.86
0.45
0.97
0.51
0.69
0.64
0.55
0.47
1.07
0.69
0.87
0.47
0.99
0.53
0.70
0.66
0.55
0.48
1.07
0.72
0.88
0.48
1.01
0.55
0.73
0.68
0.57
0.50
1.07
0.74
0.90
0.50
1.03
0.56
0.75
0.70
0.58
0.52
1.10
0.76
0.90
0.51
1.04
0.58
0.77
0.72
0.59
0.53
1.11
0.78
0.92
0.53
1.05
0.59
0.79
0.74
0.60
0.55
1.13
0.81
0.93
0.54
1.06
0.61
0.80
0.76
0.64
0.56
1.13
0.83
0.94
0.56
1.08
0.62
0.84
0.79
0.67
0.58
1.15
0.85
0.95
0.57
1.09
0.64
0.85
0.81
0.70
0.59
1.16
0.87
0.95
0.59
1.10
0.65
0.87
0.83
0.72
0.61
1.17
0.90
0.97
0.60
1.11
0.67
0.90
0.85
0.76
0.62
1.18
0.92
0.98
0.62
1.13
0.68
0.92
0.87
0.79
0.64
1.19
0.94
0.99
0.63
1.14
0.70
0.94
0.89
0.82
0.65
1.20
0.96
0.99
0.65
1.15
0.71
0.96
0.91
0.85
0.67
1.21
1.00
1.01
0.66
1.16
0.73
0.99
0.93
0.89
0.68
1.22
0.68
1.17
0.75
1.01
0.95
0.92
0.70
1.24
0.69
1.18
0.76
1.04
0.98
0.95
0.71
1.25
0.71
1.20
0.78
1.06
1.00
0.99
0.73
1.26
0.72
1.21
0.79
1.07
0.74
1.28
0.74
1.22
0.81
1.11
0.76
1.28
0.75
1.25
0.82
1.13
0.77
1.31
0.77
1.26
0.84
1.15
0.79
1.32
0.78
1.27
0.85
1.17
0.80
1.33
0.80
1.30
0.87
1.21
0.82
1.36
0.81
1.31
0.88
1.23
0.83
1.37
0.83
1.31
0.90
1.26
0.85
1.39
0.84
1.33
0.92
1.28
0.86
1.40
0.86
1.36
0.93
1.31
0.88
1.43
0.87
1.39
0.95
1.33
0.89
1.44
0.89
1.40
0.96
1.36
0.91
1.46
0.90
1.40
0.98
1.38
0.92
1.47
0.92
1.43
1.00
1.43
0.94
1.50
0.93
1.46
0.95
1.52
0.95
1.47
0.97
1.53
0.96
1.50
0.98
1.55
0.98
1.50
1.00
1.55
1.00
1.53
107
108
Appendix B
Horizontal Image Magnification

Air
0
n0
Tank
Wall
1
nT
2
Liquid
nL
t
T
3
L
a.
b.
dm
R
db
c.
Figure A.1 - Snell's-Law magnification of a silhouette: a. Refraction through a curved
pane b. Traces of 1 Critical, 2 Supercritical, and 3 Subcritical light rays c. Dependence of
image magnification on distance
109
As the diameter is calculated from the imaging result through its relationship to
area when sphericity is assumed [2], there is a desire to estimate the result of
magnification in the horizontal plane on the final result. The area of a sphere‘s silhouette
is a level set of the volume having equivalent radius, for which the area is of course r2.
If the silhouette of a sphere is magnified in one and only one plane, a regular ellipse is the
result, and the new area measurement would be ab, where a = r and b = (1+m)r. ‗m‘ is
the magnification ratio. The change in diameter of such an ellipse, referenced to the
original dimension is
db = 2r[(1+m)(1/2) – 1]
(A.1)
where the equivalent bubble diameter is classically calculated as in equation 3.5.2.1. If
the magnification was, say, 5.00% of the total area value, the resultant change in diameter
would then be 2.47%. In addition, the image will be calibrated using a physical reference,
often the orifice tip itself. The orifice tip lies in the horizontal plane, which has been
magnified, causing the image scale measured in pixels per unit area to increase slightly.
This slight increase diminishes the additional area induced by magnification and
complicates the estimation process somewhat, while at the same time explaining why the
size reported by researchers is not impacted significantly by visualization surface
curvature, while the aspect ratio decreases by nearly 20% in some cases.
110
Appendix C
Pixel Analysis Script (Matlab)
%Bubble Attributes - by GBW
close all; % Close all figures (except those of imtool.)
clear all; % Erase all existing variables.
workspace; % Make sure the workspace panel is showing.
fontSize = 20;
files = dir('*.tif'); %Calls all TIFF files in the working directory
dout = 2.1031;%input('Enter the outer diameter of the orifice : ');
d0 = 1.7532;
% imCal = imread('~cal.jpg'); %Use this for Aravind's images
% imCal = imcrop(imCal,[0 0 600 600]); %Use this for Aravind's images
imCal = imread('~cal.tif'); %Use this for Gabriel's images
imCal = imCal(:,:,1);
imCal = im2bw(imCal,graythresh(imCal));
% %Rotate Aravind's image 90 degrees clockwise
% imCal = transpose(imCal);
% imCal = fliplr(imCal);
imCal = ~imCal;
Cal = numel(nonzeros(imCal));
[rows columns numberOfColorBands] = size(imCal);
CalHt = find(imCal(round(rows/2)));
LnCal = max(find(imCal(:,2)))-min(find(imCal(:,2)));
LnScale = dout/LnCal; %Conversion factor for units/pixel
for k=1:numel(files)
grayImage = imread(files(k).name);
% grayImage = imcrop(grayImage,[0 0 600 163]); %for Aravind's images
grayImage = imcrop(grayImage,[0 0 600 246]); %for GBW images
grayImage = grayImage(:,:,1);
grayImage = im2bw(grayImage,graythresh(grayImage)); %Convert the Gray Image to binary data
%
%
%
%Rotate Aravind's image 90 degrees clockwise
grayImage = transpose(grayImage);
grayImage = fliplr(grayImage);
[rows columns] = size(grayImage);
grayImage = imfill(grayImage,'holes');
grayImage = ~grayImage;
grayImage = imfill(grayImage,'holes');
if grayImage(round(rows/2),columns) == 1
grayImage = ~grayImage;
columns = find(grayImage(round(rows/2),:),1,'last')+1;
grayImage = ~grayImage;
%
figure
%
imshow(grayImage)
end
% Parametrize the edge
111
%Note: in my case, the IMAGE is rotated 90 degrees ccw and justified left
%Resets the object start position
obj = 1; %Initialatize object counter
top = CalHt; %Initialize the vertical limit
CtrY = 0; %Initialize the Centroid measure
while top<columns
if max(grayImage(:,top)) ~= 0
%Trace an individual object
width = max(find(grayImage(:,top)))-min(find(grayImage(:,top)));
if round(width/2)-(width/2) < 0
ctr = round(width/2)+1;
elseif round(width/2)-(width/2) > 0
ctr = round(width/2)-1;
else
ctr = width/2;
end
for w=ctr:-1:0 %draws the object bottom line
lhsX(obj,ctr+1-w) = min(find(grayImage(:,top)))+w;
lhsY(obj,ctr+1-w) = top;
end
pos = ctr+1;
for m=top:columns
%Get the centroid
CtrY = CtrY + numel(nonzeros(grayImage(:,m)));
%We didn't want to start on a corner
if max(grayImage(:,m)) ~= 0
%
if m-top == 0 && max(grayImage(:,m+4)) ~= 0
%
OmgaL(m) = (min(find(grayImage(:,m+4)))-min(find(grayImage(:,m))))/3; %1-element
root-angle calculation
%%
OmgaR = ( min(find(grayImage(:,m+6)))-min(find(grayImage(:,m-1)))/5 ); %1element root-angle calculation
%%
omg(k,obj) = (OmgaR+OmgaL)/2
%
end
lhsX(obj,m-top+pos) = min(find(grayImage(:,m))); %Obtains the left edge X's & Y's
lhsY(obj,m-top+pos) = m;
rhsX(obj,m-top+pos) = max(find(grayImage(:,m))); %Obtains the left edge X's & Y's
rhsY(obj,m-top+pos) = m;
else
%Calculate Object's Attributes
if obj == 1 && ( numel(nonzeros(grayImage(:,top:m)))-Cal ) < 3
dsph(k,obj) = 0; %Sets the diameter to zero in the case of slight variance
elseif obj == 1
Aobj(k,obj) = ( numel(nonzeros(grayImage(:,top:m)))-Cal )*(LnScale^2);
dsph(k,obj) = real(sqrt(Aobj(k,obj)/pi))*(2/d0);
else
Aobj(k,obj) = ( numel(nonzeros(grayImage(:,top:m))) )*(LnScale^2);
dsph(k,obj) = real(sqrt(Aobj(k,obj)/pi))*(2/d0);
end
boxW(k,obj) = (max(rhsX(obj,:)) - min(lhsX(obj,:)))*LnScale;
boxH(k,obj) = (m-top)*LnScale;
CentroidtY(k,obj) = CtrY/(m-top);
CtrY = 0;
%
alpha(obj) = boxW(obj)/boxH(obj);
%
formA(obj) = pi*((boxW(obj)*LnScale/2)^2);
%draw the object top line
112
pos = m-top+pos;
top = m-1; %saves the terminating vertical position
width = max(find(grayImage(:,top)))-min(find(grayImage(:,top)));
if round(width/2)-(width/2) < 0
ctr = round(width/2)+1;
elseif round(width/2)-(width/2) > 0
ctr = round(width/2)-1;
else
ctr = width/2;
end
for w=0:ctr
lhsX(obj,pos+w) = lhsX(obj,pos+w-1)+1;
lhsY(obj,pos+w) = top;
rhsX(obj,pos+w) = rhsX(obj,pos+w-1)-1;
rhsY(obj,pos+w) = top;
end
break
end
end
%
pos = 1; %Resets value for the object trace index
top = top+1;
clear lhsX lhsY rhsX rhsY
if obj < 2
obj = obj+1;
else
break
end
%
Atot(k) = numel(nonzeros(grayImage))*(LnScale^2);
else
top = top+1;
end
end
clear lhsX rhsX lhsY rhsY
end
% S = cellstr(['t [msec]' 'db [mm]'])
myFile = char(files.name);
myFile = cellstr(myFile);
myHead = cellstr({'PictureFrame'; 'db* object1'; 'db* object2'; 'Height obj1'; 'Height
obj2'; 'Width obj1'; 'Width obj1'});
myHead = transpose(myHead);
xlswrite('Spreadsheet_Data.xls', myHead, 'Temp-Spat', 'A1')
xlswrite('Spreadsheet_Data.xls', myFile, 'Temp-Spat', 'A2')
xlswrite('Spreadsheet_Data.xls', dsph, 'Temp-Spat', 'B2')
xlswrite('Spreadsheet_Data.xls', boxH, 'Temp-Spat', 'D2')
xlswrite('Spreadsheet_Data.xls', boxW, 'Temp-Spat', 'F2')
113
Appendix D
Measurement Uncertainty
As Moffat suggests [55], several expressions of uncertainty exist for values reported by
the experimentalist. Here, a brief treatment of the measurement uncertainty is included to
lend perspective to the results presented. The primary experimental system relied on
measures of interfacial pixel area, volumetric gas flow per unit time, time per bubble
departure event, and maximum pixel length for interfacial silhouette dimensions. The
measurement uncertainties associated with each of these quantities are included in Table
D.1 along with a general range of values encountered over the experimental domain.
114
Table D.1 – Measurement uncertainties of the principal variables measured in the present study
Estimated at a typical
high flow rate
Estimated at a typical
low flow rate
0.03 - 1.6
0.00049%
0.00250%
ms
4465 - 53
0.00890%
0.00000%
Length
pixels
66 - 382
0.09183%
0.07506%
Area
pixels2
17938 - 55939
0.05800%
0.01865%
Measure
Units
Range of Values
Volumetric gas
flow per time
mL/s
Bubble Interval
Measurement Uncertainty
115