A Parametric Investigation of Gas Bubble Growth and Pinch-Off Dynamics from Capillary-Tube Orifices in Liquid Pools

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 and Applied Sciences

2012

Deepak Saagar Kalaikadal

B.E. Mechanical Engineering Anna University, Chennai, Tamil Nadu, India, 2007

Committee Chair: Dr. Raj M. Manglik

ABSTRACT

The air-bubble dynamics phenomena in adiabatic liquid pools has been studied so as to present a better understanding of the parameters which that govern the process of ebullience, bubble growth and departure from a submerged capillary-tube orifice. The orifice diameter is found to directly dictate the bubble departure diameter, and the pinch-off is controlled by a characteristic neck-length.

To study the role of orifice size on the growth and departure of adiabatic single bubbles, experiments were performed with different diameter capillary tubes submerged in of distilled de-ionized water as well as some other viscous liquids. A correlation has been developed based on the experimental data of this study along with those reported by several others in the literature. The predictions of this correlation agree very well with measured data for water as well as several other more viscous liquids. It is also found that the bubble departure diameter is the same as the orifice diameter when the latter equals twice the capillary length.

The phenomenon of bubble necking and departure was explored experimentally and through a scaling analysis. Experiments were performed with five different liquids (water, ethanol, ethylene glycol, propylene glycol, and glycerol) to extract the departure neck-lengths for isolated gas bubbles at pinch-off from the capillary orifice. A scaling analysis of the experimental data indicated that the bubble neck-length at departure or pinch-off was predicted by a balance of buoyancy, viscous and surface tension forces. These were established to be represented by the Galilei and Morton numbers, and a power-law type predictive correlation has been shown to be in excellent agreement with the available data over a wide range of liquid properties.

To characterize and model the growth and departure of single bubbles in different liquid pools, a theoretical model has been established. The motion of the gas-liquid interface has been modeled as a scaled force balance involving buoyancy, gas-momentum, pressure, surface tension, inertia and drag. With one-dimensional scaling of these forces, the model captures the incipience, growth, necking and departure of a bubble as it emerges from the orifice. Here necking and pinch-off is modeled based on the newly developed neck-length correlation. The results are compared with experimental data and are found to be in excellent agreement for a range of liquids, orifice sizes and flow rates. The predictions highlight the variations in bubble equivalent diameters at departure with orifice sizes, flow rates and fluid properties, and they further reiterate the well-established two-regime theory of bubble growth. The latter involves (a) the constant volume regime, where the bubble volume remains near constant and relatively independent of flow rate, and (b) the growing bubble regime, where the size of the bubble increases proportionately with the gas flow rate.

Finally, the complex nature of ebullience in aqueous surfactant solutions has been studied using the reagents FS-50, SDS, and CTAB. The influence of the modulated liquid surface tension or more specifically, the role of the time dependent dynamic surface tension on the formation and departure of adiabatic bubbles has been investigated. Comparative studies have been undertaken to investigate the effect of time-dependent surface tension relaxation in surfactant solutions as opposed to ebullience in pure liquids with the same equilibrium surface tensions. Results highlight the effects of the surfactant’s molecular weight on the adsorption-desorption kinetics, and the consequent influence on ebullience. It has been established that the bubbling characteristics in surfactant solutions are, in the first order, governed by the dynamic surface tension of the solute-solvent system.

iii

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my advisor, Dr. Raj Manglik, for the invaluable guidance, motivation and most importantly, patience, extended during my graduate studies at the University of Cincinnati. I am thankful to him for his numerous suggestions, the unwavering support, and the timely advices over the years I spent on my

Master’s research. I am equally thankful to Dr. Milind Jog for his availability and counsels throughout my studies. I would also like to thank Dr. Yuen-Koh Kao for taking time and serving on the committee. It was an enriching experience to work under them and I am honored to continue doing the same as I proceed towards my doctoral degree.

Special thanks are also due to my roommate – Mr. Prassanna Sai Ramesh, for sticking through the thick and thin, and to several friends, who were as much a source of inspiration, support, and joy. I am also indebted to all my lab mates at the TFTPL, especially Mr. Vishaul

Ravi, Mr. Rupesh Bhatia, and Mr. Utkarsh Verma, for all their timely assistance in conducting experiments, and the many lively lab discussions and talks which were equally fun and informative.

And lastly, I would love to thank my mother and all of my family and friends back home in India, for their unconditional love, support, and motivation, which have always helped me to keep my spirits high and my goals within sight.

TABLE OF CONTENTS

ABSTRACT i

ACKNOWLEDGEMENTS iv

TABLE OF CONTENTS v

LIST OF TABLES viii

LIST OF FIGURES ix

NOMENCLATURE xiii

CHAPTER 1, Introduction and Scope of Study 1

CHAPTER 2, Effect of Orifice Size on Bubble Departure Sizes 5

2.1 Introduction 5

2.2 Experimental Method 8

2.3 Results and Discussion 10

2.3.1 Bubble departure in water 11

2.3.2 Bubble departure in viscous liquids 19

2.4 Conclusions 23

CHAPTER 3, Bubble Necking in Isolated Bubble Regime 24

3.1 Introduction 24

3.2 Experimental Procedure 28

3.3 Results and Development of Correlation 29

3.4 Conclusions 36

CHAPTER 4, Theoretical Modeling of Single Bubble Dynamics 37

4.1 Introduction 37

4.2 Mathematical Model and Force Balance 44

4.2.1 Aiding Forces 45

4.2.2 Retarding Forces 49

4.2.3 Force Balance and Departure 54

4.3 Results and Discussion 57

3.4 Conclusions 62

CHAPTER 5, Dynamic Surface Tension Effect on Bubble Growth Dynamics 64

5.1 Introduction 64

5.2 Materials Used 69

5.3 Surface Tension Measurements 70

5.4 Experimental Setup 70

5.5 Results and Discussion 72

5.5.1 Dynamic Surface Tension Measurements 72

5.5.2 Effect of Temporal Surface Tension Relaxation 74

5.5.2 Effect of Surfactant Molecular Weight 81

5.6 Conclusions 85

CHAPTER 6, Recommendations for Future Work 86

Bibliography 88

Appendix A, Image Processing and Error Analysis 95

A.a. Gray Scale Pixel Analysis 95

A.b. Experimental Precision and Uncertainty 96

Appendix B, Experimental Data 97

B.a. Bubble Departure Diameters (Effect of Orifice Size) 97

B.b. Bubble Neck Lengths at Departure 99

B.c. Bubble Departure Diameters (Theoretical Modeling) 100

B.d. Bubble Departure Diameters in Surfactant Solutions 107

Appendix C, Algorithm for Solution of Mathematical Model 109

vii

LIST OF TABLES

Table No. Description Page No.

Table 2.1 Properties of distilled, deionized water at 23oC 8

Table 2.2 Bubble diameters in the constant-volume regime for the 11 orifices sizes considered

Table 2.3 Physical Properties of Liquids at 23oC 19

Table 3.1 Effect of Orifice Size on Neck Length 32

Table 5.1 Physicochemical Properties of the Surfactants used 69

viii

LIST OF FIGURES

Figure No. Description Page No.

Fig.2.1 Bubble departure diameters in the constant volume regime 7 observed over a range of orifice sizes

Fig.2.2 Effect of gas flow rate at various orifice sizes as observed by 7 Subramani et al. (2007, 2008)

Fig.2.3 Schematic diagram of experimental apparatus 9

Fig.2.4 Spatial and temporal evolution of bubble from two different 13 orifices at an air flow rate of 20 ml/min

Fig.2.5 (a)-(b) Schematic representations of bubble incipience and departure 15 for (a) dO < 2lC and (b) dO ~ 2lC

Fig.2.6 A comparison of the buoyancy and surface tension forces at 16 incipience for various orifice sizes

Fig.2.7 Visualization and bubble departure diameters for the orifices 17 under study

Fig.2.8 Variation of normalized bubble departure diameter with orifice 18 diameter – experimental data and empirical correlation

Fig.2.9 A comparison of Eq.(2.3) with experimental data for viscous 20 liquids

Fig.2.10 Variation of normalized bubble departure diameter with orifice 22 diameter for inviscid and viscous liquids – experimental data and empirical correlation

Fig.3.1 A comparison of the experimentally observed neck-lengths 27 with the empirical assumptions proposed by Tsuge and Hibino (1983) and Räbiger (1984)

Fig.3.2 Schematic of a departing bubble and its neck-length 29

Fig.3.3 Experimental neck-lengths for the fluids considered, at 30 different orifices

Fig.3.4 Experimental neck-lengths for glycerol, propylene glycol and 31

water, do = 1 mm, Q = 20 ml/min

Fig.3.5 A schematic representation of the dominant forces acting on 33 the elongated neck

Fig.3.6 Experimental neck-lengths for different fluids at corresponding 34 Galilei numbers

Fig.3.7 A comparison of the predictions of the developed correlation 35 and the available experimental data

Fig.4.1 Representation of the forces governing the growth of an 38 adiabatic single bubble from a submerged orifice

Fig.4.2 Comparison of Air : Water experimental data with the 42 predictions obtained from correlations given by other investigators

Fig.4.3 Comparison of Air : Glycerol experimental data with the 43 predictions obtained from correlations given by other investigators

Fig.4.4 A schematic representation of the pressure forces acting on the 48 growing bubble

Fig.4.5 Representation of the parameters involved in the formulation 49 of the surface tension force

Fig.4.6 Superposition of the Stokes Correlation and the Miyahara 52 Correlation to give a universal formulation for CD over a wide range of Reynolds Numbers

Fig.4.7 Representation of bubble growth as modeled in the theoretical 55 approach - through inception, spherical growth, development of a cylindrical neck, and departure.

Fig.4.8 Comparison of the theoretical model results with experimental 58 data from Subramani et al. for water, ethanol, glycerol and propylene glycol.

Fig.4.9 Comparison of the theoretical model predictions with the air- 58 water ebullience data from Subramani et al. (2007, 2008) (1 mm) and the current study (0.66 mm)

Fig.4.10 Comparison of the theoretical model predictions with 59 Räbiger’s (1984) experimental data

Fig.4.11 Comparison of the theoretical model predictions with the air- 60 propylene glycol ebullience data of Subramani et al. (2007, 2008)

Fig.4.12 Comparison of the theoretical model predictions with the air- 60 glycerol ebullience data of Subramani et al. (2007, 2008)

Fig.4.13 Scatter plot for comparison of theoretical predictions with 61 experimental data from literature for various fluids

Fig.5.1 Surface tension relaxation in FS-50 compared the relaxation in 68 the surfactants used by Loubiére and Hébrard (2004)

Fig.5.2 Surface tension relaxation in FS-50 at 1400 wppm as compared 73 to the static surface tensions of water and ethanol

Fig.5.3 Surface tension relaxation in FS-50 at 400 wppm, CTAB 74 (CMC) and SDS (CMC)

Fig.5.4 Departure dynamics of a single bubble growing in pure liquids 75 (do = 1 mm) – water and ethanol

Fig.5.5 Departure dynamics of a single bubble growing in FS-50 (1400 78 wppm), water and ethanol at different bubble departure frequencies

Fig.5.6 A comparison of the departure bubble sizes for FS-50 (1400 80 wppm), water and ethanol over the entire range of experimental bubble frequencies

Fig.5.7 Departure dynamics of a single bubble growing from a 82 submerged orifice (do = 0.8 mm) in FS-50 (400wppm), SDS (CMC) and CTAB (CMC) at distinct bubble departure frequencies

Fig.5.8 A comparison of the departure bubble sizes for FS-50 (400 84 wppm), SDS (CMC) and water over the entire range of experimental bubble frequencies

xii

Nomenclature

A cross-sectional area [m2]

CD drag coefficient [-] d diameter [m]

F force [kgm/s2]

Fr Froude Number g acceleration due to gravity [m/s2]

Ga Galilei Number

hib height of incipient bubble [m]

lc capillary length [m]

ln neck-length [m]

M mass [kg]

Mo Morton Number

P absolute pressure [Pa]

Q volumetric flow rate [m3/s]

3 Qb bubble volume [m ] r bubble radius [m] r* distance from the center of bubble to orifice [m]

We Weber Number

Re Reynolds Number, [-] t time [s]

tb bubble interval [s]

xiii

td growth time [s] v velocity [m/s]

V volume [m3]

3 Vib volume of incipient bubble [m ]

Greek Letters

 added mass coefficient

 viscosity [Pas]

 density [kg/m3]

 angle of contact

 surface tension [N/m]

Subscripts b bubble

B buoyancy

D drag g gas

G gas momentum

I inertia l liquid o orifice

P pressure

xiv

CHAPTER 1

Introduction and Scope of Study

The phenomena of multiphase fluid flow and heat transfer are the core aspects of several engineering applications. The contact and the subsequent interactions between liquid- gas phases is a very frequently observed occurrence in many process industries, the petro- chemical industry, sanitation systems and mining industry, among others. It is either achieved by the injection of a denser medium into a rarer medium as in the case of droplets, or by the introduction of a rarer fluid into a denser fluid in the form of bubbles. On a more specific note, systems where the gas-liquid contact occurs in the form of bubbles find widespread applications in many common processes such as aeration, froth-flotation, bio- processing, and boiling, to name a few. In all these processes, it is of critical importance to comprehend the transport of heat, momentum, and mass across the gas-liquid interface. A better understanding of these mechanisms would aid in engineering more efficient and economic designs of the equipment involved.

The dynamic complexity of gas-liquid ebullient systems has attracted the attention of many researchers in the past. Traditionally, the problem is approached by investigating adiabatic single bubble dynamics, which acts as a first-order approximation for understanding the complex thermal and hydrodynamic transport processes in such systems.

Several numerical, computational/theoretical, and experimental studies have been conducted to explore and understand the interfacial interactions between the gas bubble and its enveloping liquid pool. A recently published review by Kulkarni and Joshi (2005) provides an extensive reassessment of much of such research reported in the literature.

Decades of extensive academic as well as industrial research has established that the process of bubbling is dependent upon several static and dynamic parameters.

Fundamentally, the growth and the departure of a single bubble emanating from a submerged surface cavity or capillary tube orifice, is the collaborative action of several liquid, gas, and surface forces. These forces, which govern the nucleation of a bubble, its growth, pinch-off, and departure generally, scale with the operating conditions (cavity size/orifice diameter, flow rate etc.) and the fluid properties (interfacial tension, viscosity, density). The magnitude and direction of these forces govern the rate of bubble formation, its size, and the eventual vertical movement of the detached bubble. At the same time, it is important to note that while there are a number of governing variables, not all of them influence the ebullience to the same extent.

Hence, it naturally follows that identifying and analyzing the relative influence of the governing parameters individually is essential for a better comprehension of this phenomena.

A survey of the existing literature provides no clear and unambiguous understanding of the relative influence and corresponding scaling of the parameters involved. Also, the relative importance of the necking phenomena, with the subsequent pinch-off and departure is often understated. Moreover, given that the bubbling process finds many diverse modern applications, it is also important to study ebullience in appropriate liquid solutions, slurries, emulsions, and colloids, as opposed to traditional pure liquids such as water, ethanol etc.

Bubbling in such liquid systems is often subject to dynamic variations in physicochemical properties depending on the time scales involved, and the nature of the reagent-like additives, thus adding several additional parametric determinants to the already complex ebullience behavior. This study addresses some of these issues through experimental and theoretical

analyses of adiabatic single bubble growth in aqueous surfactant solutions. It focuses on the effects of the orifice size, dynamic changes in fluid properties, the phenomena of necking and departure, and its contribution to ebullient dynamics.

The study has been divided into four-sections, each dedicated to one of the above mentioned objectives. In the first section, the effect of orifice diameter on the departure sizes of bubbles has been experimentally examined in five different pure liquids (water, ethanol, ethylene glycol, propylene glycol, and glycerol), representing a broad range of values for liquid-gas interfacial tension and viscosity. Experiments were performed using a wide spectrum of capillary tube sizes with diameters ranging from 0.32 mm to 7.00 mm. Based on these results, as well as those in the literature, a predictive correlation has been developed that highlights the effects of the orifice diameter and liquid properties on the bubble departure size.

The second part deals with the experimental and theoretical analysis of the necking phenomenon prior to and at bubble departure. Necking is influence by the simultaneous action of several forces, whose relative governing magnitude changes during the incipience and growth phases. Experimental data of the departure neck-lengths were obtained in different pure liquids so as to determine the relative impact of fluid properties (viscosity, surface tension and density) on the neck-elongation and subsequent pinch-off. Once again, based on these data, a design correlation has been devised, which is capable of predicting the departure neck-lengths of bubbles for a wide range of liquids and orifice diameters.

The third part is an attempt to create a simple one-dimensional model which would be capable of providing first-order estimates of bubble size and frequency in an adiabatic isolated-bubble system. The effects of fluid properties and operating conditions (as derived

and obtained from previous literature) have been incorporated mathematically, together with the departure conditions studied in the previous section, to obtain estimates that closely match experimentally observed values.

The final part is an effort to understand the effect of surface-active additives on ebullience in their respective aqueous solutions. Adiabatic single bubble growth has been recorded in aqueous solutions of three different surfactants (FS-50, CTAB, and SDS) which have different molecular weights, ionic nature, and diffusion characteristics. The experimental results illustrate the influence of dynamic surface tension on bubble shape, size, and frequency. Also, the effects of the surfactant’s ionic nature, molecular weight, and diffusion time scale have been characterized so as to provide a fundamental understanding of bubble growth in reagent solutions.

CHAPTER 2

Effect of Orifice Size on Bubble Departure Size

2.1. Introduction

A detailed survey of the literature on bubble growth and its dynamics in liquid pools has been published by Kulkarni and Joshi (2005). Much of the past research has dealt primarily with the bubble growth process in an effort to characterize bubble departure size, growth time, bubbling frequency and bubble rise velocity, etc. Several factors have been identified that are known to contribute significantly to these fundamental characteristics.

These include the liquid properties of surface tension, viscosity, and density, and the operating conditions such as the flow rates, gas pressure and orifice size. Each of these variables has a distinctive effect on the growth and departure pattern of the bubble. For instance, with all other properties and parameters held constant, an increase in the surface tension of the liquid increases the departure diameter of the emerging bubble and vice-versa.

Similarly, the effects of viscosity, density, gas flow rates, and pressure have also been identified.

Of special interest in the present research is the influence of orifice size on the departure bubble diameters. Besides bubble reactors and aeration chambers, this has an indirect bearing on many two-phase heat transfer applications as well. In boiling, for example, nucleation and subsequent departure of bubbles are strongly influenced by the size, shape, and distribution of heater-surface cavities that act as nucleation sites. Typical cavities on a heater surface, which facilitate ebullience, are of the order of several microns, which are hard to replicate and study. Nevertheless, an understanding of the effect of the size of

comparatively larger orifices on the ebullience would provide additional insights from which it would be plausible to model similar behavior at much smaller scales.

Again, the literature lists several researchers who have experimented with a wide range of orifice sizes in various liquids. Figure 2.1 depicts the data that have been obtained by Benzing and Myers (1955), Datta et al. (1950), Krevelen and Hoftijzer (1950) and

Subramani et al. (2007, 2008), working with a range of orifice sizes. It has been established that the effect of orifice diameter in smaller orifices is negligible, while for large diameter orifices, the bubble diameter increases with the flow rate. The previous studies also unanimously agree with the fact that the size of the bubbles increases with an increase in orifice size, at any given flow rate for a particular liquid, as can be inferred from Fig.2.2, which shows the effect of gas flow rate on the bubble departure diameter in orifices of different sizes, as reported by Subramani et al. (2007, 2008). Additionally, since the contact angle of the bubble varies with the thickness of the orifice, the orifice external diameter is found to have an impact on the bubble departure volume.

It can be seen that while a general idea of the effect of the orifice size is already well established, the relative importance of the orifice size in controlling the bubble departure diameter has not been studied independent of the other parameters. In the present study, the complex relationship between bubble departure diameter and orifice size has been revisited from an experimental and scaling approach in an attempt to establish its exact nature, and further extend the understanding of its relative influence on several ebullience parameters.

Fig.2.1. Bubble departure diameters in the constant volume regime observed over a range of orifice sizes

2 Air-Water Ebullience

do = 0.32 mm

do = 1.00 mm do = 1.76 mm 0 101 102 103 Re O

Fig.2.2. Effect of gas flow rate at various orifice sizes

as observed by Subramani et al. (2007, 2008)

2.2. Experimental Method

The experimental setup for generation and observation of adiabatic bubbles in a quiescent pool of water is depicted in Fig.2.3. Distilled and de-ionized water, with the properties as stated in Table 2.1, was maintained at room temperature in a rectangular container constructed from clear and transparent acrylic slabs. The container has a capacity of over 7000 ml with a rectangular cross section measuring 195 mm x 165 mm. The generation of bubbles was facilitated by a constant supply of compressed air through a blunt dispensing needle tip attached vertically to the bottom of the container. The air flow rate was controlled using flow valves and was quantitatively measured with flow meters and an inverted funnel-burette system. Care was taken to isolate the pool from the surroundings by sealing the mouth of the container with clear protection sheets. To emhasize on the orifice size, needles of five different diameters (0.66 mm, 0.80 mm, 4.89 mm, 6.05 mm and 7.02 mm) were chosen for the experiments. A high-speed high-resolution digital camera was used to capture the bubbling phenomena in real time. The acquired videos and images were then analyzed to extract the desired parameters.

Table 2.1. Properties of distilled, deionized water at 23oC

Density [kg/m3] 1027

Viscosity [Ns/m2] 0.911 x 10-3

Surface Tension [N/m] 72.1 x 10-3

Reflective White Screen

PAR Lighting System Funnel

Graduated Burette

High Speed High Resolution Camera Beaker Dispensing Needle Data Acquisition System

Gas Flow Controller

Air Supply

Fig.2.3. Schematic diagram of experimental apparatus

Relevant parameters such as bubble shape, size, departure frequency, and growth time, were obtained through high-speed high-resolution image processing techniques. The

HSHR digital camera (Hi-DCam II version 3.0 - NAC Image Technology) was set-up perpendicular to the needle tip to obtain a clear view of the bubbles in the frontal plane. All images were recorded at a frame rate of 3000 fps with the camera set at a focal distance of

1.75 ft. from the needle tip. Clear and high contrast images of the growing and departing bubbles were obtained with the help of PAR (Parabolic Aluminized Reflector) lighting systems focused on a white surface in a plane parallel to the plane of the needle. The high resolution images thus obtained were then analyzed using an image processing software,

Image-Pro Plus 4.0 from Media Cybernetics. The equivalent departure diameter of the bubble was obtained from the area occupied by the bubble in the image, assuming the bubble to be a perfect sphere and hence a perfect circle in the plane of the image. The measurements were based on a calibration of the saturation density using the outer diameter of the orifice as the base for the calibration. The growth and departure times were obtained from the frame rate of the camera and the recording time elapsed between two consecutive departures of a bubble from the orifice tip.

The air flow rates were varied to different levels depending on the size of the orifice in use. Bubble growth was recorded in both the constant volume regime and the growing bubble regime, though for this study only the data acquired in the constant volume regime have been taken into consideration.

Further, the errors in the measurement were also estimated. At a frame rate of 3000 fps, the precision in measuring the bubble interval and frequency was ±0.33 ms. Similarly, the precision in the measurement of the equivalent bubble diameter was 0.026 mm, based on the number of pixels in the image and the corresponding saturation density.

2.3. Results and Discussion

The aim of this section is to evaluate and understand the effect of the orifice diameter on the ebullience characteristics of both inviscid and viscous liquids alike. To this effect, this section has been divided into two parts. Firstly, the results of the experimental runs in pure water are analyzed and examined. In the subsequent part, the conclusions obtained from the analysis of the water data are extended to viscous liquids to provide a comprehensive idea of the influence of the orifice diameter on the bubble departure sizes.

2.3.1. Bubble departure in water

The primary data obtained from the experiments with pure water are bubble departure diameter, db, and gas flow rate, Q , at five different orifice sizes characterized by the orifice inner diameter, do. Table 2.2. gives a summary of the bubble departure diameters obtained for each orifice at the lowest air flow rate in their respective constant volume regimes.

Table 2.2. Bubble diameters in the constant-volume regime

for the orifices sizes considered

Orifice Diameter, Bubble Departure Diameter, Normalized Bubble

do (mm) db (mm) Departure Diameter, db/do

0.66 3.36 5.01

0.80 3.41 4.26

4.89 5.12 1.05

6.10 5.23 0.86

7.05 5.98 0.84

The acquired experimental data reiterate the general trend that has been observed by past researchers, i.e. the bubble diameters show a clear increase with an increase in the orifice size. However, a relatively less observed trend is the variation of the rate of change of departure diameter with orifice size. The tabulated results clearly indicate that while the bubble size increases with orifice diameter, the rate at which it does so decreases progressively in the same direction.

A possible explanation for this trend can be arrived at by considering the interplay of the growth-aiding and growth-retarding forces at the orifice tip. Due to the inviscid nature of air-water system, the bubbling characteristics can be assumed to be predominantly governed by the liquid-gas interfacial surface tension, liquid density, and gas flow rate. For a small orifice, during the early stages of bubble development immediately following incipience, the pressure forces and the gas momentum forces are very weak. Additionally, because of the small size of the incipient bubble, the buoyancy forces are also very small. The weak aiding forces result in the bubble being attached to the orifice tip for a comparatively longer period of time before it attains a large enough volume. Then, the buoyancy force overpowers the restraining surface tension force, and the bubble is lifted away from the orifice. The increased time during which the bubble remains attached to the orifice facilitates a larger volume of gas to flow into the bubble, thus resulting in an increased size. On the other hand, bubbles emanating from larger orifices are aided by a comparatively greater buoyancy force right from incipience, which can be attributed to the larger volume of the incipient bubble. The pressure force is also much greater in large orifices and offsets the corresponding increase in the surface tension forces (The pressure force varies as the square of the orifice diameter while the surface tension varies linearly with the orifice size). This increase in the magnitude of the growth-aiding forces quickens the attainment of equilibrium and subsequent pinch-off, resulting in much smaller bubbles.

Figure 2.4 illustrates the spatial and temporal evolution of a bubble at a flow rate of

20 ml/min from orifices of two different sizes (0.32 mm and 1.00 mm). It can be seen that for the smaller orifice, the bubble growth is gradual - particularly near incipience, with the growth time occupying a large fraction of the bubble interval. On the other hand, the bubble

from the larger orifice grows much faster, consuming a relatively smaller fraction of the bubble interval for departure. The plot clearly reveals the slow growth of the aiding forces in the smaller orifice as compared to the larger one, thus leading to a comparatively larger bubble.

d 4

2 Air : Water db = 0.32 mm db = 1.00 mm

0 0 0.2 0.4 0.6 0.8 1  /BI g Fig.2.4. Spatial and temporal evolution of bubble from two different orifices

at an air flow rate of 20 ml/min

Since the volume of the incipient bubble plays an important role in determining the growth time and the departing diameter, its quantification is of paramount importance. Apart from the volume, the shape of the incipient bubble, its curvature, and the corresponding height also show a pronounced dependence on the orifice diameter.

The bubble, at incipience, tends to take the shape of an oblate hemispheroid with its major and minor axes along the radial and the longitudinal directions of the orifice respectively. At low flow rates, a reasonable estimation of the volume and other related parameters can be obtained by considering the incipient bubble to be analogous to a water meniscus rising inside a capillary tube.

The depth and curvature of a capillary meniscus have been mathematically modeled by a number of researchers such as Adams and Bashforth (1883) and Rayleigh (1916), amongst others. The curvature of the meniscus has been shown to decrease with an increase in the diameter of the orifice, progressively approaching a relatively flat free-surface. Using a similar mathematical approach, it is possible to obtain the volume of the incipient bubble, which would then give an estimate of the buoyancy force at the beginning of the growth cycle. The volume and the depth of the meniscus can be expressed in terms of orifice diameter and capillary constant, as has been shown by Gupta (2004). Equations .(2.1) and

(2.2) are modified forms of Gupta’s equations, employing capillary length in place of capillary constant.

3 dO Vib  2 (2.1) 24 4(dlOC / 2 )

3dO hib  2 (2.2) 6 (dlOC / 2 )

3 From Eq.(2.1) it can be seen that the non-dimensional bubble volume (Vdib/  o ) decreases with the orifice diameter. Similarly, the non-dimensional height of the incipient

bubble ( hdibo/ ) also decreases with the orifice diameter, thus reflecting the bubble’s approach towards a flat surface in larger orifices. Figure 2.5 gives a schematic representation of bubble incipience and departure from orifices of two different sizes.

It should however be noted that the bubble volume, by itself, increases with the orifice size, facilitating a larger buoyancy force and a comparatively earlier departure. The trends followed by the aiding and restraining forces immediately after incipience, for specific orifice sizes, based on volumes calculated from Eq. (2.1) are shown in Fig.2.6.

(a) (b)

Fig.2.5. Schematic representations of bubble incipience and departure for

(a) dO < 2lC and (b) dO ~ 2lC

10-2

10-4

10-6

)

(

s -8

e 10

F 10-10

10-12 Buoyancy Surface Tension

10-14 10-3 10-2 10-1 100 101

dO (mm)

Fig.2.6. A comparison of the buoyancy and surface tension forces

at incipience for various orifice sizes

Equations (2.1) and (2.2) very clearly indicate the strong dependence of both the

incipient bubble volume and the meniscus depth on the parameter dlOC/2 . In Fig.2.7, the normalized bubble departure diameter is plotted against the parameter . A nonlinear regression analysis was performed on the data of bubble departure diameters and orifice sizes available in the literature, to yield a correlation between the normalized bubble diameter

ddbO/ and the parameter , which is of the form shown in Eq.(2.3). A comparison

between the correlation’s predictions and the experimental data has also been depicted in

Fig.2.8.

15.9 7.4(dloc / 2 ) ddbo/  (2.3) 1 22.3(dloc / 2 )

l l l l l dO/2 C = 0.187 dO/2 C = 0.329 dO/2 C = 0.913 dO/2 C = 1.123 dO/2 C = 1.311

db/dO = 3.381 db/dO = 2.411 db/dO = 1.108 db/dO = 0.867 db/dO = 0.855

Fig.2.7. Visualization and bubble departure diameters for the orifices under study

It is evident from Fig.2.8 and Eq.(2.3) that the normalized bubble departure diameter

decreases with an increase in the orifice size and reaches unity when dlOC/ 2 1. Beyond

this point, the bubbles emerging from the orifice were observed to be smaller than the orifice

opening. This orifice diameter, which equals twice the capillary length, can be considered to

be the critical orifice size for water. A visual comparison of the bubbles departing from

orifices of different sizes is shown in Fig.2.7. It is interesting to note that for orifices larger in

size than twice the capillary length bubble necking and subsequent pinch-off take place

below the level of the orifice opening due to the flatter geometry of the incipient free surface.

Fig.2.8. Variation of normalized bubble departure diameter with orifice diameter –

experimental data and empirical correlation

2.3.2. Bubble departure in viscous liquids

The effect of liquid viscosity on the size of bubbles emanating from submerged orifices has been explored in several experimental and computational investigations reported in the literature (Datta et al. (1950), Davidson and Schüler (1960), Snabre and Magnifotcham

(1998) and Subramani et al. (2007, 2008)). Though a consensus on the precise effects of viscosity has not been reached, it can be safely claimed that for low to moderate flow rates and orifice diameters, the liquid viscosity has a positive effect on the bubble departure diameter, i.e. bubbles growing in more viscous liquids tend to be comparatively larger than those in liquids with lower viscosities.

Figure 2.9 depicts the normalized bubble departure diameters in four pure liquids with largely different viscosities, from orifices of different sizes, as compared to the values predicted from Eq.(2.3). The liquids and their physical properties are listed in Table 2.3. As in the case of water, it can be seen that while the departure sizes of the bubbles increase with the orifice size, the rates of their increase progressively decrease in larger orifices, the reasons for which are similar to those stated in Section 2.3.1.

Table 2.3. Physical Properties of Liquids at 23oC

Liquid Density Surface Tension Viscosity (kg/m3) (N/m) (x 10-3 Ns/m2) Glycerol 1261 65.0 749.3 Propylene Glycol 1036 35.0 49.9 Ethylene Glycol 1133 48.0 17.0 Ethanol 789 22.7 1.08

Fig.2.9. A comparison of Eq.2.3 with experimental data for viscous liquids

A closer look at Fig.2.9 reveals that though the experimental values are very different from those predicted by the empirical correlation, they tend to follow a similar trend. It can also be noted that the deviations of the experimental data from the correlation are in the increasing order of the liquid viscosity.

Taking into consideration the similar trend of the viscous liquid data and the order of their deviations from the empirical correlation obtained for water, a non-linear regression analysis of the experimental data was carried out with the liquid viscosity as a second

0.15 independent variable. The analysis yielded a viscosity correction factor, (/)w , which when introduced into Eq.(2.3), resulted in a modified correlation as given in Eq.(2.4).

0.15 15.9 7.4(dlo / 2 c )( w / ) ddbo/  0.15 (2.4) 1 22.3(dlo / 2 c )( w / )

The predictions of the modified correlation obtained above were found to be in excellent agreement with the experimental data. The correlation together with the experimental data have been graphed in Fig.2.10 to illustrate the universal applicability of the developed correlation.

14 Glycerol,  = 749.3 cP Propylene Glycol,  = 49.9 cP Ethylene Glycol,  = 17 cP 12 Ethanol,  = 1.08 cP Aqueous Glycerine Solutions, 7 cP <<800 cP Modified Correlation 10

d / 8

6 0.15 15.9 + 7.4(do/2lc)(w/) db/do = 1 + 22.3(d /2l )( /)0.15 4 o c w

0 0.2 0.4 0.6 0.8 1 (d /2l )( / )0.15 o c w  Fig.2.10. Variation of normalized bubble departure diameter with orifice diameter for

inviscid and viscous liquids – experimental data and empirical correlation

2.4. Conclusions

The effect of orifice diameter on the growth and departure of bubbles has been studied and the findings have been compared with previous results obtained in water and viscous liquids alike. Experiments were performed with orifices of five different diameters to analyze the individual contribution of orifice size to bubble growth. It has been observed that though the orifice size has a positive effect on the departing bubble diameter, this effect is progressively smaller in orifices of larger sizes. Further, the bubble departure volume in

water is found to be a strong function of the parameter dlOC/2 , and an empirical correlation was obtained which related the final bubble diameter to the parameter . Likewise, it was established that for viscous liquids a similar relationship exists between the departure

0.15 bubble size and the parameter (dlo / 2 c )( w / ) . A critical orifice diameter, equaling twice the capillary length, has been established for water. The departing bubbles tend to be larger than the orifice when its size is lesser than the critical diameter and become smaller than the orifice at larger orifice sizes.

CHAPTER 3

Bubble Necking in Isolated Bubble Regime

3.1. Introduction

Given the importance of bubbles, drops, and sprays in several industrial processes and natural phenomena, knowing and understanding the generation and growth mechanisms of bubbles and droplets is one of the key essentials to gain a better insight into the complex behavior of multi-phase flows. Invariably, the formation of a bubble takes place in several stages culminating in a pinch-off of the interface between two immiscible fluids. Pinch-off begins with the formation of a neck between the growing bubble and the orifice opening/cavity. The newly formed neck then extends gradually; thinning out to extremely small dimensions, and finally breaks off to release the bubble, thus imitating an approach towards singularity. The mechanics of bubble necking and subsequent bubble detachment are issues that have attracted a lot of focus from several researchers in the past decades.

In the early years of bubble pinch-off research, due to lack of proper techniques and equipment to capture the micro-scale pinch-off process, investigators attempted to characterize necking through mathematical and computational means. In one of the earliest works, Lonquet-Higgins et al. (1991) studied the detachment of bubbles from underwater orifices based on the balance of pressure and capillary forces. Oguz and Prosperetti (1993) used a numerical approach involving boundary-integral methods to study the growth and departure of bubbles in inviscid liquids. Subsequently, Wong et al. (1998), and Sierou and

Lister (2003) also used boundary-integral methods to study pinch-off in viscous liquids.

Gerlach et al. (2007) utilized a combined volume-of-fluid and level-set method to simulate

the process of bubble formation and detachment from a submerged orifice with constant gas flow rate. More recently, Quan and Hua (2008) have numerically modeled and investigated the pinch-off of gas bubbles in viscous liquids through an improved front tracking method.

Both Gerlach et al. (2007) and Quan and Hua (2008) have independently examined the individual influence of each of the fluid properties (density, surface tension, and viscosity) on bubble pinch-off. Their investigations unanimously agreed upon the fact that the shape and dynamics of the pinching neck depend solely several parameters based on fluid properties.

It was only in the recent years that much of the experimental investigations in the field were undertaken, aided by developments in high-speed and high-resolution photography. These advancements revolutionized the ongoing research by offering better techniques to record, capture, and analyze the process regardless of the microscopic scales involved. Peregrine, Shoker and Symon (1990), Burton et al. (2005), Keim et al. (2006) and

Thoroddsen et al. (2007) were some of the many researchers who had put to use the advantages offered by high-speed photography in their investigations of bubble pinch-off.

Interestingly, while most of the previous works have focused exclusively on the phenomenon of pinch-off, not much has been studied about the simultaneous neck elongation accompanying the pinch-off process. In a buoyancy-driven pinch-off of a growing bubble, upward forces of buoyancy and gas velocity tend to increase the length of the attached neck, while surface tension and viscosity oppose the growth, ultimately resulting in a shearing process that frees the bubble. Consequently, in the time elapsed between onset of necking and bubble departure, the neck experiences an extension along the longitudinal axis to a certain fixed length (henceforth termed as ‘departure neck-length’). The departure neck- length has been utilized as an important parameter in a number of existing analytical bubble-

formation models, and is often used as the absolute departure criterion for a fully grown bubble. However, there have been no studies that have attempted to exclusively characterize the neck-length. Analytical bubble growth models employ a number of loose empirical values for the neck-length at departure to arrive at reasonable predictions of several other important ebullience parameters such as bubble size, growth rate, frequency etc. For instance, both Snabre and Magnifotcham (1998), and Gaddis and Vogelpohl (1986) have made use of a departure criterion which assumes the neck-length to be one-fourth of the bubble diameter

(ldnb /4). The assumption was based on the experimental observations of Räbiger (1984) for bubbles growing from a 2mm diameter orifice in water and aqueous glycerol solutions.

Likewise, Nahra and Kamotani (2003) have assumed their departure neck-lengths to be equal to the bubble diameter at force-balance, while Ramakrishnan et al. (1969) have taken it to be the force-balance bubble radius. Both these parameters are extracted from the analytically modeled bubble at the instant when upward bubble forces of buoyancy, pressure, and gas momentum balance downward forces of surface tension, drag, and inertia.

However, the most commonly used departure criterion was proposed by Tsuge and

Hibino (1983), and was subsequently used by Zhang and Shoji (2001) in their modeling of aperiodic bubble growth, and by Kasimsetty et al. (2008) in the theoretical modeling of adiabatic single bubbles. They proposed that the bubble can be assumed to pinch-off when

the neck-length equals the orifice diameter (ldno ). A comparison between Räbiger’s assumption, Tsuge and Hibino’s departure criterion, and the experimentally obtained neck- lengths from Subramani et al. (2007, 2008), is shown in Fig.3.1. It can be clearly noticed that while the empirical correlations make fairly reasonable estimates for inviscid fluids like water and alcohols, their predictions are often erroneous in the case of more viscous fluids.

In this study, neck-length data derived from several experiments have been analyzed to formulate a universal correlation that would be capable of predicting the bubble departure neck-lengths in liquids with a wide range of physicochemical properties. The developed correlation has been extensively compared with existing data and they are found to be in excellent agreement. The new correlation has also been employed as the departure criterion in an analytical bubble-growth model, and the results obtained from the model are seen to be in good match with previous experimental observations.

Fig.3.1. A comparison of the experimentally observed neck-lengths with the empirical

assumptions proposed by Tsuge and Hibino (1983) and Räbiger (1984)

3.2. Experimental Procedure

Much of the experimental data that have been used in this study were obtained from the work of Subramani et al. (2007, 2008). However, additional experiments were conducted with a select few liquids to encompass a wider range of fluid property values into the analysis. Tables 1.1 and 1.4 list the physicochemical properties of the fluids that were used in the study.

The experimental setup that was used for the additional experimental runs with water, glycerol, propylene glycol, and ethylene glycol is similar to the one described in Section 2.2, with some minor changes that have been recorded below.

(i) The orifice sizes used are different from the ones mentioned in Section 2.2. Five different orifices, with inner diameters of 0.32 mm, 0.66 mm, 1.00 mm, 1.35 mm, and 1.76 mm were used to capture the effects of orifice size on necking behavior.

(ii) The high resolution images captured from the experiments were then analyzed using the image processing software Image-Pro Plus 4.0 from Media Cybernetics. The neck- lengths were obtained by simply measuring the distance between the orifice tip and the upper end of the neck. A schematic diagram illustrating the measurement of neck-length is depicted in Fig.3.2. All the measurements were based on a calibration of the saturation density using the outer diameter of the capillary as the base for calibration.

(iii)The air flow rates were varied between 1ml/min and 200 ml/min depending on the size of the orifice used. Bubble growth was recorded in both the constant volume regime and the growing bubble regime, though for this study only the constant volume regime has been taken into consideration.

(iv) Further, the errors involved in measurement were also estimated. At a frame rate of

3000 fps, the precision in measuring bubble interval and frequency was ±0.33 ms. Similarly, the precision in the measurement of neck-length was 0.026 mm based on the number of pixels in the image and the corresponding saturation density.

Fig.3.2. Schematic of a departing bubble and its neck-length

3.3. Results and Development of Correlation

Departure neck-lengths extracted from the experimental images are presented in

Fig.3.3. These values correspond to a wide range of liquid properties (789 kg/m3  1261

3 -3 -3 -3 -3 kg/m ; 22x10 N/m  72x10 N/m; 1 x10 Pa-s  750 x10 Pa-s) and orifice

diameters (0.32 mm do 1.76 mm). Further, clear images of the bubble necks for water, propylene glycol, and glycerol, at near departure time frames are presented in Fig.3.4.

1.5

)

m 1

(

0.5 Water Ethanol Ethylene Glycol Propylene Glycol Glycerol

0 0 0.5 1 1.5 2 d (mm) o

Fig.3.3. Experimental neck-lengths for the fluids considered, at different orifices

Glycerol Propylene Glycol Water

Fig.3.4. Experimental neck-lengths for glycerol, propylene glycol and water,

do = 1 mm, Q = 20 ml/min

From Figs.3.3 and 3.4, it is evident that at a given orifice, the neck-length at departure increases with an increase in the viscosity of the liquid. This observation could explain the inability of the empirical assumption made by Tsuge and Hibino to provide a fair estimate of the neck-length. The assumption did not take into consideration the interfacial properties of the background liquid and equated the neck-length to the orifice diameter regardless of the liquid under study. On the other hand, the experimental results clearly indicate a strong dependence of the neck-length on the interfacial properties of viscosity and surface tension.

Table 3.1 portrays the experimental neck images for three different fluids at two distinct orifice sizes (do = 1 mm and do = 1.76 mm) each. Once again, from Fig.3.3 and Table

3.1, it can be seen that for any particular liquid, the neck-length increases with an increase in the orifice size

Table 3.1. Effect of Orifice Size on Neck-length

Liquid do = 1.00 mm do = 1.76 mm

Water

Propylene Glycol

Glycerol

From these experimental observations, we can conclude that viscosity and orifice size are two important parameters which govern the departure neck-length. Apart from these, it has also been observed by earlier investigators that the liquid surface tension plays an important role in the preliminary stages of necking by setting the initial size of the bubble.

However, its effect decreases with the growth of the neck, when the viscous forces begin to

dominate the dynamics of pinch-off (Thoroddsen et al. (2007)). Finally, and perhaps most importantly, the entire process of neck-elongation and pinch-off is largely driven by the buoyancy of the fully grown bubble. Hence, the neck-length is also dependent on the density of the background liquid and the prevailing gravity conditions. Figure 3.5 is a schematic diagram that summarizes the nature and directions of the major forces that act on the bubble neck during its growth and subsequent pinch-off.

Buoyancy

Drag Force

Surface Tension

Fig.3.5. A schematic representation of the dominant forces acting on the elongated neck

With the identification of the parameters that directly control the pinch-off process, the relationship between the bubble departure neck-length and these parameters can be expressed as in Eq.(3.1).

ln f(,,,,) l   d o g (3.1)

A scaling analysis of these variables is performed to find a functional relationship for the neck-length in terms of several dimensionless parameters. In the scaling analysis fluid

density ( l ), acceleration due to gravity ( g and orifice diameter ( do ) are taken to be the

repeating variables, while liquid viscosity (  ), surface tension ( ), and neck-length (ln ) are considered to be the non-repeating variables.

Fig.3.6. Experimental neck-lengths for different fluids

at corresponding Galilei numbers

The analysis yields three dimensionless parameters – the dimensionless neck-length

2 (ldno/ ), the square root inverse of the Bond number ( 1/Bo / gdo l c / d o ), and the

32 Galilei number (Ga gdo / ). It is to be noted that while the Bond number signifies the balance between buoyancy and surface tension forces encountered at the initial stages of necking, the Galilei number corresponds to the viscosity-buoyancy balance predominant

during the later stages. Figure 3.6 shows a scatter-plot of the experimental neck-lengths against the Galilei number

As can be observed from Fig.3.6, the slopes of the individual graphs are approximately the same. It is therefore possible, through the use of a third scaling parameter, to collapse these curves into a single curve that is capable of characterizing the neck-lengths for a range of liquids. A global scaling analysis of the available data and the established dimensionless variables presented a third parameter in the form of the Morton number

42 3 ( Mo g()  /l  g  l  ). Non-linear regression analysis of the experimental neck-length data, the capillary length, the Galilei number, and Morton number yielded a generic correlation between the variables as in Eq.(3.2).

0.25 0.145 lnc/ l 1.018 Ga Mo (3.2)

Fig.3.7. A comparison of the predictions of the developed correlation and the available

experimental data

Figure 3.7 is a comparison between the experimentally obtained neck-length values and the corresponding predictions obtained from Eq.(3.2). As can be seen, there is an excellent match between the observed and theoretically calculated values. The errors were found to be less than 10% over the entire range of fluid properties. As stated previously,

Eq.(3.2) was also used as the departure criterion in the development of a theoretical model for adiabatic bubble growth from submerged orifices, and the results from the model very close matched the existing experimental figures.

2.4. Conclusions

Experimental observation and scaling analysis of necking behavior in single bubble growth were implemented from the perspective of neck elongation rather than from the traditional viewpoint of neck-thinning and pinch-off. It was established that the neck-length is solely dependent on the size of the orifice and the fluid properties of viscosity, density, and surface tension. A scaling analysis was then performed between the dependent (neck-length) and independent variables (orifice diameter and fluid properties) to develop a relation that would be able to predict the departure neck-length over a large range of orifice sizes and fluid properties. The correlation has been validated with the available experimental data and has been found to be highly accurate with less than 10% error in all instances. The predictions were also applied to several theoretical bubble development models and the outcomes from the models have also been very encouraging.

CHAPTER 4

Theoretical Modeling of Single Bubble Dynamics

4.1. Introduction

On a fundamental basis, the phenomena of bubble incipience, growth, and departure from a submerged orifice is dictated by a number of forces, each depending on the various liquid properties and the operating conditions. The forces of buoyancy, pressure, and gas momentum which aid the growing bubble are opposed by the inertia, viscous drag, and interfacial tension forces which tend to restrain its growth. Figure 4.1 is a schematic representation of the forces involved and their combined effect on the growing bubble. These six forces can be characterized and analyzed through six independent variables – liquid/gas density, pressure gradient across the orifice, momentum from the gas flow rate, liquid surface tension, liquid viscosity, and orifice diameter. A study of these forces provides a basic understanding of the complex dynamics involved during the inception and growth of a bubble, for it is the interplay between these forces that governs the bubble departure diameter, a crucial property that is often indispensable while computing several associated design parameters.

With the establishment of the key parameters that influence bubble growth, several models were developed to predict the size and growth times of bubbles in liquid pools. All the models unanimously made use of the various forces in determining the departure conditions of the incipient bubble. In one of the first of the many existing models, Hughes et al. (1955) considered the process of bubble formation to be continuous without any distinct separation of stages. They applied the force-balance to the bubble and solved for drag

coefficient, bubble velocity, and bubble volume. Davidson and Schuler (1960) followed with the proposal of a single-stage model by considering the balance between the buoyancy and inertial forces alone. However, the single-stage models could not explain the domination of the different forces at different instants of time during the incipience and growth of the bubble.

Inertia

Pressure Buoyancy Drag

Surface Tension

Gas Momentum

Fig.4.1. Representation of the forces governing the growth of an adiabatic single bubble

from a submerged orifice

In their paper, Siemes and Kaufmann (1970) put forth a two-stage mechanism for bubble-formation in inviscid liquids. The first stage was assumed to be similar to the growth of a bubble when the gas flow rate approaches zero. The second stage commenced when the buoyancy and surface tension forces balanced each other and the bubble began to rise.

However, the same mechanism did not hold for viscous liquids and the model failed to

predict the departure time reliably. Kumar et al. (1970) then came up with a more acceptable two-stage model that can be used to explain the process of bubble incipience and growth. In this model, as the bubble grew in the first stage, the buoyancy force was balanced by the viscous drag, inertial drag, and surface tension forces. In the second stage, the buoyancy force exceeded the downward forces and the bubble detached when its center had travelled a distance equal to the radius of the force-balanced bubble. Wraith (1971) further simplified the two-stage model of Kumar by applying the concept of velocity potential to a hemispherical expanding bubble, thus neglecting the viscous and surface tension forces.

Wraith’s proposed correlation took the form of a power law equation, as shown in Eq.(4.1), where only the buoyant forces came into consideration.

Q6/5 V 1.09 (4.1) b g 3/5

In the very first attempt to model bubble growth as a multistage process, Räbiger and

Vogelpohl (1986) developed an equation to predict the bubble departure diameter by applying the force balance without the gas momentum force.

1/3 2 Kv2 d 33ddoogo db    (4.2) g ll

Later, Tsuge and Hibino (1983) developed a modified two stage model that considered all the forces of buoyancy, pressure, gas momentum, surface tension, liquid inertia, and liquid drag. The forces were grouped into various dimensionless groups to predict the bubble departure volume as given in Eq.(4.3).

22 2 4gQt dX 22 M' X2 0.5 CDl r t  do  dt Vb  (4.3) l gV b gt

In various subsequent studies, Tsuge et al. (1986) further modified the above model for specific operating conditions such as a downward facing orifice, and in high pressure systems. They also studied the growth of non-spherical bubbles using a modified Rayleigh’s equation to develop a simplified formula for bubble volume.

Apart from theoretical modeling, considerable experimental work has been undertaken to study the incipience and departure of gas bubbles from submerged orifices. In a very early study, Eversole (1941) investigated the growth of nitrogen bubbles from glass orifices submerged in water and alcohol, and observed that the bubble size varied solely with the gas flow rate. Van Krevelen and Hoftijzer (1950) then proposed a predictive equation for the variation of the bubble diameter, based on the buoyancy-surface tension force balance, through a series of experiments with water. Datta et al. (1950) then observed the existence of two regimes of bubble growth based on the inlet gas velocity and orifice diameter. Following this observation, both Coppock and Meikeljohn (1951), and Benzing and Myers (1955), have individually characterized the isolated bubble growth from submerged orifices to be divided into two distinct regimes – a constant volume regime at low gas Reynolds numbers, and a growing volume regime at higher Reynolds numbers. Also, experiments by Datta et al.,

Hoftijzer and Krevelen, and Coppock and Meikeljohn, all coincide in characterizing the constant volume region as

1/3 6do db   (4.4) l g

Many of these experimental studies have provided further insights for the development of generalized equations to predict and trace bubble properties during various phases of its growth. Tadaki and Maeda (1963) worked on the experimental data of bubble

formation in various liquids, orifice diameters, and flow rates to devise a set of equations, Eq.

(4.5), which would predict the departure bubble diameter.

1/3   do We We db 6 2.5, 16 gl FrFr (4.5) 1/3 1/9  d Fr We d 1730,o 16 b  gWel Fr

In a semi-empirical approach, Gaddis and Vogelpohl (1986) developed a widely accepted correlation for the determination of bubble diameter by introducing an assumption

for the neck-length at departure to be db /4based on Räbiger’s (1984) experiments.

1/3   2  6dQolWe81 Q 1 13527g 1 db 1     2  2 2  (4.6) (  )g 4  (    ) g  d  4    (    ) g d  l g l g  b  l l g b 

More recently, Jamialahmadi and other coworkers have statistically arrived at yet another correlation based on selected data from the literature.

0.36 1/3 1 9.261Fr 0.51 dbo d1.08   2.147 Fr (4.7) Bo Ga

Though these models have been in widespread use in the investigations of bubble growth, they fail to predict the bubble departure diameter under the entire range of flow rates and orifice sizes. Fig.4.2 clearly depicts the divergence of the predicted values from the experimental data at either extreme ends of the flow regime for an air-water system.

Air:Water d = 1.00 mm 5.5 o

4.5

b 4

3.5

Subramani et al. (2007) 3 Tadaki and Maeda (1963) Wraith (1971) Rabiger and Vogelpohl (1986) 2.5 Gaddis and Vegelpohl (1986) Jamialahmadi et al. (2001)

2 0 50 100 150 200 Re o

Fig.4.2. Comparison of Air : Water experimental data with the predictions obtained

from correlations given by other investigators

Further, a number of these models have been validated with experimental data derived from studies in water and alcohols alone. Therefore, the extent of their applications in systems using fluids that differ from water in their interfacial and viscous behavior is quite questionable. The large error between the theoretically obtained bubble diameter values and the experimental data for an air-glycerol system as shown in Fig.4.3, is a clear indication of the inability of the existing correlations to predict the ebullience parameters in non-water-like systems.

Air:Glycerol

do = 1.76 mm

d 4

Subramani et al. (2007) 2 Tadaki and Maeda (1963) Wraith (1971) Rabiger and Vogelpohl (1986) Gaddis and Vegelpohl (1986) Jamialahmadi et al. (2001) 0 0 50 100 150 200 Reo

Fig.4.3. Comparison of Air: Glycerol experimental data with the predictions obtained

from correlations given by other investigators

In the current investigation, the phenomena of bubble growth from the tip of an orifice submerged in a liquid pool is revisited. A theoretical model that continuously represents the bubble growth dynamics from incipience, through growth to necked detachment through a balance of aiding and retarding forces is developed.

The model’s validity has been established by comparing the predicted results with data from experiments conducted in adiabatic pools of five different liquids (water, ethanol, ethylene glycol, propylene glycol and glycerol), with orifices of different diameters over a wide spectrum of inlet gas velocities. The fluids were chosen so as to cover a range of

-3 -3 -3 surface tensions (22x10 N/m  72x10 N/m) and viscosities (1 x10 Pa-s  750 x10-3 Pa-s). The model has also been matched with other experimental data found in the literature and has been found to predict the departing bubble diameters with reasonable accuracy for a wide range of orifice diameters, flow rates, and fluid properties.

4.2. Mathematical Model and Force Balance

As seen in Fig.4.1, the growth and subsequent departure of a bubble from a submerged orifice is governed by six different forces which vary in magnitude over different phases of the bubble’s growth. These forces also determine the time taken for the bubble to grow from inception to departure, and hence the size of the departed bubble. By considering the bubble to be spherical throughout its growth, it is possible to mathematically model this complex bubble-growth process using a first-order approach. With this assumption, the volumetric growth rate of the bubble can be mathematically captured through a balance of the six forces.

FFFFFFBPGSDI     (4.8)

An understanding of each of these forces, their scope, their mathematical formulations, and their subsequent effects on the bubble size is an integral exercise in development of the theoretical model.

4.2.1. Aiding Forces

(a) Buoyancy Force

The buoyancy force comes into play when a bubble emerging from a submerged orifice displaces the surrounding liquid in the pool. It is perhaps the most dominant of the aiding forces, and increases in magnitude as the bubble grows from the orifice. The earliest formulation of the buoyant force can be traced back to the Archimedes Principle which correlates the weight of a submerged body to the volume of the liquid that it displaces.

Mathematically, the equivalent volume of a bubble can be given as

4 Vr  3 bb3

Where is the bubble volume and is the bubble radius obtained by taking the non- spherical bubble to be an equivalent sphere.

Multiplying the bubble volume by the liquid density, we obtain the weight of the displaced liquid.

Wb  l V b g

4 3 Wb  l r b g 3

It is this weight that constitutes the buoyancy force which acts in the upward direction aiding the bubble growth. Therefore, it can be expressed as

4 F  r3 g B3 l b

Teresaka and Tsuge (1993) further accounted for the downward drag of the bubble due to its inherent weight, and proposed a modified equation, as presented in Eq.(4.9), for the determination of the buoyancy force.

4 Fr g ()   3 (4.9) Blg3 b

(b) Gas Momentum Force

The gas momentum force arises from the incoming flow of gas through the orifice that imparts a force in the direction of the gas flow. Since the gas momentum force depends solely on the velocity of the incoming gas, its magnitude remains constant throughout the entire growth cycle. At low Reynolds numbers, the gas momentum force is often ignored in many models, since it has the least magnitude compared to the other five forces.

From Newton’s second law, we know that

FG = Rate of change of momentum

Or, in terms of the mass flow rate and velocity,

FG = mass flow rate of gas*gas velocity

FAG v g o g 

Since the gas flow rate and the orifice cross-sectional area are related through

Qo vg  Ao

It follows that the gas momentum force can be expressed as

 Q2 F  go G A o

Substituting for the orifice cross-sectional area, we get

2 4goQ FG  2 (4.10)  do

The gas momentum force is generally very low in magnitude compared to the other five forces and has often been ignored in many of the previously proposed models.

The pressure difference across the orifice gives rise to a pressure force which can be simply stated as

FAPgl P0 () P

In a bubble, due to its sphericity, the pressure inside the bubble is quantified on the basis of the Young-Laplace equation, taking into consideration the surface tension of the surrounding liquid. Also, the additional static pressure imposed by the height of the bubble

(Fig.4.4) is taken into account while formulating the pressure force.

A simple pressure balance on the spherical bubble, based on the Young-Laplace equation, and subsequent rearrangement of the terms gives us

2 PPPhyd() g  l  rb 2 ()PPPg l   hyd rb 2 ()()Pg P l    l gh rb

Fig.4.4. A schematic representation of the pressure forces acting on the growing bubble

Approximating the height of liquid across the bubble, h, to the bubble diameter, as shown above, the Young-Laplace equation becomes

2 (Pg Pgr ll )  b  (2 ) rb

Substituting the above derived equation for the pressure difference across the orifice into the expression for pressure force, we get

2 FP Agr ol b (2 ) rb

2 2 FP d o(2 l gr b ) (4.11) 4 rb

The pressure force is particularly high in cases of large orifice diameters and also small bubble diameters as can be inferred from Eq.(4.11). The pressure force is the last of the three upward forces aiding the growth of the bubble.

4.2.2. Restraining Forces

(a) Surface Tension Force

The solid-liquid-gas interface at the orifice tip brings into play the surface tension of the liquid which acts around the circumference thus keeping the bubble attached to the orifice tip. The resultant force can be mathematically expressed as

FdSo  sin (4.12)

Where,  is the angle between the orifice and the tangent to the bubble at the orifice.

Fig.4.5. Representation of the parameters involved in the formulation of the

surface tension force

From the above representation, θ and φ are related as

  2

From trigonometry,

r cos sin o rb

Substituting for sin φ in Eq.(4.12)

r Fd  o Sor b

2 do  F  (4.13) S 2r b

Many of the previous models have ignored the variation of the surface tension force with the dynamic change in the angle of contact, and have simply taken the force to be constant throughout the period of growth, as in Eq.(4.14)

FdSo  (4.14)

However, in reality, the angle of contact varies from 90o to a minimum value and then rapidly increases again with the onset of necking.

(b) Drag Force

Any object moving in a viscous medium encounters an opposition due to the viscosity of the fluid. The growing bubble too experiences a viscous drag by virtue of its motion, which is described by its moving center of mass as it grows.

Lord Rayleigh is attributed with having formulated an expression for viscous drag as

FCAPD D c f

Where CD is the coefficient of Drag,

Ac is the cross sectional area perpendicular to the flow, and

Pf is the free stream fluid dynamic pressure.

Since the free stream dynamic pressure is dependent on the velocity of the moving bubble,

1 2 Pvf  l b 2

Further, expressing the velocity of the growing bubble as the rate of change of its radius, we obtain

2 1 drb Pfl   2 dt

Also, the area of cross-section of the spherical bubble is easily expressed as

2 Arcb 

Substitution of the above parameters into Rayleigh’s equation yields,

2 1 drb 2 FCrDD lb  (4.15) 2 dt

It can be seen that the drag force is dependent on the coefficient of drag, which in turn depends on several factors including the shape of the moving object and it’s acceleration among others. Various analytical expressions have been developed for predicting the drag coefficient for a growing bubble. For low Reynolds numbers, by ignoring the internal air circulation within the bubble- as it is still attached to the orifice - many authors use Stoke’s correlation for a solid sphere and report the drag coefficient to be

24 C  (4.16) D Re

For higher Reynolds numbers, experimental results by Takahashi et al. (1976) predict the coefficient of drag to be given by Eq.(4.17). The expression yields a value of unity for very high Reynolds numbers.

16 C 1 (4.17) D Re

Snabre and Magnifotcham (1998) suggested that though attachment of the bubble to the orifice limits internal circulation, large bubbles behave very differently from solid

spheres at high Reynolds numbers. Therefore they considered the bubble to be a solid sphere in the creeping regime and a freely circulating sphere in the inertial regime, and derived a semi-empirical expression by superimposing Eqs.(4.16) and (4.17), to give a single expression that provides a good fit for the drag coefficient over a wide range of Reynolds numbers, as can be seen from Fig.4.6.

24 C 1 (4.18) D Re

105

104

103

Stokes Flow Takahashi et al. (1976) d 102 Snabre & Magnifotcham (1998)

101

100

10-1 10-4 10-2 100 102 104 106 108 Re

Fig.4.6. Superposition of the Stokes Correlation and the Miyahara Correlation to give a

universal formulation for CD over a wide range of Reynolds Numbers

Since the growing bubble’s center of gravity accelerates away from the orifice, it brings into action an inertial force that tends to oppose the upward growth of the bubble. This inertial force can be formulated, from Newton’s second law of motion, as shown in Eq.(4.19)

d FMv () (4.19) Ibdt

The accelerating bubble also displaces a certain quantity of the surrounding liquid as it moves, since the bubble and the liquid cannot occupy the same physical space simultaneously. This liquid moving along with the bubble adds an additional mass to the bubble which is termed as the added mass or apparent mass. The total mass which contributes to the inertial force is a sum of the actual mass and the apparent mass of the bubble. Substituting for the total mass in Eq.(4.19) we obtain

d FIa b M b M v dt d  FvIl V g b b   dt 

The apparent mass is often calculated using an apparent mass coefficient, which is the ratio of the added mass to the displaced fluid mass. Considering a potential flow, it can be assumed that the virtual mass added to the bubble is the same as that for a sphere moving perpendicular to a wall in an inviscid liquid. Horace Lamb (1932) derived an expression for the apparent mass index for a sphere, as in Eq.(4.20), that can be directly applied to the model.

3 13rb  1 3 (4.20) 28s

Where rb is the bubble radius and s is the vertical distance from the center of the bubble to the orifice tip. For small orifices, since the total distance travelled by the bubble center from the orifice, before detachment, is very small, s approximates to rb and the added mass coefficient is approximated to be 11/16. Substituting this approximated value into the expression, we obtain

d 114  3 FrIl v gb b     dt 163 

4 11 d 3 drb FrIl gb    (4.21) 3 16 dt dt

The above expression is an excellent approximation of the inertial force encountered in inviscid liquids, and gives a fairly good estimate of the inertial force in viscous liquids too.

4.2.3. Force Balance and Departure

At inception, an isolated bubble, which is free of any wake effects of the preceding bubble, is assumed to be hemispherical with a diameter equal to that of the orifice. At this stage, both the bubble interface and the surrounding fluid are at rest and the internal pressure of the bubble is balanced by the hydrostatic pressure and the pressure due to surface tension.

Hence, the initial conditions at t = 0 can be taken as

* * dr 2 r roo,  0 and P (0)  P  (4.22) dt ro

Also, the radial growth of the bubble can be related to its volumetric growth rate, which is governed by the mass conservation of the incoming gas flow,

* dQb  2*4 3 dr do V r  (4.23) dt43 dt

Inception Growth Necking Departure Fig.4.7. Representation of bubble growth as modeled in the theoretical approach - through inception, spherical growth, development of a cylindrical neck, and departure.

Beyond inception, the bubble growth is taken to be a continuous process that can be described in entirety by the Eqs. (4.9-4.11), (4.14-4.15) and (4.21). As mentioned earlier, the forces vary considerably over the growth period and hence, different phases of the bubble growth cycle see the dominance of different forces. During the static or expansion stage of growth, which immediately follows incipience, the buoyancy and gas momentum forces are comparatively weak, and the pressure force dominates among the aiding forces. On the other hand, growth phases immediately preceding departure see the dominance of the buoyancy force over all the other five forces. Also, since the forces are characterized by the fluid properties and operating conditions they tend to change considerably with different liquids, orifice sizes, and flow rates. For example, at high gas flow rates, the gas momentum plays a larger role in the growth process as compared to low Reynolds number flows. Also, fluids with higher viscosities tend to have a more dominating drag force as compared to less

viscous fluids. On the other hand, a higher surface tension imposes a larger restraining force, thus increasing the growth time and departure diameter.

At inception, the restraining forces dominate over the aiding forces. As the bubble grows, the buoyancy increases and at maximum bubble volume the restraining forces are almost balanced by the aiding forces. Beyond this stage, the lifting forces become so large that the bubble can no longer be held to the orifice by the resisting forces. The bubble begins to form a neck which grows with the lifting bubble, contracts, and eventually shears out, leading to detachment. At this stage, the bubble is no longer taken to be spherical and growth is modeled as

dr* dr dl r*  r  l and  n (4.24) n dt dt dt

FFFFFFBGPSDI     (4.25)

Here is the neck-length. Many recent investigations (Kasimsetty et al. (2008),

Tsuge and Hibino (1983), and Zhang and Shoji (2001)) have considered the bubble departure to occur when . This iteration has also been validated by several experimental measurements in water. However, it is not universal in its application, as it fails to hold in other liquids. A more recent investigation has proposed a generalized correlation (Eq.(4.26)) for the neck-length at departure through experimental and statistical analyses of data from a wide range of liquids.

0.25 0.145 lnc1.018 l Ga Mo (4.26)

Thus, the entire phenomena of bubble growth and departure can be modeled by simultaneously solving the force balance equation with the specified initial conditions, while marching in space and time, until the criterion for detachment, given by Eq.(4.26) is satisfied.

The final time of departure gives the bubble interval, from which the bubble departure diameter can be calculated as,

6Qt d  3 (4.27) b 

The complete methodology for solving the first-order mathematical model is summarized in Appendix C.

4.3. Results and Discussion

The predictions of the theoretical model developed in this investigation have been compared and analyzed with several sets of experimental data that have been reported in the literature, and are found to be in excellent agreement. A brief representation of the comparison with the experimental data of Subramani et al. (2007, 2008) is shown in Fig.4.8 to demonstrate the robustness of the developed model over a wide range of liquid properties.

Comparisons for individual fluids are also shown in subsequent figures. These experiments encompass a wide range of fluid properties (22x10-3 N/m  72x10-3 N/m; 10-3 Pa-s

 0.75 Pa-s), flow rates (2 ml/min ≤ ̇ ≤ 120 ml/min) and orifice diameters (do = 0.32 mm, do = 1 mm and do = 1.76 mm). The developed model also shows a clear delineation between the constant volume and the growing volume regimes of bubble growth, which is a reflection of the model’s general applicability.

do = 0.32 mm Water 8

o Ethanol

b 6

d Water (do = 0.66 mm) Glycerol 4

Propylene Glycol d = 1.76 mm 2 o

0 200 400 Re o

Fig.4.8. Comparison of the theoretical model results with experimental data from

Subramani et al. for water, ethanol, glycerol and propylene glycol.

12 Air-Water (Mo = 1.72 x 10 -11) 10 do = 1.00 mm do = 0.66 mm 8 Theoretical Model

b 6

0 50 100 150 200

Reo

Fig.4.9. Comparison of the theoretical model predictions with the air-water ebullience

data from Subramani et al. (2007, 2008) (1 mm) and the current study (0.66 mm)

A comparison of the predictions from the current model with Räbiger’s (1984) experimental data, which were obtained for air-water ebullience from an orifice of 2 mm diameter, has been graphed in Fig.4.10. To attest the general validity of the model in predicting ebullience parameters in highly viscous liquids, a comparison of the theoretical results with the experimental results obtained by Subramani et al. (2007, 2008) in propylene glycol and glycerol have also been shown in Figs.4.11 and 4.12 respectively. The excellent agreement of the results with the experimental data clearly depicts the effectiveness of the first order model.

8 Air-Water (Mo = 1.72 x 10 -11)

do = 2 mm 6 Rabiger (1984) Theoretical Model

d / 4

0 101 102 103 Re o

Fig.4.10. Comparison of the theoretical model predictions with Räbiger’s (1984)

experimental data

12 Air-Propylene Glycol (Mo = 1.738 x 10 -3) d = 0.32 mm 10 o do = 1.00 mm

do = 1.76 mm 8 Theoretical Model

b 6

0 101 102 103 Re o

Fig.4.11. Comparison of the theoretical model predictions with the air-propylene

glycol ebullience data of Subramani et al. (2007, 2008)

12 Air-Glycerol (Mo = 8.89)

10 do = 1.00 mm do = 1.76 mm Theoretical Model 8

b 6

0 100 101 102 103 Re o

Fig.4.12. Comparison of the theoretical model predictions with the air-glycerol

ebullience data of Subramani et al. (2007, 2008)

To further extend the validity of the model to a broader range of flow rates, orifice diameters, and liquids, an extensive comparison of the predicted bubble sizes with additional experimental data from the investigations of Räbiger and Vogelpohl (1986),

Davidson and Schuler (1960), and Jamialahmadi et al. (2001) is presented in Fig.4.13.

These include orifices of sizes ranging from 0.32 mm to 3 mm, and flow rates of up to 150 ml/min. These comparisons further attest the model’s general ability to provide reliable predictions of bubble diameters for a wide range of air-liquid ebullience conditions.

Fig.4.13. Scatter plot for comparison of theoretical predictions with experimental data

from literature for various fluids

The theoretical model that has been developed in this study provides predictions that

match experimental data to a greater extent than several other models and correlations that

have been developed earlier. One possible explanation for this relatively better agreement

could be the criterion for departure that has been used in this model. This statistically

derived criterion is very different from the assumptions of Gaddis and Vogelpohl (1986) (ln

= db/4), and Kasimsetty et al. (ln = do). These semi-empirical assumptions could possibly

explain the over-prediction and the under-prediction of the bubble diameters that is often

seen their respective models. This is especially true in cases of liquids with high viscosity,

where the neck-length is not only a function of the orifice diameter but also depends

considerably on the liquids’ inherent properties.

4.4. Conclusions

A theoretical model has been developed to predict the final departure diameters of bubbles emanating from orifices submerged in liquid pools. The model takes into consideration the dynamic effects of the six different forces of buoyancy, gas momentum, pressure, surface tension, drag, and inertia, which govern the incipience, growth, and departure of a bubble. The model has been compared and validated with several sets of experimental data obtained from various investigations reported in the literature. The developed model is found to be in very good agreement with all the data that include a wide range of flow rates, liquid viscosities, surface tensions, and orifice diameters. Accuracies between +15 % and -15 % have been obtained for all the reported data, some even approaching very closely to the experimental values. This agreement of the present model

with a variety of data attests the general validity of the model in predicting ebullience in various air-fluid systems.

Chapter 5

Dynamic Surface Tension Effects on Bubble Growth Dynamics

5.1. Introduction

An important parameter that plays a dominating role in the growth and departure of gas-bubbles in liquid media is the gas-liquid interfacial tension. The liquid surface tension manifests itself as surface forces that help the bubble to remain adhered to the orifice tip, thus delaying detachment (Hughes et al. (1955)). A number of researchers have investigated the effect of surface tension on bubble formation dynamics in pure liquids (Datta et al. (1950),

Benzing and Myers (1955), Ramakrishnan et al. (1969), Liow (2000) and Subramani et al.

(2007)). Their results unanimously agree on the positive effect of liquid surface tension on the departure size of bubbles from the orifice, i.e. when all other fluid properties and operating conditions are held constant, an increase in the surface tension of the liquid phase results in a larger bubble emanating from the orifice. It is important to note that in these studies involving pure liquids the surface tension remains constant with time. In contrast, many industrial applications involve liquid phases that are often complex in their composition with spatial and transient variations in their properties. To analyze the impact of dynamic changes in liquid properties, recent studies in boiling heat transfer, gas-liquid reactors, and bubble formation have attempted to explore the effect of additives on bubble growth dynamics. Notable changes in ebullience behavior have been reported for even small concentrations of these additives, thus presenting a passive method with an enormous potential to control heat and mass transfer processes. These changes are attributed to the alterations in the interfacial properties of the liquid that are brought about by the additives.

Surfactants or surface active reagents are one of the many categories of additives which are found to be particularly effective in the control of liquid-gas interfacial properties.

Essentially, they are soluble organic amphiphilic long-chained compounds with hydrophobic tails and hydrophilic heads (Rosen (1989)). Based on the chemical composition of the molecular heads, surfactants are generally classified into anionic, nonionic, cationic, and zwitterionic surfactants. In the presence of a liquid-vapor interface, the surfactant molecules tend to adsorb at the interface with their heads oriented towards the liquid phase and their tails towards the vapor. This aggregation of the surfactant molecules significantly changes the interfacial properties of the liquid-vapor system, often resulting in a reduced surface tension. It has been noticed that the resultant decrease in surface tension, brought about by the addition of a surfactant, facilitates changes in the ebullience dynamics of surfactant solutions, exhibiting a pronounced variation in the size and frequency of the emergent bubbles.

Surface tension relaxation due to the presence of surfactants is a time dependent process, resulting in a dynamic variation of the surface tension that approaches an equilibrium value over a finite period of time. As pointed out by Rosen (1989), and Manglik,

Wasekar and Zhang (2000), this transient behavior of surface tension relaxation is typically affected by the surfactant’s bulk concentration, the type of surfactant, its diffusion-adsorption kinetics, molecular weight, and mobility amongst several other factors. On the other hand, the equilibrium surface tensions of surfactant solutions decrease with the bulk concentrations up to a certain threshold value beyond which there is little change and possible degradation.

This threshold value of concentration is referred to as the Critical Micellar Concentration and is a characteristic property of the surfactant. There are several reviews and studies available

which have compiled the effects of surfactants, their concentrations, and the time scales involved on the surface tensions of aqueous surfactant solutions (Rosen (1989), Hirt (1990),

Chang and Franses (1995), Eastoe and Dalton (2000), Wu et al. (1995), and Ferri and Stebe

(2000)).

Though extensive investigations have been undertaken to characterize the surface tension dynamics of surfactant solutions, and their subsequent effects on heat transfer patterns in boiling and other two phase flows, relatively fewer experimental studies have been performed on the dynamics of adiabatic single bubble formation in surfactant solutions.

As observed by Wasekar and Manglik (2001, 2002), and Zhang and Manglik (2003, 2004), bubble ebullience in aqueous solutions of surface-active agents is quite complex because of the time-dependent adsorption-desorption behavior of surfactant molecules. Owing to their dependence on the dynamics of surface tension relaxation, the bubble departure diameter and frequency are influenced by the reagent’s bulk concentration, diffusion time, ionic nature and molecular weight, bulk convection, interface deformation, and mobility among others, further adding to the complexity of the phenomenon.

Early in the last decade, the ebullience behavior in several concentrations of Sodium

Dodecyl Sulfate and Triton-X 100 was explored by Shu-Hao Hsu et al (2000). From their experimental observations, they reported smaller bubble diameters in surfactant solutions as compared to water at low flow rates. However, the emergent bubble diameters were found to gradually approach the corresponding water values as the incoming gas flow rates were increased. They also used a modified version of the Ruff-model to obtain theoretical predictions of bubble departure diameters in surfactant solutions.

Later, Loubiére and Hébrard (2004) worked with aqueous solutions of anionic

(Sodium Lauryl Sulfate), nonionic (Fatty Alcohol C-12/18-10 EO), and cationic (Lauryl

Dimethyl Benzyl Ammonium Bromine) surfactants, and investigated the growth of bubbles from rigid and flexible orifices submerged in the respective surfactant solutions. Similar to the conclusions of Shu-Hao Hsu et al., they also observed that the bubbles generated in surfactant solutions had smaller sizes and higher frequencies as compared to water, though the difference gradually decreased as the gas flow velocity was increased. They were also able to conclude that both static surface tension and surfactant-dependent diffusion kinetics play an important role in determining the final bubble departure sizes. However, the surfactants used in the study had significantly high Critical Micellar Concentrations ( O(10 mol/l)), which bring into question the possible effects of such concentrations on the other liquid properties of viscosity and density.

On the other hand, in recent times, several high quality surfactants are commercially available, which show a much more pronounced surface tension relaxation at comparatively lower concentrations. Figure 5.1 depicts the surface tension relaxation of the surfactants used in the study by Loubiére and Hébrard (2004) compared to FS-50 (a relatively new surfactant), at their respective Critical Micellar Concentrations. The principal objective of this study is to investigate the effect of such surfactant additives and their dynamic surface tension relaxation on the mechanics of bubble formation in submerged orifices. A zwitterionic fluorosurfactant, DuPontTM CapstoneTM FS-50, an anionic surfactant, Sodium

Dodecyl Sulfate (SDS), and a cationic surfactant, Cetyl Trimethyl Ammonium Bromide

(CTAB), were chosen for the study based on their CMCs, ranges of surface tension relaxation, and the time-dependent adsorption behavior. To provide a relative reference for

analyzing the roles of dynamic surface tension and molecular adsorption during the bubble growth cycle, adiabatic bubble dynamics were also observed in two other pure liquids – water and ethanol. Bubble shapes and sizes, at various stages of growth, were captured using high speed visualization techniques and analyzed to establish the role played by surface tension (equilibrium and dynamic) in the growth of bubbles from submerged orifices, both in pure liquids and aqueous surfactant solutions alike.

80  = 72.4 mN/m (water)

)

N 40

(



20 Anionic Surfactant Cationic Surfactant Non Ionic Surfactant Zwitterionic Surfactant

0 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 C (mol/cc) Fig.5.1. Surface tension relaxation in FS-50 compared the relaxation in the surfactants

used by Loubiére and Hébrard (2004)

5.2. Materials Used

As mentioned, studies were carried out with three different surfactants and two pure liquids to evaluate the effects of dynamic and static surface tension on the growth and departure of single bubbles in quiescent pools. The surfactants, FS-50, CTAB, and SDS, were of different ionic natures with varied molecular weights, CMC’s and physicochemical properties. The various properties of the surfactants are given in Table 5.1. Water and ethanol were the two pure liquids which were used as a base for analyzing the effects of dynamic surface tension.

Table 5.1. Physicochemical Properties of the Surfactants used

CTAB SDS Surfactant (Cetyl Trimethyl (Sodium Dodecyl FS-50 (Chemical Name) Ammonium Sulfate) Bromide) Chemical C12H25SO4 Na C19H42BrN - Formula

Ionic Nature Anionic Cationic Zwitterionic

Appearance White Powder White Powder Clear Liquid

Molecular Weight 288.3 364.5 -

Manufacturer Fisher Sigma-Aldrich Dupont

Purity >99% 99% -

Melting Point 206°C 230°C -

5.3. Surface Tension Measurements

The measurements of the dynamic surface tensions of several samples of carefully

prepared aqueous surfactant solutions were made using the SensaDyne QC6000

tensiometer. The tensiometer operates on the principle of the Maximum Bubble Pressure

Method (MBPM). Two orifices of unequal sizes are immersed into the test liquid and dry

air is allowed to flow through them. The air flow creates a differential pressure drop

between the two orifice tips, which by the principle of MBPM, is directly proportional to

the interfacial tension. By controlling the air flow rate through the two orifices, a range of

bubble frequencies and surface ages can be achieved. Surface age is the total duration of an

interface’s existence, and hence acts as a time scale for the adsorption-desorption dynamics

of the surfactant molecules. By sufficiently altering the bubble frequency/surface age, it is

possible to extract the equilibrium and dynamic surface tensions of the test solution. High

surface ages or low bubble frequencies lead to static conditions and hence yield the

equilibrium surface tension. Details of the calibration and measurement validation

procedures are described by Bahl et al. (2003), Manglik, Wasekar and Zhang (2000), and

Iliev and Dushkin (1992). The maximum uncertainties in solution concentration,

temperature, and surface tension measurements were found to be ±0.4%, ±0.5%, and

±0.7%, respectively.

5.4. Experimental Setup

The setup that was used for the experimental runs with water, ethanol, FS-50, CTAB, and SDS is the same as the one mentioned in Section 2.2, with some minor changes that have been recorded below.

(i) The orifice sizes used were different from those mentioned in Section 2.2. A single orifice with an inner diameter of 0.8 mm was used for experiments with all the test liquids, both surfactant solutions and pure liquids.

(ii) The high resolution images captured from the experiments were then analyzed using the image processing software Image-Pro Plus 4.0 from Media Cybernetics. The only parameter extracted from the images was the bubble departure diameter, by assuming the bubble to be a perfect sphere. All the measurements were based on a calibration of the saturation density using the outer diameter of the capillary orifice as the base for calibration.

(iii)The air flow rates were regulated in terms of bubble departure intervals and bubble frequencies. The bubble frequency was varied from as low as 0.03 bubbles/s to as high as nearly 50 bubbles/s. The experiments were performed in both constant volume and growing volume regimes. The bubble intervals were also cross-verified with the precision time control available in Image-Pro.

(iv) Further, the errors in the measurement were also estimated. At a frame rate of 3000 fps, the precision in measuring the bubble interval and frequency was ±0.33 ms. Similarly, the precision in the measurement of the equivalent bubble diameter was 0.026 mm based on the number of pixels in the image and the corresponding saturation density.

(v) The surfactant solutions were all prepared using a precision electronic weighing scale and magnetic stirrers to obtain a uniform, homogeneous solution of the desired concentration.

The solutions were stirred for at least 24 hours before they were used in the experiments.

5.5. Results and Discussion

5.5.1. Dynamic Surface Tension Measurements

Dynamic surface tension measurements were conducted on two solutions of FS-50 at concentrations of 1400 wppm (CMC) and 400 wppm, and on one solution each of SDS and

CTAB, at their respective Critical Micellar Concentrations. The dynamic surface tension data have been fitted with the correlation proposed by Hua and Rosen (1988) to give smooth curves that could be used for surface tension determination at any required surface age.

Figure 5.2 shows the dynamic surface tension behavior of FS-50 at CMC, as compared to the upper and lower limits of water and ethanol respectively. The enhanced effectiveness of the surfactant is very evident from the plot. At a considerably low concentration, the surfactant reduces the surface tension of the solution from that of pure water at 72.4 mN/m to as low as

22 mN/m. However, it is to be noted that the time scale of diffusion is comparatively much larger than other surfactants that have often been used in the past. The difference in the adsorption times is clearly discernible from Fig.5.3, which illustrates the dynamic surface tension behavior of SDS (CMC), CTAB (CMC), and FS-50 (400 wppm). Though the three surfactant solutions have approximately the same equilibrium surface tensions, the time taken by the respective surfactants to achieve complete surface tension relaxation varies by an order of magnitude.

It is understood that, while equilibrium surface tension is a distinct characteristic of the chemical composition of a surfactant and its bulk concentration in the aqueous solution, the rate of relaxation and the resulting dynamic surface tension are heavily dependent upon the molecular weight of the surfactant molecule (Manglik, Wasekar, and Zhang (2000)).

Heavier molecules tend to diffuse slowly and thus surfactants with higher molecular weights

have a much higher relaxation time as compared to lighter ones. This explains the comparatively larger relaxation time of FS-50 (MW  450g) at 400 wppm than that of SDS

(MW= 288.3g) or CTAB (MW = 364.5g) at CMC.

 = 72.4 mN/m (water) 70

) 50

N FS-50 1400 wppm

m ( 40



20  = 22.1 mN/m (ethanol)

10 10-2 10-1 100 101 Surface Age (s)

Fig.5.2. Surface tension relaxation in FS-50 at 1400 wppm as compared to the static

surface tensions of water and ethanol

) 55

N ( 50



40 CTAB - CMC (400 wppm) SDS - CMC (2500 wppm) 35 FS-50 (400 wppm)

30 10-4 10-3 10-2 10-1 100 101 102 Surface Age (s)

Fig.5.3. Surface tension relaxation in FS-50 at 400 wppm, CTAB (CMC) and SDS (CMC)

5.5.2. Effect of Temporal Surface Tension Relaxation

For analyzing the role played by dynamic surface tension on the growth mechanics of an adiabatic single bubble from a submerged orifice, experiments were performed in three fluids – FS-50 (1400 wppm), water, and ethanol. As was observed in the relaxation behavior of FS-50 at CMC, the surface tension varied from nearly 72.4 mN/m (water), at small surface ages ( 1 s), to 22 mN/m (ethanol) at equilibrium. To provide a baseline for comparison, the two limits were referenced using pure liquids whose surface tensions corresponded to the higher and lower limits of the surfactant’s dynamic behavior.

The departure images of bubbles from a submerged orifice in the two pure liquids are shown for various bubble frequencies in Fig.5.4 (Subramani et al. (2007, 2008). The pure liquids under consideration are water (  = 1027 kg/m3; = 72.4 mN/m and  = 0.911 mPas) and ethanol ( = 789 kg/m3; = 22.1 mN/m and = 1.08 mPas). Since their viscosities and densities are approximately the same, the difference in their bubble departure diameters can be exclusively attributed to the role played by surface tension in bubble evolution and departure.

Q = 30 ml/min Q = 50 ml/min Q = 70 ml/min Q = 90 ml/min Q = 100 ml/min

BI = 0.086 s BI = 0.06 s BI = 0.052 s BI = 0.0397 s BI = 0.0297

db = 3.85 mm db = 4.02 mm db = 4.11 mm db = 4.11 mm db = 4.31 mm

(a) Water

Q = 30 ml/min Q = 50 ml/min Q = 70 ml/min Q = 90 ml/min Q = 100 ml/min

BI = 0.041 s BI = 0.029 s BI = 0.024 s BI = 0.022 s BI = 0.021

db = 3.12 mm db = 3.29 mm db = 3.58 mm db = 3.79 mm db = 3.97 mm

(b) Ethanol

Fig.5.4. Departure dynamics of a single bubble growing in pure liquids (do = 1 mm)

At a very fundamental level, for inviscid pure liquids, it is the balance between the surface tension and buoyancy that governs the size and frequency of bubble growth. In liquids of low surface tension, the buoyancy of the growing bubble encounters a much smaller surface tension force and hence the force balance is reached at an earlier time, causing the bubbles to depart at smaller sizes (Gaddis and Vogelpohl (1986)). Hence ethanol, with a much smaller surface tension as compared to water, sees bubbles departing at smaller sizes and higher frequencies than the latter. However, an important point to be noted is that in pure liquids, the surface tension is static and hence the growing bubble interface sees a uniform interfacial force throughout the bubble growth cycle.

Compared to the bubble growth in pure liquids, ebullience in surfactant solutions is a complex phenomenon owing to the time dependent surface tension relaxation brought about by the surfactant molecules. Figure 5.5 illustrates the departing bubbles in FS-50 (1400 wppm), water, and ethanol at different gas flow rates corresponding to different regions of the surface tension relaxation curve. It can be observed that at lower bubble frequencies

(higher surface ages of approximately 8-10 s), bubbles emerging from the aqueous solution of FS-50 are almost the same size as those growing in ethanol. Contrastingly, at the higher reaches of bubble frequencies, the bubbles departing in FS-50 are a closer match to those growing in water.

The results very clearly illustrate the role played by the time-dependent adsorption/desorption behavior of the surfactant molecules in the growth and departure of the bubble. Higher bubble frequencies give very little time for the surfactant molecules to migrate towards the gas-liquid interface and hence the interface encounters a surface tension that is almost equal to that of water. As a result, the bubbles have shapes and sizes that are

characteristic of those in water. On the other hand, at lower frequencies, the surfactant molecules are given adequate time to move and align themselves at the gas-liquid interface, thus causing surface tension relaxation to set in. The surface tension gradually decreases with decreasing departure frequencies until it approaches an equilibrium value at high surface ages, when the bubbles attain sizes that match with the ones in ethanol.

BI = 12 s BI = 1.14 s BI = 0.45 s BI = 0.106 s

db = 3.38 mm db = 3.42 mm db = 3.45mm db =3.58 mm (a) Water

BI = 9.4 s BI = 1.19 s BI = 0.45 s BI = 0.101 s

db = 2.08 mm db = 2.27 mm db = 2.78 mm db = 3.41mm (b) FS-50, 1400 wppm

BI = 10.35 s BI = 1.33 s BI = 0.49s BI = 0.098 s

db = 2.081 mm db = 2.12 mm db = 2.16 mm db = 2.22 mm

Fig.5.5. Departure dynamics of a single bubble growing in FS-50 (1400 wppm), water

and ethanol at different bubble departure frequencies (do = 0.8 mm)

Since the bubble frequencies encountered while moving from equilibrium to near- water surface tensions span the constant volume as well as the growing bubble regimes, experiments were run over the entire range of frequencies in all the three liquids to provide a surface-age based comparison of the bubble sizes. The bubble departure diameters over the entire frequency range for all the three liquids are shown in Fig.5.6. The trends followed the three liquids, particularly the s-shaped curve of the FS-50 solution is characteristic of the diffusion-time dependent dynamic surface tension of the surfactant solution. It is very clear from the experimental observations that the dynamic surface tensions of the surfactant solutions are higher than the equilibrium value, and hence, it would be more appropriate to employ the dynamic surface tension as the scaling factor in the study of physical processes involving ebullience, such as boilers, bubble reactors, and froth-flotation tanks

Fig.5.6. A comparison of the departure bubble sizes for FS-50 (1400 ), water and

ethanol over the entire range of experimental bubble frequencies

5.5.3. Effect of Surfactant Molecular Weight

Figure 5.7 illustrates the departing bubbles in FS-50 at a concentration of 400 wppm,

CTAB (400 wppm), and SDS at CMC (2500 wppm), at several distinct bubble departure frequencies/surface ages. All the three solutions have approximately the same equilibrium surface tension, however, it can be noticed that the bubbles departing from the FS-50 solution are comparatively larger than those in the SDS or CTAB solutions. The figure clearly explains the effect of molecular weight of the surfactant on surface tension relaxation, and hence on the departure bubble diameters. Both SDS (MW = 288.3g) and CTAB (364.5g) are lighter molecules as compared to FS-50 (MW  400g). As was discussed in Section 5.5.1, the

FS-50 molecules take a longer time to diffuse towards the gas-liquid interface due to their lower mobility. Hence, at any particular instant of time, in the non-equilibrium surface tension region, the gas-liquid interface in FS-50 solution sees a lower interfacial concentration of the surfactant molecules. Therefore, the surface tension is higher as compared to either the CTAB or SDS solutions, whose gas-liquid interfaces encounter a larger number of molecules, as a result of which the bubbles arising in them tend to be smaller at the same surface age. Like in Section 5.5.2, experiments were conducted in the entire range of frequencies that encompass the equilibrium and near-water surface tension regions.

BI = 6.43 s BI = 1.47 s BI = 0.522 s BI = 0.11 s BI = 0.08 s BI = 0.025 s

(a) FS-50 400 wppm

BI = 5.95 s BI = 1.44 s BI = 0.59 s BI = 0.1 s BI = 0.084 s BI = 0.026 s

(b) SDS 2500 wppm

BI = 5.4 s BI = 1.71 s BI = 0.55 s BI = 0.12 s BI = 0.088 s BI = 0.025 s

Fig.5.7. Departure dynamics of a single bubble growing from a submerged orifice (do =

0.8 mm) in FS-50 (400wppm), SDS (CMC) and CTAB (CMC) at distinct bubble

departure frequencies

Figure 5.8 shows the bubble departure diameters of the three surfactant solutions over the experimental frequency range, with water being used as an upper reference limit. It can be noticed that while FS-50 reaches water-like behavior at comparitively lower frequencies (higher surface ages), SDS and CTAB do not do the same until much higher surface ages. This clearly indicates that because of the time-dependent diffusion and adsorption-desorption of surfactant molecules at the growing liquid-air interface, the dynamic surface tension at any instant of time tends to be higher for a heavier surfactant molecule, thereby producing larger bubbles. However, all the three solutions do exhibit the same behavior at either ends of the frequency spectrum (equilibrium surface tension and water-like surface tension).

5.5

d o = 0.8 m m W ater 5 F S -50 400 w ppm S DS 2500 w ppm C TA B 400 w ppm

4.5

d / 4

3.5

2.5 10 -2 10 -1 10 0 10 1 10 2 B ubble Frequency (s -1)

Fig.5.8. A comparison of the departure bubble sizes for FS-50 (400 wppm), SDS (CMC)

and water over the entire range of experimental bubble frequencies

5.6. Conclusions

High speed visualization techniques were used to record and analyze adiabatic single bubble dynamics in surfactant solutions and compare them with those in pure liquids. In the case of pure inviscid liquids, it has been established that the surface tension plays a major role in determining the size of the departing gas-liquid interface. Ebullience in surfactant solutions is however, much more complex, due to the time dependent adsorption-desorption behavior of the surfactant molecules at the gas-liquid boundaries, which manifests itself as the dynamic surface tension. The dynamic surface tension of surfactant solutions is always higher than the equilibrium value, thus resulting in larger bubbles as compared to a pure liquid with a surface tension equivalent to the equilibrium surface tension of the surfactant solution. The time dependent diffusion of the surfactant molecules towards the interface is also found to be heavily dependent upon the molecular weight and hence the mobility of the surfactant molecule. In general, heavier surfactant molecules, like FS-50 for example, tend to take a longer time to move towards the gas-liquid boundary and hence have longer relaxation times as compared to the lighter ones. This results in them having water-like ebullience behavior at much lower frequencies compared to the lighter surfactants, which show water- like trends only at very high ebullient frequencies. Due to the heavily time-dependent nature of surface tension relaxation encountered in surfactant solutions, it is always appropriate to consider both the equilibrium and dynamic values of surface tension while studying and designing processes that involve ebullience.

CHAPTER 6

Recommendations for Future Work

Effect of Orifice Size

 Experimental investigation of adiabatic single bubble growth from large orifices in

viscous liquids ( )

 Using orifices made from glass, plastic, or other transparent materials to observe the

dynamics of necking and bubble departure, which occur below the orifice surface, in

the case of large orifices.

Necking and Pinch-off

 Correlating the calculated neck-length with the dynamics of bubble neck-thinning and

subsequent pinch-off. This would help in accurately predicting the time taken for

necking and pinch-off.

Theoretical Modeling

 Including the effects of the bubble’s angle of contact with the orifice tip. The surface

tension force has been approximated as a constant throughout the growth cycle, while

in reality, the force changes with the bubble’s angle of contact, which by itself is a

function of the bubble’s size. Understanding the dynamics of the gas-liquid-solid

contact angle is still in its nascent stages and very limited experimental data are

available in the literature. Incorporating this coupled relationship in the theoretical

model might provide a more accurate prediction as compared to the current

methodology.

Ebullience in Surfactant Solutions

 Investigating the effect of contact angle dynamics on adiabatic bubble formation.

Surfactant solutions, apart from exhibiting a dynamic surface tension behavior, also

exhibit a dynamic variation in their contact angles with an available interface.

Coupled with the varying gas-liquid-solid contact angle of the physical system, the

surfactant’s contact angle behavior may play a very dominating role in surfactant

ebullience.

 Theoretical modeling of adiabatic bubble growth in surfactant solutions, based on the

model suggested previously. The variation of the fluid’s surface tension with time (as

derived and fitted from surface tension measurements) can be fed as an input to the

model to aid in the computation of the dynamic surface tension forces.

Bibliography

Bahl, M., Zhang, J., Manglik, R. M., 2003. Measurement of Dynamic and

Equilibrium Surface Tension of Surfactant Solutions by the Maximum Bubble Pressure

Method, ASME Conference Proceedings 2003(36932), 765-771.

Bashforth, F., Adams, J. C., 1883. An Attempt to Test the Theories of Capillary

Action : By Comparing the Theoretical and Measured Forms of Drops of Fluid. with an

Explanation of the Method of Integration Employed in Constructing the Tables which Give the Theoretical Forms of such Drops. Cambridge [Eng.]: Cambridge University Press.

Benzing, R. J., Myers, J. E., 1955. Low Frequency Bubble Formation at Horizontal

Circular Orifices, Industrial & Engineering Chemistry 47(10), 2087-2090.

Burton, J. C., Waldrep, R., and Taborek, P., 2005, Scaling and instabilities in bubble pinch-off, Physical Review Letters, 94, 184502.

Chang, C., Franses, E. I., 1995. Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms, Colloids and

Surfaces A: Physicochemical and Engineering Aspects 100, 1-45.

Coppock, P. D., Meikeljohn, G. T., 1951. The Behaviour of Gas Bubbles in Relation to Mass Transfer, Transactions of the Institution of Chemical Engineers 29(a), 75.

Datta, R. L., Napier, D. H., Newitt, D. M., 1950. The Properties and Behavior of Gas

Bubbles formed at a Circular Orifice, Transactions of the Institution of Chemical Engineers

28, 14.

Davidson, J. F., Schüler, B. O. G., 1960. Bubble formation at an orifice in a viscous liquid, Transactions of the Institution of Chemical Engineers 38(a), 144.

Davidson, J. F., Schüler, B. O. G., 1960. Bubble formation at an orifice in an inviscid liquid, Transactions of the Institution of Chemical Engineers 38(a), 335.

Eastoe, J., Dalton, J. S., 2000. Dynamic surface tension and adsorption mechanisms of surfactants at the air–water interface, Advances in Colloid and Interface Science 85(2-3),

103-144.

Eversole, W. G., Wagner, G. H., Stackhouse, E., 1941. Rapid Formation of Gas

Bubbles in Liquids, Industrial & Engineering Chemistry 33(11), 1459-1462.

Ferri, J. K., Stebe, K. J., 2000. Which surfactants reduce surface tension faster? A scaling argument for diffusion-controlled adsorption, Advances in Colloid and Interface

Science 85(1), 61-97.

Gaddis, E. S., Vogelpohl, A., 1986. Bubble formation in quiescent liquids under constant flow conditions, Chemical Engineering Science 41(1), 97-105.

Gerlach, D., Alleborn, N., Buwa, V., and Durst, F., 2007, Numerical simulation of periodic bubble formation at a submerged orifice with constant gas flow rate, Chemical

Engineering Science, 62(7), 2109.-2125.

Hirt, D. E., Prud'Homme, R. K., Miller, B., Rebenfeld, L., 1990. Dynamic surface tension of hydrocarbon and fluorocarbon surfactant solutions using the maximum bubble pressure method, Colloids and Surfaces 44, 101-117.

Hsu, S., Lee, W., Yang, Y., Chang, C., Maa, J., 2000. Bubble Formation at an Orifice in Surfactant Solutions under Constant-Flow Conditions, Industrial & Engineering Chemistry

Research 39(5), 1473-1479.

Hua, X. Y., Rosen, M. J., 1988. Dynamic surface tension of aqueous surfactant solutions. I: Basic parameters,124(2), 652-659.

Hughes, R.R., Handlos, A.E., Evans, H.D., Maycock, R.L., 1955, The formation of bubbles at simple orifices, Chemical Engineering Progress, 51 (12), pp. 557 - 563.

Iliev, T. H., Dushkin, C. D., 1992. Dynamic surface tension of micellar solutions studied by the maximum bubble pressure method. 1. Experiment, Colloid & Polymer Science

270(4), 370-376.

Jamialahmadi, M., Zehtaban, M. R., Mu ller – Steinhagen, H., Sarrafi, A., and Smith,

J. M., 2001, Study of Bubble formation under Constant Flow Conditions, Transactions of the

Institution of Chemical Engineers, 79, Part A.

Kasimsetty, S.K., Manglik, R. M., and Jog, M. A., 2008, Theoretical modeling and correlational analysis of single bubble dynamics from a submerged orifice in liquid pools,

Master’s Thesis, University of Cincinnati

Kasimsetty, S.K., Subramani, A., Manglik, R. M., and Jog, M. A., 2007, Theoretical modeling and experimental measurements of single bubble dynamics from a submerged orifice in a liquid pool, Proceedings of HT, ASME-JSME Thermal Engineering Summer

Heat Transfer Conference, Vancouver, British Columbia, Canada.

Kasimsetty, S.K., Manglik, R. M., and Jog, M. A., 2008, Theoretical modeling and correlational analysis of single bubble dynamics from a submerged orifice in liquid pools,

Master’s Thesis, University of Cincinnati.

Keim, N.C., Møller, P., Zhang, W.W., and Nagel, S.R., 2006, Breakup of Air Bubbles in Water: Memory and Breakdown of Cylindrical Symmetry, Physical Review Letters,

97(14), 144503.

Kulkarni, A. A., Joshi, J. B., 2005. Bubble Formation and Bubble Rise Velocity in

Gas-Liquid Systems: A Review, Industrial & Engineering Chemistry Research 44(16), 5873-

5931.

Kumar, R., Kuloor, N. K., 1970. The Formation of Bubbles and Drops, in: Thomas B.

Drew, Giles R. Cokelet, John W. Hoopes,Jr. and Theodore Vermeulen (Ed), Advances in

Chemical Engineering, Academic Press, pp. 255-368.

Lamb, H., 1932, Hydrodynamics, Dover Publications, 130.

Liow, J., 2000. Quasi-equilibrium bubble formation during top-submerged gas injection, Chemical Engineering Science 55(20), 4515-4524.

Longuet-Higgins, M. S., Kerman, B. R, and Lunde, K., 1991, The release of air bubbles from an underwater nozzle, Journal of Fluid Mechanics, 230, 365.

Loubière, K., Hébrard, G., 2004. Influence of liquid surface tension (surfactants) on bubble formation at rigid and flexible orifices, Chemical Engineering and Processing 43(11),

1361-1369.

Manglik, R. M., Wasekar, V. M., Zhang, J., 2001. Dynamic and equilibrium surface tension of aqueous surfactant and polymeric solutions, Experimental Thermal and Fluid

Science 25(1-2), 55-64.

Nahra, H.K, and Kamotani, Y., 2003, Prediction of bubble diameter at detachment from a wall orifice in liquid cross-flow under reduced and normal gravity conditions, Chemical Engineering Science, 58(1), 55.

Oguz, H., and Prosperetti A., 1993, Dynamics of bubble growth and detachment from a needle, Journal of Fluid Mechanics, 257, 111.

Park, Y., Lamont Tyler, A., de Nevers, N., 1977. The chamber orifice interaction in the formation of bubbles, Chemical Engineering Science 32(8), 907-916.

Peregrine, D.H., Shoker, G., and Symon, A., 1990, The bifurcation of liquid bridges, Journal of Fluid Mechanics, 212(1), 25.

Quan Hua, Shaoping Jinsong, 2008, Numerical studies of bubble necking in viscous liquids, Physical Review E, 77(6), 066303.

Räbiger, N., 1984. VDI-Forschungsheft , 625.

Räbiger, N. and Vogelpohl, A., 1986, Bubble formation and its movement in

Newtonian and Non-Newtonian liquids, Encyclopedia of Fluid Mechanics: Gulf Publishing:

Houston, 59.

Ramakrishnan, S., Kumar, R., Kuloor, N. R., 1969. Studies in bubble formation—I bubble formation under constant flow conditions, Chemical Engineering Science 24(4), 731-

747.

Rayleigh, L., 1916. On the Theory of the Capillary Tube, Proceedings of the Royal

Society of London. Series A 92(637), 184-195.

Rosen, M. J., 1989. Surfactants and Interfacial Phenomena. New York, NY: Wiley.

Gupta, S.V., 2004. Capillary action in narrow and wide tubes—a unified approach,

Metrologia 41(6), 361.

Siemes, W., Kauffmann, J. F., 1956. Die periodische Entstehung von Gasblasen an

Dusen, Chemical Engineering Science 5(3), 127-139.

Sierou, A., and Lister, J.R., 2003, Self-similar solutions for viscous capillary pinch off, Journal of Fluid Mechanics, 497(1), 381.

Snabre, P., Magnifotcham, F., 1998. I. Formation and rise of a bubble stream in a viscous liquid, The European Physical Journal B - Condensed Matter and Complex Systems

4(3), 369-377.

Subramani, A., Manglik, R. M., and Jog, M. A., 2008, Experimental Characterization of bubble dynamics in isothermal liquid pools, Master’s Thesis, University of Cincinnati

Subramani, A., Kasimsetty, S. K., Manglik, R. M., Jog, M. A., 2007. Computational and Experimental Characterization of Single Bubble Dynamics in Isothermal Liquid Pools,

ASME Conference Proceedings 2007(42894), 1481.

Tadaki, T., and Maeda, S, The Size of Bubbles from Single Orifices, Chemical

Engineering, Japan, 1 (1), pp. 55-60

Takahashi, T., Miyahara T., Izawa, H., 1976, Drag coefficient and wake volume of single bubble rising through Quiescent liquids, Kagaku Kogaku Ronbunshu, 2, pp 480.

Terasaka, K., and Tsuge, H., 1993, Bubble Formation under Constant-Flow

Conditions, Chemical Engineering Science, 48(19), pp. 3417-3422.

Thoroddsen, S. T., Etoh, T. G., Takehara, K., 2007. Experiments on bubble pinch-off,

Physics of Fluids 19(4), 042101.

Tsuge, H., Rudin, P. and Kammel, R., 1986, Bubble formation from a vertically downward facing nozzle in liquids and molten metals, Journal of Chemical Engineering of

Japan, 19, 326.

Tsuge, H., and Hibino, S., 1983, Bubble formation from an orifice submerged in liquids”, Chemical Engineering Communications, 22, pp. 63-79.

Van Krevelen, D. W., Hoftijzer, P. J., 1950. Studies of Gas-Bubble Formation:

Calculation of Interfacial Area in Bubble Contactors, Chemical Engineering Progress 46, 29.

Wasekar, V. M., Manglik, R. M., 2002. The influence of additive molecular weight and ionic nature on the pool boiling performance of aqueous surfactant solutions,

International Journal of Heat and Mass Transfer 45(3), 483-493.

Wasekar, V. M., 2001. Nucleate Pool Boiling Heat Transfer in Aqueous Surfactant

Solutions,.

Wong, H., Rumschitzki, D., and Maldarelli, C., 1998, Theory and experiment on the low-Reynolds-number expansion and contraction of a bubble pinned at a submerged tube tip, Journal of Fluid Mechanics, 356(1), 93.

Wraith, A.E., 1971, Two Stage Bubble Growth at a Submerged Plate Orifice,

Chemical Engineering Science, 26(10), pp. 1659-1671

Wu, W., Yang, Y., Maa, J., 1995. Enhancement of Nucleate Boiling Heat Transfer and Depression of Surface Tension by Surfactant Additives, Journal of Heat Transfer 117(2),

526-529.

Zhang, J., Manglik, R. M., 2003. Visualization of Ebullient Dynamics in Surfactant

Solutions, Journal of Heat Transfer 125(4), 547.

Zhang, J., 2004. Experimental and Computational study of Nucleate Pool Boiling

Heat Transfer in Aqueous Surfactant and Polymer Solutions.

Zhang, L., and Shoji, M., 2001, Aperiodic bubble formation from a submerged orifice, Chemical Engineering Science, 56, 5371-5381.

Appendix A : Image Processing and Error Analysis

A.a. Gray Scale Pixel Analysis

The images captured, using the high-speed high-resolution camera, were analyzed using an image-processing software, Image-Pro Plus 4.0 from Media Cybernetics. The software provides accurate measurements of various length, area, and angle parameters based on Gray-Scale Pixel Analysis. A gray scale pixel value represents a particular level of greyness which may range from black to white. It is often referred to as monochrome.

Depending on the size of the image being analyzed, pixels are given values which range from

0 for a completely black pixel to a higher number (for example, 255 in the case of an 8-bit image or 65,535 in a 16-bit image) for a white pixel. Pixel values in between represent a variety of grey shades.

Image-Pro provides an inbuilt feature which allows one to calibrate and scale an image to fit any coordinate system of choice. Initial calibration is very important for accurate measurement of subsequent parameters. Essentially, calibration helps in assigning a particular length scale to one pixel, which would be used as the base measurement in other length/area computations. One way to perform the calibration would be to manually assign a value to each pixel. For example, the measurement scale can be calibrated so that each pixel measures one centimeter in length, that way, a line spanning across 10 pixels would be 10 cm long and an area encompassing 25 pixels would measure 25 cm2. Another more efficient way is to use of an object present in the image as the scaling-length. Provided that the objects

physical dimensions are known, the measurement scale can be directly calibrated from the object.

A.b. Experimental Uncertainty and Precision

In determining the uncertainties in the measurements and subsequent calculations of various parameters used in experiments and data analysis, the Single Sample Error

Propagation Method, developed by Moffat (1988), was employed.

In this method, the overall uncertainty in the measurement/calculation of a parameter

R can be calculated by:

2 1/2 n R  RXi  X i1 i where, R = R(X1, X2,… Xn)

Each term in the equation is a representation of the contribution made by the uncertainty in the measurement of a single variable Xn, to the overall uncertainty in the measurement of parameter R. The estimated uncertainty in the result has the same probability of occurrence as have the uncertainties in the individual measurements (Moffat (1988))

At a frame rate of 3000 fps, the precision involved in measuring the bubble interval was ±0.33 ms. The bubble diameter could be measured to a precision of ±0.026 mm precision based on the saturation intensity and pixel density. Using the propagation error analysis explained above, the maximum overall uncertainty in the measurement of bubble diameter/neck-length was found to be 1.04% and the maximum uncertainty in the measurement of air-flow rate was determined to be 1.32%.

Appendix B : Experimental Data

B.a. Bubble Departure Diameters (Effect of Orifice Size)

B.a.1. Water

Investigators Orifice Diameter (mm) do/2lc db/do 0.320 0.060 7.844 Subramani et al. 1.000 0.187 3.830 1.760 0.329 2.411 0.360 0.066 6.655 0.600 0.111 4.887 0.970 0.179 3.460 1.410 0.260 2.710 1.980 0.365 2.141 Datta et al. 2.700 0.498 1.828 3.880 0.716 1.476 4.320 0.797 1.330 5.200 0.959 1.103 6.300 1.162 0.935 1.500 0.277 2.642 Krevelen & Hoftijzer 9.400 1.752 0.816 2.010 0.372 2.144 2.430 0.449 1.942 2.930 0.542 1.747 3.260 0.603 1.653 Coppock et al. 3.610 0.667 1.584 4.000 0.740 1.488 4.380 0.810 1.397 4.770 0.882 1.338 0.610 0.113 4.744 0.910 0.168 3.580 1.020 0.188 3.337 Benzing and Myers 1.810 0.334 2.209 1.880 0.347 2.108 2.930 0.540 1.650

Investigators Orifice Diameter (mm) do/2lc db/do Takahashi and Miyahara 1.260 0.238 2.914 0.840 0.155 4.235 Tadaki and Maeda 1.640 0.306 2.650 Ramakrishnan et al. 3.670 0.547 1.418 0.660 0.123 5.255 4.890 0.914 1.109 Present study 0.800 0.149 4.238 6.100 1.140 0.858 7.000 1.308 0.856

B.a.2. Other Liquids

Orifice Viscosity Investigators Liquid Diameter d /2l d /d (mPa-s) o c b o (mm) 0.32 0.094903 5.90625 Ethanol 1.08 1 0.198111 3.799401 1.76 0.2967 3.47 0.32 0.086218 8.240625 Subramani et al. Propylene Glycol 49.9 1 0.269433 4.126 1.76 0.474202 2.702841 0.32 0.0698 14.09375 Glycerol 749.3 1 0.218125 6.346 1.76 0.3839 3.925 Räbiger 7 2 2.25 0.842198 Davidson and Schüler 515 0.67 6.716418 0.162611 Davidson and Schüler Aqueous Glycerin 1040 0.67 8.208955 0.147634 Snabre and Solution 430 0.6 5.748333 0.203965 Magnifotcham Snabre and 800 0.6 9.255333 0.131557 Magnifotcham 0.32 0.079032 6.242424 Present Study Ethylene Glycol 17 1 0.237095 3.171717 1.76 0.421503 2.436932

B.b. Bubble Neck-length at Departure

Liquid Orifice Diameter (mm) Neck-length (mm) 0.32 0.32 0.66 0.56 Water 1.00 0.84 1.35 1.03 1.76 1.20 0.32 0.40 Ethanol 1.00 0.99 0.32 0.35 Ethylene Glycol 0.99 0.90 1.76 1.51 0.32 0.51 0.66 0.71 Propylene Glycol 1.00 1.05 1.35 1.31 1.76 1.67 0.32 0.76 Glycerol 1.00 1.50 1.76 1.81

B.c. Bubble Departure Diameters (Theoretical Modeling)

Bubble Departure Diameters vs. Flow Rates and Orifice Reynolds Numbers, Subramani et al. (2007,2008) , do = 0.32 mm Water Ethanol d d Flow Rate (ml/min) Re b Flow Rate (ml/min) Re b o (mm) o (mm) 2.000 8.523 2.510 2.000 8.518 1.890 4.000 17.045 2.507 4.000 17.037 1.890 8.000 34.090 2.520 8.000 34.073 1.930 12.000 51.135 2.520 12.000 51.110 1.956 16.000 68.181 2.530 16.000 68.146 1.978 20.000 85.226 2.560 20.000 85.183 1.978

Propylene Glycol Glycerol d d Flow Rate (ml/min) Re b Flow Rate (ml/min) Re b o (mm) o (mm) 4.000 17.045 2.634 2.000 8.518 4.510 8.000 34.090 2.641 4.000 17.037 4.520 12.000 51.135 2.652 6.000 25.555 4.670 16.000 68.181 2.670 12.000 51.110 4.650 18.000 76.703 2.714 20.000 85.184 4.630 20.000 85.226 2.758

100

Bubble Departure Diameters vs. Flow Rates and Orifice Reynolds Numbers, Subramani et al. (2007,2008) , do = 1.00 mm Water Propylene Glycol Glycerol Flow Rate d Flow Rate d Flow Rate d Re b Re b Re b (ml/min) o (mm) (ml/min) o (mm) (ml/min) o (mm) 20.000 27.272 3.830 8.000 10.909 4.126 8.000 10.909 6.346 30.000 40.908 3.850 20.000 27.272 4.222 12.000 16.363 6.386 44.000 59.999 3.970 40.000 54.544 4.330 16.000 21.818 6.449 52.000 70.908 4.030 80.000 109.089 4.534 20.000 27.272 6.420

56.000 76.362 4.040 100.000 136.361 4.814 24.000 32.727 6.509 60.000 81.817 4.090 120.000 163.633 4.960 60.000 81.817 6.847 70.000 95.453 4.120 140.000 190.906 5.087 80.000 109.089 7.100 80.000 109.089 4.230 160.000 218.178 5.242 120.000 163.633 7.388 92.000 125.452 4.240 180.000 245.450 5.451 160.000 218.178 7.706 96.000 130.907 4.290 200.000 272.722 7.953 100.000 136.361 4.320 240.000 327.267 8.232 106.000 144.543 4.350 110.000 149.997 4.380 120.000 163.633 4.510

101

Bubble Departure Diameters vs. Flow Rates and Orifice Reynolds Numbers, Subramani et al. (2007,2008) , do = 1.76 mm Water Propylene Glycol Glycerol Flow Rate d Flow Rate d Flow Rate d Re b Re b Re b (ml/min) o (mm) (ml/min) o (mm) (ml/min) o (mm) 4.000 3.099 4.240 8.000 6.198 4.776 8.000 6.198 6.908 10.000 7.748 4.260 12.000 9.297 4.757 16.000 12.396 6.956 20.000 15.496 4.260 16.000 12.396 4.801 20.000 15.496 6.972 30.000 23.243 4.300 20.000 15.496 4.827 32.000 24.793 7.069

40.000 30.991 4.320 60.000 46.487 5.133 80.000 61.982 8.023 46.000 35.640 4.350 80.000 61.982 5.252 120.000 92.973 8.446 50.000 38.739 4.380 100.000 77.478 5.424 160.000 123.965 8.986 56.000 43.388 4.370 120.000 92.973 5.500 200.000 154.956 9.538 66.000 51.135 4.440 160.000 123.965 5.812 240.000 185.947 9.908 70.000 54.235 4.470 180.000 139.460 6.044 240.000 185.947 9.908 76.000 58.883 4.460 200.000 154.956 6.128 240.000 185.947 9.908 80.000 61.982 4.500 86.000 66.631 4.510 90.000 69.730 4.600 96.000 74.379 4.620 100.000 77.478 4.660 106.000 82.127 4.750 110.000 85.226 4.740 116.000 89.874 4.810 120.000 92.973 4.850

102

Departure Bubble Diameters vs. Flow Rates and Orifice Reynolds Numbers, Davidson and Schüler (1960), do= 0.668 mm Water Ethanol Flow Rate Flow Rate Re d (mm) Re d (mm) (ml/min) o b (ml/min) o b 30.000 61.209 3.520 30.000 61.209 2.538 60.000 122.418 3.600 35.000 71.411 2.605 90.000 183.627 3.710 40.000 81.612 2.939 122.007 249.056 4.628 45.000 91.814 3.073 50.000 102.015 3.200

Departure Bubble Diameters vs. Flow Departure Bubble Diameters vs. Flow Rates and Orifice Reynolds Numbers Rates and Orifice Reynolds Numbers in Water, Park et al. (1977), do= 3.2 in Water, Räbiger (1984), do= 2.00 mm mm Flow Rate d Flow Rate d Re b Re b (ml/min) o (mm) (ml/min) o (mm) 3.118 1.329 4.529 120 81.817 5.857 11.174 4.762 4.495 180 122.725 6.429 17.100 7.287 4.666 210 143.179 5.857 39.312 16.752 4.915 288 196.36 6.214

73.737 31.422 5.178 378 257.722 6.929 112.842 48.085 5.217 185.949 79.238 6.055 404.415 172.333 7.516 630.445 268.650 8.658 947.105 403.588 9.610

103

Departure Bubble Diameters vs. Flow Rates and Orifice Reynolds Numbers in Water, Benzing and Myers (1955),

do (mm) Flow Rate (ml/min) Reo db (mm) 2.010 1.380 0.936 3.530 2.430 1.998 1.121 3.990 2.930 2.460 1.145 4.280 3.260 2.898 1.212 4.520 3.610 3.378 1.276 4.750 4.000 3.960 1.350 5.010 4.380 4.440 1.382 5.210 4.770 4.962 1.418 5.400

Departure Bubble Diameters vs. Flow Rates and Orifice Reynolds Numbers in Water, Jamialahmadi et al. (2001)

do = 3 mm do = 2.5 mm do = 2.0 mm Flow d Flow Rate d Flow Rate d Rate Re b Re b Re b o (mm) (ml/min) o (mm) (ml/min) o (mm) (ml/min) 61.404 27.910 5.396 63.708 34.749 5.148 62.556 42.651 4.826 122.556 55.706 5.805 121.404 66.219 5.586 121.404 82.774 5.308 176.784 80.355 6.302 177.942 97.057 6.083 180.246 122.893 5.791 234.480 106.580 6.624 236.784 129.153 6.463 235.632 160.655 6.200 295.632 134.376 6.945 296.784 161.879 6.784 297.942 203.138 6.521 354.480 161.124 7.383 352.170 192.089 7.164 352.170 240.111 6.901 412.170 187.347 7.661 412.170 224.816 7.471 409.860 279.445 7.179 468.708 213.045 7.997 471.018 256.914 7.763 468.708 319.568 7.471 527.556 239.794 8.216 529.860 289.009 8.026 525.246 358.116 7.705 586.404 266.542 8.435 586.404 319.851 8.231 585.246 399.024 8.026

104

Departure Bubble Diameters vs. Flow Rates and Orifice Reynolds Numbers in Water, Jamialahmadi et al. (2001)

do = 1.5 mm do = 1.0 mm Flow Rate d Flow Rate d Re b Re b (ml/min) o (mm) (ml/min) o (mm) 60.246 54.768 4.446 64.860 88.444 4.125 120.246 109.313 5.016 119.094 162.398 4.651 176.784 160.710 5.528 180.246 245.785 5.133 236.784 215.254 5.893 234.480 319.739 5.542 295.632 268.751 6.244 291.018 396.835 5.937 354.480 322.249 6.594 354.480 483.373 6.200 413.322 375.740 6.887 468.708 426.090 7.193 525.246 477.487 7.383 585.246 532.032 7.690

Departure Bubble Diameters vs. Flow Rates and Orifice Reynolds Numbers in Water, Present Study

do = 0.66 mm do = 0.80 mm Flow Rate d Flow Rate d Re b Re b (ml/min) o (mm) (ml/min) o (mm) 15.000 30.991 3.390 0.810 1.381 3.383 18.000 37.189 3.410 1.500 2.557 3.395 21.000 43.388 3.450 2.000 3.409 3.388 27.000 55.784 3.490 5.500 9.375 3.399 30.000 61.982 3.500 9.000 15.341 3.417 36.000 74.379 3.510 14.000 23.863 3.430 45.000 92.973 3.530 18.000 30.681 3.459 24.000 40.908 3.455 40.000 68.181 3.496 60.000 102.271 3.516 85.000 144.884 3.561 110.000 187.497 3.588

105

Departure Bubble Diameters vs. Flow Rates and Orifice Reynolds Numbers in Ethylene Glycol, Present Study

do = 0.32 mm do = 1.00 mm do = 1.76 mm Flow d Flow Rate d Flow Rate d Rate Re b Re b Re b o (mm) (ml/min) o (mm) (ml/min) o (mm) (ml/min) 13.000 53.753 2.062 15.000 20.674 3.514 45.000 34.887 4.289 15.000 62.022 2.087 18.000 24.809 3.515 54.000 41.865 4.307 18.000 74.427 2.087 21.000 28.944 3.621 64.000 49.618 4.415 21.000 86.831 2.132 24.000 33.078 3.643 72.000 55.820 4.574 24.000 99.235 2.144 30.000 41.348 3.709 80.000 62.022 4.733 27.000 111.640 2.208 36.000 49.618 3.744 88.000 68.224 4.822 45.000 62.022 3.831 96.000 74.427 4.930 48.000 66.157 3.893 56.000 77.183 3.965 64.000 88.209 4.034 72.000 99.235 4.132 80.000 110.262 4.261 88.000 121.288 4.439 96.000 132.314 4.645

106

B.d. Bubble Departure Diameters in Surfactant Solutions

Water FS 50 (1400 wppm) Ethanol Bubble Bubble Bubble d (mm) d (mm) d (mm) Frequency (s-1) b Frequency (s-1) b Frequency (s-1) b 0.028 3.333 0.070 2.072 0.047 2.600 0.048 3.336 0.106 2.084 0.097 2.601 0.059 3.360 0.182 2.090 0.121 2.606 0.083 3.383 0.308 2.125 0.152 2.610 0.147 3.395 0.526 2.185 0.250 2.613 0.213 3.388 0.840 2.268 0.273 2.620 0.543 3.399 1.010 2.339 0.442 2.621 0.877 3.417 1.176 2.409 0.571 2.631 1.333 3.430 1.429 2.511 0.752 2.644 1.754 3.459 1.859 2.680 1.064 2.663 2.222 3.455 2.183 2.777 1.370 2.674 3.774 3.496 3.155 3.002 2.041 2.694 5.348 3.516 4.831 3.220 4.167 2.725 7.463 3.561 6.024 3.301 6.135 2.760 9.434 3.588 8.065 3.378 12.346 2.776 9.901 3.409 20.408 2.824

12.610 3.486

107

Departure Bubble Diameters vs. Bubble Frequencies the test liquids, Present Study, do = 0.80 mm SDS (2500 wppm) FS 50 (400 wppm) CTAB (400 wppm) Bubble Bubble Bubble d (mm) d (mm) d (mm) Frequency (s-1) b Frequency (s-1) b Frequency (s-1) b 0.111 2.660 0.111 2.445 0.185 2.561 0.174 2.662 0.156 2.455 0.404 2.567 0.244 2.662 0.290 2.593 0.583 2.582 0.376 2.672 0.452 2.706 0.763 2.607 0.696 2.672 0.680 2.921 1.135 2.632 0.962 2.679 0.862 3.195 1.652 2.643 1.053 2.687 1.064 3.309 1.893 2.658 1.351 2.699 1.272 3.393 3.030 2.703 1.695 2.707 1.433 3.430 5.347 2.774 2.041 2.706 1.916 3.474 7.142 2.839 2.421 2.716 2.825 3.481 8.333 2.865 2.703 2.722 4.348 3.552 11.363 2.960 3.333 2.737 6.250 3.553 11.904 2.991 4.348 2.760 9.091 3.577 14.492 3.017 5.556 2.774 15.873 3.795 14.705 3.023 10.000 2.851 22.222 3.973 19.230 3.131 29.412 3.037 22.222 3.973 38.461 3.497 29.412 3.037 22.222 3.973 40.000 3.539 38.462 3.386 22.222 3.973 43.472 3.744 41.667 3.670 40.000 4.101 45.455 3.790 45.455 3.894

108

Appendix C : Algorithm for Solution of Mathematical Model

1. Initialize at t = 0

* * dr 2 r rPoo,  P 0 and (0) dtr o

2. Begin Iteration

3. Calculate FB, FG, FP, FS, FD and FI

4. Calculate AF = FB + FG + FP

5. Calculate RF = FS + FD + FI

6. If AF < RF

ttt  

Return to Step 3

7. If AF ≥ RF

dr* dr dl r*  r  l and   n n dt dt dt

8. Solve equations simultaneously

t t   t

0.25 0.145 9. If lnc1.018 l Ga Mo

Return to Step 8

0.25 0.145 10. If lnc1.018 l Ga Mo

End Iteration

11. Calculate departure bubble diameter

6Qt 3 db  

109