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

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 By 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. i 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. ii 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. iv 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 v 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 vi 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 ix 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

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