T H E UNI V E R S IT Y O F M I C H I G AN COLLEGE OF ENGINEERING Department of Mechanical Engineering Heat Transfer and Thermodynamics Laboratory THE DYNAMICS OF GAS-VAPOR BUBBLES IN BINARY SYSTEMS Poul S. Larsen John A. Clark ORA Project 04268 under contract with~ NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GEORGE C. MARSHALL SPACE FLIGHT CENTER CONTRACT NO. NAS-8-825 HUNTSVILLE, ALABAMA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR November 1963

This report was also a dissertation submitted by the first author in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1963.

ACKNOWLEDGMENTS The authors wish to express their appreciation to Professors Gordon J. Van Wylen, Joseph J. Martin, Vedat S. Arpaci, Herman Merte, Jr., Richard E. Sonntag, and Wen-Jei Yang, for their interest and cooperation in serving as members of the Doctoral Thesis Advisory Committee. iii

TABLE OF CONTENTS Page LIlST OF TABLES,...e......................................... evii LIZST OF FIGURES.....................e..e.e..ee............e...e.e... ix LIST OF APPENDICES.............................................. xiii Nom~mABTRA eAeURE o... e o o o. o o o o o e a a 6 Is o e o e o 0 o o o e o o o eo o e a o e. 0a o o. e a x ABST RCT.................................,................,....,.. xix I INTRODUCTION AND LITERATURE SURVEYe......,,................. 1 A. Introduction........................................... 1 B. Literature Survey..................o................ 7 1. Isothermal Bubble Dynamics...........,...... 7 2, One-component Boiling........,,,................... 8 35 Two-component Boiling...................,,,,,..... 9 4. Gas Absorptioni o,.,,,,,,,,,,.,,,..,,..,.,,,, 10 5o Gas-vapor Bubbles. o........,.....,....,..,... 12 6. Interface Equilibrium,.,,...,,,,.,,.,..,,,...,,,, 14 II THEORETICAL ANALYSIS.,,................................ 16 A. Statement of the Probleme..,,,eo,.,o..,,eeeo..,o..o,. 16 Bo General Formulation,...........,.,..................... 20 1. Gas Phase (O < r < R).............................. 21 2. Liquid Phase TR < r < oo).,D........................ 22 3. Interface Conditions (r = R)...,.,,............... 24 C, Simplifications of the General Formulation............. 25 1. Gas Phase (O < r < R).......coo................... 31 2. Liquid Phase TR < r < ~oo),,,,,, oo,. o.o..o........ 31 3. Interface Conditions (r = R),,......,....... 33 D. The Case of a Nonsoluble Gas.......................... 34 1, Solutions........................................... 34 a. Solution for Small Values of Time............... 40 b. Join Point (T = 71).............................. 44 c. Solution for Intermediate Values of Time (T > T1) 45 d. Asymptotic Solution,,...,.....ee.,.e...,.., 49 v

TABLE OF CONTENTS (CONT'D) Page 2. Discussion of Analytical Results - Parameter Study..... 56 E. The Case of a Soluble Gas............................... 65 1. Solution for Small Values of Time..................... 66 2. Solution to Bubble Collapse in Nonvolatile Liquid...... 73 III EXPERIMENT.................................................... 80 A. Introduction......................................... 80 B. Apparatus.................................................. 81 C. Experimental Procedure...............................95 D. Reduction of Data.................................. 98 1. Properties........................... 98 2. Transient Bubble Size................ a.a............ 104 3. Computer Program..................................... 108 E. Results..............................110 1. Discussion of Experimental Results..................... 110 a. Case of a Nonsoluble Gas........................... 110 b. Case of a Soluble Gas.......................... 120 c. The Case of Bubbles Having Translatory Motion..... 130 d. Bubble Oscillations.................138 2. Experimental Uncertainty............................. 139 IV SUMMARY OF RESULTS........................................... 142 APPENDICES................................................... 144 BIBLIOGRAPHY..........................e.00* ****.~~ 183

LIST OF TABLES Table Page I Significant Effects for Various Bubbles Dynamic Processes... 4 II Parameter Range in Experiments for the Nonsoluble Gas Case.. 111 III Calculated Experimental Uncertainties for Run 523........... 141 vii

LIST OF FIGURES Figure Page 1. Model for General Formulation of the Bubble Dynamic Problem... 17 2, Isobars for Binary Phase-equilibrium (Schematic)............... 18 3. The Influence of Inertia Effects on Bubble Growth (Schematic). 26 4. Composition and Temperature Profiles. Nonsoluble Gas Case..... 37 5. Approximation to Interface Potential for Asymptotic Bubble Growth................................................ 54 6. Analytical Solutions for Relative Dimensionless Bubble Growth. Solution for Small and Intermediate Values of Time and Asymptotic Solution. Nonsoluble Gas Case...,.................. 57 7. Parameter Study, Q(Ja,T). Intermediate and Asymptotic Solutions. Nonsoluble Gas Case................................ 58 8. Parameter Study, Q(Lu,T). Intermediate and Asymptotic Solutions. Nonsoluble Gas Case...................................59 9. Parameter Study, Q(xO,T). Intermediate and Asymptotic Solutions. Nonsoluble Gas Case.............6o............... 60 10. Parameter Study, Q(xo,T). Intermediate and Asymptotic Solutions. Nonsoluble Gas Case...... o..........o....... 61 11o Time Constant for Relative Dimensionless Bubble Growth vs x. and Ja..........................................o............ 63 12. Time Constant for Relative Dimensionless Bubble Growth vs xx and Lu................ e ooo..... s o e o o............ 64 13. Parameter Study of Initial Phase-growth, X/Ja vs Ja/1u. Soluble Gas Case............................................. 72 14. Approximating Function,(P), Equation (118)........,........ 77 15. Dimensionless Bubble Collapse in Nonvolatile Liquid........... 78 16. Experimental Apparatus. General View.......................... 82 17. View Tank with Bubble Injector................................ 82 18. Schematic Diagram for Gas Flow, Pressure Control and Instrumentation.... e o o o ee......................... 83 19. Bubble Injector (Assembled and Disassembled)............... 85 ix

LIST OF FIGURES (CONT'D) Figure Page 20. Section of Bubble Injector.................................. 86 21. Close-up of Dynafax Camera in Position for High Speed Film Recording..o... e0~*eo... 89 22. Close-up of 35 mm Camera in Position for Recording of Single Frames. ooa aa a0a0 a. ~........................ 89. 23. High Speed Film Record and Single Frame from Run 523.......... 91 24. Schematic of Wiring of Camera, Timer and Injector Circuit..... 92 25. Schematic of Synchronization of Film Exposure and Bubble Formation.............................................. 94 26. Oscilloscope Records of Solenoid Current During Bubble Formation..............................96............ 96 27. Calibration Curve for the Electrical Resistivity of NH3 -Solutions................ o.........1oo............. 101 28. Ja-number vs Saturation Temperature for Water................ 102 29. Lu-number vs Saturation Temperature for Water-Nitrogen, WaterHelium and Water-Ammonia.............................0.00. 103 30. Characteristic Dimensions of Injector Tip and Bubbles......... 105 31. Bubble Formation, Run 511, P vs T......,,,..,,,......... 113 32. Bubble Growth, Run 523, Q vs T........................O... 114 335 Bubble Growth, Run 504, Q vs T....o...... oo......O........ 115 34. Bubble Growth, Run 702, Q vs T..........................O.. 116 35. Bubble Growth, Run 524, Q vs T......o......o o............ 117 36. Bubble Growth, Run 517, Q vs T........................... 118 37. Bubble Growth, Run 521, Q vs Tee.....ooooo.....06000* 0.. 119. 38. Bubble Growth, Run 601, Q vs To........................... 121 39. Bubble Growth, Run 609, Q vs T..~~..r...,.@.......oooo 122 40. Bubble Growth, Run 613, Q vs T..................,..Q000~oo@ 123 x

LIST OF FIGURES (CONT 'D) Figures Page 41o Bubble Collapse, Run 951, R vs t............................. 125 42. Bubble Collapse, Run 953, R vs t............................. 126 435 Bubble Collapse, Run 954, R vs t....................o o 127 44. Bubble Collapse, Run 955, R vs t............................. 128 45. Growth of Moving Bubble, Run 805, Q vs T........e.e. oe....... 132 46. Growth of Moving Bubble, Run 806, Q vs T..................... 133 47. Bubble Position vs time, Runs 805 and 806.................... 134 48. Collapse of Moving Bubble, Run 901, R vs t................... 135 49. Collapse of Moving Bubble, Run 903, R vs t..ooo.............o 136 50. Bubble Position vs Time, Runs 901 and903..................... 137 APPENDIX II-1 Relative Equivalent Radius vs Relative Polar Amplitude for Second Zonal Harmonic Bubble Oscillation..o.................. 152 III-1 Comparison Between Exact Solution and Solution by Integral Technique. Bubble Growth in One-component Superheated Liquido.o.o.o. aeo o o o e....................................... 156 xi

LIST OF APPENDICES Appendix Page I DEPARTURE FROM INTERFACE EQUILIBRIUM DURING MASS TRANSFER... 145 II BUBBLE OSCILLATIONS........................................ 150 III APPLICATION OF THE INTEGRAL TECHNIQUE TO A PROBLEM OF SPHERICAL PHASE-GROWTH...................................... 153 IV CRITERION FOR LINEARIZATION OF EQUATION (65)............... 157 V DATA REDUCTION BY DIGITAL COMPUTER....................... 159 V-1 Program for Nonsoluble Gas Case........................ 160 V-2 Nomenclature for Computer Program for the Case of a Nonsoluble Gas....................................... 167 V-3 Data Input for Run 523................................. 171 V-4 Print-out for Run 523.................................. 172 VI SUMMARY OF EXPERIMENTAL DATA........................... 179 VI-1 Data for the Case of a Nonsoluble Gas.................. 180 VI-2 Data for the Case of a Soluble Gas................... 181 VI-3 Data for Bubble Oscillations........................... 182 xiii

NOMENCLATURE Roman a coefficient, polar amplitude Ao constant, Equation (56) b constant, Equation (81> c molal specific heat at constant pressure C constant Equation (119) CH coefficient,Equation (69) CM coefficient, Equation (69) mass diffusivity E functional notation E() = exp()2 erfc(), Equations (106) and (107) f frequency, parameter function, function F ratio of heats of phase transition, F = hfgB/hfgA h enthalpy I integral,Equation (69) INT integral, Equation (127) Ja Jakob number, defined by Equations(50) and (111) k thermal conductivity K expression, Equation (66)', Henry law constant, Equation (94) Lu Lukomskiy number, defined by Equations(50) and (101) L1 reference length, Figure 30 L2 reference length, Figure 30 M expression, Equation (66), molecular weight in Appendix I n order of harmonic N mole flux, expression in Equation (66) xv

p pressure, partial pressure when subscripted P dimensionless bubble size, P = R/Ro spherical surface source Q relative dimensionless bubble size, Q - (P - 1)/(PO - 1) r radial independent space variable R bubble radius, universal gas constant in Appendix I Re Reynolds number for bubble motion, Re = 2'R V/v R(I) equivalent bubble radius from I-th frame of film t time T temperature U variable, U - PP/Lu v velocity V velocity VMAX asymptotic bubble volume VO initial bubble volume VR volume, defined. in Figure 30 V(I) visible volume from I-th frame of film, defined in Figure 30. w relative bubble size, w P/Po x molefraction of component A y independent. space variable measured from bubble interface in the direction of the liquid phase Y spherical zonal harmonic, dimensionless similarity variable, Y -: y/l t Greek ~a thermal diffusivity 6 uncertainty on xvi

E density deficienty, E = 1 - p"/p' 9 dimensionless temperature, defined by Equations(50) and (112) X phase-growth constant, defined by Equation (97) =L cos* v kinematic viscosity of the liquid e intermediate time between t' and t p partial molal density ra surface tension T dimensionless time, T = l't/Ro function, defined by Equation (118) polar angle Xw cyclic frequency Subscript s A component A B component B c bubble center, collapse eq equivalent fg liquid-gas phase transition H heat M mass r radial R at the bubble surface s interfacial T temperature x composition xvii

o initial, reference value 1 join point between initial and intermediate solution 2 join point between intermediate and asymptotic solution 00 asymptotic, at a distance from Superscripts () )*saturation, equilibrium liquid ()" gas ( ) time derivative Operators V. Nabla operator..* convolution xviii

ABSTRACT The dynamic problem of the growth or collapse of a single stationary spherical bubble suddenly exposed to a quiescent liquid is analyzed for a binary system by means of intra-phase mass and heat transport subject to thermodynamic phase-equilibrium at the bubble interface, Two cases are studied, For the nonsoluble gas case, in which the liquid constitutes a single component, approximate solutions to bubble growth are obtained by means of the integral technique for the initial and intermediate stages and the theory of sources for the asymptotic stage of the transient. For the soluble gas case, the initial bubble behavior is predicted and a solution is given to bubble collapse in a nonvolatile liquid, in which case the gas phase constitutes a single component. Dimensionless bubble growth is governed by the Jakob number, Ja = p-(dT/dp)*. (plc'/p"hfg), which characterizes the rate of heat transport when phase change occurs, and the Lukomskiy number, Lu = /a' which gives the ratio of mass to heat diffusivity. The significant ratio, Ja2/Lu, is shown to represent a measure for the relative significance of mass to heat transport for the rate of the process. The growth and collapse of single bubbles is studied experimentally from high-speed film recordings. For the nonsoluble gas case nitrogen and helium bubbles were injected into water, and for the soluble gas case ammonia was used. For stationary nitrogen and helium xix

bubbles in water the analysis is shown to describe the average bubble growth well over a parameter range corresponding to the temperature range 150-206"F and initial bubble radii of 0O7 - 3.0 mm. The effect on the transient heat and mass transport of the observed bubble oscillations is concluded to be negligible. Translatory bubble motion is shown to cause a significant increase in the growth rate when the rising bubble has been accelerated and a flow transition has occurred. The rate of collapse of ammonia bubbles is shown to become greater when the governing rate-parameters and solubility of the gas increases and the liquid volatility decreases. xx

CHAPTER I INTRODUCTION AND LITERATURE SURVEY Ao Introduction The purpose of the present study is to analyse a particular case of the comprehensive subject of the dynamics of binary gas-liquid systems, namely the dynamics of a single gas bubble suddenly brought into contact with a volatile liquid, where either one or both of the phases may comprise two components. The process of injecting a soluble or nonsoluble gas into a liquid has several applications. These can be associated either with the resulting mass and/or heat transfer and/or with the resulting mechanical agitation of the liquid. In gas injection-cooling the primary effect is that of heat removal from the liquid facilitated by its evaporation into injected gas bubbles until these become saturated in this component. In principle, this process is similar to stream distillation from the aspect of forced transfer of a more volatile component from the liquid phase to the gas phase, and it has found application in the cooling of cryogenic systems, especially when local and short duration sub-cooling (35) is desired. A lumped analysis of the injection cooling process based on the instantaneous attainment of the equilibrium composition in the gas phase, predicts satisfactorily the transient cooling for the average system. For certain system geometries, however, - and in particular when the gas is soluble in the liquid - this approach is inadequate and analysis of the distributed system becomes necessary. Accordingly -1 -

-2 -attention need be given the discrete processes, namely the dynamics of the bubbles, when appearing individually or in clusters in the liquid. In the present work the dynamics of the single bubble has been investigated, excluding interference from neighboring bubbles. Although translatory bubble motion will always persist whenever the gas-liquid system is subject to a gravitational field, most attention has been given the case of the stationary bubble in the later justified anticipation that the dynamic process is rapid enough to come practically to an end before appreciable motion has occurred. The effect of translatory motion invariably results in an increase in the rate of the process thus rendering the findings for the stationary bubble conservative. To identify the present problem among the many types of binary gas-liquid dynamic problems, it is pertinent to examine the physics of the process in light of the following characterizing aspects, dispersed liquid phase (drops, fog) separated phases (plane, nonplane (i) system geometry interface) dispersed gas phase (bubbles) (thermodynamic non-equilibrium pressure gradient (ii) disturbance temperature gradient (ii) disturbance concentration gradient non-equilibrium shape imposed body force field mass transport momentum transport heat transport (iii) significant process viscous dissipation viscous dissipation (inter/intra-phase) | capillary effects gravity effects

-3 -Although the injection cooling process in fact represents the case of a dispersed gas phase, the study of the single bubble implies separated phases. When a stream of gas is introduced into the liquid, it breaks into individual bubbles representing a more stable configuration. Related to the hydrodynamics of this process the formed bubbles may oscillate, interfere with neighboring bubbles, coalesce or further break up into smaller bubbles. The present study considers the single bubble having a spherical interface of separation, representing the thermodynamic equilibrium shape in the absence of a gravity field. The nature of the disturbance initiating the dynamic process is primarily the thermodynamic state of non-equilibrium at the interface prevailing at the moment the bubble is introduced. This disturbance causes non-equilibrium temperature and composition distributions to be established by means of which the process is analyzed. The presence of a gravity field represents a secondary disturbance to the shape of the stationary bubble, while being a primary initial disturbance for acceleration of translatory bubble motion. Among the listed significant processes, inter-phase and intraphase heat and mass transport become important for the bubble size and for the types of gases and liquids under study. The inertia effects become important only in the very initial stage when the dynamic process is accelerated, covering a negligible part of the transient for the stationary bubble. For the case of the moving bubble, however, inertia, viscous, gravity, and surface effects determine the motion. Next, the present case of bubble dynamics is identified among the many particular cases previously studied as they relate to diverse applica

tionso The summary given in Table I has been prepared for the frequently encountered process labels and listed are the effects governing the particular bubble dynamic transient. For completeness the effects being important only in the initial stage have been included and marked with I, while the otherwise controlling effects are denoted by Xo TABLE I SIGNIFICANT EFFECTS FOR VARIOUS BUBBLE DYNAMIC PROCESSES Significant effects Liquid Gas phase phase 4' 0 P\ U U)oll U I 4-' U) 4' 4-) 4 4' o Co 0 0 0 ) ) Bubble dynamic P4 F PI P4 P4 process label a) a) C\ U4 U)O ' C C Cd o \ k k k C) m> vaporous X X X Cavitation 0 O | g gaseous I X X X o 0 one-component liquid IX I o ~ two-component liquid X I X I o Boiling Hd O presence of permanent ~ r X X I 4 a' U gas nuclei H U) - c~ | Gas-vapor bubbles (injection cooling) X X X P4 M cJ Gas absorption X X ~ r O ~. O q Gas desor~tion X

-5 -The cavitation and boiling processes are characterized by an initial microscopic bubble size, while gas injection cooling, gas adsorption and desorption are characterized by initial visible (macroscopic) bubble sizes. In cavitation and boiling the dynamic growth starts from nuclei, small bubbles of the thermodynamic equilibrium size corresponding to liquid superheat and the state of the interface. The transient is triggered by a sudden disturbance in pressure (cavitation), gaseous diffusion into the nuclei (gaseous cavitation) or a temperature disturbance (boiling). While in all cases the inertia and surface effects play a significant role in the very first stage of the growth, the heat transport in the liquid phase to the bubble interface required for the evaporation associated with the growth may be omitted in the study of "cold" cavitation. For "hot" cavitation, however, which, aside from the externally imposed pressure variations, physically is identical to boiling, heat transport controls growth subsequent to its initiation. The reason for this difference in behavior, alone determined by the temperature level, is found in the difference in slope of the vapor pressure curve (dT/dp)*. Consider a nucleus of a certain size in thermodynamic equilibrium at the uniform.bulk liquid temperature, such that the excess vapor pressure is balanced by the surface tension forces. When, started by some disturbance, the bubble has grown a small amount, the change in interface saturation temperature, AT, is related to the pressure change by AT = Ap(dT/dp)*. For identical nucleation size and subsequent growth, the pressure change is the same but the temperature difference which governs the rate of heat transfer depends on the vapor pressure curve. Low temperature implies

-6 -large values of the Ja-numberl Ja = /T CI'c which, as shown later is the governing parameter for nondimensional bubble growth controlled by heat transport in the liquid alone. Large Ja-number implies very rapid growth if heat transport were the only effect. This in turn implies that growth is governed and limited by inertia effects in such cases, and heat transport becomes insignificant. For moderate or small values of the Ja-number heat transport dictates the growth rate to be slow enough that inertia effects are of no importance. The presence of dissolved gases in a cavitating or boiling fluid and the case of boiling of binary mixtures may necessitate the consideration of the intra-phase mass transport. As shown later in the analytical treatment, the Lu-number2 L - is convenient for measuring the relative significance of mass diffusion to heat diffusion. Mass diffusion is significant in bubble dynamics of Jakob number, after the late Max Jakob, referred to by Savic(53) A dimensionless group appearing when phase change occurs in connection with the molecular diffusion of heat, change in density and release or absorption of heat. Lukomskiy number, a dimensionless group appearing when mass and heat transfer by molecular diffusion occur simultaneously 38ntroduced in the Russian literature (see e.g. Lykov and Mikhaylov( ) ) With the same notation the group is called the Luikov-number by Mikhaylov(39), and is the inverse of the so called Lewis number, criticized by Jo Ho Arnold in the paper by Klinkenberg(32) as an inappropriate notation, although used later in the American literature,

-7 -of boiling mixtures, important for the greater part of the dynamics of gas-vapor bubbles, controls gas absorption and desorption, but is relatively unimportant for the cases of gaseous cavitation and boiling liquids containing permanent gas nucleio B. Literature Survey The dynamic phase-growth in gas-liquid systems has primarily been studied for the application to the processes of underwater detonations, cavitation, boiling, and chemical engineering unit operations involving mass transfer to or from a dispersed gas phase. As might be expected, the growth or collapse of a gas cavity in a liquid has been chronologically explored in steps of increasing complexity. The present survey of the literature is therefore conveniently divided into sections accordingly. Firstly, in isothermal bubble dynamics the sole effect of an initial pressure disturbance is considered. Next, in one-component boiling the effect of heat transport in the liquid is added, and in twocomponent boiling the effect of mass transport in the liquid is included, while this effect alone controls gas absorption. In the dynamics of gasvapor bubbles heat transport in the liquid is coupled to mass transport in the gas phase at the moving boundary. Finally, attention is given the problem of interface equilibrium when mass transfer occurs. 1. Isothermal Bubble Dynamics (4) Bubble dynamic studies were initiated by Besant who formulated the problem of bubble collapse. The solution to this problem was not given until much later by Raleigh(52) after whom the governing equation is named. A summary of the formulation and the solution to the problem of bubble

-8 -collapse neglecting the internal pressure and to isentropic bubble expansion neglecting the external pressure is given by Lamb(33) Later, Plesset(44) modified the Rayleigh equation to account for surface tension effects and Barlow and Langlois(2) included and made specific use of the viscous dissipation in the liquid in the study of the early stage of growth of gas bubbles in a viscous plastic. 2. One-component Boiling Because bubble growth in boiling is related to the rate of vapor formation at the bubble interface facilitated by heat transfer from the surrounding liquid, the initial phase of growth is a coupled thermal and dynamic problem. Considering surface tension and relating the excess pressure in the bubble to the temperature difference between the interface and the bulk of the liquid by ClapeyronYs equation, Plesset and Zwick(46) formulated the coupled problem for a uniformly superheated liquid. It was shown that the evaporation rate did not violate the assumption of interface equilibrium, and an approximate solution was obtained. The intermediate stage of growth in which the inertia terms become less important is difficult to treat. It is, however, like the initial stage of growth of very short duration compared to the asymptotic stage of growth, and is thus generally omitted from consideration. In the asymptotic stage of growth the inertia effects are negligible and bubble growth is governed alone by heat transfer in the liquid as shown by Forster and Zuber(l9) who also evaluated the magnitude of the inertia terms. For the asymptotic stage, solutions to bubble growth in a superheated liquid were obtained by Plesset (446and Zwick,47) and Zwick assuming the temperature drop in the liquid is localized

to a thin film next to the interface. Forster and Zuber(17,18,19) obtained nearly identical solutions by the use of source theory and by approximating the fluid and boundary motion by successive differential processes of convection and conduction. Chambre analyzed, by means of similarity variable, the coupled dynamic problem of fluid flow and heat transfer in a viscous liquid surrounding a growing solid of plane, cylindrical and spherical geometry. (22) Griffith analyzed the growth of hemi-spherical bubbles on a heated surface into a liquid with a linearly decreasing superheat. Bankoff and Mikesell (1) (46) Mikesell() extended the Plesset and Zwick solution to the growth of spherical bubbles in a liquid initially having a linear or exponential (9) temperature distribution. Clark, et al,, formulated a simple quasisteady model for the collapse of vapor bubbles in a boiling liquid subsequent to sudden pressurization. Collapse by condensation was controlled by heat transfer from the bubble interface into the liquid by conduction. The bubble size was assumed constant and hence convective contribution to the heat transport was ignored. Bubble growth rates in various superheated liquids have been studied experimentally by Dergarabedian(13) who found consistency with the predictions by Plesset and Zwick(46) for the asymptotic stage of growth and small superheats. Ellion(l5) measured radii and radial velocities of bubbles growing on a heated surface. 3. Two-component Boiling Scriven(55) formulated the asymptotic stage of growth for a single spherical vapor bubble in a superheated binary mixture. Growth was controlled by heat and mass transfer in the liquid phase. By means

-10 -of the technique of similarity variables an exact solution to the postulated model was obtained analogous to the solution presented earlier by Birkhoff, et alo and again used by Barlow and Langolis(2) for the similar problem of the diffusion of a gas into an expanding bubble. A significant result of the analysis was that interface temperature and composition remained constant throughout growth. The effect of finite mass diffusivity in reducing the growth rate was shown. For the special case of a one-component boiling liquid the exact similarity solution also apply, and permit comparison with the approximate solutionso Bubble growth rates in binary mixtures of water and ethylene glycol were experimentally studied by Benjamin and Westwater(3) who found qualitative agreement with the predictions of Scriven The smaller bubbles due to slower bubble growth in boiling binary mixtures as compared to those in any of the pure components constituting the mixture, have been observed in the study of boiling and burn-out in binary mixtures by van Wijk, Vos, and van Stralen(62,61,57) and by Grigorlev and Usmanov(25) 4o Gas Absorption Gas absorption from a bubble rising through a non-volatile liquid was treated by Higbie (26) based on the "penetration theory" assuming that the dissolved component penetrates only a short distance from the interface into the liquid~ As the bubble rises, the gas in the bubble undergoes a toroidal motion such that there is essentially no slip between the gas flow and the liquid at the interface, and fresh liquid is encountered at the top of the bubble at all times. Under these assumptions the transfer of gas to the liquid is treated as the transient diffusion into a

-11 -falling film, the exposure time taken to be the bubble diameter divided by the bubble rise velocity. Average absorption rates from rising gas bubbles (0.3-0.5 cm diameter) based on this theory has been verified by Hammerton and Garner(25) while Groothuis and Kramers(24) successfully predicted mass transfer rates during drop formation on a capillary tip. The presence of surface active agents caused a considerable decrease in (25) (10) transfer rates. Coppock and Meiklejohn studied bubble formation from horizontal circular orifices and measured mass transfer coefficients for the absorption of oxygen in water fom 0.15-0,3 cm diameter rising bubbles. Datta, Napier and Newitt(12) reported on carbon dioxide absorption in air-saturated water from rising bubbles formed from circular capillary orifices submerged under mercury to minimize absorption during bubble formation. Pattle(41'42) studied experimentally aeration of water by measuring the absorption of oxygen from air injected in various bubble sizes, and formulated the theoretical problem of gas diffusion into the liquid in terms of a coefficient of per cent uptake per unit length travelled by the bubble. The height through which small bubbles will rise through air free water before being fully absorped was measured and the average adsorption coefficient was determined. The efficiency was shown to depend strongly of liquid height over injection location, initial bubble size, and agitation. Danckwerts(ll) made a more general formulation of gas absorption in agitated liquids treating the phenomenon of surface renewal statistically by defining a fractional surface age distribution function. Lightfoot(37) employed the "stagnant liquid film" theory to predict the

-12 -effect of the chemical. reaction between solvent and solute on the absorption rate. Each gas bubble is assumed surrounded by a thin stagnant liquid film through which the gas diffuses in steady state to the assumed constant bulk concentration outside the filmo It is shown that only high reaction rates can change the absorption rate significantlyo 5o Gas-vapor Bubbles Lienhard(36) in a theoretical study of inception of cavitation and flashing of liquids in hydraulic equipment treats the thermodynamic stability in a superheated liquid of vapor bubbles with and without the presence of a permanent gas. He proceeds to give first a comprehensive formulation of the dynamic growth of a gas-vapor bubble containing a constant mass of gas and growing entirely owing to evaporation. Inertia, surface effects, heat transfer and liquid viscous dissipation are included in this formulation. With the attention focused on incipient cavitation the heat transfer is neglected and representative numerical approximate solutions based on the direction field method are presented. The second case treats isothermal growth resulting entirely from mass diffusion of dissolved gas in the liquid into the bubble. Neglecting liquid motion a simplified quasistatic model analogous to that of Reference 9 is postulated and the solution given. In another recent fundamental study of the cavitation process Plesset(48) compares boiling and cavitation bubbles and gives an account of the thermodynamics of oscillating gas bubbles, including the first order damping effect of heat diffusion in the liquid. Criteria in terms of diffusion lengths, thermal properties and frequency of oscillation are

-13 -given for the thermodynamic behavior of an oscillating gas bubble in a liquid. As expected, isothermal behavior describes the bubble well for low frequencies while for higher frequencies the behavior is adiabatic. For very high frequencies, however, the behavior again becomes isothermal. Mention is made of the second order rectified diffusion phenomenon, and the stability of a growing or collapsing spherical gas bubble is analyzed. Bubble growth is stable while bubble collapse is unstable. The instability is unaffected by surface tension and becomes evident when the bubble radius has been reduced by a factor of ten from its initial value. This agrees with experimental observations of the fragmentation of collapsing bubbles. For bubbles with an appreciable amount of permanent gas the decrease in size may be insufficient to cause instability. The problem of rectified diffusion of mass into a spherical gas bubble in an oscillating pressure field was studied in detail by Hsieh and Plesset(27)o This problem is identical to that of a spherical gas bubble carrying out an undamped harmonic oscillation. The second order effect of a net flow of mass to the gas phase by diffusion from the liquid phase is named rectified diffusion, and it arises from the greater average transfer area available during the expansion half-cycle, when the mass flow is inward, than during the compression cycle, when the mass flow is outward. The Mathieu equation of stability was furthermore investigated, indicating that the second mode of harmonic oscillation is most probable. The analytical expression for the rate of mass transfer was given and sample calculations confirmed the effect to be secondary. Strasberg(58) studied experimentally the pulsation frequency of gas bubbles formed on a

capillary tip and rising through a liquid~ The zero-th order pulsation frequency was found in accord with the standard frequency of an oscillating gas bubble as calculated using the acoustic approximation, and this frequency was not significantly affected by the lower frequency oscillations (second and higher order) carried out by the bubble while departing and rising. Approximate solutions for stationary gas bubble growth or collapse owing to mass diffusion in a nonvolatile liquid respectively oversaturated and undersaturated with the soluble gas were obtained by Epstein and (17) Plesset using a quasistatic model neglecting liquid motion, similarly to the model used by Clark, et alo(9) The same isothermal problem of (2) bubble growth was solved by Barlow and Langlois for the considerably more general case including the inertia of the viscous liquids surface tension, and the changing gas concentration. A numerical computer solution was carried out for the complete formulation, while approximate analytical solutions for the initial and asymptotic stages were obtained in closed form. 6. Interface Equilibrium In the analysis of bubble dynamics in one and two component boiling systems, in the analysis of gas absorption, and in general in mass and heat transfer processes across phase boundaries, it is tacitly assumed that thermodynamic phase equilibrium exists at the interface. Strictly speaking this is incompatible with the observation that a process takes place, since the driving potential in the first approximation is proportional to the departure from equilibrium0 However, if the "reaction rate" is small, the departure will be small0 This will, in general, be

-15 -the case if the rate of the controlling processes - in the above cases heat and mass diffusion - are slow compared to the rate at which thermodynamic equilibirum locally can be restored. The assumption of interface thermodynamic equilibrium has been studied experimentally by Tung and Drickamer(60) and Emmert and Pigford(16) and by Scriven and Pigford(56) who measured the rate of absorption of carbon dioxide into a liquid laminar jet of water flowing at high velocity. Comparison with analytical predictions based on the assumption of interface equilibrium justified the latter, A theoretical study by Schrage(54) based on the kinetic theory seems to support the experimental studies.

CHAPTER II THEORETICAL ANALYSIS In this chapter the general problem is stated of the dynamics of a single bubble having no translatory motion. Based on an account of the detailed physics of the problem the corresponding general mathematical formulation is given. Re-evaluating the physics of the process a simplified - mathematically tractable formulation is deduced. Solutions to this formulation are investigated for two cases. For the nonsoluble gas case, in which the liquid constitutes a single component, approximate solutions to bubble growth are obtained by integral technique for the initial stages and the theory of sources for the asymptotic stage of the transient, and the analytical results are presented in dimensionless form in a parametric study. For the soluble gas case the initial bubble behavior is predicted and a solution is derived for bubble collapse in a nonvolatile liquido Ao Statement of the Problem A spherical bubble of radius Ro containing a binary mixture of gas components A and B of uniform composition x" (molefraction of the more volatile component A) and temperature To is suddenly introduced into a quiescent binary liquid mixture of components A and B of uniform composition xo and temperature T, o The initial pressure distribution is assumed uniform throughout each phaseo Figure l(a) shows schematically this initial condition for the system. -16 -

-17 -GAS PHASE (") LIQUID PHASE (') Ti I To. T, I To l T ~I T ~~o.~ r x ___I_ __ x'. x s.. I (b) INITERMEDIATE STATE OF TRANSINTTION (t> ) Figure 1. Model for General Formulation of the Bubble TI Too r (b! INTERMEDIATE STATE OF TRANSIENT (t>o) Figure 1. Mod-el for General Formulation of the Bubble Dynamic Problem.

DEW CURVE T.. -18 -TI — l - I./ ~ I \BOILING CURVE. 0. o X0 I! MOLE FRACTION X' AND X OF COMPONENT A Figure 2. Isobars for Binary Phase-equilibrium (Schematic).

-19 -Owing to the initial thermodynamic non-equilibrium gas-liquid compositions and temperatures at the bubble interface, transfer of components A and/or B from one to the other phase will begin. The necessary heat liberated or absorped in this phase change is transferred to or taken from the liquid and gas, thereby changing the interface temperatures. This process precedes rapidly at the beginning of the transient until at the interface a state of thermodynamic phase-equilibrium is established, whereby local values of pressure, temperature and compositions become functionally related. This initial state is considered attained instantaneously since it affects only a few molecular layers in each phase. The intra-phase non-equilibrium distributions in composition and temperature in conjunction with the pressure distribution govern the subsequent bubble growth and/or collapse. The net flow of heat to the interface satisfies at all times the requirements for the instantaneous net phase change of components A and B Insight into the asymptotic bubble behavior is gained by considering a typical non-azeotropic, binary phase-equilibrium diagram shown in Figure 2. Noting that the liquid phase is considered unbounded in extent, the approach to thermal equilibrium in the asymptotic stage implies a uniform system temperature identical to the initial liquid temperature, TX, The persistance of two phase therefore requires the liquid to be at the saturation state, x*. For the general case of complete miscibility The departure from phase equilibrium owing to interface mass and heat transfer is small as discussed in Appedix I, and omitted from the analysis.

-20 -of the two components in each phase, a non-zero asymptotic bubble size can only exist when the liquid is initially at the saturation composition x* o The resulting gas composition is correspondingly xX as shown in Figure 2o For the case of an initially super-saturated liquid, no finite equilibrium size exists. From the physics of the problem it may furthermore be seen that for the case of a soluble gas, the bubble may grow initially or collapse, depending on the initial conditions and the governing parameters for mass and heat transport. Thus, if the liquid component A is very volatile the tendency for this to evaporate may exceed the tendency for the nonvolatile gas component B to go into liquid solution, and a net growth may occur initially. For the case of a nonsoluble gas the boiling curve of Figure 2 coincides with the line x = 1 e The liquid may be interpreted to be saturated with respect to the nonsoluble gas component B irrespective of its temperature. Hence, a finite asymptotic bubble size of composition X0 is approached in all cases of a subcooled liquid, TX < T* o In the limiting case, the liquid is also saturated with respect to component A, T. = T*, and the corresponding equilibrium composition in the gas phase, x = 1, is approached only by unlimited growth of the bubble. B. General Formulation The following general mathematical formulation takes into account the simultaneous processes of mass, energy and momentum transport in both phases9 all coupled at the moving bubble interface to satisfy here mass, energy and momentum balances as well as thermodynamic phase-equilibrium. Cross coupling is neglected in intra-phase transport between the transport

-21 -of one of these quantities and the driving potential for the others (Dufour and Soret effects and pressure forced mass diffusion), as predicted by the Onsager linear relations~() of the theory of the thermodynamics of irreversible processes. The liquid is considered inviscid and incompressible and the system geometry is spherically symmetrical. In the notation used, extensive properties are on a partial molal basis. Thermal and transport properties are assumed constant. 1. Gas Phase (O < r < R) Intra-phase mass transport by diffusion and convection is governed by ~ — (x) (jp"x" l/") -= "a W (pa"7 /) (1) subject to the initial and boundary conditions X"(r, O), xO' x"(Y,t)o x"(R. t) - X' where subscript s refers to the interface condition. Combining Equation (1) with the corresponding equation for component B, noting that x = xA = l-xB, the equation of continuity in terms of the molal density of the gas mixture, p", becomes p V. (ep 0 O (2)

=22 -With the assumption of constant average thermal properties the energy transport is given by — ', V~lpTY a"~'T~ i, O (5) at p"c+ " D subject to T" (r,O) - To )T"(Ot) VT"(R,t)- = T Momentum transport is given by at (v.) v = -.p/ (4) subject to the initial conditions p("(S,o)- l v" o)= o In addition, an equation of state relates the density of the mixture to pressure and temperature locally p~"C~r, t) = f (e~ T") ~(5) 20 Liquid Phase (R < r < oo) The corresponding formulation for the liquid phase, noting its incompressibility, becomes, for mass transport

-23 -t + V. = v2x' (6) X'(t, I) = x'O i'(R,t) = xs continuity V,.?' o (7) and heat transport t +. vr -- = 2, (8) T'(rO) = To T (cc)t) TO T'(R,) i) T In light of the incompressible, irrotational flow, the momentum transport may be written V () = ( (9) subject to the initial conditions

-24 -The formulation is completed with the statement of the conditions at the moving bubble interface, separating the two domains and initially subject to R (O) 1 ' 0. ~~~~~~~(10) where R denotes the interface velocity. 35 Interface Conditions (r = R) In terms of mole fluxes NA and NB of respectively component A and B transferred from the gas phase to the liquid phase the interface mass balances, considering each phase separately, become NA = (v- i-) pX5 -: P (11) NA - (V' 5 — R) q'xS a' L SDXr | (12) Combining Equation (11) with the similar form for component B, the net mole flux results NAtN (= (/-R)) (13) With the contributions from enthalpy flux and molecular diffusion the interface heat balance becomes AIA ~d 4 NE _ = NA 1 Ni, ) (14) // -;+N ' a31 Nfil Nl ' C

-25 -Including the momentum flux associated with the interface mass transfer and accounting for the surface curvature by considering surface tension, the momentum balance becomes ~"'7 - t' - ( v' v/'~) ( NAt,N) t (15) Finally, thermodynamic phase-equilibrium provides two relations (dew curve and boiling curve) for temperature and compositions (16) xs' = f (IS /o')(17) C. Simplifications of the General Formulation The following additional assumptions, discussed in detail below, are made to bring the general formulation into a mathematically tractable, yet physically meaningful, form. (i) Inertia effects in both phases are neglected (ii) Surface tension effects are negligible (iii) The gas phase is adiabatic (iv) The binary mixtures in both phases are ideal (v) The assumed constant thermal and transport properties are evaluated at the state of the bulk liquid, p, and T (vi) The thermodynamic phase-equilibrium relationship prevailing at the interface can adequately be approximated linearly.

-26 -w I// aorm~ | / 2I INCLUDING INERTIA EFFECTS ~a],~ i/ ~ --- — EXCLUDING OSCILLATIONS --— EXCLUDING INERTIA EFFECTS Ro TIME Figure 3. The Influence of Inertia Effects on Bubble Growth (Schematic).

-27 -(i) Inertia effects, primarily in the liquid phase, become important in two aspects of the bubble dynamic process as shown in schematic in Figure 3. Firstly, the initial rate of volumetric expansion of the bubble owing to mass transfer becomes finite insteady of infinite as predicted by models based on mass and heat transport alone, following the initial step-change in potential distributions. The initial acceleration of the fluids, however, takes place in a very short time (fractions of milliseconds) and may rightfully be omitted in the present case where the controlling factors for bubble growth are mass and heat diffusion. Secondly, inertia effects become important when comparing the idealized analytical model of an initially static spherical bubble shape to the actual case of the sudden introduction of the gas phase into the liquid. Owing to the initial non-equilibrium shape in the latter case, bubble oscillations invariably will occuro The sudden introduction of the gas phase from a circular capillary may thus be visualized to give rise to a predominant mode of oblate-oblong oscillation, persisting until damped out by viscous dissipation and thermal effects. Because bubble oscillations are experimentally observable, yet neglected in the analysis, an account of this behavior is given in the following for the case of undamped oscillations governed by inertia and surface effects. Assuming axis-symmetry the shape of the bubble surface may be expressed as a sum of spherical zonal harmonics, Yn ) superimposed the equilibrium shape which in the absence of gravity effects is a sphere of radius R

-28 -where E is the polar angle and Cn the cyclic frequency of the n-th harmonic~ Based on a stability analysis Hsieh and Plesset(27) showed that the second zonal harmonic is the most unstable mode of oscillation, hence, the dominating. Because the second zonal harmonic is identical to the Legendre polynomial of first kind and second order, the first approximation for the bubble shape for the case of an undamped harmonic oscillation with cyclic frequency cu and polar amplitude a may be written i L l (SC) 3C _ 1 iwt ] (19) The frequency, f = c/2T, of this mode of oscillation may be obtained from hydrodynamic considerations, showed by Lamb(33), as f: I,- R (20) and the polar amplitude, a,is given from the initial conditions particular to the injection process. A dominating mode of oblate-oblong oscillation having a frequency consistent with Equation (20) has been experimentally evidenced. The similarity of this mode to that given by Equation (19) has furthermore been substantiated by comparing from experiment the magnitude of the relative polar amplitude a/R to the magnitude of the amplitude of the relative equivalent radius Req/R, where Req is the radius of the sphere having the same volume as the deformed bubble. The analytical relationship, obtained from Equation (19) as shown in Appendix II, is

-29 -- [ 5 (R) Sint + 7 (R)) Sin-3t (21) As discussed in Chapter III under the evaluation of the experimental results the magnitude of the relative polar amplitude a/R varies between experiments in a random manner since it is determined by the initial displacement during formation. Typical values range from 0.01-0.10 in the first part of the transient damped out to 0.001-0.05 in the later and asymptotic stage. For the mode of oscillations of zero-th order (spherical oscillater) the analytical study by Hsieh and Plesset(27) of rectified mass diffusion indicates that amplitudes comparable to those in the present study have a negligible perturbing effect on the mean bubble size over short periods of time. Thus, even for an initial bubble size of 1 mm and a relative pressure amplitude of 0.25 the time for doubling of the bubble radius exceeds 106 sec. In light of these results and owing the experimentally observed small amplitudes of oscillations in most cases, only the zero-th approximation, r = R, to the bubble shape will be considered in the present study. Whenever the oscillations in shape play a small role for the heat and mass transfer processes the solutions thus produced represent the time history of the equivalent mean bubble radius. (ii) Surface tension effects are disregarded in the present study because only visible (macroscopic) bubble sizes are considered. (iii) For relatively slow transients as well as for low frequency oscillations, as pointed out by Plesset(48) the thermodynamic behavior of the bubble is isothermal In view of the comparatively small change in inter

-30 -face temperature experienced during the transient and because of the low heat capacity of the gas phase, it is appropriate to neglect heat transfer to or from the gas hence to consider it adiabatic. This assumption implies that only cases for which the initial gas temperature is near the liquid bulk temperature are considered. (iv) Ideality of the binary system implies that the heat and volume change of mixing are zero and that properties for the mixture can be obtained from those of the pure substances. When experimental phase-equilibrium relations are not available, these can thus be expressed analytically in terms of system pressure, poo, and respective vapor pressures, p*, for the pure components A and B (see e.g. Dodge(14) Eqs. (XII.3) and (XII.4)). X A' ( T.l ) = — /9(T) (22) - vPa(r)(22) (T ) - ~(T) (23) Ideality in the liquid phase is generally expected when the components have similar chemical structure and in particular for dilute solutions. A requirement for considering the gas phase ideal is low pressure and high temperature of the system, and particularly that its state is far removed from the critical state of either component. (v) The evaluation of thermal properties at the bulk liquid temperature primarily introduces an error in the heat of phase transition. This error is small, however, during most of the transient because of the small temperature difference between the bubble interface and the bulk liquid.

(vi) Linear approximations to the dew curve and boiling curve of the phase-equilibrium relations are adequate for interface temperatures in the neighborhood of the liquid bulk temperature. With the above assumption the bubble dynamics is considered governed by mass diffusion in the gas phase and mass and heat diffusion and radial convection in the liquid phase. The general formulation is accordingly simplified as follows. io Gas Phase (o < r < R) With a constant molal density p" of the gas mixture, the equation of continuity, Equation (2), infers that the gas phase is stagnant and assumptions (i) and (ii) then implies that the pressure is uniformly p, throughout the two phases. In terms of the potential difference, x"-x, intra-phase mass transport from Equation (1) becomes a I -f~r2 r x xj (24) subject to x"1 (r,O)- x = O -r = 0 -"(R,t)X = - ' 2. Liquid Phase (R < r < oo) In terms of the composition difference, x'-x1 9 Equation (6) for spherical symmetry becomes

-32 -cat ) ~r ~r )r = 1 -2r r-. (25) dr 1~a subject to -x' (r,) - 0 X4 - x'(0,t) = /a- t'(Rt) - CxO-A where v+(r,t) is the radial liquid velocity, satisfying the continuity equation, Equation (7), now written r2 a (r =0 (26) Omitting the notation ('), the heat transport from Equation (8) may be written ( Too _ _ _ _r__ _z O - T = < d( c [r 2 (T - T) ( 2 7 ) C r(27) subject to / Tz- T(r,O) - 0 To- T(0,t) = O To - T(R,t) - Tx-Ts

-33 -3. Interface Conditions (r=R) Noting that the gas phase is stagnant, Equations (11), (12), and (13) for the interface mass balance simplify to NA = -R "x" - p (k,t) ~NA ellp CZ (28) NA = r(R,) (29) NAN = - " =- [ (R)t)-Re' (30) The energy balance of Equation (14) with assumptions (iii) and (iv) may now be written kA,A '"8 J je, k (31) where hf indicates the heat of evaporation of the pure component, ideng tical to the change in partial molal enthalphy when going from liquid to gas solution because the heat of mixing is zero. The phase-equilibrium relations of Equations (16) and (17) simplify to xs" = ("1$)S p) (32) xs e f(Thse l) (33) where the functional relationship is linear and p~o is constant.

D. The Case of a Nonsoluble Gas The case in which the gas component B is nonsoluble or has a negligible solubility in the volatile liquid component A, represents the particular case of most interest for the application to injection cooling. The bubble dynamics is then governed by mass diffusion in the gas phase and heat transport in the liquid phase. For simplicity the notation (') and (") for respectively the gas and liquid phases is left out where not needed for clarity. lo Solutions Although the physics of the problem has been simplified considerably in the formulation given above, an exact solution to this is not readily obtained. The mathematical difficulties may be seen to arise from the following characteristics of the problem. The transient is sought for the two spherical domains, 0 < r < R and r > R of which one is semi-infinite, the other finite. The two domains are coupled at the moving interface, r=R(t), where the time-wise change of the potentials for transfer is in turn coupled to the interface motion. It is worth noting that similarity solutions exist for the chosen model either for the case of one dimensional plane geometry(34) (i.e., two semi-infinite domains as shown in part E. 1. of this chapter), or for spherical geometry for the case of initial zero bubble size(55)o In the present case, however, similarity solutions can be shown not to exist, primarily because of the finite, initially non-zero, domain of the bubble interior, but also because of the arbitrary phase-equilibrium relations imposed at the bubble interface, even when these relations are assumed to be linear.

-35 -For large values of time an asymptotic solution is obtained using source theory under the quasisteady approximation of uniform gas composition. The solutions for small and intermediate values of time, however, are obtained by the use of the mass and energy integrals of the governing differential equations employing one-parameter parabolic distributions of mass concentration and temperature in respectively the gas and liquid phases. In applying the integral technique to a problem, suitable profiles for the dependent variables are assumed, satisfying the initial and boundary conditions, while the governing differential equations subsequently are satisfied only in their integrated forms, that is on an average over the interval in question. First introduced by Pohlhausen(49) this method has found extensive use in the solution of boundary layer problems of the viscous fluid flowo The similarity between the boundary layer concept of these problems and the concept of thermal boundary layer growth in transient heat conduction problems have lead to an increasing use of the Pohlhausen technique. Thus, Yang (63) employed the integral technique to a heat conduction problem taking into account the effect of variable thermal properties, and Goodman(21) and Poots(50'51) applied it to problems of solidification. It should be mentioned that one of the main advantages of the approximate integral technique lies in its ability to handle analytically difficult geometrieso Inherent nonlinearities of a problem, however, are not eliviated by the method. The usefulness of applying the integral technique to a problem of spherical phase-growth is illustrated in Appendix III for the case of bubble growth in one-component boiling having an exact solution. The profile for mass concentration and temperature distributions are chosen in accordance with the physics of the problem. Referring to

-36 -Figure 4(a), the shown initial conditions suggest the transient growth of a concentration and thermal boundary layer from the interface into respectively the gas and liquid phases during the first part of the transient. For the first time domain, 0 < t < tl, terminated when the composition boundary layer has penetrated to the bubble center, the profiles are therefore suitably expressed in terms of the time dependent penetration depths, fT(t) and fx(t), as shown in Figure 4(b). The parabolic one-parameter forms are TCO - Ts T -:/j: () - y/c~) ( X -Xo ~~ u /;) (35a) satisfying the following set of boundary conditions TX-T(O),t)- O x(-fx,)-X = X f 0 -y y where y=r-R is the independent space variable with respect to the interface. In the second time domain, t > t1, the gas composition at the bubble center, xc, becomes affected by the mass diffusion process as the transient proceeds and the uniform equilibrium composition, x,, is asymptotically approached throughout the gas phase. With xC(t) as a suitable

-37 -GAS PHASE (COMPONENT A AND B) LIQUID PHASE (COMPONENT A) xT r (a) INITIAL CONDITION (t O) x o I I cx r,T I T R(t) (c) INTERMEDIATE VALUES OF TIME (t tI1) Figure 4. Composition and Temperature Profiles. Nonsoluble Gas Case.

-38 -parameter function for this time domain, the parabolic profile for composition is written x = XC t (X5-Xc)( +R) (35b) satisfying the following set of boundary conditions = 0 x (-Rt) x to and coinciding with Equation (35a) for t = t1, when Xc = -X and fx = R. Noting that the need for two different profiles for the composition in the gas phase was prompted by the finiteness of this domain, it follows that the temperature profile of Equation (34) may be continued unaltered in the second time domain. Before proceeding to obtain solutions for small and intermediate values of time, the differential formulation is further reduced. Thus, integration of Equation (26). using the liquid velocity at the bubble interface, vA, as reference gives 2 r R (r)" (36) Combining Equations (36) and (30) a relation is obtained between the liquid velocity at R and the interface velocity tR = (t1 - em) R= e (37)

-39 -where C is referred to as the density deficiency. The liquid velocity distribution then becomes,iii = ER ( (38) With the use of Equations (30) and (38) the interface mass balance for the gas phase, Equation (28), may be reduced to, d x (t,) (39) and the interface heat balance, Equation (31) becomes / 'C' _T(,) (4) where now hfg represents the heat of evaporation for component A Because component B is nonsoluble in the liquid phase, the interface equilibrium relation, Equation (22) reduces to Xs = (41) In summary the differential formulation is now given by Equations (24), (27), (38), (39), (40), and (41) subject to the initial condition R(O) = Ro Referring to the discussion of the asymptotic bubble behavior under the statement of the problem (Section IIA) the finite asymptotic bubble size, R, may be related quantitatively to the initial bubble size, Ro, the initial gas composition, xo, and the final equilibrium composition, x0(T0), by continuity consideration. The result is

-40 -(R)3 /-xo (42) a. Solution for Small Valuesof Time Substituting the radial liquid velocity from Equation (38) into the energy equation, Equation (27), subsequent integration over the thermal (29) boundary layer, R < r < R+fT, and using Leibnitz' rule for the differentiation of integrals with variable limits, gives R+fT RTf, Rff R+fT -d J(TiT)r~dr (r-T)r| + RR~ (T- T)I r2 ((,-T) ( act R T)d - fT-T) r R (43) Similarly the integral over the mass concentration boundary layer, R-fx < r < R, applied to Equation (24) yields dl( ar = - (44) R- x d R-fx R-f Introducing the profiles of Equations (34) and (35a) into Equations (43) and (44), noting that r = R+y, and performing the spatial integration, gives with the use of Equations (39) and (40) (T 2 t2(fr~ 1' — 6)(T-T$)] d (1:() t { -s f ) 2dtdt (45) dtd 2r R + dt 6(R)

-41 -Introducing the profiles Equations (34) and (35a) into the interface heat and mass balance, Equations (40) and (39), these reduce to R R 2z f" xI- ((7) XRs 2 $ X0EXo R (48) Dimensionless quantities may now be introduced to facilitate a clear evaluation of the governing parameters. The use of a linear approximation to the equilibrium relation, Equation (41), suggests that the expression (6T/6x)Toop be applied as a convenient and meaningful reference temperatureo Introducing Equation (41) into this expression, it becomes b7\ Xr tdT) (49) The initial bubble radius, Ro, being a natural reference length, the dimensionless quantities may then be summarized as oIt Xz~~~T - 7 & = R' (50) PO (d/d)TO 5 Le = - l LI,. -- -

-42 -where Ja is the Jakob number and Lu the Lukomskiy number.1 The linear form of Equation (41) then becomes X a 0s ( K- x) (51) representing the closest approximation to the dew curve in the neighborhood of T (i.e., G=1), the value approached by Ts as t -4 oo 00 Introducing the dimensional quantities of Equation (50) the fomulation for this time domain of dimensionless bubble growth, P = P(T), in terms of four governing parameters, Ja, Lu, xcx and xo may be summarized to -3 3 it el. { Ja (4-6) P Y it 1J = A( J -3 /.:'' { ( X-s-,) l() ("2() i (v)() (52b) P ~~P 2 LJ t (1Q)) (52c) 1sI= 0-(''- Xoo) See footnote on page 6.

subject to P(O) = 1, fT(O) and fx(O) = Oo A solution to this nonlinear formulation may be obtained for small values of time when 'R and ) (gT) 2 and (f,{) 2 (53) that is, when the terms 1/2(R/f) and. 1/10 in Equations (52a) and (52b) can be neglected compared to the term (R/f)2. Physically this corresponds to omitting only the curvature effect in the terms for the time-rate of change in mass and energy over the respective boundary layers, hence the conditions of Equation (53) may be undue stringent. It may be added that in most cases the transport properties for respectively mass and heat diffusion are such that fx > fT. Hence, in view of the definition of the time domain under consideration the criterion for validity of the subsequent solution may be simplified to >X > I (54) With this limitation the solution leaas zo constant interiace potentials, xso and Gs, and integration is readily carried out leading to a cubic equation in bubble size P o The explicit solution of this, satisfying the conditions f > 0 for t > 0, may be written x P(T) t 2 A s - Arcos( -Aor)2 (55 where the constant A0 is given by

A0 = At (4L)2 (56) ~ 1 (4- ~)cJa1 (1 -XO)) Equation (56) is also the implicit equation for determing the initial interface condition (Xso, 0so) subject to Equation (52e). The criterion for validity, Equation (54), may now be written explicitely in terms of this solution as P3- 1 < a- X0 (57) -- XSO b. Join Point (T = T1) To continue the solution from the first time domain, 0 < T < T ' into the second domain, T > T1, the join point P1 = P(T1) must be established. When at T = T1 the concentration distribution has penetrated to the bubble center, R/fx = 1 and the state of the transient is evaluated by simultaneous solution of the five equations of Equation (52). Because in most cases not close to the critical state (,-E) 3J (I-ao) ~ d (58) particularly owing to the smallness of the density ration, 1 - C, this term is omitted, and Equations (52a) and (52b) can be integrated. Rearranging the result, the five governing equations can be summarized to

-45 -JQ (i# -&) =ir( R) ( T)+ 2 o lo C (59a) Ksf ~('I 5 4_ XO ) (59b) pX -- (- iy1 ) (59e) While Equations (59) readily give P1, P1, (R/fT), sl and xsl the exact value of time T1 can only be obtained from exact integration of Equations (52), An approximate value may, however, be obtained from the solution for small. values of time, Equation (55), by extrapolating its validity range to fx = Ro The result is 1 -e (~ P (60) c. Solution for Intermediate Values of Time (T > T1) In this time domain the temperature profile of Equation (34) is unchanged implying that the form of the energy integral remains that given by Equation (52a). Introducing, however, Equation (35b) for the concentration profile, the integral of Equation (44) over all of the gas phase in dimensionless form gives

-46 -T { p ( 3 XC tr 5 k5) _ (p3) (61) The interface mass balance, Equation (39)9 accordingly becomes *Xs -X PP - 2 LX d X; (62) The nonlinearities of this formulation, Equations (52a), (61), (52c), (62) and (52e), again restrict the solutions that may readily be obtained. With the limitations R(') ~ 2 and (1-E) JR (e-es) 4< (63) however, Equation (52a) can be integrated despite the time varying Gs o Integrating also Equation (61) and rearrranging, the five governing equations may be summarized to jo (d- i..(. f P-4 (64a) Pi(42X -S- 3 X) =4- x. (64b) PP =2 Ja (64e) PP = 2- L -X c d (64d) xs = is - ~(bxy) (64e)

-47 -Eliminating the four variables, (R/fT), s' Xs and xc, Equations (64) are reduced to a cubic equation in the variable U = PP/Lu U t (- - A ) + (25+ t) - 50 = - (65U) U3 ~ (l- A)Z + ~ ~(65) where K= Ja/LC M = (x-xO)- ( x)(4-/P3) ( (66) N. I - 4/p3 A numerical evaluation indicates that over a wide range of values of the four parameter, Ja, Lu, x,~ and xo, and variable bubble size P, Equation (65) may be linearized by retaining only its two last terms. As shown in Appendix IV, the criterion for the validity of this approximation may be written KM L50 KL + (SO-2 N )(25+ 20 L)] (!J KLI"I 31 <~ '(67) (25 + Zo0 K)3 Because of the complexity of Equation (67) the limination on the variable bubble size P in terms of the governing parameters is not readily obtained explicitely. Instead, the inequality Equation (67) need be checked as the calculations proceed. Retaining the last two terms of Equation (65) the following differential equation results

-48 -_p = K( P _ ) (68) 2 1 ( where for simplicity P3 = (l-xo)/(l-x.) according to Equation (42) has been substituted. Separating variables and integrating from the. join point T = T1 to T > T1 gives explicitely time as function of the more conveniently chosen bubble size w = P/Po (i.e. wo = 1/P ). Separating the distinct contributions to elapsed time for a given bubble growth from respectively the process of mass and heat transport, the result may be written T - - (' ' (I-,C e (69) where ~CH 2 JaY ( f x,)Z wo8 2 am rna LL wv2 and IH =2 (-w4) + 3 (X -W) (- W3 4- -W - 4 ( W-23) (Z I) W1- [5('t rc - Arctg W 4 (6-2di~h j~W4 -t 'I iT

-49 -Im = - ~_ (W2w 3 f ( w - r )i3 - Ag rctg 211 / 4+) Noting that the rate of heat transport depends on Ja and the rate of mass transport depends on Lu, the terms CHIH and CMIM are recognized to represent the contributions to elapsed time beyond the join point T = T1 from respectively the heat and mass transport mechanisms. A measure for the relative significance of heat transport to mass transport for retarding bubble growth is therefore obtained in the ratio CHI 2 f (W)w1,, W) (70) CM IM J The function f(wo,wl,w) in Equation (70) is monotonous and increasing as w increases. Hence, the significance of mass transport for the transient decreases and heat transport becomes the controlling factor as the asymptotic stage is approached. The latter observation supports the choice of a quasi-steady model for the asymptotic stage of growth, based on the assumption of uniform composition distribution in the gas phase. d. Asymptotic Solution For large values of time the composition distribution in the gas phase approaches uniformity and growth of the bubble is primarily controlled by heat transport in the liquid phase.

-50 -The quasisteady model postulated for this time domain assumes a thermally insulated spherical gas phase having uniform composition x growing at the rate at which the heat of evaporation can be supplied to the bubble interface by heat transport in the liquid phase. The gas phase is ideal as previously assumed, and thermodynamic equilibrium at the bubble interface provides a unique relation between temperature Ts and gas composition x o The average composition, x, is directly related to bubble size, P, and initial composition, xo, X-l = _ - XO (71) P3 as seen from continuity considerations or from Equation (64b) for s = Xc = x Hence, a unique relationship exists between interface temperature and instantaneous bubble size. From Equations (42), (51), and (71) this relation becomes I- O (- )( P.- 1) (72) An approximate solution to the model outlined above is obtained by first formulating the heat transport problem with the aid of the theory of heat sources, and at the bubble interface imposing the condition of Equation (72). Next, this formulation is linearized to facillitate integrationo The formulation of the heat transport problem follows an approach similar to that introduced by Lightfoot (reviewed page 293 of Reference 7)

and used by Forster(18) for the related problem of spherical phase growth in one-component boiling. The approach takes its start in identifying the heat liberated or absorped at the bubble interface, associated with the interface mass transfer, as a uniformly distributed surface source or sink. In the present case, for a bubble of size R growing at a rate R, the instantaneous spherical source, q, on its surface has the strength q = -hA) NA 4R2 - 42R " (75) Next, the solution to the equation of linear flow of heat in the domain bounded internally by and insulated at the sphere r = R is written for the case that the domain is at zero temperature prior to the release at t = 0 of a unit instantaneous spherical surface source on the sphere r = R o The result (Equation XIV-16 of Reference 7) is the Green's function for the problem in question. Evaluating this at r = R the temperature, Ts, of the bubble interface can be written = 47TR2 R7 L 'R exp( (V) erfc(t)] (74) The solution for a time varying heat source of strength q(t) is obtained by convolution in the time domain, 0T = 2 R2_ (t-C) expa [ tU(t -ter (75)

-52 -Forster(19) demonstrated how this result could be extended to account approximately for the interface motion by considering successive differential processes of linear flow of heat (conduction) and radial convection. Thus, resulting from the instantaneous differential heat release, dq(tv ) at time tI, the bubble grows in size from R(t') to R(t) during the time interval t to t o This sequence of events is then repeated. The corresponding differential change in interface temperature, dTs, is obtained again from the Green's function for the linear flow of heat, but now accounting for the change in bubble size by selecting an intermediate value of radius, R(O) where t' < < t. Forster(l9) evaluated R( ) approximately as R(I) = Rlti) P(L) (76) Summing up. the differential contributions over the time interval (O,t) Equation (75) with R = R(S) results. As discussed by Forster(l9) the second term in the integral of Equation (75) is small when the contribution to the differential change in surface temperature is large. By the time the second term becomes important the contribution to the surface temperature change is small owing to the first factor in the integral. Retaining thus only the first term of Equation (75) and substituting the source term from Equation (73) the result may be written T- S - - (R') 7( i) _t (77) t'~t) /' —'T S L., /., i.

-53 -when the inital uniform temperature is T o Nondimensionalizing according to Equations (50) and using for convenience the notation * for the convolution, Equation (77) becomes 2 Jal P(1-&s) = (P)* 5 (78) Equation (78) is linearized by approximating the left hand side by the form 2 Ja P (I-i) b (a - p2) (79) where the constants a and b are chosen to satisfy Equation (72) for P -- 1 and P = P Hence 00 a= P2 (80) b= 2J- P. c (81) The approximation Equation (79) now written _-65 ~ i P- -P (82) Xo- Xo P Pz -1 is compared to the similar form of Equation (72) -is, Pi - P3:CV_ -X 0 P poll~~,- 1(83) xi-xo Pd P3 (81 in Figure 5 for different values of the parameter P o Representing

- EQ. (83) - EQ. (82) P i1.2 1.5 2.0 DIMENSIONLESS BUBBLE SIZE P2 DIMENSIONLESS BUBBLE SIZE P Figure 5. Approximation to Interface Potential for Asymptotic Bubble Growth.

-55 -the normalized thermal potential for bubble growth the quantity (l-s )/(x -xo) is seen to approach zero as the bubble size approaches its asymptotic value, P. o The approximation is good in the asymptotic stage, particularly for small values of P. o The linearized form of Equation (78) is operated on by Laplace transformation giving the closed" form solution pm()= V p2_ - (Fz).6xp[bZ(u-To)]erwC(b bTo) (84) where the hypothetical inital condition, P(TO) = 1, is chosen such that Equation (84) satisfies P(T2) = P2 at the join point T2, beyond which this asymptotic solution is valid. Analogous to the continuation of the solution valid for small values of time at the join point T = T1, the join point T = T2 is approximately obtained by extending the intermediate solution to the value P2 which justifies the use of the quasisteady solution, Equation (84). The criterion for determining P2 is obtained by comparing the interface compositions xs, assuming a distributed gas phase, to the average composition, x, given by Equation (71). Although in principle arbitrary, a maximum deviation of 0.5% has been chosen suitable for the criterion, which then gives x( - P o.0 (85) A (P2) The interface composition xs for the case of a distributed gas phase is calculated from the intermediate solution, Equations (64a), (64c), and (68),

-56 -to Xs = XOO - ______________. ___ 14-X0 (86) r 5 1 /P3 X t/P3 p: which is seen to reduce to Equation (71) for Lu -4 o Having established P2 from Equation (85) the corresponding approximate time T2 is calculated from Equation (69). 2. Discussion of Analytical Results - Parameter Study The form of the analytical solution is shown in Figure 6, where the results of Equations (55), (69), and (84) are plotted in terms of relative dimensionless bubble size Q = (P-l)/&,-l) vs T for the case Ja = 100,, Lu -= 100, x = 0O9 and xo = 0O The graph shows the joining of the solution for small values of time, the intermediate solution and the asymptotic solution at respectively join points T1 and T2 o The solution valid for small times, being less important, is omitted in the subsequent four plots. Figures 7, 8, 9, and 10, respectively show the irnfluence of the parameter Ja, Lu, x, and xo, while the remining three parameters are kept constant. It is particularly worth noting that for the representation chosen, xo is a parameter of very weak importance0 This observation justifies the omission of x from consideration in determining the growth rate. However, decisive in conjuction with xX for the asymptotic bubble size (cfo Equation (42)) xo remains an important overall parameter. The effect of initial bubble size, Ro, on the rate of bubble growth is reflected in the dimensionless time parameter T o As expected

1.0 If iQ,, JOIN POINT t2 z 0.4 J 100 x 0.4 -' Lu = I00 LrliIX~ = 0.9.E Xo= O. a0.2 t JOIN POINT ri;I.1 0 5 10 15 20 DIMENSIONLESS TIME IOCT= 10 dt/RO Figure 6. Analytical Solutions for Relative Dimensionless Bubble Growth. Solutions for Small and Intermediate Values of Time and Asymptotic Solution. Nonsoluble Gas Case.

1.0 0. -I 0.8 (:13 cn rn 0 052 Cn z O ----- DE Lu 100 _ Xo= 0 a 0.C~~~~~~~X2 Xw= 0.9 -J O c, 0 0 5 10 15 20 DIMENSIONLESS TIME IOT-_ Io 0't/R0 Figure 7. Parameter Study, Q(Ja,T). Intermediate and Asymptotic Solutions. Nonsoluble Gas Case.

1.0 - Cr) o Go Islim Or) "' Zi w o 0.4 Ja =1 00 U=0 z W Xo0= 0.9 C3, 0.2 I-J z I I cr 0.4J 0 0 5 10 15 20 w XCO= 0~~~~~~~~~~~. 2 DIMENSIONLESS TIME T 103 = 103a t/R0 Figure 8. Parameter Study, Q(Lu,T). Intermediate and Asymptotic Solutions. Nonsoluble Gas Case.

m V LO.aL I a', II -, X a. CE] I I Lu 1~~~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~00 1 0.6 09 oo w -, o-..] o |E Z 04. 0 z Jo = 50 w Lu 100 a0Q2 ~ ~ ~ ~ DMNSOLS X= l0 w w 0 5 10 15 20 3 3.2 DIMENSIONLESS TIME lOT'' IOa:t/R. Figure 9. Parameter Study, Q(X.,T). Intermediate and Asymptotic Solutions. Nonsoluble Gas Case.

0. I0 aP w N. 0.8 C,) ~~~0.6 ~ ~ ~ ~ 0 CD C,) z O 004 H4 Z JO =50 w M II I Lu:100 0.2 Xca 0.9 wa. Cr 0 0 5 10 15 2 3 3 2 DIMENSIONLESS TIME IO T= Iacit/R2 Figure 10. Parameter Study, Q(x0,T). Intermediate and Asymptotic Solutions. Nonsoluble Gas Case.

-62 -the growth rate increases as Ro becomes smaller, As seen from Figures 7 and 8, the effect of increasing the Jaand Lu-numbers lead to increased growth rates. The case Lu -4oo is represented by the asymptotic solution throughout. Over the range of parameter values of practical interest the influence of increasing Janumber is more significant than that of increasing Lu-number, in accordance with the relative significance of these two parameters as expressed by the characteristic group Ja /Lu o For fixed initial gas composition, xo, as the liquid approaches the saturation state (i.e. xo approaches unity), the rate of relative bubble growth decreases, as shown in Figure 9. The growth rate for the actual bubble radius for constant initial bubble size, however, increases as xo becomes greater, because the normalizing asymptotic bubble size, PO, also becomes greater. The net heat transfer rate from the liquid consequently also increases. Another useful representation of the transient bubble growth process in terms of a pseudo time constant is shown in Figures 11 and 12. The time for 63.2% (= 1 - l/e) completion of the relative growth is plotted versus equilibrium composition x for particular values of the set of significant parameters Ja and Lu, and xo = 0. In the dimensionless representation of bubble growth derived in this section, the combined property group, Ja2/Lu, appears whenever the simultaneous processes of heat and mass transport are considered, In the solution for small values of time and for 6 = 1, as assumed in the subsequent solutions, the initial state at the interface is determined by Ja /Lu as seen from Equation (56). As this property group increases, the

-63 -O - 14 a 0 Lu.: 100 -r- 12 -X: 0~~~X 0 II-._J w c. 8F 0 CM I — o Z 6 C,) C, w 0 z w 0 0.2 0.4 0.6 0.8 1.0 EQUILIBRIUM MOLEFRACTION Xm Figure 11. Time Constant for Relative Dimensionless Bubble Growth vs x. and Ja. Nonsoluble Gas Case.

-6 -o 14 SI )o 0 Ja 100 12 X- 0O )I I0 1r - I0. LJ U.I N 0 w 8 -- z 2 0 "' L U) Z 24 - v o~ z 0 0 W CP 0 0.2 0.4 0.6 0.8 1.0 EQUILIBRIUM MOLEFRACTION Xoo Figure 12. Time Constant for Relative Dimensionless Bubble Growth vs xc and Lu. Nonsoluble Gas Case.

-65 -driving potential, 1-eso, for heat transport decreases, while the one for mass transport, xso-xo, increases. The relative significance of the two mechanisms in determining the rate of growth is therefore shifted in favor of the mass diffusion process, although the growth rate increases when either the Ja-number or the Lu-number becomes greater. The same result, now in addition a function of bubble size, was found in the solution for intermediate values of time, as expressed by Equation (70). In the limit as Ja2/Lu becomes either very large or very small, the growth rate is controlled respectively by mass transport or by heat transport. E, The Case of a Soluble Gas The injection of a soluble gas into a volatile liquid -having little application for the injection cooling process- is treated in less detail in the present study. First, the initial behavior of the bubble is predicted by neglecting the curvature and considering the gas and liquid phases as semi-infinite domains. The results predict whether the bubble tends to grow or collapse in the very fist part of the transient. Secondly, by the use of source theory, the approximate solution is derived to the bubble collapse governed alone by heat and mass transport in the liquid phase. This represents the limiting case of a soluble gas bubble in a nonvolatile liquid. Compared to the case of a nonsoluble gas, the present general case is increased in complexity. Two additional parameters enter the problem, namely the initial liquid composition, x, and the property for mass diffusion in the liquid through the dimensionless group, Lu1= l'/ao

-66 -Furthermore, the density deficiency, E = l-p"/p, cannot be taken to unity as for the case of a nonsoluble gas, because this would violate the interface mass balanceo The added degrees of freedom are reflected in the possibility for initial growth or collapse, eventually followed by collapse in all cases of a nonsaturated liquid~ lo Solution for Small Values of Time Neglecting the curvature effect and considering the gas and liquid domains semi-infinite the formulation given by Equation (24), (25), and (26), in terms of the independent space variable, y, measured from the interface in the direction of the liquid phase, becomes (a) Gas phase: x, x "= 0 '0'" c)T (',x" - x o")(87) d: dy ay2 f "(-0,t)- o': o; "(O t) - Xo X: -X (b) Liquid phase:,, _ d(-x'),)a (x-k 9 At +~ -x) _ _ = ~211 (88) a~d~ ~ ay dy2 I XI -x00' () i) -~0 0)

-67 -(T_- T) + a S(T_-T), d2(T,-T) (89) at ay dy2 Tc - T'(7-,t)= 0 oT,-T(CO) 0 Too -T(O,t) = s where the assumption of plane geometry and incompressible fluids imply that the gas and liquid velocities, v" and v', are spatially independent. (c) Interface conditions: The mass balance given by Equations (28), (29), and (30) becomes N~ ~,t N vYIt -, (90) A, +NB = 411"" QZ~ Xs - Xs where the moleflux to the liquid phase of component A and B expressed in terms of the composition distribution in the gas phase is respectively NA - e y s(91) 'and N8 = "." ({-X') + e"",y (92) The heat balance, Equation (31), simplifies to NAf hrA t NB bh = k' a

-68 -Interface equilibrium is given by Equations (22) and (23). Introducing the Henry law constant, KB = (l-x')/(l-x"), which for dilute solutions of component B may be calculated directly from Equations (22) and (23) to Ks -(94) the linear phase equilibirum relations become '".-....... - (() (95) X = 4 - K (I-x5) (96) where dimensionless temperature, e, is defined by Equation (50). The formulation given by Equations (87), (88), and (89) of onedimensional mass and heat transfer with convection is satisfied by a set of similarity solutions (see Bird, Steward and Lightfoot(5), Example 19.1-1) in such a way that the interface potentials, xs, x6 and Ts, remain constant during the transient. Each of the partial differential equations is reduced to an integrable ordinary differential equation of second order by introducing the independent similarity variable, Y = y/fgt and by expressing the gas velocity by ~ ' =, i> (97) hence by continuity, the liquid velocity by the corresponding form u" = (I.-a:),; ' (98) t

-69 -In Equations (97) and (98) X is the phase-growth constant, to be determined from the interface conditions. Thus Equation (89), for example, reduces to the form dY2(ze) t [ Y _(-e~)2] j (0) 0 (99) subject to 1-0 10 - for Y= 0 0 for Y = a where dimensionless temperature has been introduced according to Equation (50). Twice integration of Equation (99) gives the solution to the transient temperature distribution in the liquid,i -_ 4~ - erf [Y/2-~-E)2,]~ (10D) '1-is,I. erf [(4-E)A] Introducing the remaining dimensionless quantities from Equation (50) with the addition of the Lu-number for mass diffusion in the liquid phase, Lu -, (101) Equations (87) and (88) integrate to x-x'i- er( L Ad)J (102) -os X + 1 r [(4-E) lI ef [ ~~-

-70 -and Y2 - Ao x 1-xb 1 erf(') (103) Substituting from Equation (100), (102), and (103) into the equations for mass and heat balance, Equations (90) and (93), these become F = 'exS -x( {lE (104) and 1-0s __ _ X//-X//__ X77 Jet (105) X' exA (,.)2erf ) FF exp L(1,-)'2 {' X er" a(I-)]} EiF i CXP + e where the ratio of heats of phase transition is given by F = hfgB/hfgA Equations (104) and (105) in conjunction with the condition of phaseequilibrium at the interface determines the growth constant X and the state at the interface. Introducing the linear approximations to phaseequilibrium, Equations (95) and (96), and eliminating x1, xv and e s S S a single implicite equation in X results, fLW 44 {C} r lLUiArj A (A t (106) + - I ~' (,t-~} - J ~ _- (___. [4 f) - 1F __ +_ where for simplicity the functional notation, E() = exp()2erfc(), is.used~ The interface temperature is accordingly given by

-71 -8/-xi- F + - 1-F c1 {l Ja E{-(_)>A E 7zOs -J.... (107) and the corresponding interface compositions are subsequently obtained from Equations (95) and (96)o As seen from the definition of the phase-growth constant X by Equation (97), X > 0 implies initial bubble collapse, while X < 0 conversely indicates a net flow of gas away from the interface, hence growth of the gas phase. Applying the equation of continuity to the incompressible gas phase of a bubble, its dynamic behavior for small values of time may be approximated by P(r) = - 2AJ r (108) That X may take either positive or negative values is seen from Equations (104) and (105)o Thus for large values of Ja2/Lu and 1 l-ua 1 N~ju?/Lu initial growth. may be anticipatedo Setting the growth parameter in relation to the Ja-number by introducing the ratio x/Ja into Equations (106) and (107), the Lu-numbers always appear in the significant groups, Ja2/Lu and Ja /Lu' As found for the case of a nonsoluble gas, the simultaneous processes of mass and heat transport are governed by these characteristic property groups, giving the relative significance of the two mechanisms in determining the rate of phase change~ Figure 1.3 shows a plot of X/Ja vs Ja/Lu for Ja,/Lu = 1000 and Ja/Lu = 100 with x" as parameter, obtained from Equation (106) for a representative set of values of the remaining para

fTJ/n sA rL/Y '(fFoJ9-GaSed Jo0 4pn4S JlauG4Gld *T GJInOg //Drf O 01 1 0 j8'0 O-j 001: or,Io x~ 3 o ~x 10'0: M I OO:~X 001-:n r____ / 6'0 -X / / S -~ SO'O-/ I'0 o GO'O!!-I Sos/0/ 010 I _ _ISl. _. I I 0a OO 1- 01'0 amm~~~~~~~~~~~~~~~~ 1

-73 -meters as follows '??' = 0 1-e = 0.0003 x0= 1 KB 0o01 Ja = 1.00 F = o08 Because the Ja-n.umber only appears in Equation. (106) with the small factor (l.c)Xk/Ja in the argument of the exponential and error functions the representation of Figure 13 is practically independent of the Ja-number when this is less than about 200 As:x"" increases, the liquid component A becomes more volatile and initial growth, rather than collapse, is obtained at lower values of the parameter Ja/f oL A one hundred fold increase of the mass diffusivity in the liquid phase, appearing as a ten fold decrease of the parameter Ja/VLu? results, as shown in Figure 13, in a shift of A/Ja towards greater values, implying a higher tendency to collapse of the gas phase, other parameters being unchanged. 2,. Solution to Bubble Collapse in. a Nonvolatile Liquid For the special case of a non-volatile liquid, the collapse of a bubble containing only component B is governed by heat and mass transport in thae liquid phaseo Th.e method of source theory, as discussed in detail in Section D, part!Ldi,9 is now applied to the spherical domain, r > R, for the simultaneous processes of mass and heat diffusion. This approach has been used by Yang(64) in predicting approximate bubble growth rates in boiling binary mixtureso Analogous to identifying the liberation of heat of phase transition as a spherical heat source, Equation (73), the instantaneous mass flux,

-74 -NB, entering into liquid solution at the bubble interface is identified as a spherical mass source of strength 4A - R 2 N _p 417R2 (109) Introducing Equation (109) into the analogous form of Equation (74) for mass diffusion and integrating, there results -(x'Xs - A i R(t) f (110) Dimensionless quantities are next introduced from Equations (50) and (101), now, however, basing the Ja-number and the reference temperature used in nondimensionalizing T on the properties of component B, such that,,,T /9 - [ (T-) T \i c' ( 78T) B 8 d/, ifB (111) and 0'= 1 - (Ti —r (112) KB dTP)'O where KB is given by Equation (94) for p~ >> p Using the notation * for the convolution, Equations (110) and (77) for mass and heat transport become -2 (xg- x2) P) (113) 4 3d (1-~) P= ( P2) *C. 1< (114)

-75 -The similarity of the two processes, as evidenced by the analogous form of Equations (113) and (114), implies that during the collapse, interface temperature and composition remain constant, as given by XO_ - Xs E) (115) subject to the phase equilibrium relation, which from Equations (22) and (23) for the case component A is nonvolatile becomes '< = e - - (-x*) (116) where x* is the liquid equilibrium composition at T (see Figure 2). The nonlinear integral equation, Equation (113), is linearized by the approximation - 2 (x,-) (P) 2) (117) where the function +(P) has the form ~(P) = aC- (4-a) p (118) and a is a constant. Operating by Laplace transformation on Equation (117), subject to the initial condition P(O) = 1, the solution for bubble collapse becomes P(T) = Ih exp [C2(1-a2r] erfc LC-a)/i (119) where C = 2 1 (- YI)

-76 -For a + 0 the collapse time, Tc, is given by the implicite equation cxp? [ci erJ Ji il-.)g:] r C CL (120) while the case a = 0 implies asymptotic collapse. With reference to the plot of +(P) vs P in Figure 14 the choice of a = 0.4 indicates that the function ~(P) of Equation (118) closely represents the relation ~(P) = P, which it approximates, over the first half of the bubble collapse. The resulting solution P(l) - exp (j2cr) cur (<\f ) 2 (121) shown in Figure 15, thus closely approximates the actual bubble collapse over the interval 1 > P> 0.5 while for P < 0.5 it indicates too rapid a collapse because V(P) exceeds P in this region. This solution represents a lower limit for the transient. For comparison the case a=O for which +(P) satisfies both initial and terminal conditions of 4(P)=P, is also included in Figures 14 and 15. Because of the inadequate approximation obtained in this case over most of the interval 1 > P > 0, as seen from Figure 14, the resulting asymptotic bubble collapse represents a very conservative upper limito As for the previous solutions, the characteristic group, Ja 2/Lu9, determines the state at the interface through Equation (115) and thus the relative significance of mass to heat transport for the rate of the process.

-77 -l ~~' ace/// / I /z3,S,~///" I / / 01 0 Fiue 1.ApoxmtnFnton(P qaon(l. Fgr14AOprxmtnFucin(P Eqaon18

1.0 0 cr. UPPER LIMIT A=O IN EQ.(118) Q. 0. -LOWER LIMIT A= 0.4 IN EQ.(118) 0 0.8 0.6 Nw CLl ZO 0.4 0 co w z v,0.2 0 ~ ~ OIFE 2IESOLSTM 3 4l MODIFIED DIMENSIONLESS TIME ____ at 2 m)(. x S 2 Figure0 15. DmninesBblColp Figure 15. Dimensionless Bubble Collapse in Nonvolatile Liquid.

-79 -2 2 For very large values of (I-e)Ja' /Lu' bubble collapse becomes mass diffusion controlled and the processes can be considered isothermal. In this case 0' 1 i and Equations (115) and (116) reduce to xs = x* S s For this case the collapse time, Tc I which according to Figure 15 is given approximately by \f - 2/C, may be compared to the quasisteady (9) solution obtained by Clark et alo Considering mass diffusion rather than heat diffusion in their model, which neglects radial convection and assumes the initial interface area, 4R2o, to be available for mass transfer throughout the process, the collapse time is given by -Tc + (4fgLu/2)TC = - /3C For small values of 4Lu' and Tc this result is approximately written.rc= q -/3C, indicating collapse times a factor 36/0 smaller than those predicted by the present study.

CHAPTER III EXPERIMENT A. Introduction This chapter describes the experimental study of gas-vapor bubble dynamics for stationary bubbles initially containing both a nonsoluble gas and a gas of high solubility. In addition, results for the case in which the bubble is permitted to detach and rise through the liquid are included to show the effect of translatory bubble motion on the dynamic bubble growth. Water was used as the volatile liquid component in all cases. The gas to be injected was respectively nitrogen, helium and ammonia. While the actual process of gas injection into a liquid is simple to repeat experimentally, the analytical idealization of the stationary bubble with its initial non-equilibrium conditions cannot be duplicated exactly by experiment. The analytical model may, however, be approached closely, a main objective in the design of the experiment~ The key features of the experiment are, firstly, to introduce the gas phase into the liquid in a time very short compared to the time for appreciable bubble growth, secondly, to maintain the bubble stationary in the liquid. This has been accomplished by a specially designed bubble injector. The bubble is formed and maintained attached to the lo5 mm I oD, tip of the injector during its transient growth, The injector tip may face downward and upward in the liquid during the experiment. In the downward position the static equilibrium shape of the formed bubble deviates from the sphere in being slightly -80 -

-81 -oblate. When the injector tip faces upward the static shape is oblong. The latter configuration being less stable was almost exclusively used. for the study of the bubble having translatory motion, by injecting a sufficient amount of gas to facilitate departure from the tip. Bo Apparatus The general view of the experimental apparatus in Figure 16 shows from left to right, gas cylinder, instrument panel, cameras, insulated view tank (test section proper) and instrument table. The close-up of the view tank in Figure 17 shows its 3 inch styrofoam insulation, the 1 kw immersion heater, bubble injector, gas preheater coil, thermocouples, and connections for gas flow and pressure regulation. The diagram for gas flow, pressure control and instrumentation is shown schematically in Figure 18. The gas used in the experiment is first passed from the cylinder equipped with pressure regulator through two silica gel absorbers in series to remove possible traces of water vapor. Before entering the bubble injector the gas is preheated essentially to the liquid temperature in a 1/8 inch stainless steel tube coil submerged in the liquid. The gas pressure in the bubble injector is measured with a standard 30 inch mercury manometer connected to the return line from the injector, To ensure purity and constant pressure of the gas, a small steady flow of about 0.2 - 0.5 liters/min is permitted to pass through the injector by bleeding the manometer line to the atmosphere through the sample valve, a flowmeter and a needle valve. The three-way sample valve also permits the gas flow upstream of the injector to be vented through the flowmeter, which can be connected to a sampling tube

figure -1i6 Exper: imen ntal Apparatus,. Generalti View. Figure '17 View Tankr wt. h iBbbl.e 1Injector.

GAS TANK VACUUM PRESSURE TAP WATER ACTIVATED ALUMINA ABSOR PER VENT ' aGAS I I EJECTOR FLOW VAC METER VACUUM PUMP TC-9 VACUUM SAMPLE REGULATOR VALVE SILICA DRAI GEL ABSORPER _- - BUBBLE INJECTOR PRESSURE TC-7 I kw IMMERSION HEATER P REGULATOR TC 5 TC- 6 VIEW TANK 115 VAC VA RIAC GAS CYLINDER Figure 18. Schematic Diagram for Gas Flow, Pressure Control and Instrumentation.

containing activated alumina. In this way the content of water vapor of the gas respectively upstream and downstream of the injector may be obtainedbyaccurate weighings of the sample tube, hereby checking for possible leaks around the injector valve causing diffusion of water vapor into the gas. The tank pressure is adjustable between atmosphere pressure and about 20-25 inch mercury of vacuum by means of an ejector pump driven by tap water. The pump suction line is connected to the vapor space of the closed view tank through a shut-off valve and can be vented either directly from the atmosphere through another shut-off valve or through a conventional inverted pressure regulator. The tank pressure is measured with a standard 30 inch mercury manometer connected to the vapor space. The view tank and bubble injector, being components designed specifically for the experiment, will be described in detail. The 17xllx2-1/8 inch view tank, holding the experimental liquid, is constructed from 2-1/8 inch stainless steel channels welded into a frame, sandwiched between the 1/2 inch Pyrex glass sides by two 1-1/2 inch welded angle frames bolted together. The tank lid, which closes a 6x1-3/4 inch opening in the tank top, serves as mounting plate for the bubble injector, gas preheater coil and thermocouple support. The bubble injector is shown disassembled in Figure 19 and in section in Figure 20. A 6 volt, DoCo, 24-oz, solenoid L with adjustable core stop K for the plunger M, is mounted on the top plug E in the cylindrical 1-3/4 inch O.D. casing H o The solenoid plunger M is connected to the valve, P, which forms a 60~ conical seat near the tip of the 1,5 mm IoDo nozzle R by means of the Spring, S

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4 3/8" 6 7/8" Figure 20. Section of Bubble Injector.

-87 -Gas from the preheater coil is supplied at the fitting, T, to the inside of the casing, which is sealed at all joints with O-rings, The 1/4 inch bleed-off line is soldered to the top plug E through which leads for thermocouples and solenoid power enters by means of insulated glass seals, The valve stroke is adjustable from outside the injector by the screw A sealed in the center hole of the top plug by an O-ring. The moving parts, N and P, are machined from aluminum to reduce the mechanical time constant of the spring-valve system. A teflon valve tip, attached to the valve stem with a press fit, is used to prevent damage to the valve seat and to ensure a perfect sealo The remaining parts are machined from brass and finished with a chrome plating. The nozzle is designed to minimize the residual gas volume exposed to the liquid prior to bubble formation by placing the valve seat as close as structurally possible to the end of the nozzle tip. In addition, it was desirable to keep the bubble far removed from the main body of the injector. Bubble formation is accomplished by energizing for a short period of time the solenoid, giving a valve lift sufficient for a flow of gas needed to form a single bubble. Adjustment of valve stroke, energizing time and voltage of the solenoid, spring tension, and gas pressure permitsthe desired initial bubble size to be attained, All temperature measurements were carried out using 22 gauge copper-constantan thermocouples each having separate cold junction at the ice-point, The EMF is measured with a Leeds & Northrup Model 8662 portable precision potentiometer. The liquid temperature and its distribution is measured with six thermocouples (TC-1... TC-6), positioned as shown in

-88 -Figure 18. The thermocouples are suspended from a 3/16 inch stainless steel tube entering the tank lid through a conax fitting with teflon seal, and are embedded in epoxy which also serves as seal. To prevent direct contact of the junction with the water, the thermocouple ends were coated with thermal hardened heat resistant paint, The temperature of the gas to be injected is recorded with thermocouple TC-7 which is installed in the gas space just above the valve seat inside the bubble injector. Thermocouple TC-9 is installed in the bleed line at the flowmeter inlet to allow for temperature corrections on the flowmeter readings. For heating and temperature control of the liquid, the view tank is fitted with a 1 kw immersion heater with stainless steel blades and thermostat. The heater power is adjusted with a variac, The transient bubble size was recorded as shown in close-up in Figure 21 with a Beckman and Whitley DYNAFAX Model 326 high-speed camera timed with the bubble injection moment to give the beginning and major part of the event. The film length in the camera being limited to 36 inches of 35 mm film, equivalent to a total of 224 successive frames, limits either the resolution (time interval between frames) or the total writing time. The film speed is adjustable between 200 and 26,000 fps (frames per second), and remains constant during a recording at a value which can be read accurately on an electronic counter connected to the magnetic camera pick-up. Depending on the particular conditions for the experiment the main part of the bubble growth event lasts 50 - 200 ms (milliseconds) and the film speed is accordingly chosen to 4,000 - 1,000 fps. This record of the transient was supplemented for greater values of

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-90 -time by taking one or several still pictures with a single frame 35 mm camera, shown in position in Figure 22, A representative photographic record for a single experiment (Run 523) is reproduced in Figure 23, showing the negatives every second frame of the high speed film and a single exposure from the 35 mm camera, Each frame shows the downward facing tip of the bubble injector on which the bubble is formed, and the hot junction of two of the upward pointing thermocouples positioned in a plane behind the axis of the injector. Frame number 0 of the high speed film shows the injector tip prior to gas injection which is completed between frame numbers 3 and 4. The wiring for the high speed camera and the optical characteristics of the system is shown in the schematic Figure 24. As evidenced from Figure 17, the view tank was completely surrounded with pieces of expanded polyethylene plate and open to the ambient only through two circular holes, one in each side, serving respectively as observation and light port, The latter was covered with a ground glass plate which made the back lighting from the 150 watt flood light uniformly diffuse, The closed tank in conjunction with the diffuse backlighting gives ideal conditions for a clear and contrasty picture of the bubble, The center portion of the bubble will transmit light while more and more is reflected as the perifery is approached. Total reflection occurs when the angle between bubble interface and line of sight exceeds Arcsin (n'/n"), where n' and n" are the indices of refraction for respectively the liquid and the gas phase. Because side light is excluded a perfectly sharp black edge of the bubble perifery is obtained. In addition these optical characteristics aid in visualizing the approximate bubble shape when this departs from that of a sphere.

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TIMER INJECTOR SWITCHING CIRCUIT S4 ll~7 S2 SI S3 R4T 7 D L S7 Ic -312v ~~- ~~ss-~L L2 RI S6 2 OSCILLOSCOPE L41 -I 1 5 VAC OG~~~~~~~RD~~' OSCILLOSCOPE i I I I I I ' ---o (TIME ADJ.) COUNTER r FILM SPEED INJECTOR GAS SHUTTER-I SOLE NO i-,~~~~~~~ CAMERA t f:4 LENS 150 W i L" f 2.8LENS 4 LENS GROUND GLASS FLOODLIGH T CAMERA PLATE DRIVE ~V J 'B ~2" -J 3" '3 17" Figure 24. Schematic for Wiring of Camera, Timer and Injector Circuit.

-93 -The electrical circuits for bubble formation and camera shutter control are shown in the schematic in Figure 24. Because the built-in timer in the high-speed camera shutter had an upper exposure time limit of 60 ms it was necessary to use an external timer for film speeds below 4000 fps. The circuit for the timer, built for this purpose, is shown in Figure 24, where L4 is the solenoid of a double pole, double throw type NE-22 Agastat adjustable time delay relay, When the manual switch S5 is closed momentarily the solenoid L4 throws S6 and S7 to the opposite positionsof those shown, which are held for the adjusted time delay before again released. Switch S6 connects the shutter solenoid to 115 VAC while S7 puts the 12 V battery voltage on the injector switching circuit where S3 is in closed position. The subsequent sequence of events with approximate delay times relative to the moment of initiation are as follows. The shutter is released by the solenoid after 3 - 6 ms and reaches fully open position after 24 - 27 ms. Switch S2, normally open, closes after 15 - 25 ms starting the current through relay coil L2 and injector solenoid Ll, which starts the opening of the valve at 30 - 35 ms. At 33 - 38 ms solenoid L2 opens the normally closed switch S1 whereby L1 is de-energized and the valve is again in closed position at 34 - 39 ms. With the shutter timer of the camera in position B, the film is exposed till the shutter solenoid is de-energized at the end of the time delay set by the external timer, namely when S6 and S7 return to their normal positions, It should be recalled that S5 was only closed momentarily to initiate the sequence of events and is thus open when S6 closes, preventing a repetition of-the cycle. The sequence of events is shown schematically in Figure 25. When

-94 -VALVE LIFT 30; 34 ms VALVE SOLENOID CURRENT MIN... I I: I I 20 30 33 34 ms t SHUTTER OPENING SOLENOID 5 25 t + 25ms t EXP OPENING SEQUENCE OF SWITCHES S3 S5 S2 g S6,S7 20 33ms t Figure 25. Schematic of Syncronization of Film Exposure and Bubble Formation.

-95 -the built-in timer in the camera shutter is used, its setting determines the exposure time, and the same circuitry is used. The desired syncronization of the events is achieved by proper adjustment of timer delay and resistors Rl, R2, R3 and R4, determining the respective time constants for the R-L circuits. The effective time for the flow of gas through the valve, from its motion starts till again fully closed, may be obtained from an oscilloscope record of the voltage drop over resistor R1, representing the solenoid current. Actual recordings of this voltage as taken of the oscilloscope with a Polaroid camera are shown in Figure 26, Referring to the schematic Figure 25 the beginning of the valve motion is identified by a minimum current, imin1 measured once for all for a given spring tension as the current necessary for starting valve motion, The mechanical impact at the closing of the valve manifests itself in a small blip of induced current super-imposed the exponential decay, The bubble formation time obtained this way is 3 - 4 ms which is in agreement with film recordings of bubble formation in case of no growth, as discussed below, C, Experimental Procedure In preparation for a series of experiments for the case of a non-soluble gas the electrical resistivity of the liquid (single distilled water in all cases) was checked to exceed 200,000 ohm-cmo For the case of a soluble gas the resistivity was read subsequent to each experimental run. Careful degassing was carried out at atmospheric pressure or below, depending on the conditions desired later. Before and after a series of experiments sampling for water vapor in the gas stream vented before and after

(b) time (2ns/unit of scale) (Necessary current for valve motion, minW 8.2 units on ordinate scale) Figure 26. Oscilloscope Record of Solenoid Current During Bubble Formation. a. En ergizing time adjusted for bubble formation without departure. Impact[, when valve seats, virsib.e as small. bl ip as current decays b. Energizing time increased for the:formation of a large departing bubble. Impact when v valve reaches fully open and fu.lly closed position visible as blip on respectively curve for current rise and decay.

-97 -the bubble injector was done by means of the activated alumina sampling tube, The mass of gas sampled was determined from the time of sampling and from gas flow meter readings, corrected for type of gas, pressure and temperature. The mass of absorbed water vapor was obtained from successive weighings of the sampling tube on a micro-balance to 0.5 mg accuracy. In all cases of a nonsoluble gas the content of water vapor was found less than 100 pm, or in terms of molefraction, less than 10-4. This is less than the uncertainty in determination of the initial composition in the bubble and in the order of magnitude of the maximum gas impurity, which according to the manufacture's catalog was respectively 0.3% and 0.01% for the nitrogen and helium used, For the present purpose the gases were therefore assumed to be pure. Comparing samples before and after the injector, no increase in water vapor content could be detected and it was concluded that the injector valve seat was leak proof for diffusion of water vapor into the gas. Leaks in the opposite direction, readily detectable visually, were not encountered. Prior to a specific run the temperature and pressure were adjusted to and kept at their desired values for a sufficient period of time to insure uniform temperature of liquid, tank and bubble injector. After the heater was shut off the system was left for several minutes to permit convective currents to decay. The drift in temperature owing to the small heat flow to the ambient amounted ot 0ol - 0.4~F/mino The lateral temperature variation as checked with the five upper thermocouples (TC-1.,, 5 of Figure 18) did not exceed O lC'F/cm, and the vertical thermal gradient obtained from TC-1 and the lower thermocouple TC-6, was no greater

-98 -than 0o15~F/cm, when an experiment was carried out. For the case of a soluble gas the liquid was thoroughly stirred prior to heater shut-off to ensure uniform composition throughout. With the high speed camera at the desired speed as read on t:he counter and the back light switches on, the event was triggered by momentarily closing switch S5 of the timer unit (see Figure 24)o At the same time a stop-watch was started to fix the time for the single frames taken with the 35 mm camera. The high intensity floodlight, not needed for the single frame exposures, was only switches on during the high speed film recording to prevent heating of the liquid. From readings of the thermocouples in the liquid no temperature change could be traced to result from the use of the flood light over periods of time up to one minute, the maximum time this light was on during an experiment. Immediately following the event, tank vacuum, hydrostatic pressure, barometer and temperature readings were made. After every other run the high-speed films and corresponding single frames were developed. For both cameras it was found satisfactory to use Kodak Plus-X negative film, normally developed and fixed, For the high speed camera triple perforated 35 mm film was used to enable subsequent slitting of the film into two 16 mm single perforated strips, suited for conventional movie projection. C. Reduction of Data 1.o Properties The state of the bulk liquid is defined by its temperature, TX, pressure, pm, and composition xo. The temperature was obtained as the mean value of TC-1 o TC-3, te the three thermocouples closest to the bubble

-99 -corrected according to the calibration for the particular gauge 22 copperconstantan wire used. The calibration curve was based on the ice-, steamand tin-poin.t The liquid pressure at the location of the bubble was obtained from barometer readings corrected for temperature, the reading of tank vacuum and hydrostatic head of the liquid in the tank corrected for its temperature. The equilibrium gas compositions x0 at TX, was calculated from steam tables(30) according to Equation (22). For the case of a nonsoluble gas the liquid was considered pure water when the electrical resistivity exceeded 200,000 ohm-cmo In the course of the sequential injection of several bubbles of the soluble gas, the liquid composition deviated slightly from that of pure water, xX = 1.0 o The measured resistivity was related directly to composition. with the use of the standard curve shown in Figure 27, obtained from resistivity measurements of solutions of ammoniahydroxide of known concentration. As seen from Figure 27 the resistivity was found not to change with temperature over the range 26-370C, in which the measurements of liquid samples were also conducted. In all experiments using ammonia as the gas, the mole fraction of ammonia, 1 x1 9 was in. the range 10 -5 10 The thermal diffusivity a (T ) is obtained from Jakob and 00 Hawkins(28) and the two important dimensionless property groups, Ja and Lu, are evaluated at T from the graphs shown in Figures 28 and 29~ 00 The plot of Ja-numbers vs temperature for water was prepared from steam tables(30) according to the definition of Equation (50). The Lu-number for the diffusion of water vapor through respectively nitrogen, helium and ammonia was evaluated from Gilliland's equation1 1 For the pressure and temperature range investigated in the present study, Equation (122) gives values of the mass diffusivity consistent with those determined when using the more accurate Chapman-Enskog formula (see Reference 5 p.o 510 ff)o

-100 -T 3/2 / 0AB ( 0.04. [ (122) where T is the absolute temperature in ~K, p the total pressure in atm, M the molecular weight, V the molecular volumes (cm3/gmole) at the normal boiling point obtained from Reference 59. In forming the Lu-number according to the definition of Equation (50), the thermal diffusivity for water was again obtained from Reference 28~ The Lu-number for the diffusion of ammonia through water was calculated according to the definition of Equation (101) using the constant value (43) ' = 15 x 10-5 cm2/sec for the mass diffusivity, which is little affected by pressure and temperature. The assumption of ideality of the binary system, prompted by the analysis, is onlypartly approximated by the ammonia-water system chosen for the soluble gas case. While equilibrium gas compositions calculated according to Equation (22) were found to be consistent with data summarized by Perry,3) the heat of evaporation of mixtures indicated some nonidealityo However, a value of F = hfgB/hfgA = 0.8, based on the gas composition, was found to adequately approximate the data of Reference 43 for 1 atm over the range O06 < x" < 1, and was used in the calculationso The (30, 43) Henry law constant KB calculated based on vapor pressure data using Equation (94), is valid only for very dilute solutions. A more accurate value, used in the calculations, was obtained from the definition, KB = (l-x*)/(l-x"), by employing the experimentally determined equilibrium molefraction x* at Too 9 as taken from Table 24, Section 3 of Reference 43

o 26 OC o 30 OC o v 370C 105 0 I- H~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ cn U, 104 1 ~~~~~6 -49 16 Id6 10 40 MOLEFRACTION AMMONIA I- X' Go Figure 27. Calibration Curve for the Electrical Resistivity of NH3-Solutions.

-102 -200 150 - o j *.. 100 o 50 150 160 170 180 190 200 210 TEMPERATURE, OF Figure 28. Ja-number vs Saturation Temperature for Water.

-103 -1600 1400 1200 1000 -I U 800 600 H20-He 400 2 0-NH3 200 H20- N2 0 1 150 160 170 180 190 200 210 TEMPERATURE, OF Figure 29. Lu-number vs Saturation Temperature for Water-Nitrogen, Water-Helium and Water-Ammonia.

-104 -2. Transient Bubble Size The remaining data, in essence equivalent mean bubble radius as function of time, is obtained from the high speed film and the single frames exposed during the transient process. The approach followed in getting the initial bubble size, and composition, xo, being important for the interpretation of the data, is subsequently explained in detail for the case of a nonsoluble gas. The spherical symmetry in the liquid is experimentally rather undisturbed by the method of suspending the bubble from the injector tip, whereby part of its surface is left unexposed to the liquid, provided the sector taken up by the injector may be considered adiabatic. The approach to symmetry also in the gas phase calls for the displacement of the volume of the spherical sector subtended to the unexposed area of the bubble surface. Although the bubble undergoes growth this correction to the system geometry is approximately accomplished by a constant sector of 600 top angle, formed by an extension of the teflon valve tip as shown in Figure 30(a)o The resulting system geometry, as shown during the growth process in Figure 30(c) and (d), simulates the analytical idealization of spherical symmetry in the two phases. The equivalent mean bubble radius is thus determined as the radius of a Qphere having a volume equal to the volume displaced in the liquid, V(I) in Figure 30(c). Because of its smallness and the nonwetting nature of teflon, the ring shaped volume Vring in Figure 36(a) will not be filled with liquid when a bubble has departed from the tip, but it will contain a mixture of water vapor and gas of the equilibrium composition, x o The same applies to any small cavity left 00

-105 -TEFLON VALVE VR VRiNG VTIP 'TIP 1.50 MM. - -'< LI =2.08 MM. (a) L2 tO0 t= t. t — 7 0 - 7 '-, Y s?0)) vAX. y I (b) (C) 3 (e) Figure 30. Characteristic Dimensions of Injector Tip and Bubbles. x~~~~,,Y (e)

on the teflon tip and visible on the film prior to the injection moment. The water vapor existing in these cavities will be present in the initial bubble formed on the injector tip and constitutes the only contribution to an initial non-zero gas composition xo. Owing to the supposed well mixing taking place during the injection, x0 is assumed uniformly distributed and is calculated in the following way, ~o = X V +v(o) (123) VO where Vo = (VMAX + V) (-x>) t- V(O) +VR ] x., (124) since 3 #-x_ VMAK (125) Here, referring to Figure 30(a), (b) and (d), VR is the difference between the ring shaped volume, Vring, and the tip volume Vtip, V(O) is the visible volume prior to bubble formation, i.e., that of the tip plus any possible remanent cavities on this, and VMAX is the asymptotic bubble volume of the equilibrium composition x,. As seen from Equation (124) the initial bubble volume, VO, is calculated from the measured volumes V(O), VR and VMAX and the gas composition xX previously determined. This is a necessary procedure because VO is in fact not observable but represents an idealization. Although the injection of the gas to form the initial bubble volume VO is very rapid (' 3 ms) the growth process is already in progress when this has been accomplished. VO can therefore not

-107 -be obtained from the film, even if the exact formation time was known. The asymptotic bubble volume, VMAX, being a critical quantity, was obtained as follows. Starting from frame number Il, chosen about 20 frames from the end of the high speed film, the first local maximum in bubble volume was located. The arithmetic mean of the volumes on subsequent frames including the next local maximum was then determined. This mean value was taken to be VMAX unless the volume on the first still picture, recorded 30-50 sec after the introduction of the bubble, exceeded this in which case the volume from the still picture was used. Bubble volumes were obtained in the following way from the twodimensional picture appearing on the films. The measurements were carried out on a 30-40 times magnified picture of the negative film frame, back projected onto an opaque glass screen in a specially designed film analyzer. With a ruler system designed for the purpose the characteristic lengths as shown in Figure 30(e) were measured. Bubble length H along the axis of assumed symmetry, five diameters yo '.. y4 with an even spacing of H/4 along the axis, and the tip diameter L2 of the injector as reference length. Expressing the diameter variation y(x) as a fourth order polynomial through the five diameters yo(O)... y4(4.H/4), a close approximation to the actual volume is obtained from 04 (126) where INTr ) C ( - (127)

-108 -L1 is the true value in desired units of the reference length L2 appearing in the magnified picture. The integral INT of Equation (127) is obtained by analytical integration of the square of the approximating polynomial, y(x), as obtained by Newton's interpolation formula for a polynomial of fourth order. The explicit form of INT appears in part (1) of the computer program, Appendix V-lo The equivalent mean radius, R(I), of the bubble having the volume V(I) is then calculated from R() V () (128) 3. Computer Program Because of the extend of arithmetic manipulation needed for the reduction of data following the above procedure, a program was prepared for the processing to be carried out on an IBM 7090 digital computer. The computer program for the case of a nonsoluble gas comprises three main parts: reduction of experimental data, calculation of the analytical solution for the particular conditions of the experiment, and plotting of both theoretical and experimental points. The actual program written in the MAD (Michigan Algorithm Decoder) language and utilizing the subroutine library MESS (Michigan Executive System Subroutines) is reproduced in Appendix V-lo The nomenclature for the program is listed in Appendix V-2, followed in. Appendix V-3 by a typical sample of the data input for a single run (RUN 523). Subsequent to general input data comes, in format, the data from each picture successively. In the process of measuring the characteristic dimensions of a bubble in the film analyser

-109 -these data cards were directly punches by the operator, made possible by locating the film analyzer, next to a conventional key-punch machine. To illustrate the results of the digital computer program, the print-out, likewise from run 523, has been included in Appendix V-4. First, in Part 1(a) of this a complete table of the input from the bubble measurements is printed, giving frame number und characteristic lengths. Then follows in Part l(b) the reduced experimental data in the form of initial and asymptotic conditions and a table of dimensional and dimensionless bubble growth. Next, in Part 2, the analytical solutions to dimensionless bubble growth are given: (a) for small values of time, (b) at the join point T1, (c) for intermediate values of time, (d) at the join point ~2, and (e) for large values of time. Part 3 gives the symbols used in the plots and the dimensionless time, TAUMS, corresponding to one millisecond, and the print-out is completed by plots of respectively P vs T and Q vs T, in which the titles list the parameter values and substances for the experiment. The analytical Part (2) of the computer program was used for the parameter study reported in Chapter II-Do For the case of a soluble gas a computer program was written to include three parts, reduction of experimental data, plot of these and calculation of the initial bubble behavior according to the analysis, Equation (106). The latter part also provided numerical results from which the parameter study of Figure 13 was prepared.

-110 -Eo Results In this section the experimental results are presented and discussed, and the uncertainty of the results is illustrated by a sample calculation Because of the nature of the process under study -an initial value, transient problem depending on many parameters- the experimental program was planned to cover a number of particular conditions for which in each case the analytical solutions were compared to experiment. In the range of agreement between experiment and theory the results can thus conveniently be presented by the parameter study based on the analysis. lo Discussion of Experimental Results ao Case of a Nonsoluble Gas In comparing experiment with theory two nondimensional representations are used, namely P(r) _? --- (t" l, ) X IZ, im )) and In nondimensionalizing time to T = o't/R2, one of the five parameters, Ro has been accommodated. The relative nondimensional bubble size,, normalizes growth to the interval 0 < Q < 1, and was found most useful in the representation of the data. For the- case of a nonsoluble gas, the range of parameter values subject to study in the experiments reported here, using nitrogen and helium as gas and water as the volatile liquid, are summarized in Table II.

-111 -TABLE II PARAMETER RANGE IN EXPERIMENTS FOR THE NONSOLUBLE GAS CASE System Ja Lu xo xo Ro(mm) H20-N2 170-570 > 015 > 0003 90-235 < - e oo65-1o5 < 0oQ94 H O-He 540-1530 The values of the Ja- and Lu-numbers are based on a system temperature range of T = 206-150~F with corresponding saturation pressures of 00 p* = 26.5-7o6 inch Hg abso The presence of remanent gas prior to bubble injection sets a lower limit for xo o The upper limit for x0o corresponds to a compromise between the desires for. considerable bubble growth (P0O = 1o8-2o4) reasonable initial size (Ro = 0o65-1o5 mm) and the bubble to stay on the injector tip when growth terminates (Ro l oL6-1o8 mm) A summary of the experimental runs conducted is tabulated in Appendix VI-i giving the conditions for each run. Plots of experimental data and calculated analytical solutions for a number of runs are shown in Figures 31-40, directly reproduced from the print-out obtained from the IBM 7090 digital computer according to the program shown in Appendix V-o. To clearly identify the sequence of the experimental points in the

-112 -plots a solid line has been drawn through successive points. In addition, the intermediate and asymptotic solutions joined at point T2 are included. The characteristic parameters for each run appears in the title of the graph, where A(mm2/sec) is the thermal diffusivity and RO(mm) the initial bubble radius. Figure 31 for Run 511, shows the bubble formation P(T) representative for a number of runs made under conditions where no growth occurred. As seen. from the title of the graph, Poo = 1.009, hence the formation size Ro, is practically identical to the asymptotic size, R., and the plot shows how the equivalent bubble radius approaches its initial value in the course of 3-4 ms. The oscillations in bubble shape, being associated with a volume oscillation as discussed in Appendix II, appears clearly in the plot. Taking these into account the actual gas injection is seen to have been terminated in less than 3 ms. Because the initial bubble size is chosen considerably smaller in the cases where growth occur, by reducing the activation time of the injector valve., it can be concluded that a formation time of 3 ms was in no case exceeded, which is in agreement with the less accurate oscilloscope measurements of injector solenoid current discussed in Section III-B. Next, relative dimensionless bubble growth Q(T) for nitrogen in water is presented in Figures 32-37. The first three graphs, Runs 523, 504 and 702, show bubble growth under practically identical values of the governing parameters, Ja, Lu, x0, xo, but with a considerable difference in the randomly introduced initial disturbance responsible for the subsequent oscillations superimposed the transient growth. A particularly large

1,20 ------- -------------------- - - - - - - N I I I I!! 0 I I I I I___~ ~ ~ ~~_ I N ~~~I I I I I I I ~~~I I I I I I E ~ ~~~ I I I I I N lco +.o.... --- —------------- c ----------- --- -- - -.. --- — I I, ~~~~I I I I I I 0.80 + — / -----------------------------------.... ~ ~.......... O ~~~I I I I I I N ~~I I I I.....I..............I A ~~I I I I I I --------------------- ----- ~ ~ ~ ~ ~ ' -~ '-'-~~ — I I I.I.......!.......... I_ 0.60 + --- —-----------— + --- —---------------------------- U I ~~~~~~~ ~~I I I I EB I I. I.I JOIN POINT AU1 I I EINTERMEIATE THEORY I I! I I X JOIN POINT TAU2I I I I + ASYMPTOTIC THEORYi 0.40 ----------------------------------— + --- EXPERIMENTAL POINT Z ~~~~~~~~I I i....iIJA = 450 A = 0.1435 I I I I LU = 1600 RO = 1.547 I I IrXO = 0.000 POO = 1 I I G I I I I INTIXOC = 0.029 WATER I I NI T ROGEN. I I C.20 ----------------— + --- —--------------- - +EPRMETLPIT --- —--------------------------------- I I I I 0.20 0. 0. 1.2 1. I I XO............L~~ I...........I — I I I X 00 WT I I I I I I I i I I I I -I I I I I I I I I~ ~~.5m I I I I I I ~~~~~~I I I I I I r2 p ms_ I _ I............ 0,4 C,~~~~~~~~~08 1.2 1.6 2.0 NONtIVEhSICKA L T'IE 1000*TAU = 1C00*A*T/RC.P.2 Figure 31. Bubble Formation, Run 511, P vs T

1 20 + --- —--------------------------------- ------------------- ---------- -------- - ----------------- + R I I I I I I E I I I I I I L I I I I I I A I I_ _ _ _ _ I __ _ _ _ _ _ _ _ _ _I __ _ _ _ _i T I I I I Iv I I I I I E 1.00 + --- —------------— ~+~~-~-~ --- —------------------------ I I I I I' INJ I~ I + __ _ _ __ _ _ I o3 i I I I IN I I I I I _ _ _ _ _ _ I O I I I I? I I I. I I I _ _ _ _ _ _ _ _ _ _ __ I N I I I I 0 I I I I I I N I I i I I I _ _ _ _ _ _ I I I I I _________ I ________ A 17__A017 I I E 0.0 +- ---------- -+ --- —------------ --------- -------------------------— ~ — + --- —------ ------- + E ri-I wI 76 I I _ _ _ _ _ _ _ _ _ _ _ _ _ I I x 0 = 0. 6 82o = 1. 8 I I S I1I I I I __ __ _ I __ _ _ _ _ _I I O ONP INTT I I L Itli I INEMEt I HOR I I __ __ _ I __ _ _ _ _ _I I X ONP INTT I I I ~I I AYPOI TER I I C.6~0 ---------------— + --- —----------------------------------------------------- I I I I U I RUN 523 I, 0.0 1C.20.0I 3J0.0 40. 50.0U I L I I 0 INTERMEDI ATE TER I E- I I N A TIM POINTTAUA I TFI u G RunM23,TIv TH s 0.40 + -------- ---------------- — + --- —------------- EXPERIMENTAL POINT - -------- ---------------— + I I I I z I I JA 9. =017 I E I 185 o e T I I I I~~~~~~~IXC.83 WAE I I I ~~~~~~~~~~I NIROE I C.20 ---------—,, ----+ ---~ --- —------------------------------------------ 0,0 ~~ ~~1C.0 20.0 30.0 4C.C50, NCNDIMUENSICNAL TIME 1000*TAU 1000*A*T/RO.P..2 Figure 32. Bubble Grow-th., Run 523, Q VS T

1.20 + + + * R I I I I I I E I I I I I I L I I I I I I A I I I I I I T I I I I I I _ I II I I I I I N I I I t I E 1.00 + --- —--------------------------------- ------- A ----------------— + N I I I N I + I I I I E 0.80 + ~~~~ --- —-- ------- 0.80 + ---- ----------------— + --- —-------- N I. I I I I I $ I T I I I I I I ITO I I I I 0 I I I I I I N I j I I I I I A I I I I QI I I I I I UI I I I RUN 504I I 0.20 + --- ~ + ---+~-~B I I I I I B I I I 0 JOIN POINT TAU1. I I L I I I INTERMEDIATE THEORY' I I E I I I X JOIN POINT' TAU2 I I I. I + ASYMPTOTIC THEORY. I I S 0.40 + --- —---- ------------ + --- —------------- - * EXPERIMENTAL POINT ----------------— + I I I I I I Z I I I JA = 93.8 A = 0.1674 I I E II I LU = 11 RO = 0.761 I 5 I I XO = 0.227 POO = 1.997 I I Q I I XOO = 0.903 WATER I I I I NITROGEN I I I I I I I 40 ms I 300I I I NONDIMENSICNAL TIME 1000*TAU = 1_000*AT/RC.P.2 Figure 33. Bubble Growth, Run 504, Q vs T

1.20 + - +- + ----------- -— + --- —+R I I I I I I E I I I I I I L I I I I I I A I. I I I I I E 0 *80 + ~ — t —"- -+~ ---+ ---~-~ --- —--- ----— 4 N I I I I S I I _ _ _ _ _ _ __I I I I _ _ _ _ _ _ _ I AI i /) I t ~~~~~~~~~~~~~~~~~~~~~~~~~~I IE C.60 + — ----------— ~L I I I {r - JII - I i U I _ _ _ _ _ _ _ _ _ _ I _ _ _ _ _ _ _ _ _ _ _ I R U N 7 0 2 I _ _ _ _ _ _ _ _ _ _ I B I I I I I H B I _ _ _ _ _ _I __ _ _ _ _ _I O_ _ JOIN POINT TAU1 -__ _I __ _ _ _ _ _I H L - -- 1| II I. "-ER?4E-IATE ThIRY I I x E 14 _ _ _ __ _ _ _ _ I _ _ _ _ _ _ _ _ __ I X JOIN POINT TAU2 I _ _ _ _ _ _ _ _ __ I ~~~~~~~~~~~~~~~~~~~'. - ~ E I I + ASYMPTOTIC THEORY I I I0~~ I I I I I Z____ I|__ I ______________ I JA= 95.4 A! =.1672 I I E I IU- 90R-O.763 I I 0__ _ I I I_= 0.203 P00= 1.935 I _.' ~I N~~~~~~~~~ Q I -I XCO = 0.890 WATER I I I I I I C 20~I I I I I I — I I I I I I I I I I I I I I I I I I I I I I I I I I N~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~O C.6 0 +- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - I B _ _ __I I I I usII I RUN I I I B~~~~~~~~~~~~~~~~~ II B~~ ~ ~ I! JOI POIN TAI O.00.L..~~ +~+~+~+.O I. 20.0 30.0 40.I 50.0 I ~~~~~~~~~~~~~~I I AYPCI THEORY C____ --- —------------------— __EXPERIMENNCNCIPENSICNAL TIME lOOO*TAU = 1POI *ANT/RC.P.2 Figure 34f. Bubble Grawth, Ruln -702, Q vs T z~~~~ I I IA = 954A=017I E~~~~ I.I IU = 19IO.6 Q~~~ I IG = I -------- I IIII 0.00 m.....+ — +L..... --- —---------------------------------------------- O.0 1C.O 20.0 30.0 40.0 5. NC-NCItPENSICNAL TIME 1000*TAU = lOO00eA*T/RC.P.2 Figure 34. Bubble Grow-th, Run '702, Q VS T.

1.20 + --- —-------------------------------- R I I I I I I E I I I I I I L I I I I I I A I I I I I I T I I I I I I I I I I I I I V I I I I I I E 1.00 + --- —------------— + --- —------------— + i I I I I R~~~~~~~~~~~ N I I I I I I I I I N I I I I D I I I I I I I I I I I I IM I I I I I O, I I I 1 I E.8 ++ 4.+ N I / I I I I I A I f op I I I I I E S I I I I I I I I //II / I - I I I I I I I A~~~~~ B I I I I I I A I f/I I I~ 2 I I s I /tI I O JOIN POlNT TAU1 I I H L I I I I If I I.I E I I I X JOIi POINT TAU2 I I I I I + ASYMPTT 2IC T4ECRY I I S.(40 +- -------------- — + --- —------------— +- — EXPER II-ENTAL PO INT --------------— + I I I I I I B I I I JAU1 I IERMEIATE TH I 1 E I I I XO J ON230 POOI = TA2 I I Q. I XIG = 0.933 WATER I I I I NITROGEN I I ZX I I JA:9.0A=0.'7 I I I I I I I I. I I I II I I I I I I I IO Is- I II I~-11 COO + —'-'-'-' --- —---------------- ------- --- ------------------- (.0.C 20 --- —G 30.0 40.0 50.0 NCNDIPENSIC-A L TIME 1000*TAU = 1000*A*T/RC.P.2 Figure 35. Bubble Growth, Run 524, Q vs T.

1.20 + ---------------------- R I I I I I I E I I I I I I L I I I I I I A I I I I I I T I I I I I I I I I I I I I V I I I I I I E I.CO + ---- + --- —--------— + --- —------------— ++ N I I T I t 0 I. I O I T I I N I I I I I 1 I A I I I I ~ I E 0.40 +- ~- _________ --- —-----------— +-~ EXPERIENTAL+POINT --- —- - + NI I I I I S I J 18.I I I I I 1 R 0.80 I I __ _ __-_ -_ _ _ _ _ _ __I I I I I I Q I i I I I I I I NA~~~ I + I I I I I At I I I I I I N I I I I I A_____ UI RUNI I I I B I; I I II I 8 II I I O JOIN PCINT TAUI I I 1 L I I I - INTERMEDIATE THEORY I ' I.E I X JOIN POIN, TAU2 I I I I + ASYMPTOTIC THEORY I I S 0.40 --------------- — + --- —------------— +- - EXPERIMENTAL POINT -----------------— + I I I I Z I I JA =184.3 A =.lh127 I I E I I LU = 421 RO = 0.8-20 I I I I XO = 0.1-55 POO = 1 932 I I ~~~~~~~Q ] I I XCE = 0.883 WATER I I I I NITROGEN I I I I I I 0.20 + -------------------------------------------- I - I __ = 2 RO = I 8 _ I I i ~ ~~.I I I0. I I i I I I I I I I I I I I I I I I I I I 10 ms I - I- I C.0 1C.O 20.C 30.0 40.0 50.0 NCNOI'ENSIENAL TIME 100O*TAU = l00*A*T/RO.P.2 Figure 36. Bubble Growth, Run 517, Q vs T.

1.20 + --- —---------------------- R I I I I I I E I I I I I I L I I I I I I A I I I I I I T I I I I I I 1 I I I I I I V I I I I I E 1.00 + —~~ -4- — +~~~~-4-~~-4.. _ _ _ _ _ _ I _ _ _ _ I _ _ _ _ I _________ _' - -. -I- I I I I R~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ O I I r 'I I I I N I_ _ I I I I O! k flrr w ~wC I I I I I _ I__ _ _I I I I I M I f/ r I I I I I E.C180 -----------------—. --- —-------- ------ --------------------- -- ---------------- N I I I I I S I I I I I I I {!fII L I I I I I I 0 +~ B i g ~ I I I i I B I I I I I L A I I........E............... T.....EORY...........I................. I LI I I I I _C.6I I A= 21 - A=010 I8 ill ~~~~i I I I U II I..RUN 521 I B I I I I F B III I I JO IN POINT TAUI I F-J L I I I. I - I I '' 8JINP I INT_ _ AU2 I I I I + ASMPTOTIC THEOR I I S C40 - -------------------------------------- - EXPERIMENTAL- _pOINT I I I I I I Z I I JA = 241 A = 0.1603 I E I............... I' LU (-....58'7.......RO" =...O.6 60...'i'....... I I XO = 03380 PO = 2C052 i 5 Q I I XCO = 0.929 WATER I I I NSI T ROG.EN I I I I G C 20 -- --- --- -+ --- —---— + I I.................i...............I..........I I I I I I I I I I -I I I I ~~~~~~~I I I I I ~~~~~~~~I I I i I I ]0 ms `I'......................... I I C.00 + - J —- -'..........+ 4 +................. C.O 1C02C.0 30.0 40.C 50.0 NGNOIPENSIIDNAL TIME 1000*TAU = IOO0eA*T/RO.P.2......... Fi'gure-'-'i. "-Bubble' Oro'~wt-,Run 5e'1, Q vs 'r

-120 -amplitude of oscillation was obtained in Run 702 which, as the only run included here, was carried out with the experimental set-up inverted such that the bubble was blown vertically upwards. Despite the considerable difference in amplitude of oscillation the transient growth is virtually identical for the three runs, indicating that the second approximation to the dynamic bubble shape, Equation (19) has only at most a secondary effect on the process studied here. Although not conclusively demonstrated, this secondary effect tends to accelerate the growth. Comparing next Figure 35 of Run 524 to eogo,, Run 523 in Figure 32, the effect of a change in xo, and xo resulting in an increase in P, of nearly 15% shows the corresponding decrease in growth rate~ Finally, Runs 517 and 521 in Figures 36 and 37 show the considerable increase in growth rate with. up to a 2 5 fold and a 3 fold simultaneous increase in respectively Ja-number and Lu-numbero Figures 38-40 show Runs 601, 609, and 613, all for helium in water and for increasing Ja and Lu. For unchanged system temperature and pressure the use of a different gas implies a change in the Lu-number only, hence a change in the initial and intermediate growth rates. Comparing thus Run 504 to Run 601 the effect of a 3 fold increase in Lu-number alone is seen for the case of Ja X 90, while the similar effect for Ja 220-240 is seen by comparing Run 521 and Run 613o b. Case of a Soluble Gas For the case of a stationay bubble initially containing a soluble gas four experiments are reported here in which ammonia gas was injected into water. The conditions for the experiments, Runs 9519 953, 954 and 955

1.20 + ------------- R I I I I I I E I I I I _ I L I I I I I I A I I I I I I T I I I I I I I I I I I I V I I.I -,. II E 1.00 + --- —------------— + --- —-------------- ----------------- ------------ N I f _II I Mi I | * I. I - I I 111. 11_C.80 + ----s- _4 / _________________________________ I I, O I I -I I S I -} wF i;. I I I I I N I I I I I I 0 I I I I I I I I I I I I M I I I I I I E C.80 + ---+- - — + --- —--— ' ---------— ' --- —------------— + --- —---— +. I.I I I I I I I I I I I I B I I I I I I.I I I JNPINTTAU1 I I L I I I fIAT I I H Ll I I + ASYMPTOTIC THEORY I I I I I I I I I U I I IRUN 601 I I I II I I XO=.325 P = 1.993 I I B I. I I 0 JOIN POINT TAU1 I I L I TEI I i ' Ni'-ERYI. Q I I I X JOIN POINT TAU2 I I I I I + ASYMPTOTIC THEORY I I I I I I I Z I I IJA 904 A I I E..'.. I I I I - I._.............._ I I I XO = 0.325 POO = 1.993 I I I I I I I I Q I I XGG = 0.914 WATER I I I HELIUM I I I I I I I 1nI I I I I c.0 1C.0 20.0 30.0 40.0 50. NCNDIMENSICNAL TIME 1000*TAU = 1000*A*T/RO.P.2 Figure. Bubble Growth, Run 601, Q VS_ Figure 38. Bubble Growth, Run 601, Q vs-T.

1.20 + —+ +.. - - - — + R I I I I I I E I I I I I I L I I I I I I A I I I I I I T I I I I I I I I I I I I I V I I I I I I E 1.00 + ~ ~ ~ + ~~~~~.. I I. I I N. I I I ":[ i I 0 I I + I I I N I I I I I S I I I I I I I I I I I I I M I + I I I I I E 0,80 + ~~~~ — ~ +~. N I I I I I I S I I I I I II I / I I I I I 0 I / I I I I I N I I I I I I A I I I I I I L I I I I T I B I I I I I U I I I RUN 609 I I E I | I I X JOIN POINT TAU2 I I I I I + ASYMPTOTIC THEORY I I S 0.40 +1. --------------— + --- —------------— +- * EXPERIMENTAL POINT ---------------- I I I I II Z I I I JA = 180.5 A = 0.1627 I I ____ IJ_ I I X= 0.404 P00 = 2089 I I I Q I I XOC = 0.934 WATER I I I I I HEOIUM I I I I I I I ZB~: _- I I JN r 18. A_ = 01 2I I TU I I I. I I I I I I I E I I I OI I I I I I I I I~OO + — ------- --— L --- — ----------- - - I _ _ _ _ _ I_ _ _ _ _ _ _ _ _ _ I_ _ II I I I I z I I I JAI8s I 0.I1 -0.0 5.0 10.0 15.0 20. 25.0 NNDIMENSICNIAL TIME 1OOO0 TAU = 1000HA'T/RC.P.2 Figure 39. Bubbe Growth, Run 609, Q vs.

1.20 + --- — -- -------------------------------------------... -~ --- —---------------------------- R R I I I I I I E I I I I I I L I I I I I I A I I I I I I T I II I I I I I I I I I ~ I I I I I I E 1.0 0 + — ---......... —I-~-~ ---. +............. ~ f~ + ----~- ---- - +-41- ~~~~~I~ —~+ I I I N I I I I V I I I I I N~ Ij' I - _ I I N I I I I I I O I I I I I I E C.80 -- ----- ----- + --- —------------------------------------------- ---------- ----- N I I _________ I __ _ __ _ I __ _ _ _ _ o I I I I I I _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I _ _ _ _ _ _ _ __I_ I __I I I O I I I I I I N I I I I I I N _ I I I I I I A I I L~~~~~~~~~ C.60 + --------------- ' --- —------------— + --- —-------------------------- 8 I I I I - _ C.*60 +~ ' 4~ + ~+-............ U I I RUN 613 I I! is11 I I I I ro B I I 0 JOIN POINT IAU1 I I L~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. L I I ' '-t-ERM'E DAE THEORY 'r I E I I I X JOIN POINT TAU2 I I I I + ASYMPTOTIC THEORY I I S C.40 -------------— +- * EXPERIMENTAL POINT ------------- + I I I I I Z I I JA 224.7 A = 0.1611 I I E { I. -" [ U:.14;...'-' i ___ _ ~I I XO = 0.385 POO = 1.884 I Q0~~~~ - ~I......I XOC - 0.908 WATER I I _______ I I HELIUM I I -- - - - - - - - - - - - - - - ----— I -------—...I - -- - - - ------ ------.- -— 1- - --- 0.20 + ++ --- —-------— +- -.-. —.-.-.....+ I I I I I I I~ I I I I I I I - r I I I I I I. I I I joins -.-. - "-.-.-I. O.OO +....J 4 —, —+ ~ +, --- —-- C.C 5.0 10.0 15.0 20.0 25.0 NCNDIlENSICNAL TIME 1000*TAU = 1000*A*T/RO.P.2 Figure 40. Bubble Growth, Run 613, Q vS T

-124 -are listed in Appendix VI-2 and the transient bubble collapse in the form of equivalent radius (mm) versus time (ms) is shown in Figures 41-44 directly reproduced from the print-out of the digital. computer program for this case. The study of the collapse, opposed to growth, of a stationary gas bubble initially of nearly spherical shape suspended on the injector tip, made it necessary to adopt a different procedure of data reduction. Thus, the radius plotted versus time is the equivalent radius of a spherical bubble having the actual volume of gas present on the tip, implying that the asymptotic bubble radius is zero. Consequently, the experimental results represents spherical bubble collapse only in the initial part of the transient when the bubble volume is large compared to the volume of the teflon tip, Vtip in Figure 30, subtracted from the visible volume Based on the resistivity measurements of liquid samples, its composition, using the standard curve of Figure 27, ranged from 1L6 - 185 x 105 for the molefraction of ammonia, hence the liquid was far from saturated, providing for eventual bubble collapse in all cases, Because of this, no remanent gas phase could be present on the injector tip prior to bubble injection, and the initial gas composition could be assumed to be pure amImonia, 0 Runs 951 and 953 in Figures 41 and 42, were conducted at essentially the same, relatively high temperature, Too 2050F but at different pressures, thus showing the effect of decreased volatility of primarily the liquid component A (water) for which the equilibrium gas composition, x1 at To, is a measure. The maximum bubble radius being nearly the same, permits direct comparison of the much slower collapse in Run 951 to that in

B 1.50 + ----------------- + --------- - -- -- + U I I I I I I B I I I I I I B I I I I I I L I I I I I I E 1 JCIC*t t I I I I I MI I I I I R I I I I I A 1.25 + —P --- —---------— + ---- ~........ ---+D I I I RUN 951 I I I I I I I U I I WATER-AMMONIA I S I I I I I I I I 04.6 F I I II II 3.7 PSIA I I R I I I I I I 1.00 +S _____ ---_____,,__________, __________ ---- __________ MHI _ I I I I M I I I I I L I 1 I I I I _ _L_ _ _ __ I I I I I I, I I I I I I M I I I I I I E 0.75 + __________________ --- —--------— __________________ — __________ ---_____ —__________________ — __________________ T I I I I I I E -tI I I I I I R I I I I I I I E I I I I I I i II I I I t I I I II _ I I I I I I I I I I I I I I I I I I I I I I I 0.50 + 50.0o +75.0 00.0125.0 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I. I I' I I I I -I I I I TIME T (MILLISECONDI S) Figure 4l. Bubble Collapse, Run 95i, R vs t.

B 1 50 -- ~-+ -..+....-f..............+ U I I I I I I B I I I I I B I I I I I I L I I I I I I E I I I I I I I I I I I I R I I I I I I A 1..25 ~++~+~+ ~~ +~, D I I I RUN 953 I I I I I I I I U I I I WATER-AMMONIA I I S I,I I I I I I I 205.3 F I I I I I 14.3 PSIA I I 1.00 f-~ -++~+ ~~ - -~+ I I I I I I M I I I I I I I I I I I I I L I I I I I I L I I I I I I I I I I I I I I M I I I I I E 0~75 + ~ - +- + -+ T- I I I I I E I I _ _ _ _ _ _ _I \I I I R I I I I I I H ' __ I I I I I O.50 + __________ I ___________ I ____________.I II I I I I I I _ I I I " ",.I~ I I a 'x I I I I I ' I _ I I _ _ __ _ I I _ I I I I I I I I I I I 0.5________________________________+___ II__________ I ______ +______ I ____________ I I I I! 1- I i I I __ I I I i I I I I I I I ~~~I I _ _ _ _ _ _ I _ _ _ _ _ _ I _ _ _ _ _ _ _ I I I I I I ____________ I I I __________________ I ____________I -I I I II 0.00,+~f~+~~ e+ __ 0.0 25.0 50.0 75.0 100.0 125.0 TIME T (MILLISECONDS) Figure 42. Bubble Collapse, Run 953, R vs t.

B 1.50 ---------------- U I I I I I I B I I I I I I B I I I I I I L I I I I I. E I I I I I I I i I I I. I R I I I: I I I U~~~ D I I i RUN 954 1 I I I I _ _ _ _ _ _ _ I I I U I I I WATER.-AMMONIA I I I I I 177.4 F I I _ _ _ _ _ _ _ L_ _ _ _ _ _ _ _ _ _ 7.8 9 P S I I R I I I ~t1 --- — -------------------------------------------- WT ANI I I I M! I _ _ _ _ I I I I I i I I I L I ____ I I I I I 1 1 I I I ______ I ______I ______ N I I I7. I I E 0.75 + ~ ~~- — ~ -~ — ~L I I~-~-.*.~~~~ - ~~~-~~-*~~~ T I I I E I _ _ _ _ _ _ I I I I R I I I I I I Is I _ _ _ _ _ _I I _ _ _ _ __I _ _ _ _ __I _ _ _ _ __I P0 I I ~ tI-I I I I I I I I I I I I I I 0 50 + ---------— ~n - - - - - -- - -- ---- + --- —----- ---------- 1 1 1 I I m~~~~~~ 0 *25 +~~~~-+ I~~~~~~ T III 0~00/ —t~~~~~ ---~ —~~~~- -~.-~ ---~~~ ----~~ --- —---------------; --- —-- 0.0 25.0 50.0 75.0 100.0 15. TINE T (MILLISECONDS) Figure 43. Bubble Collapse, Run 954, R vs t

B 1.50 + --- —------------— + --- —---------------— + U I I I I I I B I I I I I I B I I I I I I L I _ I I I I I E I I I I I I I I I I I R I I I I I I A 1.25 +-~ ~ --- —— + — +. --— ~. --- —----—. ---+ D I I RUN 955 I I I I _I I I I U I I I WATER-AMMONIA I I S I I I I I I + I I 178.0 F I I I______|____ _I I 9.47 PSIA I I 1.00 + ----------------------------- --- -- + R_ _ I I I I I I M I I I I I I L I II I I I M I I i I I T I \I I I I I I I\ I I I I I R I t I I I I I H E I I I I I __I_ _ I II I I I t I I I I II I I I I \ I I I I 0 225 5 0750~0.++++ ~+ _ _ _I I I I I I I I I I 1 I I I I _ _I I I I I I I I I I 0__ 0.250 50.0 75.0 100.0 125.0 TIME T (MILLISECONDS) Figure 44. Bubble Collapse, Run 955, R vs t

-129 -Run 9535 According to the single 35 mm pictures taken subsequent to the high speed film, collapse was completed in the order of minutes after initiation of Run 9519 while for Run 953 the gas phase had vanished when the first picture was taken 48 seconds after bubble injection. Runs 954 and 955 were conducted to show the same trend, but at a relatively low temperature of T: 17'8~F, where all of the dimensionless properties for the rate of the process are greater and the solubility, x* of the gas higher (see Appendix VI-2)o The results, shown in Figures 43 and 44, indicate the corresponding increase in the rate of collapse with decreasing volatility of the liquid, and when compared to Run 591 and 953 the much more rapid collapse is evident. The first 35 mm frame from Run 954, taken 7y8 seconds after bubble initiation, shows that collapse has been completed, while this is seen to occur at 50-75 ms directly from the high speed film in the more extreme case of subcooling (x" = 0~758) for Run 955o The analytical prediction. of the initial bubbles behavior, calculated according to Equation (106), is given in Appendix VI-2 in terms of the phase-growth parameter X The results, indicating initial growth in all cases, cannot be compared directly to the experimental results because the initial bubble size, unlike the case of a nonsoluble gas, may not be calculated from its asymptotic size. The observed bubble formation times measured from the high speed film for the case of no growth, Run 511, and measured from solenoid current, Figure 26, are in tie order of 3 ms + 1 ms~ Although their smallness have been conclusively established, their exact values are not known in each experiment, and it becomes impossible to con

clude from the plots of transient bubble size if initial bubble growth has in fact been experienced. The tendency for growth, however, compared to the tendency for collapse is clearly demonstrated by comparing, for example, Run 951 to Run. 953, and noting the calculated values of the growth parameter listed in Appendix VI-2 of respectively -8.18 and -6o 04 c. The Case of Bubbles Having Translatory Motion A number of experiments were conducted for the injection of respectively nitrogen and ammonia into water with the view tank in an inverted position from that shown in Figure 17. A sufficient amount of the gas was injected to enable bubble departure and subsequent rise through the liquid under the influence of buoyancy forces. By removing the simple 1 inch objective lens from the optical system shown in Figure 24, the view field of the camera was enlarged to cover the bubble rise up to 40 mm above the injector tip. This provided for an observation time of about 140-170 ms for the case of a growing bubble and more than 200 ms for the collapsing bubble rising more slowlyo Firstly, two experiments, Runs 805 and 806, are included to show the effect of translatory motion on the rate of growth for nitrogen bubbles injected into water. The nearly identical conditions for the two experiments, listed in Appendix VI-1, were chosen to provide growth-rates great enough to ensure completion of the growth during the available observation time. In this way the initial bubble size could be calculated and comparison made with the analysis for the case of a stationary bubble. The resulting bubble growth, plotted in dimensionless representation as Q vs T, is compared to the analytical solutions for the stationary

131 bubble in Figures 45 and 46, where for clarity the time scale has been expanded compared to Figures 32-40. The effect of translatory motion as an increase in the growth.rate does not appear significant until some time after departure and acceleration of the bubble when a flow transition. has eventually occurred. This observation may be verified by comparing the bubble growth in Figures 45 and 46 to Figure 47, showing the position of the bubble center versus time for Runs 805 and 806~ Bubble departure occurs at A, while the flow transition takes place over the region marked B - C As seen from Figure 47, which also gives bubble velocity and Reynolds number, Re = 2oR'V/v, for a number of bubble positions for Run 805, the bubble is practically accelerated at the time of departure to a velocity which remains nearly constant until transition occurs. Although the Reynolds number is very large the liquid motion is believed to be laminar over this part of the bubble rise. In the transition region B - C marked by an increase followed by a decrease and again an increase in bubble velocity to a nearly constant value beyond point C, a new stable flow regime is established believed to be turbulent. In the laminar region the small departure of the measured growths, rate from the analytical prediction in case of no motion supports the "stagnant film" theory,(5) which. assumes the bubble to be surrounded by a stagnant liquid film of a th.i.ckness large compared to the penetration of the potential distribution associated with the mass transfer. In the transi= tion and turbulent regions, on the other h.and, the significant departure from the analytical prediction supports the "penetration" theory(5) which assumes full slip between the liquid and the gas phase, the latter being

1.20 + - 4. 4. +-+-. _.. R I I I I I -_ - E I.I....._... I II L III I I I A I I IAIII T I I I I I I............. _ _ _ _ _ _ _ II V I II E 1.*00 +.....L -......... - + -~+ I I- -- I- I I --- _ __ _ _ _ _ _ _ _ _ _ _ _ _ E 0 I 80 I~ I ~ ~~ ~"~I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I M ~ ~~~~~~~~~~~~ I INEIEIT THOR I IR) E 0.40~ 4.~ ++ + XEIETLPIT N I ~~~~~~~~~~~~I I I s_______ I _ _ J 13.2. A...0.... I I........_ I ~I I....2...O.3..0 I I' I _ _ _ _ _ __0 I O = 0. 0 0 P 0 I. 0 I _ _ _ _ _ I N II I I I A~ I I......I I __I ~I I I I I I I I I I I U_ I I........I......... IU ___- I.._I B I I I I I F0.4 0 1.0_ 2..0. -— + - 4.0.EPRMNA PON 5.0 __...__ NOIENIONAL TIM 130.2TAU = 100*ATR.16 8 I........ I~ ~ ~ ~~~~Fgr I5 GroLt o= MoinB 1be Run = 8.05, Q vs

1.20 + ------------ - + — R I I I I I I E I I I I I I L I I I I I I A I I I I I.I T I I I I I I I I I I I I. I V I I I I I I E 1.00 +-~+.-..-~...... ~.. --- —__ _ __~~ --- —- +~ N I I I I I I 0 T I N I I. I I I D I I I I I I I SI i - i I I I / I I I 0.60 + --- --- --------------- + --- —------- -------— + --- ~ ~.. --- ——. --- —------- --. + N I I I I I U I I RUN 806 I I N I I I I I A I I I JOINPOINTTAU I I H LI. I I I I NTERMEDIATE THEORY I I 0E I 60I X JOIN.P. INTAU2 I-...........I.. I..S 0.40 + --- —-------— +- * EXPERIMENTAL PINT -------------— + B I tl7I I I I U I I I RUN 806 II a I I I I I 0.20 + --- -+ --- —---- - --------------------- ------ ------------- ----- + IN I I I I I I I I N TERMEDIATE I I I JOIN I I U I I I ASYMPTOTIC THEORYI I I I I I I I ZI I JA = 1374 A =0.1647I 0.0 1. 2.0 3.0 4.0 1.892 I I5.0 NONDIMENSIONAL TIME 1000*TAU = 00085*AT/R.P.2 Qe__ _ __ _Figure 46. Grwth of Moving Bubble, Run 806, Q... Figure 46. Growth of Moving Bubble, Run 806, Q VS T.

40 30 I — 0 o RUN 806 v = 20.3 CM/SEC Z A BUBBLE DEPARTURE V 15.6 CM/SEC w B-C FLOW TRANSITION > 20 - 0 cn (I) V: 20.1 CM /SEC 0 A 0 20 40 60 80 100 120 140 160 180 TIME (MS.) Figuare 47. Bubble Position vs Time, Run 805 and 8o6.

B 3.00 + —..... + —+ — - + U I I I I I I B I |- _ I I I I I 8 I <a<" L I I I I I EIr I I I II RI I I I I I S I ~~~~~~~~~I I I __ ____I I R I~~~~~~~~~~~~~~~~~~ A 2.50 +- _________________+__ ------------------------------------------- - --------- - + DI I I RUN 901I I _1 I I \ I I I UI I WATER-AMMONIA I 1 ~S ~I I I I I I~ ~I\ 1 181.3 F I I I I. 9.01 PSIA I I R_ _I_ _I_ _\_ I I I I I 2.00 +.... -. --- —.- + M I I I I I I _ _ _ _ _ _ _ _ _ I _ _ _ _ _ _ _ _ _ _ _ _ __I II,, I I I I ~L I' I I I I 1_ _ _ _ I _ _ _ _ _ I \ I I I I MT~~~~ I ~I I I I I E 1-50 + t;j it iI I I _ _ _ _ _ _ _ I I I I I I I RE~~~~~ I ~I I I I R~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ S ~ ~~ ~ ~ ~~ ~~~ I I I I k }-J1 t _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ I _ _ _ _ _ _ _ _ _ __I I I I -I I ~~~~~~I I I II 1.00 - ------------— ' ----l --- —------------— l — -- -- ---------— l --- —------— X --- —l --- —-— ~ --- —-— ~~ l~ ~ ~ ~~~~ i_ I! I - ---------- -- -I I I I I I I 1. 1. I I 0.0 25.0 50.0 75.0 100.0 125.0 TIME T (MILLISECONDS) Figure 4~8. Collapse of Moving Bubble, Run 901, R vs t I I I I I I ~~~~I I I I -~~~~~~~~~~~~~ i ~~~~~~~~~~~~~~~~I IIII ] ~~~~~~~I I I II I I I I I I I I I I ~ I I I I I 0.0 ~ ~~~~ ZS. S0. 75. 10.120 TIME T {MILLISECONDS) Figure. 48. Collapse of Moving Bubble, Run 901, R vs t.

B 300 + --- ------------------------------------------ ---- -- U I I I I I I B I I I I I I 8 I I I I I I L I I! I I I E I I I I I I I I I I I I R I I I I I I A 2.50 + --— ~ ---— ~ ---~ --- —--— + — +-~+ 0 I I I RUN 903 I I! I I I I I U I I I WATER-AMMONIA I I S I I_ I I I I I I 197.8 F I I 2_ OO___ # _- I I 1182 PSIA I I R I I I I 2.00 ---- ~ ---------------------------------------- MI - I I I I MI I I I I I I I \I I I I L I I I I I L I I I I I I I I ___ _ _ _ + __ _ ___________ _____ I M I I I I I I E 1,50 --— ' - ~-~+~~-~ T 1 I I I I I E II\II.I I R I I I I I1 I _I I I O I I I I I I I I I I --------------— + --- —-------------- ---------------------------— + --- — I I I I I I I I I I ___ _ I ________ _I__ I I - I _ I I I I I I I 0.50 -i _ _ __ --- —— _ --- —--------— _- ______________ --- —-— + I I I I I I -....... t.__...__. I I _ I i I I I I I _- _I I I I I I 0.0 25.0 50.0 75.0 100.0 125.0. I I I Fu...... C s of M n__-_ TIMg T ( - MILLISEC ON DS. Figure 49. Collapse of Moving Bubble, Run 903, R vs t

40 I t3. m, - RUN 901 Q:: o RUN 903 0 30 A BUBBLE DEPARTURE B-C FLOW TRANSITION _z 0 co 20 -w I.I. 0 Z A 0 I 0 20 40 60 80 100 120 140 160 180 TIME (SEC.) Figure 50. Bubble Position vs Time. Runs 901 and 903.

well mixed by its induced toroidal motion, hence predicting higher rates of mass transfer. Secondly, Run 901 and 903 are included in Figures 48 and 49 to show the collapse of an injected ammonia bubble rising through water. The conditions for the experiments are given in Appendix VI-2. Despite the greater maximum bubble sizes and the higher volatility of the liquid, the resulting collapse times for Run 901 and 903 are comparable to that of Run 954 for the stationary ammonia bubble. The plot of bubble position versus time in Figure 50 for Runs 901 and 903 shows the characteristic flow transition marked B - C > The initial velocity oscillations over the region A - B are ascribed to the observed bubble oscillations. d. Bubble Oscillations For a number of the experiments involving the study of a stationary bubble, measurements were made of frequency and amplitude of the bubble oscillations. The results are summarized in the table in Appendix VI-3, giving Run number, the mean radius, R, for which the measurements were made, experimental frequency and the theoretical frequency calculated from Equation (2) for the spherical zonal harmonics of second mode. The agreement between observed ad analytical frequency indicates that the second harmonic is the dominating mode of oscillation, hence that the bubble shape could be taken to that of Equation (19) in the first approximation. Based on measured amplitudes in radius and bubble length along its assumed axis of symmetry, the relative equivalent radius, Req/R, and relative polar amplitude, a/R, of the oscillation were calculated. As seen from Figure II-1 of Appendix II only a qualitative agreement is observed with the analytical relationship given by Equation (II-5).

-139 -2. Experimental Uncertainty A Careful evaluation of the uncertainty of the experimental results is particularly required in the present investigation because of the extensive calculations involved in the data reduction, tending to obscure the entries of uncertainty. The basic data serving as input are, liquid temperature, liquid pressure at bubble position, and bubble volumes versus time. The uncertainty of a single temperature measurement using a Leeds & Northrup portable precision potentiometer Model 8662 is estabt lished to 0.3~F. The liquid temperature is obtained as the mean value of three readings, hence 6T = Ool7F. Despite the diverse corrections, the uncertainty on the liquid pressure remains that associated with the reading of the 30 inch mercury manometer, established to bp. = 0ol inch Hgo The reading accuracy of the bubble dimensions has been established to one or two units of the scale in the ruler system used, depending on sharpness and grain size of the filmo Sample calculations of the change in bubble volume resulting from a small change in either of the dimensions indicate it reasonable to employ 5V(I) - V(I) aH/H as a measure for the uncertainty on bubble volume, H is the measured bubble length and 5H = 15 o Based on an accuracy of 00001 inch in measuring the dimensions leading to the 3 residual volume, VR, it is found that 6VR = 0o011 mm The uncertainty of the time coordinate is related to the film speed, SPEED, read on an electronic counter connected to the camera magnetic pick-upo With a one second counting period, as used here, the accuracy for the film speed is one count in a second, or + 16 fps, hence bt/t = 16/SPEED

-140 -In executing the plots of P(T) and Q(T) as part of the digital computer program for data reduction, the plot resolution -defined as one printer space horizontally and one printer line vertically- was chosen to correspond to the uncertainty in the resultso The true nature of the data is thus not obscured by the method of reproduction. Referring again to Run 523 for P = 1o.8 for example, 6P/P = 1.4 implies that 6Q = 0~04, which should be compared to the resolution of 0.025 per line in the ordinate direction. In the absissa direction one printer space corresponds to 0.5 on 103 T while 6(103 T) = 0.12 for 103.T = 4.0 The general expression for the accumulation of error or uncertainty, Sq, on a quantity, q, given as a function of a number of observables, ai, j- (a,,a2, -+,(- 'l) each encumbered with an error or uncertainty bai is (1) g Z[ (i (Da)j $ da. (129) where the latter form of Equation (129) applies when the order of the individual contribution is nearly the same. Based on the approach followed in the data reduction and employing Equation (129) the uncertainties on the ultimate results are calculatei. Table III shows representative values for the uncertainties for an experiment (Run 523).

-141 -TABLE III CALCULATED EXPERIMENTAL UNCERTAINTIES FOR RUN 523 Quantity i Uncertainty P 2.1 Ja o03 Lu 0.5 X0 4.9 x0o 0.5 POO 1.4

CHAPTER IV SUMMARY OF RESULTS In Chapter II the bubble dynamic problem was stated for a stationary spherical bubble, and approximate solutions derived for the transient growth of the bubble containing a nonsoluble gas, growth being governed by heat transport in the liquid and mass transport in the gas. The initial growth or collapse was predicted for the case of a soluble gas bubble, including the effect of mass transport in the liquid, and a solution for its collapse in a nonvolatile liquid was derived. The governing parameters were established for dimensionless bubble growth or collapse as the Ja-number and Lu-numbers, respectively characterizing the dimensionless rate of heat and mass transport. The ratios Ja2/Lu and Ja2/Lu' were shown to represent the relative significance of mass to heat transport for the rate of the processes. In Chapter III the experimental simulation of the analytical model of a stationary bubble was described and the bubble formation time, although not known exactly in each experiment, was established to be about 3 ms, a time small compared to the duration of the transient growth or collapse studied. For the nonsoluble gas case of nitrogen and helium bubbles in water the intermediate and asymptotic solutions derived in the analysis were shown to describe bubble growth adequately within the range of parameters given in Table II. The observed bubble oscillations, consistent in frequency with the second spherical zonal harmonics, could -142 -

-143 -not be concluded to cause a detectable departure from the analytical prediction of bubble growth in which oscillations were neglected. The influence of bubble oscillations on the transient heat and mass transport processes is therefore concluded to be a second order effect. The effect of translatory bubble motion was shown to appear as a significant increase in growth rates when the rising bubble had been accelerated and a transition in the fluid motion had occurred. For the case of a soluble gas increasing collapse rates were experimentally demonstrated for increasing values of the governing rate-parameters, Ja, Lu, Lu', for decreasing liquid volatility, x 00, and for increasing solubility of the gas, x*. The collapse rates of gas bubbles having translatory motion were shown to exceed those for the stationary bubbles.

APPENDICES -144 -

APPENDIX I DEPARTURE FROM INTERFACE EQUILIBRIUM DURING MASS TRANSFER The macroscopic concepts of temperature and concentration of a component in a mixture indicate these to be continuous functions in a continuum, A non-equilibrium state owing to the transfer of heat or mass inside the continuum is manifested in gradients in these continuous functions for the potentials for transfer. Near a gas-liquid boundary, however, the gas kinetic theory associates the non-equilibrium state owing to interface transfer with a temperature discontinuity (jump) proportional to the mean free path of the gas molecules, and with a discontinuity in partial pressure of the component transferred. In the present analysis where the intra-phase energy and mass transport will be related to the interface transport in terms of macroscopic properties and potentials, the primary assumption is that the interfacial temperature discontinuity is negligible, as in all cases of a dense gas phase. The discontinuity in concentration distribution is retained and the corresponding departure from the phase-equilibrium state at the interface is evaluated for the case of the nonsoluble gas suddenly exposed to a volatile liquid. Firstly, for a pure substance exposed to its vapor the net interfacial mole flux leaving the liquid may be written in terms of the Knudsen condensation coefficient a1 as(54) 2' 1 (R )(I-1) l- ~'

-146 -where M is the molecular weight, R the universal gas constant, and Ts is the liquid surface temperature, ps the corresponding equilibrium (saturation) vapor pressure, p is the actual pressure of the vapor, and a1 may be related to the conventionally defined condensation coefficient a approximately by(40) 2c (I-2) 2 + 1 Introducing the Clapeyron equation integrated between ps and p Equation (I-l) may approximately be written as N = h ('T-T) (-3) F27T1RT (-, - (T5FTT where T* is the saturation temperature corresponding to the actual vapor pressure p and the average temperature T may be taken to T with 00oo S little error. The effect of interfacial non-equilibrium on interface mass transfer in a one-component system may then be described in terms of macroscopic properties simply by the mass flux dependent deviation from the saturation condition at the interface given by Equation (I-3). This effect causes the so called "interface resistance to transfer," which does not manifest itself physically and mathematically as a resistance at the interface, but rather is a result of the changed conditions for the intra-phase energy transport. Consider thus evaporation at a rate N > O

governed by heat conduction in the liquid. The increase in Ts above the saturation value T*, reduces the driving potential for heat diffusion to the interface, hence facilitating an evaporation rate less than that anticipated, were the interface temperature taken to Ts a Secondly, for the case of a binary ideal gas-liquid system the above analysis applies when Equation (I-1) is replaced by equations of the form N =) ( i=A2B (I-4) where xis is the molefraction of component i in the gas at the interface and Pis is the vapor pressure of component i that would be in equilibrium with the liquid at temperature Ts and liquid composition xi o For the nonsoluble gas case, NB = 0 and the mole flux for the is volatile liquid component A becomes NA (xiAXS) 1 ('-5) NW MA RT where xs is the actual gas composition at the interface, and xs is the equilibrium composition corresponding to the actual temperature of the liquid XS= M$T) I ) (I-6)

-148 -Accounting for the departure from equilibrium at the interface the simultaneous intra-phase heat and mass transport in respectively the liquid and gas is subject to Equations (I-5) and (I-6), rather than the phaseequilibrium relation, xs = f(Ts,P ). The resulting actual values of the interface potentials, T and xs, are such that the driving potentials for both intraphase heat and mass transport are reduced, The departure from a state of phase-equilibrium discussed above is small, except for extremely high rates of interface mass transfer. The mathematical idealization - physically approached in the gas injection process - of a sudden step change in thermodynamic state in a gas-liquid system predicts unbounded mass transfer rates during the first moments of the transient when a state of phase-equilibrium is assumed to prevail at the interface, The latter assymption is indirectly demonstrated to be satisfactory for the present study by evaluating the encountered departure from phase-equilibrium according to Equation (I-5) from the mass transfer rates predicted from the analytical model. Thus, from Equation (30) the evaporating mole flux for the expanding bubble is NA R (I-7) Employing Equation (55) for small values of time when 4 A, < 1 Z 3 Equation (1-7) becomes NAt - e 4,,, + 7. (I-8) L

-149 -Equating this result to the expression in Equation (I-5), the departure from equilibrium is given by * xT o -t A3 1 f (I-9) A representative numerical evaluation of Equation (I-9) is based on the following data for Run 523 taken from Appendices V-4, Part 2(a) and VI Ro = 0.0877 cm l' = 0.1675 x 10-2 cm2/sec T =T = 205.4 OF = 369.7 ~K Ao = 247.8 RHO0 = 4,615 x 106 erg/grams/OK using a lower value of al,H20 = 0.5 as measured for condensation by Nabavian!40) The result Xj-.. 7.= 0.384 X o '/ + Aj where the time unit is seconds, is valid for t << 14 ms. At one millisecond after the start of the process this expression indicates a deviation in interface composition of xs - xs = 0.0018, which is negligible compared to the driving potential for mass transfer, xs - xo = 0.5544 (see Appendix V-4, Part 2(a)).

APPENDIX II BUBBLE OSCILLATIONS For the case of axisymmetrical potential flow in the incompressible liquid surrounding a gas bubble, Laplace's equation becomes (see e.go, Lamb (33) ))) d L(4A2) l n (n+ 4) Y = 0 (II-1) in terms of spherical zonal harmonics Y of order n, where = cos 4 and r is the polar angle. Equation (II-1) is Legrendre's equation satisfied by the Legrendre polynomials of first and second kind (e,) = A Pn(,) B Qn (11-2) Since for n = 2 the second term of Equation (II-2) diverges for 1 = + 1, the second zonal harmonic becomes P2(p), and the bubble shape for a harmonic undamped oscillation is given by r = rf) = (a) Sin ( (II-3) where a/R is the relative polar amplitude. The variable bubble volume, V(t) is obtained by integration of Equation (II-3) V(t) = 2 T r2sinyf r sin ldyt (II-4) V~~~~~~~~~~~Cz4

-151 -Introducing the equivalent radius, Req, as the radius of a sphere having the same volume as that of the oscillating bubble, V(t), the relative amplitude in equivalent radius becomes Re9 = 1 5 () skin6t + 7 (R S1 n ad i 3R)t.1 ( Figure II-1 shows a plot of Req/R versus a/R, indicating that the bubble of oblong shape has a smaller volume than that of the oblate shape, Hence, the effective mean radius is always less than the arithmetic mean of the maximum and mininum values. For relative polar amplitudes less than 0.2, however, this deviation is negligible.

-152 -R q/R 1.2 0 EXPERIMENT - THEORY, Eq.(I- 5) -0.4 a\ A 2 0X1 - 0o 0.9 Figure II-1. Relative Equivalent Radius vs Relative Polar Amplitude for Second Zonal Harmonic Bubble Oscillation.

APPENDIX III APPLICATION OF THE INTEGRAL TECHNIQUE TO A PROBLEM OF SPHERICAL PHASE -GROWTH The usefulness of applying the integral technique to the problem of spherical phase-growth may be illustrated by the problem of bubble growth in a boiling one-component liquid for which an exact solution exists, The model used considers the growth of a single stationary spherical vapor bubble, initially of zero size, growing in an inviscous infinite liquid of initial uniform superheat, Under the assumption of a single component system and in the absence of surface effects and pressure gradients, the interface temperature remains constant at the saturation value throughout the process of growth, which is now governed solely by the heat transport in the liquid surrounding the bubble. Based on the use of the parabolic one-parameter profile of Equation (34) the formulation is analogous to Equations (45) and (47) leading to the form d 3 R, pjf, j-3 R + R2.. t. {0a=)J (.R3 (III-1) RR - 2~ jc ()R (III-2) The characteristic temperature in the Ja-number is in this case the liquid superheat, hence Ja = (T.-T*)o(p'c'/p1"hfg) o Since the Ja-number is -153 -

-154 -constant, Equation (III-1) may be integrated in its entire form, subject to the initial condition R(O) = 0. The result, R 2 R 4 l d- (- i0 (f ) -2 (III-3) indicates the constancy of (R/fT), hence the similarity nature of the solution, namely, the thermal boundary layer thickness, fT ' growing proportional to the bubble radius, R. The form of the solution as obtained from Equation (III-2) is R = 2 A t (III-4) where the growth constant A (Jamb) EJe (III-5) is an implicit function of the Ja-number and the density deficiency E = 1 - p"/p', through (R/fT) given by Equation (III-3). The growth-constant X as function of the Ja-number with c as parameter is compared in Figure III-1 to the exact solution by Scriveno(55) Evaluating the Ja-number at the interface condition this solution is written 2 p L- 2E 3i A = 2 A2~e? [A2(12 2 ex2 2X C1x (iii-6) A

-155 -Despite its simplicity, the integral technique produces a rather close approximation to the comparatively much more complicated exact solution of Equation (III-6), While the vertical asymptotes for e < 1 are identical in the two solutions, the case, E = 1, X ->oo, deviates slightly For completeness the asymptotic solution, =- 1, X - oo as obtained by a number of methods is listed in the table in Figure III-1o It may be noted that neither of the investigators, References 18 and 46, included the effect of a density deficiency different from unity, rendering their findings less general.

-156 -BUBBLE GROWTH CONSTANT X(Ja,f) FOR ONE COMPONENT BOILING R= 2X A-t - EXACT SOLUTION(SCRIVEN 55) INTEGRAL TECNIQUE(PARABOLIC PROFILE) COMPARISON BETWEEN ASYMPTOTIC SOLUTIONS 10 f:I, a-o / — o METHOD X/Ja EXACT (SCRIVEN 55) i/7;:0.987 I /, APPROX. (PLESSET, ZWICK 46): 0.97 SOURCE THEORY (FORSTER,ZUBER 48)=/2 0.8871 // INTEGRAL TECHNIQUE 1.00 / I, / // Ph L // ~~~O*'~~~~~~~ /~ ~/,,// oo=(TO=-T) pc P"hfg Figure III-1. Comparison Between Exact Solution and Solution by Integral Technique. Bubble Growth in One-component Superheated Liquid.

APPENDIX IV CRITERION FOR LINEARIZATION OF EQUATION (65) The criterion, Equation (67), for the validity of the linearization of Equation (65) by retaining its last two terms is obtained as follows, Writing for simplicity the polynomial form, Equation (65), as LU3 + Ad Lt -t 2 L a 3 0 (IV-1) the coefficients can be expressed in terms of the three roots, U1, U2 and U3, of Equation (IV-1) as follows ad = 10 - 2 K1/N - -a 4 2- -U3 a? =-25 1 10 KLIf/N U- +U l.l i2 j l,3 I. (IV-2) a3 - 50 K M/N = 12 L13 The linearized solution, U* = a3/a2, in terms of the three actual roots become s U1i + + Uh (IV-3) U_ + U SL2 U3 Substituting from Equation (IV-2) to eliminate U2 and U3, Equation (IV-3) is written N1X -l )- (Iv-4h) -157 -

-158 -From Equation (IV-4) it is seen that U* = U1 whenever U~ (Ua+ta~).... <'<' "l (IV-5) U1 (Ult L1) + aIntroducing in this case U1 a3/a2, the criterion becomes |0 (a(< d (IV-6) IV2 ar2 which is the form of Equation (67) when the coefficients from Equation (IV-2) are substituted.

APPENDIX V DATA REDUCTION BY DIGITAL COMPUTER -159 -

APPENDIX V-1 -.,.PROGRAM FOR NONSOLUBLE GAS CASE **** DYNAMICS OF NONSOLUBLE GAS BUBBLE - INJECTION COOLING **** PART (1). REDUCTION OF EXPERIMENTAL DATA.. INTEGER _I,N RUN, I1, NUM,J,JO,J1, I2. __ __ *001 DIMENSION V(300), R(300), Tt300) *002 DIMENSION P(300),TAU(300),Q(300),TAU1(10) Pl(10),Ql(10),.003 1 TAU2(25) P2(25),Q2(25) *003 VECTUR_VALUES INPUT_ =$.4tBF4OS _ _ _,_$ *004 VECTOR VALUES. OUTPUT = $14, 6F15.4*$' *005 BEGYND READ DATA *006 PRINT COMMENT $1BUBBLE GROW[H NONSOLUBLE GAS$ *007 PRINT RESULTS RUN *008 PRINT COMMENT $OINPUT$ *009 PRINr COMMENT SO I S HI L2 Y4 Y3 Y2 YI YOS *010 PRINT COMMENT $ ~~~. --- —------------------------—.S$ *011 EXECUTE ZERO.(T(O)...T(300)) *013 BEGIN READ FORMAT INPUT, IS,H1,L2Y4,Y3,4Y Y2,YLYO *014 PRINT FORMAT INPUT I,StHI,L2Y,Y4Y3,Y2,YI,Y0 *015 BO = YO *016 B8 = -3.*Y4 + 16.*Y3- 36.*Y2 + 48.*Y, - 25.*YO *017 2 = 11.*Y4 - 56.*Y3 + 114.*Y2 - 104.*Yl + 35.*YO *018 83 = -3.*Y4 +14.*Y3 - 24.*Y2 + 18.*Y1 - 5.*YO *019 B4 = Y4 - 4.*Y3 + 6.*Y2 - 4.*Yi + YO *020 INT -- (B4*32./9).P.2 + 4*83*128./9 + 84*B2*128./63. *021 1 + (83*16./3.).P.2/7. + 84*B1*32./27. + B3382*32./27. *021 2 + B4*B0*64./15. + B3*B1*32./45. + B2.P.2*4./45. *021 3 + B3*B0*8./3. + B2*81/9. +.B2*B04../9. + *B / 1.P.2/27. *021 4 + 81*BO/3. + BO.P.2 *021 H = HI/S *022 V(I) = (3.141593*INT*H/4.)* (LI/LZ).P.3 *023 R(I) = (0.75*V(I)/3.141593).P.O.3333 _024 T(I) = Si*I/SPEL -*025 WHENEVER I.L. N, TRANSFER ro BEGIN *026 THROUGH SW6, FOR I = I, 1, I.G. (N-1) *027 WHENEVER T(I).NE. (S1*1/SPEED), TRANSFER TO SW6 *028 WHENEVER VIMI.G.VIM2.AND. VIMi -G. V(I).AND. VIM2.NE. 0. *029 WHENEVER NUM.E. 0 *030 TRANSFER TO SW6A *031 OTHERWISE *032 TRANSFER TO SW7 *033 END OF CONDITIONAL *034 OTHERWISE *035 WHENEVER NUM-.E. Ot TRANSFER TO SW6B *036 END OF CONDITIONAL *037 SW6A NUM = NUM + 1 *038 SUM = SUM + V(I) *039 SW6B VIM2 = VIML *040 VIMI = V(I) 041 SW6 CONTINUE *042 SW7 VMAX = SUM/NUM *043

DV = V(N)/VMAX - 1..044 WHENEVER DV.L.E5.AND. OV.GO..ODR. VMAX.E. 0.t VMAX=V(N) *045 VO = (VMAX+VR)*(1.-XOU) + XOU*(V(O)+VR) *046 XO = XOU*(VR+V(O))/VO *047 PTH = (VMAX/(VO-VR)).P.O.3333. *048 PO =((1.-XO)/(L.-XOO)).P.0.3333 *049 RO = (.75*VMAX/3.141593).P.O.3333/POO -050 TAUDEL = TDELAY*A/(RO.P.2) -051 TAUMS = A/(RO.P.2) *052 ALPHA = A *053 PRINT COMMENT $1REDUCED EXPERIMENTAL DATA$ -.*054 PRINT RESULTS SPEEDONRUN *055.. PRINT RESULTS AL1,VR.*056 PRINT RESULTS VO, RUOt XO *057 PRINT RESULTS XOO, PO, PTH... _058 PRINT RESULTS VMAX, NUMI1 *059 PRINT RESULTS E5,TDELAYTAUDEL.*060 PRINT CUMMENT $01I T(I) MSEC R(I)' MM V(I) -061 I CU.MM TAU * S P Q$ *061 PRINT COMMENT $ --— ~ --- — ----- ------— 062 JI D *062 1 -------------— 7 —7. —._ --- —.-7 --- —--------- 7-6 J= 0 -63 THROUGH BETA, FOR I = 0, I, I.G. N _064 WHENEVER T(I).NE. (SI*I/SPEED), TRANSFER TO BETA *065 TAU = A*T(I)/(RO.P.2) *066 P = R(I)/'RO *061 q = (P - 1.)/(POO - 1.) t068 PRI'NT FORMAT OUTPUT, i,T(I), R(I)-,'V(I),T"pAU;'P-Q..069 J = J + *_070 TAU(J) = TAU + TAUDEL *071 P(J) = P m072 Q(J) = Q *073 BETA CONTINU- *074 JO = J................*075.... PART (2) ANALYTICAL SOLUTION -'*.. PART '2A) SMALL VALUES OF TAU INTEGER I, E2, RUN, J *076 DIMENSION XS(IO), (100L) -077 VECTOR VALUES FIRST = $ 4F2-0.4'*$ *078 VECTOR VALUES SECOND = $ 8F14.4 *$ *079 PRINT COMMENT SITHEORETICAL BUBBLE GROWTH - NONSOLUBLE GAS$ *080 PRINT RESULTS J-ASL,E4 *081 PRINT RESULTS LU,EItE7.........*082 PRINT RESULTS XOO,E2 *083 PRINI RESULTS XOE3 *084 PRINT COMMENT $OSMALL VALULS OF TAU$ *085 D(O)= IXOO - XO)/2. *086 C = LU/((1.-XO)*JA.P.2) *087 THROUGH SWL, FOR I = O, It I.G. E2 *088 F = D(I).P.3 + (l.-XOO-C)*D(I).P.2 + 2.*C*(XOO-XO)*0(I) - *089 1 C*(XOO-XO).P.2 *089 FF = 3.*D(I).P.2 + 2.*(1.-XOO-C)*D(I) + 2.*C*(XOO-XO) *090 D(I+L)= D(I) - F/FF *091 SWI WHENEVER (.ABS.(D(I+1)/D(I)-I.)).L. El, TRANSFER TO ALFA *092 ALFA 0(0) = D(I) *093 XS(O) = XUO - 0(0) *094 AO = (JA*(XOO - XS(O))).P.2 *095

-162 -UX = 10. _096 P1 = (1.-3.*(XS(O)-XO)/(5.*(l.-XO))).P.-O.3333 *097 DTAU = Sl*({(P.P.3+2.)/PI-3.)P/(4**AO)/E3 098 PRINT RESULTS XS(O),AO,I *099 PRINT COMMENT $0 1000*TAU P *100 1 UX QS *100 PRINT COMMENT S --------------------------------- 101 1.................................$ *101 THROUGH SW2, FOR TAU = OTAU,OTAU, UX.LE. 1. *102 A = 4.*AOeTAU/3../S 1 *103 P = 2.*SQRT.o(I+A)'*COS.(1.O472-ATAN.{{(I.+A).P.3-1.).P.O.5) *104 1 /3.) *104 UX = (XS(O)-XO)/((I.-XO)*1.-l./P.P.3))L *105 Q = (P-1.)/(POO-1.) *106 SW2 PRINT FORMAT FIRST, TAU, P, UX, Q *107 *** * PART (2B) JOIN POINT TAUL......L.._,,... _....... PRINT COMMENT $OJOIN POINT TAUIS._..__.__.... __*108 C = C*0.6 *109 THROUGH SW3. FOR I = O, 1, I.G. E2 *110 F = D(I).P.3 + (1.-XOO-C)*D{I).P.2 + 2.*C*(XOO-XO)*D(I) - *111 I C*(XOO-XO).P.2 *111 _ FF = 3.*O(I).P.2 + 2.*(1.-XUO-C)*D(I) + 2.*C*(XOO-XO) *112 D(I+1) = D(I) - F/FF *113 SW3 WHENEVER (.ABS.(D(I+1)/D(I) - 1.)).L. El, TRANSFER TO BETAL *114 BETA1 XS(1) = XOO - D(I) *115 Pl = (1.-3.*(XS(1)-XO)/(5.*(l.-XO))).P.-O.3333 *116 P1OOT = 2.*LU*(XS(l)-XO)/(PI*(1.-XS(1)))__ *117 UTL = Pl*PlOOT/(2.*JA*(XOO-XS(1))) *118 TAUI = Sl*((P1.P.3+2. )/P1-3. )./(4.*AO) __ *119 POO = ((1.-XO)/(l.-XOO)).P.O.3333 *120 Qi = (P1-l.)/(P0-1.) *121 PRINT RESULTS TAUlP1,PlOOT *122 PRINT RESULTS QlLXS([1LUT.*123 TAU1 = TAUl + TAUDEL *124 **** PART (2C) INTERMEOIATE VALUES OF TAU PRINT COMMENT SOINTERMEDIATE VALUES OF TAUS *125 PRINT COMMENT $OTAUH/TAUM = RELATIVE SIGNIFICANCE OF HEAT TRA *126 I NSPORT TO MASS TRANSPORT FOR TAUS *126 PRINT COMMENT $SO S1*TAU P PDOT *127 1 UT OXS Q TAUH/TAUM CR *127 2 ITI$ *127 PRINT COMMENT ------------------------------------ *128 1 _ _ *128 2 ----------— $ *128 WO = I./P0 *129 CH = S1/(2.*(JA*(l.-XO)).P.2.WO'.P.8) *130 CM = 2.*S1/(5.*LU*WO.P.2) *131 W1 = P1/POO *132 DW = (I.-W1)/E4.. _- -----------.133 J = O *134 TAU2 = 0. *135 THROUGH SW4, FOR W = W1+OW, OW, W.G. (1.- oW) *136 J = J + 1 *137 TAUH = (W.P.2 - W1.P.2)/2. +*138 1 (1.-WO.P.3)*(W.P.2/(l.-..P.3) - W1.P.2/(1.-Wl.P.3))/3. - *138 2 ((5.-2.*WO.P.3) /9.)* *138

-163 -3 (ELOG. _SQRT.(( 1.+W+W.P.2)/(1.+Wl+WI.P.2))*(I.-Wl)/(1.-W)) - *138 4 SQRT.(3.)*(ATAN.((2.*W+1.)/SqRT.(3.)) - ATAN.((2.*WI+l,)/ '138 5 SQRT.(3.)))) *138 TAUM = -(W.P.2 - WL.P.2)/2. + (1./3.)* *139 1 (ELOG.(SQRT.(( 1.+W+W.P.2)/(l.+WI+WI.P.2))*(1.-Wl)/(l.-W)) - *139 2 SQRT. (3.)*(ATAN.((2.*W+1.)/SQRT.(3.)) - ATAN.((2.*Wl+1.)/ *139 3 SQRT.(3.)))) *139 TAU = TAUl + CH*TAUH + CM*TAUMH 140 RATIO = TAUH*CH/CM/TAUM *141 P = W/WO *142 PPDOT = SL*(1.-W.PP.3).P.2/(CH*W.P.3*(W.P.3-WO.P.3)+ *143 1 CM*W.P.3*(1.-W.P.3))/WO.P.2 *143 PDOT = PPDOT/P *144 UT = SQRT.(PPDOT/(2.*(1.-1./P.P.3))) *145 XS = XOO - S4RT.((1.-1./P.P.3)*PPDOT/(2.*JA.P.2)) *146 Q = (P-1.)/(POO-1.) *147 DXS = XS -- 1. + (1.-XO)/P.P.3 *148 WHENEVER TAU2.E. O. *149 WHENEVER.ABS.(DXS/XS).L. L7 *150 TAU2 = TAU - (E7-DXS/XS)*(TAU-TAU21)/(DXSI/XS1-DXS/XS) *151 P2 = P - (E7-DXS/XS)*(P-P21)/(DXS1/XS1-DXS/XS) *152 Q2 = (P2 - 1.)/(POO - 1.) *153 OTHERWISE *154 DXS1 = DXS *155 XS1 - XS *156 TAU21 = TAU *157 P21 = P *158 END OF CONDITIONAL *159 END OF CONDITIONAL *160 K = JA.P2/LU *161 L = 1. - XoO *162 NN = 1. - 1./P.P.3 *163 M = (XOO-XO) - NNe(1.-XO) *164 Al= 10. - 2.*K*L.P.2/NN *165 A2 = 25. 4 20.*K*M*L/NN *166 A3 = 50.*K*M.P.2/NN *167 CRIr1 = A3*(A3+AI1A2)/A2.P.3 *168 TAUl(J) = TAU + TAUDEL *169 P1(J) = P *170 Q1(J) = Q *171 SW4 PRINT FORMAT SECOND,TAUPPDOTUT,DXS,Q,RATIOCRIT1 *172 J1 = J *173 PRINT RESULTS CH, CM *174 '*** PART (2D0) JOIN POINT TAU2 PRINT COMMENT $OJOIN POINT TAU2$ *175 TAU2 = TAU2 + TAUDEL *176 PRINT RESULTS TAU2,P2,Q2 __ *177 "*** PART (2E) ASYMPTOTIC SOLUTION VECTOR VALUES ARG - O.,.1t,.2.3.4,.5,.75,1.,1.5,2.,2.5, *178 1 3..,3.5,4.,4.5,5.,7.,8.,10.,15.,20.,30. *178 VECTOR VALUES ER = 1.,.8965,.8090,.7346,.6708,.6157,.5069, *179 I.4276,.3216,.2554,.2108,.1790,.5529,.13699,.122485, *179 2.110705,.08059,.070524..05642,.037613,.02821,.018806 *179 VECTOR VALUES THIRD -S 3F2(i.4$ ** 180 PRINT COMMENT $SASYMPTOTIC SOLUTION $ *181 PRINT COMMENT SO SI*TAU P *182

-164 -Q$ *182 PRINT COMMENT ------------------------------------ *183 1 ------------— $ 183 G = 2.*JA*(XOO-XO)/(POO.P.2-1.) *184 THROUGH SW5At FOR J=1, 1, J-.G21 _185 P2(J) = SQRT.(POO.P.2 - (POO.P.2-1.)*ER(J)) *186 TAU2(J) = S1*(ARG(J)/G).P.2 *187 WHENEVER P2(J).G. — P2.AND. J.G. 1 *188 TAUNUL = TAU2(J-1) + (P2-P2(J-1))*(TAU2(J)-TAU2(J-1))/ *189 1 (P2(J)-P2(J-1)) *189 TRANSFER TO SW58 *190 END OF CONDITIONAL *191 SW5A CONTINUE _192 SW5B I = 1 *193 THROUGH SW5,FOR J = 0, 1, J.G. 21 *194 TAU = S1*(ARG(J)/G).P.2 + TAU2 -TAUNUL *195 P = SQRT.(POO.P.2 - (POO.P.2 -1.)*ER(J)) *196 Q = (P-1.)/(POO-l..) *-197 WHENEVER P.G, P2, I = I + 1 *198 TAU2(I1 = TAU + TAUDEL *199 P2(I) = P *200 Q2(1) = Q *201 SW5 PRINT FORMAT THIRD, TAU, P. Q *202 12 = I *203 =""* PART (3) PLOT OF ANALYSIS AND EXPERIMENT INTEGER ORA., 204 INTEGER IMAGE *205 INTEGER LINEL,IVCtIHC, IRUN,INOM 206 INTEGER IJA,ILU,IXO,IXOO, IAIROIPOO,CRUNCACU RO,GAS,GAS2 *207 DIMENSION IMAGE(1000) *208 VECTOR VALUES NSCALE = 1,O,2,Oi1 *209 VECTOR VALUES ORDP = $ NONDIMENSIONAL BUBBLE SIZE PS *210 VECTOR VALUES ORDQ = $ RELATIVE NONDIMENSIONAL BUBBLE SIZE Q$ *211 PRINT COMMENT S8PLOT OF P(TAU) AND Q(TAU)$ S212 PRINT COMMENT $00 JOIN POINT TAUIS *213 PRINT COMMENT $. INTERMEDIATE THEORY$ *214 PRINT COMMENT S X JOIN POINT TAU2$ *215 PRINT COMMENT $ + ASYMPTOTIC THEORYS *216 PRINT COMMENT $ * EXPERIMENTAL POINTS$ *217 PRINT RESULTS TAUMS *218 PRINT COMMENT $+ *219 1 1 MILLISECOND = TAUMS$ *219 PRINT COMMENT $1$S 220 PRINT COMMENT $-$ *221 EXECUTE PLOTI.(NSCALE,NHL,HSBHtNVL,NSBV) *222 EXECUTE PLOT2.(IMAGEXMAXXMINYMAXtYMIN) *223 EXECUTE PLOT3.($SO$,TAUlP1,L) *224 EXECUTE PLOT3.($.$,TAU( 1),P( 1),.J1) *225 EXECUTE PLOT3.(S+$S,TAU2(1),P2(1).tI2) *226 EXECUTE PLOT3.($X$,tTAU2,P2,.1) *227 EXECUTE PLOT3.($*$,TAU(1),P(1),JO) *228 EXECUTE TITLE. *229 EXECUTE PLOT4.(29,ORDP) *230 PRINT COMMENT $0 *231 I NONDIMENSIONAL TIME 1000*TAU = 1000*A*T/RO.P.2$ '231 PRINT COMMENT $1$ *232 PRINT COMMENT $-$ *233 EXECUTE PLOT1.(NSCALEtNHL1,NSBH1,NVLINSBV1) *234

-165 -EXECUTE PLOT2.(IMAGE,XMAXl1XMINLYAXlMAX,YMIN1) 235 EXECUTE PLOT3.(SOS,TAUl,Ql,l) *236 EXECUTE PLOT3.($.TAU1(1)Qll)t 1),J1) *237 EXECUTE PLOT3. ($+STAU2( 1),Q2( 1), 12) 238 EXECUTE PLOT3.($XS,TAU2,Q2,1) *239 EXECUTE PLOT3.(S*S,TAU(l),Q(1),JO) *240 EXECUTE TITLE. *241 EXECUTE PLOT4.(389ORDQ) *242 PRINT COMMENT $0 *243 1 NONDIME.NSIONAL TIME. 1000*TAU = 1COO*A*T/RO.P.2$ *243 e+*e. T.ITLE FOR PLOT INTERNAL FUNCTION *244 ENTRY TO TITLE. *245 IA = 10000*ALPHA *246 IRO = RO *247 IRO = (9*IRO + RO)*1000 *248 IXO = 1000*XO *249 IPOO = POO *250 IPOO = (9*1POO + PO0)*1000 *251 IXOO = 1000*XOO *252 CRUN = $ 000) *253 CA = $0.00000 *254 CRO = $ 0.000$ *255 LINEL = (NSBV*NVL+1)/6 + 1 *256 THROUGH SW8,FOR I = IVC, LINEL, I.G.(IVC+2*(HSBH-1)*LINEL) *257 SW8 IMAGE(I) = $ $ *258 THROUGH SW9, FOR I = IHC, 1 I.G. (IHC+NSBV/3) *259 SW9 IMAGE(I) = $ S *260 IMAGE(IRUN) = $RUN $ *261 IMAGE(IRUN+1)' = ORA.(BNBCD.(RUN),CRUN) *262 IMAGE(INOM) = $SO JOIS *263 IMAGE(INOM+1) = SN POINS *264 IMAGE(INOM+2) = ST TAU1S *265 I = INOM + LINEL *266 IMAGE(I) = $. INTS #267 IMAGE(I+1)'= SERMEDIS *268 IMAGE(I+2) = SATE TH$ *269 IMAGE(I+3) = SEORY $ *270 I = INOM + 2*LINEL *271 IMAGE(I)'= $X JOIS *272 IMAGE(I+1) = SN POINS *273 IMAGE(I+2) = ST TAU2S *274 I = INOM + 3*LINEL *275 IMAGE(I) = $+ ASYS *276 IMAGE(I+1) = SMPTOTIS *277 IMAGl(I+2) = SC THEOS *278 IMAGE(I+3) = SRY $ *279 I = INOM + 4*LINEL *280 IMAGE(I) = $* EXPS *281 IMAGL(I+I) = SERIMEN$ *282 IMAGE(I+2) = STAL PUS *283 IMAGE(I+3) = SINT $ *284 = INOM -1 + 6*LINEL *285 J = I *286 IMAGE(I) = S JA = S *287 IMAGE(I+1)= IJA *288 IMAGE(I+3) = $ A = S *289 IMAGL(I+4) = ORA.(BNBCD.(IA),CA) *290

-166 -I = J + LINEL _291 IMAGE(I) = $ LU = $ *292 IMAGE(I+1) = ILU *293 IMAGE(I+3) = $ RO = $ +294 IMAGE(I+4) = ORA.(8NBCD.( IRO),CRO) _ ____ *295 I = J + 2*LINEL *296 IMAGE(I) = $ XO.= $ 297 IMAGE(I+1) = ORA.(BNBCD.(IXO),CKO).298 IMAGE(I+3) = $POO = $ *299 IMAGE(I+4) = ORA.(BNBCD.(IPOO),CRO) *300 I = J *+3*LINEL *301 IMAGE(I) = $XOO = $ *302 IMAGE(I+1) = ORA.(BNBCD.(IXOO)CRO) ___ *303 IM.AGE(I+3) = $WATER $ *304! = J + 4*LINEL *305 IMAGE(I+3) = GASI *306 IMAGE(I+4) = GAS2 *307 FUNCTION RETURN *308 END OF FUNCTION ___ *309 TRANSFER TO BEGYND -310 END OF PROGRAM *311

APPENDIX V-2 NOMENCLATURE FOR COMPUTER PROGRAM FOR THE CASE OF A NONSOLUBLE GAS AO Ao in Equation (56) Al Coefficient in Equation (IV-2) A2 Coefficient in Equation (IV-2) A3 Coefficient in Equation (IV-2) ALPHA l' (mm2/sec) ARG b NT in Equation (84) (table) A Ol' and 4 AOT in Equation (55) BO4,B4 intermediate coefficients in INT CH CH in Equation (69) CM CM in Equation (69) CRIT1 Equation (67) C Lu/(Ja2(1-xo) ) D x0X - X DXS xs x in Equation (85) DTAU AT, time increment DV v(N)/vMAX - 1 DW increment for w in Equation (69) ER eb2Terfc b IT in Equation (84) (table) F function from Equation (56), F = 0 FF aF/ 6D G b in Equation (84) -167 -

-168 -HH1 bubble length in Equation (126), H - H1/S I frame number Ii frame number from which VMAX is averaged INT integral of Equation (127) JA Ja-number K Equation (66) L1 actual reference length of injector tip L2 measured reference length of injector tip L Equation (66) LU Lu-number M Equation (66) N maximum number of pictures for a run NUM number of frames used in obtaining the average value of VMAX P1DOT P1 P1 P1 P2 P2 PDOT P POO P 00 PPDOT PP P P, dimensionless bubble size P = R/Ro PTH theoretical value of Po including the effect of residual volume Q1l Q2 Q2 Q Q, relative dimensionless bubble size Q = (P-l)/(P -l) RO Ron initial radius in mm O~

-169 -RATIO CMIM/CHIH in Equation (70) R R, bubble radius in mm RUN run number S1 scaling factor for time (Sl=1000) SPEED film speed (fps.) S scaling factor for ruler system used in measuring bubble dimensions SUM sum of volumes used for determining VMAX TAU1 T1 in Equation (60), join point TAU2 T2 join point TAUDEL time delay, (here zero) TAUH TH in Equation (69) TH = CH IH TAUM TM in Equation (69) TM = CM IM TAUMS nondimensional time corresponding to 1 ms TAUNUL T0 in Equation (84) TAU T dimensionless time T = c't/R2 TDELAY time delay (here zero) T time in milliseconds UT1 (R/fT)1 in Equation (64) UT (R/fT) in Equation (32) UX (R/fT) in Equation (52) VO Vo initial bubble volume in mm3 VMAX asymptotic bubble volume in mm.3 -VR residual volume Figure 30(a) V bubble volume in mm3

-170 -WO wo = 1/Poo W1 W1 = P1/po W w = P/P 0 XO x XOO x XSI Xsl XS Xn interface composition Yl.o Y4 bubble diameters, Yo.. Y4 in Figure 30(e)

-171 -APPENDIX V-3 DATA INPUT FOR RUN 523 RUN=523, El= 005,E2'=20. rE3= 1. ~ E4=10. E5=. 1, VR=-. 306, =2. 082 TIELAY,= S11.:= iO0.,_. NHL=6 HSBH='8,NVL=5,NSBV=20,XMI N=0.XMAX= 50,YMIN=l., YMAX=2. 2, NH L1=6. NSRH 1=8 NVI 1 = 5 NS V 1 = 20, XMT N1 =0,XMAX1 =50o.,YM T N =0,YMAX 1 =12. IVC=435, IHC=551, I RUN=451, I N OM=484 GAS= $N T ROG $.GAS2=$N $, 7=.7., -05. RUN=523 SPEED=1008 *,JA=91 * 7,LU=185 * XOO=.8931,A=16751, I 1-170.N=195, I J A =$ 9.77$. Tl j=$ 1.$* 000 001 058. 127 094 0 8 055. _40.000__ 001 001 068 127 160 167 163 130 000 002 00.1 084 127 160 203 205!.7._6 -000 003 001 096- 127 128 207 109 1'60 000 004 001 115 127 128. 179_198_160 000..__ 005 001 128 1'27 127 165 186 180 000 006 -.01 31 9 127 125 146 179 186 000 -007 001 159.127 126 1.49 169 175 000 008 001 177 127 126 156 160 160 000 009 001 195 127 125 160 1'56 138 000 010 ool 20_7 127 126 161 154 127 000 01'1 001 222 127 125 163 150 119 000 012 001 230 127 125 165 155 114 000 013 001 204 127 125 165 166 136 000 014 001 179 127 126 166 173 164 000 015 001 163 127 126 164 188 176 000 016 001 154 127 126 176 193 194 000 01'8 001 141 127 126 194 221 191 000 020 001 134 127 150 220 220 179_Q. 00 182 001 205 1_2L127 19_8_82 0.3T 4. 000 184 001 189 127 129 200 206 173 000 186 001 173 127 129.200 215 19 Q..._._ "188 001 161 127 129 208 226 191 000 190 O01 155 127 190 223 23L1.'1 000 192 001 150 127 170 233 231 186 000 1 93 001 09'4 067 067 1 14 120 1 03 o00 195 001 -093 067 0'67 113 118 099 000

-172 -"Mi APPNI PRINT-0UT FOR RUN 523 PAM 1 BUBBLE GROWTH NONSOLUBLE GAS RUN = 523 (4)~ INPUT I S HI L2 Y4 Y3 Y2 Y1 YO ( '$I-~L Co' I0~ D)....... 0 L 58 127 94 84 55 40 0 88 L 171 127 127 200 214 194...0......!! 68 127.160 167 163 130 0 90 1I:159 127 125 204 227 199 0 2! 84 127 160 203 2 0.5__1_76....0............92 1 ]149 127 145 2'15 230.199 0.3! 96 127 128 207 109 160 0 94 I 144 127 170 2':-) 2-28 1 83'......0...... 4 1 115 127 128 179 198 160 0 96 1 144 127 183 230 224 1'76 0 5! 128 127 1.27 165 186 ].80 0 98 ]. 143 127 1.60 226 224 1.86 0 6 i 139 1.27 125 1.46 179 186 0 1O0 1 1.52 127 130 216 230 1.86 0 7! 159 127 126 1.49 / 69 75 1 02 ]. 161 127 1.28 204 220 197 0 8 [1177 127 126.15.6..1.60_ 160....0..............104! 176 127 128 201 213 187 0 9 ]. 195 1.27 1.25 160.156 138 0 tO6 t 200 127 L28 196 204t60.... 0 _L207..[7 1.6.__~_ 54 127.0.........108 1 208 127 129...19.4...98.165......0... L!! 222 127 125 163 150 119 0 11.2 [ 179 127 129 193 211 186 0 12 1 230 12'7 1.25 '165 [55 11]4 0 2 —2-7i',",'...... 2I1t15 13 ]. 204 1.27 125 165 1.'66 136 0 116 1 153 127 1.27 21.7 228 190 0 14 ]. 179 1.27 1.26 166 173 16.4......O................ It 1 46 127 1.60 23;1-2 3.i6.....o'... i5 I 163 1.27 126 164 t88 176 0 120 ]. 140 1-27 1S5 235 232 191 0 1 6 1 154 127 126 176_]-9_3,.194......0 1............ 22 [ 14/4 1..27"?'177 235 234 ].88 0 i8 ]. 141 1.27 1.26 1.94 221. 1.91 0 124 ]. 152 1.27 ].50 228.234 190 0 20 ]. t34 L27 IS0 220 220 t79 0 ].~~~~~~~~~~~26 I 161. 12-7"'130 2].2 228 201 0 ' 22 ]. 135 1.27 LB0 225 212 163 0 128 1 176 1.27 129' 199 217 1.93 0 24 t 129 1.27 170 224 218..166....0.............130 1. 194 1.2712].9''-.'C4...~5-...0 26! 130 127 140 20.9 221. Lee 0 132 t 206 1.27 127 1.92 196 1.62 0 28 1 144 127 1.26 194 211. 19.4......0..........3....')4 -' -204 1-2'7'-12'7 193 199 161 0' 30 1 163 1-27 1.30 190 19g6 t85 0 16 110171719 0 8 32! 190 1.27 1.26 184 1-96 161 013710262120 34! 21'6 L27 126 ].8! 189 is0 0 140 1 ].55 127 1.30 220 234 1.97 0 36 1 201. 1.27 1.27 ].80 187...6.7 0............. 42! 150 127 165 '2-33"~2'3'i5-!90o....0...... 3 8 1 185 12 7 1 2 8 -~~~~~~~~;-i 1_[4_4_.....__.146 12.7...19.0. 2.38..232' 190. 0 40 t.t58 1.27 128.191 209.1.8.6.... 1.......... 46.I 148 L27 175 237 230 185 0 42 i 146 127 1.28 203 217 194 0 148 1 15L 127 145 227 23.3 191 0 46! 133 127'190 228 222 179 0....... ' "l -I-SO.....1"'5"6'..7 13'4-209 225 2000 44 1 138 1.27 135 222 228 Is! 0 152! 170o 127 1.27 ]199 216 190 0 48 1. 135 127 /85 230 223 179 0/5!].0.216182017'".... 501 1.44 1.27 155.231 227 178 1 56 1 20312 1819 20 16 52! 152 127 127 208 229 200 0! 8 20 2?128.19 '0I i7] o 54 1'164 1.27 126 ].95 21.8 201. 0 160 1. 1.85 127 127 197 211 L87 0 56 1 187 1.27 127 1.90 200 ].80 0 ]. 62! 1.71.1 27 ].27 - — 2C)'4O~" -2 2 0-/ 9'5..... 58 [ 208 1.27 1.28 186 1.89 152 0 164 1 160 127' ]28 216 228 1-90 0 60! 213 127' 127 184 185 148...._0..............L6!10.]7.0 25 2 -"T'3........ 62 I 1-84 '127 127 183 1.97 180 0 168! 146 1.27 '172 230 232 190 0 64!168 1.27 ]126 1.91 210 1.95 0..........10!/717/0~3i"20i' 66! 157 127 127 209 228 193 0 172 1. 149 1.27 160.227 228 192 0 68 ]. 145 127 150 231. 236 19~. 0 174! 1.59 127 130 2'18-"229 ].88 O' 70 I 134 127 200 239 231 192 0 176 1 169 127 127 206 222 193 0 72 ]. 138 127 200 243 234 /80 0 178 1 1.84 127 126 199 213 189 0'74! 139 i27 170 234 232 195 0.180 1 199 127 128 1.98 207 173 0 76 I 149 1.27 127.214 229 200 0 1 8 0.7 12-7 198 203.60 7 8 I 165 127 1.26 196 215 193'.....0' 184 1 189 12 7 129 200 206 173 0 80 1ab17]7].0 ------ 1 7 0.. —4 186 1 173 127 129 200 21.5 191 0 82 I 202 127 127 189 1.96 163 0 188 [ 161 127 129 208 226 197 0 84 ]. 211 127 128 187 ]194 I58 0 ]190 1 1t55 127 190 22'3'2-3'i-t' —..." 86 [ ]194 127 126 ]191 201 174 0 192 t ]150 127 170 233 231 1186 O' ]193 1 94 67 67 1114 120 103 0 195 ]1 93 67 67 1.13 ]118 99 0

-173 -(C) 'REDUCED EXPERIMENTAL DATA SPEED - 1008.0000001 N = 195, RUN = 523 A =.167510. LI = 2.0820009 VR - -.306000 -VO = 2.785530, RO =.876954. XO a.168971 XOO -.893100t POO = 1.980842, PTH 1.922179 VAX a 21.9604081 NUNM 511 I 170 E5 =.1000001 TOELAY -.000000, TAUDEL -.000000 I T(I) MSEC R(I) MM V(I) CU.MM TAU * SI P Q 0.0000-.5837.8330.0000.6656 -.3409 1.9921 1.0608 5.0000.2161 1.2096.2137 2 i.9841 1.3133 9.4887.4322 1.4976.5073 3 2.9762 1.2774 8.7323.6483 1.4567.4656 4 3.9683 1.3644 10.6411.8643 1.5559.5667 5 4.9603 1.4058 11.6388 1.0804 1.6031.6148 6 5.9524 1.4106 11.7594 1.2965 1.6086.6205 7 6.9444 1.4449 12.6378 '1.5126 1.6477.6603 8 7.9365 1.4687 13.2729 1.7287 1.6748.6880 9 8.9286 1.4739 13.4132 1.9448 1.6807.6940 10 9.9206 1.4816 13.6236 2.1609 1.6894.7029 11 10.9127 1.4998 14.1336 2.3769 1.7103.7241 12 11.9048 1.5214 14.7519 2.5930 1.7348.7492 13 12.8968 1.5196 14.7019 2.8091 1.7329.7472 14 13.8889 1.5229 14.7963 3.0252 1.7366.7510 15 14.8810 1.5158 14.5901 3.2413 1.7285.7427 16 15.8730 1.5543 15.7294 3.4574 1.7723.7874 18 17.8571 1.5744 16.3485 3.8895 1.7953.8108 20 19.8413 1.5880 16.7755 4.3217 1.8108.8266 22 21.8254 1.5825 16.60'2.7 4.7539 1.8045.8203 24 23.8095 1.5632 16.0014 5.1861 1.7825.7978 26 25.7937 1.5614 15.9473 5.6182 1.7805.7957 28 27.7778 1.5777 16.4531 6.0504 1.7991.8147 30 29.7619 1.6011 17.1966 6.4826 1.8258.8419 32 31.7460 1.6225 17.8930 6.9148 1.8501 *.8667 34 33.7302 1.6537 18.9455 7.3469 1.8857.90.30 36 35.7143 1.6428 18.5732 7.7791 1.8733.8903 38 37.6984 1.6256 17.9957 8.2113 1.8537.8703 40 39.6825 1.6044 17.3018 8.6434 1.8295.8457 42 41.6667 1.6114 17.5286 9.0756 1.8375.8538 44 43.6508 1.6147 17.6386 9.5078 1.8413.8577 46 45.6349 1'.6205 17.8290 9.9400 1.8479.8645 48 47.6190 1.6316 18.1977 10.3721 1.8606.8774 50 49.6032 1.6590 19.1293 10.8043 1.8918.9092 52 51.5873 1.6691 19.4800 11.2365 1.9033.9209 54 53.5714 1.6722 19.5903 11.6686 1.9069.9246 56 55.5556 1.6702 19.5183 12.1008 1.9045.9222 58 57.5397 1.6502 18.8280 12.5330 1.8818.8990 60 59.5238 1.6441 18.6195 12.9652 1.8748.8919 62 61.5079 1.6414 18.5274 13.3973 1.8717.8888 64 63.4921 1.6553 18.9998 13.8295 1.8875.9048 66 65.4762 1.6756 19.7087 14.2617 1.9107.9285 68 67.4603 1.6920 20.2944 14.6938 1.9294.9476 70 69.4444 1.6815 19.9169 15.1260 1..9174.9353

-174 -(HPAT I(B) COtMM-) ___ __ 72 71,4286 1.6920 20.2932 15.5582 1.9294.9475 74 73.4127 1.6842 20.0135 15.9904 1.9205.9385 76 75.3968 1.6699 19.5090 16.4225 1.9042.9219 78 77.3810 1*6586 19.1138 16.8547 1.8913.9087 80 79.3651 1.6627 19.2586 17.2869 1.8960.9135 82 81.3492 1.6722 19.5892 17.7191 1.9068.9245 84 83.3333 1.6796 19.8511 18.1512 1.9153.9332 86 85.3175 i 1.6822 19.9446 18.5834 1.9183.9362 88 87.3016 1.6878 20.1428 19.0156 1.9246.9427 90 89.2857 1.6811 19.9023 19.4477 1.9169.9348 92 91.2698 1.6784 19.8078 19.8799 1.9139.9317 94 93.2540 1.6744 19.6665 20.3121 1.9093.9271 96 95.2381 1,6628 19,2597 20.7443 1*8961.9136 98 97.2222 1.6556 19.0114 21.1764 1.8879.9052 100 99.2063 1.6639 19.3000 21.6086 1.8974.9149 102 101.1905 1.6762 19.7316 2200408 1.9114.9292 104 103.1746 1.6921 20.2982 22.4729 1.9296.9477 106 105.1587 1.6994 20.5605 22.9051 1.9378.9561 108 107.1429 1.7085 20.8930 23.3373 1.9482.9667 110 109.1270 1*6941 20.3683 23.7695 1.9318.9500 112 111.1111 1.6801 19.8675 24.2016 1.9158.9337 114 113.0952 1.6684 19.4550 24.6338 1.9025.9201 116 115.0794 1.6724 19.5976 25.0660 1.9071.9248 118 117.0635 1.7031 20.6961 25.4981 1*9421.9605 120 119*0476 1.6906 20.2432 25.9303 1.9278.9459 122 121.0317 1.7007 20.6075 26.3625 1*9393.9577 124 123.0159 1.7084 20.8909 26.7947 1.9482.9667 126 125.0000 17.111 20.9890 27.2268 1.9512.9698 128 126.9841 1.7049 20.7631 27.6590 1.9442.9626 130 128.9683 1*6940 20.3646 28.0912 1.9317,9499 132 130.9524 1.6885 20.1692 28.5234 1,9255.9435 134 132.9365 1.6880 20.1510 28*9555 1.9249.9429 136 134.9206 1.6839 20.0044 29.3877 1.9202.9382 138 136.9048 1.6983 20.5225 29.8199 1.9366.9549 140 138.8889 1.7067 20.8280 30.2520 1.9462.9647 142 140.8730 1.7201 21.3212 30.6842 1.9614.9802 144 142.8571 1.7215 21.3725 31.1164 1.9630.9818 146 144.8413 1,7103 20.9582 31.5486 1.9502.9688 148 146.8254 1.7009 20.6160 31.9807 1.9396.9579 150 148.8095 1.6830 19.9727 32.4129 1.9192.9371 152 150.7936 1.6776 19.7801 32.8451 1.9130.9308 154 152.7778 1.6928 20.3236 33.2772 1.9304.9485 156 154.7619 1.7054 20.7801 33.7094 1.9447.9632 158 156.7460 1.7140 21.0972 34.1416 1.9545.9732 160 158.7302 1.7085 20.8922 34.5738 1.9482.9667 162 160.7143 1.7061 20.8064 35.0059 1.9455.9640 164 162..6984 1.6959 20.4328 35.4381 1.9338.9520 166 164.6825 1.6975 20.4924 35.8703 1.9357.9540 168 166.6667 1.6966 20.4596 36.3024 1.9347.9529 170 168.6508 1.6914 20.2736 36.7346 1.9288.9469 172 170.6349 1.6951 20.4039 37.1668 1.9329.9511 174 172.6190 1.6952 20.4105 37.5990 1'9331.9513 176 174.6032 1.7027 20.6810 38.0311 1.9416.9600 178 176.5873 1.7159 21.1675 38.4633 1.9567.9754 180 178.5714 1.7206 21.3390 38.8955 1.9620.9808 182 180.5556 1.7144 21.1110 39.3277 1.9550.9736 184 182.5397 1.6949 20.3978 39.7598 1.9327.9509 186 184.5238 1.6909 20.2529 40.1920 1.9281.9462 188 186.5079 1.6928 20.3213 40.6242 1.9303.9485 190 188.4921 1.7246 21.4878 41.0563 1.9665.9854 192 190.4762 1.7099 20.9445 41.4885 1.9498.9684 193 191.4683 1.7662 23.0821 41,7046 2.0140 1.0338 195 193.4524 1.7371 21.9604 42.1368 1.9808 1.0000

PART2 THEORETICAL BUBBLE GROWTH - NONSOLUBLE GAS... JA = 91.700000,.1 = S 1000.000000, E4 = 10.000000 LU = 185.000000, El 5.OOOOOOE..-03,. E7 =5.OQOOOOOE-03 XOO =.89Eo0o, Z_ E...20 ___.... XO =.16d97L, E3 = 1.000000 (a SMALL VALUES OF TAU _ XS(0) =.721422, Ao 247.838074, __.3. 1000*TAU P UX Q.0927 1.1849 1.6668.1885.1853 1.2669 1.3080.2721.2780 1.3319 1.1526.3384.3707 1.'3880 ~ 1.0619.3956.4633 1.4383 1.0013.4469.5560 1.4846.9574.4940 (b) JOIN POINT TAUI F-' TAUlO) =.105204..P1(0O) 1.976. PlDOT = 708.997482.... QI(0) =.20i502, XS(I) =.747786, UTI = 31.861320 (c) INTERMEDIATE VALUES OF TAU ---TAUH/TAUM = RELATIVE SIGNIFICANCE OF HEAT TRANSPORT TO MASS TRANSPORT FOR TAU SI*TAU P POOT UT DXS Q TAUH/TAUM CRITl.1980 1.2760 682.9116 28.9842.1292.2814.3711.749..3428 1.3543 434.1106 22.1823.0832.3612.4836.5862.5733 1.4326 269.8927 17.1163.0526.4411.6325.4321.9502 1.5109 162.3652 13.1431.0323.5209.8334.3019 1.5926 1.5892 93.0303 9.9223.0189.6008 1.1132.1983. 2.7573 1.6676 49.4700 7.2517.0103.6806 1.5231.1198 5.0896 1.7459 23.2723 5.0016.0050.7605 2.1754.0636 10.6663 1.8242 8.7036 3.0829.0019.8403 3.3823.0267 30.6338 1.9025 1.8416 1.4316.0004.9202 6.5338.0063 CH= 20.407853,..CM = 8.483750 (d) JOIN POINT TAU2 TAU2(0)= 6.395873, P2(0) = 1.764227, Q2(0). 779154

-176 -( _ _ASYMPTOTI.C SOLUTION SI*TAU P. Q 4.7370 1.0000.0000 4.7418 1.1413.1441 4.7564 1.2484.2532 4. 7806 1.3327.339 1 4.8145 1.4009.4087 4.8581 1.4573....4..........662. 5.0096 1.5626.5736 5.2216 1.6351.6475 5.8275 1.7273.7415 6.'6757 1.7824.7977 7.7662 1.8186.8346 9.0990 1.8440 $8605 10. 6742 1.8627.8796 12.4917 1.8770.8942 14.5515 1.8883.9056 16.8537 1.8974.9149 28.4858 1.9204.9384 35.7558 1.9281.9462 53.2039 1.9388.9571 113.7875 1.9529.9115 198.6046 1.9599.97.87 440.9390 1.9669.9858 PART 3_ PLOT OF P(TAU) AND Q(TAU) O JOIN POINT TAUL INTERMEDIATL THEORY X JOIN POINT TAU2 + ASYMPTOTIC THEORY - EXPERIMENTAL POINTS 1000*TAU MS =.217815 1 MILLISECOND = TAUMS

2,20 +..................+ ---..............+. N I I I I I I 0o. I_ _I I I I _ I N I I I I I I D I I I I I I I I I I I I I M [ I I I I I E I I I I I * I N 2.00 ++-+ ----------- S I I 1 I I I * I I I I I * * *** I ** **** ~***I * I 0 1 I t***** I***.. *. *4*4* *4* 4** I N I 1 * * * I1*** ** I. I I A I * I+ ** I I I I L I I. +._ I_ I I I I * +~*I I I I I 8 1.80 + ~...*** --- —-— + --- —*.~~.....~ + U I * ** X I I I I I 8 I. I I I I I B I ** + I I I I I I L I ** I I I I I E I ** I I I I I I * I I I 1 I S I I I I I I I 1.60 +-tt --- —---------— + --- —----— _ _+_ +_ I Z I I I I I I E I * I I RUN 523 I I I I I I I -4 P I*. I I 0 JOIN POINT TAU1 I I I I I INTERMEDIATE THEORY I I 1* I I X JOIN POINT TAUZ. I I I. I I + ASYMPTOTIC THEORY I I 1.40 + --- —------------— + --- —------------— +- * EXPERIMENTAL POINT --------------— + I I I I I I. I I JA = 91.7 A = 0.1675 I I I I I LU - 185 RO = 0.876 I I I I I XO = 0.168 PO = 1.980 I I I I XOO = 0.893 WATER I I I I I NITROGEN I I I I I I I 1.20 *.. + + __.- ~~ +~+..,._.... I I I I I I I I I I I_ I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1. 00 +~+ ~ ~++~+ 0.0 __ 10.0 20.0 30.0 40.0 50.0,_...-.......-.L......... NONDIMENSIONAL TIME 1000*TAU = 1000*A*T/RO.P.2

1.20 + +-4 +~+~~~ R I I I I IE. I I I I I I L I I I I I I A I I I I I T I I I I I I _...... I I I I I V I I I I I * I EA!00___ +_............... + ~ + ~+ ---*~ ~+ I I I * ** I'** **** **'I I N I.__*.__ I. *?__ **. *..... ______ I O. I I *** ** *+*** *** ** *** I. ** I I._ -!..................-.-.... - +** I * ****+ * I *.........._I_ _....-...-..- -........ I 0 I * * ** I I I I _.__..._.................. **-I. I. -.......I.I......I...... M I +** + I I I I I E 0.80 +~.....*-* *-+ ~~~~-.7.-+~+~ t~~~ + N I * X I I I I S *_..._ __~_~~____r + I I I I ___I 1 I. I I I I I o 1 **.I I 1... I I N I *I I I I I A.I _. I I. I I __ L I * o I I I I I 0.60 + —.-+~+~+~+~+ B 1* I - I I I I U _I I I RUN 523 I 1 — 4 8 1-, 1 I I I CO 8 _____ 1*_I * I 1 0 JOIN POINT TAUI I I 1 I* i I INTERMEDIATE THEORY 1 I E I. I I X JOIN POINT TAU2 I I I I I + ASYMPTOTIC THEORY I I S 0.* 40 +~-~++- * EXPERIMENTAL POINT I I I I I I Z I. I I JA = 91.7 A = 0.1615 I I E I I ' I LU = i85 RO = 0.876 I I I I I XO = 0.168 PO = 1.980 I I Q I I Xoo = 0.893 WATER.- I I.. I I NITROG-N I - * I I I i 0 *20 0 -.~~~~~ —++ + I I I I I I I I I I I I -..... I_...i I I I I I I I I I I I I I I I I I I I 0 00 + —4~-+~~+~~ 0. 0 20.C 30.0 40.0 50.0 NONDImENSIUNAL TIME '000*TAU = 1000*AmT/RO.P.2 -- r --- —----------- -~~~~~~~~~~~~~~~~..................

APPENDIX VI SUMMARY OF EXPERIMENTAL DATA -179 -

APPENDIX VI-1 DATA FOR THE CASE OF A NONSOLUBLE GAS SPEED p ' Run Liquid Gas (fps) T (~F) (inch Hg) (mm2/sec) Ro (mm) Ja Lu x, xo P, Comment 504 H20 N2 1000 204.1 28.15 0.1674 0.761 93.8 191 0.903 0.227 1.999 Stationary 511 H20 N2 4000 75 29.4 0.1435 1.548 450 1600 0.030 0.001 1.010 Formation 517 H20 N2 1024 163,.8 11. 97 O.,1627 O..820 184,.3 421 O0.883 O0.156 1.933 Stationary 521 H20 N2 1024 149.2 7.98 0.1603 0.661 241.0 587 0.929 0.387 2.053 Stationary 523 H20 N2 1008 205.4 29.37 0.1675 0.877 91.7 185 0.893 0.169 1.981 Stationary 524 820 N2 1088 205.1 27.90 0.1675 0.747 92.0 186 0.933 0.231 2.257 Stationary 601 H20 He 1008 206.3 29.17 0.1676 0.839 90.4 560 0.915 0.326 1.994 Stationary 609 H20 He 2640 164.0 11.35 0.1627 0.709 180.5 1124 0.935 0.405 2.090 Stationary 613 H20 He 2672 153.2 8.92 0.1611 0.803 224.7 1430 0.908 0.386 1.884 Stationary 702 H20 N2 1008 202.8 27.93 0.1672 0.764 95.4 190 0.890 0.203 1.935 Injector Facing up 805 H20 N2 1072 180.7 18.14 0.1648 3.010 136.2 291 0.856 0.010 1.902 Moving 806 H20o N2 977 180.3 18.00 0.1647 2.994 137.4 297 o.854 0.010 1.892 Moving

APPENDIX VI-2 DATA FOR THE CASE OF A SOLUBLE GAS Run 901 903 951 953 954 955 Liquid H20 H20 H20 H20 H20 H20 Gas NH3 NH3 NH3 NH3 NH3 NH3 SPEED 1008 1008 992 1008 1008 1008 fps TO~F 181.3 197.8 204.3 205.3 177.4 178.0 p~(psia) 9.01 11.8 13.7 14.27 7.87 9.48 at (mm2/ sec) 0.1649 0.1667 0.1674 0.1675 0.1644 0.1645 p x103 (ohm-cm) 30 30 38 36 35.5 36 P*B (psia) 642 779 835 846 613 617 KB ref. (47) 0.0578 o.0588 o.o600 0.0788 0.0526 0.0624 Ja 135.4 104.0 93.2 91.5 144.0 142.5 Lu 4o5 285 258 252 312 308 Lu' 9 x 10-3 9 x 10-3 9 x 10 30 9 x 10-3 x"00 0.858 0.932 0.920 0.901 0.901 0.758 (1-xcf)x105 2.5 2.5 1.6 1.8 1.85 1.8 1-x* 0.0082 o.oo4o o.oo48 0.0078 0.0052 0.0151 F = hfgB/hfgA o.8 o.8 o.8 o.8 o.8 o.8 (1-E)x0L4 3.3 4.5 ".0 5.0 2.9 2.9 X, Eq.(106) -8.76 -9.00 -8.18 -6.04 -8.97 -4.63 Comment Moving Moving Stationary Stationary Stationary Stationary

APPENDIX VI- 3 DATA FOR BUBBLE OSCILIATIONS Mean. Measured Theoretical Relative Relative Radius Frequency Frequency Polar Equivalent Eq. (20) Amplitude Radius RUN R f fth a/R Req/R (mm) (sec-l) (sec-1) 504 1.49 77 76.073.973 510 1o46 79 83.194.972 511 1o51 78 79.116 o934 513 1o61 70 66.142.987 522 1o 30 96 93.122.992 523 1o69 56 63.177.991 601 1057 67 69.115.974 609 1o48 77 78.130.970 613 1.50 74 76.138 o 960 702 1o 47 80 77 o061.973 X182 -

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