Growth Of Spherical Bubble Research Articles

The dynamics of spherical bubble growth

Spherically symmetric bubble expansion in uniformly superheated infinite pools of liquid have been simulated numerically. Bubble growth curves have been generated for a range of Jakob numbers, 3⩽ Ja⩽3167, by altering the initial metastable state of the system facilitated by changes in the initial superheat and system pressure. The detailed physics of vapour bubble growth is studied through delineation of the parameters governing the changes in the growth dynamics from surface tension, to inertia dominated, to diffusion controlled, and the domains between them.

Dynamics of Spherical Bubble Growth

A model for the dynamics of spherical bubble growth in a quiescent viscous liquid is presented. The gas inside the bubble is a van der Waals fluid, and the viscous liquid outside the bubble is a Flory–Hugins solvent–polymer solution. The growth of the bubble in the viscous liquid is assumed to be controlled by momentum, heat and mass transfer. Using the integral method, the transport equations were transformed into ordinary differential equations, which were numerically solved. An analytical criterion of when it is justified to make the usual isothermal assumption is also derived. The relevance of this work to the processes of polymer melt devolatilization and the production of foamed plastics is discussed.

Growth and application of large spherical bubbles using reactive gases

Under certain conditions, bubbles composed of reactive gases may be placed in liquids. When the gases are ignited, the bubbles will rapidly grow, producing large bubbles (up to almost 3 cm in diameter) that can be essentially spherical prior to and during the growth phase, even in normal gravity. Experiments and analyses regarding this type of bubble growth are presented in this paper. It is suggested that this method of bubble growth provides unique opportunities to study phenomena associated with rapid growth of bubbles in liquids, for example growth of bubbles in droplets burning in microgravity and bubble growth during microgravity pool boiling.

I. Formation and rise of a bubble stream in a viscous liquid

The continuous emission of gas bubbles from a single ejection orifice immersed in a viscous fluid is considered. We first present a semi empirical model of spherical bubble growth under constant flow conditions to predict the bubble volume at the detachment stage. In a second part, we propose a physical model to describe the rise velocity of in-line interacting bubbles and we derive an expression for the net viscous force acting on the surrounding fluid. Experimental results for air/water-glycerol systems are presented for a wide range of fluid viscosity and compared with theoretical predictions. An imagery technique was used to determine the bubble size and rise velocity. The effects of fluid viscosity, gas flow rate, orifice diameter and liquid depth on the bubble stream dynamic were analyzed. We have further studied the effect of large scale recirculation flow and the influence of a neighbouring bubble stream on the bubble growth and rising velocity.

Bubble growth in a variable diffusion coefficient liquid

A model for the mass-transfer-controlled spherical bubble growth in a quiescent liquid is presented. The liquid is of variable diffusion coefficient, which depends exponentially on the concentration. Using the method presented in Scriven [L.E. Scriven, On the dynamics of phase growth, Chem. Eng. Sci. 10 (1959) 1–13], for a constant diffusion coefficient liquid, the differential binary mass balance was transformed into an ordinary differential equation, trough a similarity transformation, which was numerically solved. The final results were also compared with approximately analytical solutions, for slow and fast growth rates, developed by Polyanin and Dil'man [A.D. Polyanin, V.V. Dil'man, The method of the ‘carry over’ of integral transforms in non-linear mass and heat transfer problems, Int. J. Heat Mass Trans. 33 (1990) 175–181] and by the authors.

The Origin of the Dynamic Growth of Vapor Bubbles Related to Vapor Explosions

An explosive type of vapor bubble growth was observed during pool boiling experiments in microgravity using R-113. Photographs reveal that the liquid-vapor interface of the explosive bubbles are wrinkled and corrugated, leading to the conclusion that some type of instability mechanism is acting. The classical hydrodynamic instability theories of Landau and Rayleigh-Taylor do not consider the effect of heat transfer, at the interface, which is believed to be responsible for the observed instability of the evaporating surface. This was confirmed by the mechanisms proposed by Prosperetti and Plesset, combined with a model of the early growth of spherical vapor bubbles.

Bubble growth in viscous newtonian and non-newtonian liquids

A model for the dynamics of hydrodynamically controlled spherical bubble growth in quiescent viscous newtonian and non-newtonian liquids is presented. Two constitutive equations were used to describe the behavior of the liquid medium: (1) a simple power law relation and (2) a truncated power law. Application of the truncated power law resulted in four differen cases: (a) at all times the liquid acts as a newtonian liquid; (b) at all times the liquid behaves as a simple power law liquid; (c) at all times the liquid close to the growing bubble obeys a power law relation and the liquid far away from the bubble behaves as a newtonian liquid; (d) all the liquid initially behaves as a newtonian liquid and at some time transforms to a two-region liquid. Analytical expressions were derived for bubble growth as a function of dimensionless system parameters. The relevance of this work to the process of polymer devolatilization is discussed.

Spherical vapor bubble growth in uniformly superheated liquids

A numerical procedure is presented for the solution of vapor bubble growth with radial symmetry from the thermodynamic critical size in an initially uniformly superheated liquid, which includes the influences of surface tension, liquid inertia and heat diffusion. Results are presented in the form of time varying interface radius, velocity, acceleration and temperature, with particular emphasis on how circumstances during the very early growth periods affect the later growth. The effect on the solution of the disturbance required to initiate the growth is examined in some detail. Comprehensive comparisons are made with previous experiments, analyses and numerical results.

Experimental simulations of explosive degassing of magma

Explosive flows of strongly gas-supersaturated liquids have been investigated experimentally in two different configurations to simulate conditions in violent volcanic eruptions. i) Decompression of water saturated with dissolved C0_2 at pressures up to 0.7 MPa causes explosive bubble growth and foaming. The large volume increase accelerates the two-phase mixture upward to velocities as large as 14 m/s at nearly constant accelerations up to more than 200 g. Constant acceleration implies that the bubble growth rate is proportional to t^(2/3), measurably larger than the t^(1/2) law expected for diffusive growth of spherical bubbles. ii) Rapid mixing of concentrated K_2CO_3 and HCl solutions generates CO_2 supersaturated to a few MPa. In this case, reaction and degassing generate an increasingly accelerating flow until the reactants become depleted at peak accelerations of about 100g and velocities up to 45 m/s. In both experiments the rapid expansion causes the foam to fragment into a heterogeneous spray. The experimental flows are comparable to explosive volcanic flows in terms of decompression ratio, velocities, accelerations and in the large range of scales present. The large accelerations cause strong extensional strain and longitudinal deformation of the foam. Tube pumice is a major component of Plinian deposits and ignimbrites, and preserves evidence of these flow conditions.

Pool boiling heat transfer of mercury in the presence of a strong magnetic field

The effect of a vertical magnetic field on the saturated nucleate pool boiling of mercury on a horizontal surface was investigated experimentally and analytically. The experimental result shows that as the magnetic field is increased, the boiling Nusselt number decreases, and at 5 T it reaches 40–60% of that at zero magnetic field. However, the decrease in the Nusselt number is much less in a magnetic field of 5–7 T. Spherical and ellipsoidal bubble growth models with the Forster and Zuber approach developed previously were applied to this boiling phenomenon. It is found that the sophisticated ellipsoidal bubble growth model greatly overestimates the boiling heat transfer, whereas the spherical bubble growth model, although based on an unrealistic spherical magnetic field, gives a relatively better prediction. By modifying the characteristic length and velocity in the bubble Reynolds number, the prediction of the ellipsoidal bubble growth model was improved.

Nucleation and flashing in nozzles—2. Comparison with experiments using a five-equation model for vapor void development

A quasi-one-dimensional, five-equation, homogeneous, nonequilibrium model has been developed and utilized on a microcomputer to calculate the behavior of flowing, initially subcooled, flashing water systems. Equations for mixture and vapor mass conservation, mixture momentum conservation, liquid energy conservation and bubble transport were discretized and linearized semi-implicitly, and solved using a successive iteration Newton method. Closure was obtained through simple constitutive equations for friction and spherical bubble growth, and a new nucleation model for wall nucleation in small nozzles combined with an existing model for bulk nucleation in large geometries to obtain the thermal nonequilibrium between phases. The model described was applied to choked nozzle flow with subcooled water inlets based on specified inlet conditions of pressure and temperature, and vanishing inlet void fraction and bubble number density. Good qualitative and quantitative agreement with the experiment confirms the adequacy of the nucleation models in determining both the initial size and number density of nuclei, and indicates that mechanical nonequilibrium between phases is not an important factor in these flows. It is shown that bulk nucleation becomes important as the volume-to-surface ratio of the geometry is increased.

Widealized Strut Geometries for Open-Celled Foams

ABSTRACTOpen-celled foam is modeled as a network of struts formed by the nucleation, growth and intersection of uniform spherical bubbles arranged in a regular three-dimensional array. The number of struts per unit cell and their characteristic geometry and orientations depend on the bubble-centering algorithm chosen to define the array and on the degree of bubble intersection. A foam model based on a body-centered cubic arrangement of uniform spherical bubbles is used to compare strut geometries over a range of bubble interferences. The model predictions are compared to the microstructures observed in amorphous carbon foam. The significance of these idealized foam results for graphitic carbon foams is discussed.

The influence of surface tension on the diffusion-controlled growth or dissolution of spherical gas bubbles

Surface tension can markedly affect the growth or dissolution of small gas bubbles but, even when spherical symmetry is maintained and the interfacial concentration assumed constant, generally valid analytical solutions for the change of size with time cannot be obtained; approximations of limited validity are therefore often used. However, accurate and efficient methods for computing the diffusion-controlled growth or dissolution of spheres in such conditions have recently been developed; the results obtained when interfacial solute concentration is assumed constant have been published and validated by comparing them, where possible, with the equivalent analytical solutions. This paper extends that work to include the effects of a constant surface tension, for both Henry’s and Sievert’s laws. The numerical results are valid for as wide a range of parameters as is likely to be needed and are compared with the few analytical solutions available for particular cases. It is shown that, when surface tension is introduced, a complete description of the problem requires an additional saturation parameter as well as the dimensionless surface tension. It is also shown that Henry’s and Sievert’s laws can have very different effects on dissolving bubbles: when Sievert’s law applies a higher surface tension can increase time for a bubble to dissolve but this paradox can be resolved by examining the competing effects involved.

Computer calculations of air and steam blowdown suppression

We describe a computer code that combines an Eulerian incompressible-fluid algorithm (SOLA) with a Lagrangian finite-element shell algorithm. The former models the fluid and the latter models the containing structure in an analysis of pressure suppression in boiling-water reactors. The code (PELE-IC) calculates loads and structural response from air blowdown and from the oscillatory condensation of steam bubbles in a water pool. The fluid, structure, and coupling algorithms are tested by recalculating problems that have known analytical solutions, including tank drainage, spherical bubble growth, coupling for circular plates, and submerged cylinder vibration. Code calculations are also compared with the results of small-scale blowdown experiments.

Vapour-bubble growth in a superheated liquid

It is shown that the approximation of a thin thermal boundary layer gives an accurate description of the growth of spherical vapour bubbles in a superheated liquid except for very small superheats. If the further approximations of a linear variation of vapour pressure with temperature and of constant physical properties are made, then scaled variables can be introduced which describe the growth under any conditions. This scaled description is not valid during the early, surface-tension dominated, portion of the growth. The rate of bubble growth for large superheats is somewhat overestimated in the intermediate stage in which both inertial and thermal effects play a role. This overestimate does not lead to a serious error in the radius-time behaviour for ranges of practical interest. The asymptotic, or thermally controlled, stage of growth is accurately described by the scaled formulation.

Growth of spherical gas bubbles by solute diffusion in non-Newtonian (power law) liquids

In this work, diffusive bubble growth in newtonian liquids and non-Newtonian liquids of the power law type is studied. Three regions are considered, the first where growth is limited entirely by the rate of arrival dissolved gas at the bubble surface, the second where growth is limited entirely by hydrodynamic forces and the third, a general case, where both diffusion and hydrodynamics play a part. Parameters which have previously been used to define regions where viscosity, inertia, surface tension and diffusion are important are shown to hold good for power law liquids. It is found that, all else being equal, bubbles in power law liquids grow more slowly the smaller the characteristic exponent.

Numerical Solution for the Spherical Bubble Growth Problem in a Uniformly Ultraheated Liquid

A multi-point, implicit-type, finite-difference method for solving the bubble growth (or collapse) problem in an ultraheated liquid is proposed. The method is applicable to both inertia-controlled growth and heat-diffusion-controlled growth. The results are compared with several asymptotic solutions, involving Plesset-Zwick and Mikic-Rohsenow-Griffith solutions. Present results strongly support the Mikic-Rohsenow-Griffith solution for the wide range of bubble growth conditions, e.g. given fluids, pressure, liquid ultraheat, etc.

Influence of radial convection on bubble growth governed by mass transfer

IN THE classical work of Epstein and PIesset [l] on the growth of spherical gas bubbles by mass diffusion, the influence of the convective transport due to the radial motion of the bubble wall was neglected. This effect was taken into account by Striven [2] who formulated the coupled problem with simultaneous heat and mass transfer and gave some asymptotic solutions. in this paper the mass-transfer problem considered by Epstein and Plesset and Striven is reexamined with the purpose of obtaining a simple criterion showing when the effect of the radial convection has to be taken into account.

Non-equilibrium effects in the growth of spherical gas bubbles due to solute diffusion—II: The combined effects of viscosity, liquid inertia, surface tension and surface kinetics

A formulation and computed results are presented for the growth of stationary gas bubbles due to solute diffusion in super-saturated solutions. In the formulation account is taken of the role played by surface kinetics, liquid inertia, viscous forces and surface tension forces. Various asymptotic regimes are defined which would correspond to diffusion control (parabolic growth rate) inertia control (constant growth rate) and control by surface kinetics. Dimensionless groups are suggested which allow the quantitative definition of the validity of asymptotic solutions corresponding to these regimes. While inertial effects predominate in the very early stages of growth, in the majority of systems inertial effects will be of practical importance at greatly reduced pressures only (vacuum distillation, vacuum degassing of molten metals). Viscous effects will be important for highly viscous fluids only, such as some polymers and molten glass. The numerical value of the rate coefficient for surface kinetics may vary over wide limits so that surface kinetics could be potentially important as a rate limiting factor over a wide range of conditions.

Spherical vapour bubble growth in superheated liquids

Experimental data has been obtained on the growth of vapour bubbles in water at constant uniform superheats up to 20°C. The data is shown to agree within the limits of the experimental errors with the predictions of a simple theory derived assuming temperature equilibrium at the bubble wall. The theory is extended to predict the conditions for which differing fluids will exhibits similar growth rates, and it is shown for example that sodium vapour bubble growth may be modelled accurately in water.