Bubble Growth Model Research Articles

Sound synchronization of bubble trains in a viscous fluid: Experiment and modeling

We investigate the dynamics of formation of air bubbles expelled from a nozzle immersed in a viscous fluid under the influence of sound waves. We have obtained bifurcation diagrams by measuring the time between successive bubbles, having the air flow (Q) as a parameter control for many values of the sound wave amplitude (A), the height (H) of the solution above the top of the nozzle, and three values of the sound frequency (fs). Our parameter spaces (Q,A) revealed a scenario for the onset of synchronization dominated by Arnold tongues (frequency locking) which gives place to chaotic phase synchronization for sufficiently large A. The experimental results were accurately reproduced by numerical simulations of a model combining a simple bubble growth model for the bubble train and a coupling term with the sound wave added to the equilibrium pressure.

A mathematical model for predicting bubble growth for low Bond and Jakob number nucleate boiling

In an effort to better understand the physical mechanisms responsible for pool boiling heat transfer, an analytical model is developed that describes the changing shape and size of a growing vapour bubble in a non-uniform temperature field. Indeed, any analysis of thermal transport due to nucleate pool boiling requires bubble frequency predictions which are intimately linked to bubble growth characteristics such as volume and time of detachment from the nucleation site. The model is developed and validated for quasi-static bubble growth due to vaporization within the heat-transfer controlled growth regime. It highlights the need to include the asymmetric nature of growing bubbles when modeling bubble growth and it accounts for the movement of the thermal boundary layer in the bubble׳s vicinity.Analysis of previously published numerical treatments of bubble shape evolution leads to simplifying bubble geometry for low Bond number applications. The geometric model accounts for bubble shape transformation throughout the bubble growth cycle. It is idealized as a segmented sphere atop a cylindrical neck. An analytical model of quasi-static diabatic bubble growth is accordingly developed and is coupled with a geometric detachment relation and a force balance detachment criterion that are dependent on the Bond number.The resultant low Bond number bubble evolution model is validated with images of n-pentane vapour bubble formations issuing from an artificial cavity. In order to solve the mass–energy balance at the vapour bubble interface, a spherical surface area is commonly assumed. This leads to the need for correction factors and provides little insight into the physical mechanism responsible for bubble shape. In this study, the transitioning shape of a vapour bubble is considered in the integral analysis of the interfacial mass–energy balance. The model predicts the following bubble growth and detachment characteristics: profile, volume, center of gravity, bubble height, Aspect Ratio and moment of detachment.

Stability of periodic bubble departures at a low frequency

The dynamics of bubble departures (at a frequency of f=3Hz) from a glass nozzle submerged in a tank filled with distilled water has been experimentally and theoretically studied. The volume of the system that supplies air to the nozzle (plenum chamber volume) and the air volume flow rate were changed in the experiment. The air pressure, bubble paths and liquid flow inside the nozzle were simultaneously recorded using a data acquisition system and a high-speed camera. It was shown that an increase in the plenum chamber volume leads to an increase in the intensity of the occurrences of chaotic changes in the subsequent waiting times. The analysis of the mechanism of the stability loss of the periodic bubble departures was based on changes in the time of the air pressure, the depth of the liquid penetration into the nozzle, the time of the bubble growth, the waiting time, and the bubble paths and their sizes, which is presented in this paper. The results of the analysis are compared with simulations that are based on the models of bubble growth and liquid flow inside the nozzle during the waiting time. It was shown that the air pressure rise, Δpl, during the waiting time is a non-linear function of the gas pressure after the bubble departure and the liquid velocity around the nozzle outlet. The nonlinearity of Δpl increases when the plenum chamber volume increases, and it decreases when the air volume flow rate increases.

A Simplified Bubble Nucleation, Growth and Coalescence Model for Coke Production Process

A Simplified Bubble Nucleation, Growth and Coalescence Model for Coke Production Process

Volcano inflation prior to an eruption: Numerical simulations based on a 1-D magma flow model in an open conduit

We numerically simulate volcanic inflation caused by magma ascent in a shallow conduit at volcanoes which repeatedly erupt, in order to understand the effect of volatile behavior on magma from geodetic data. Considering magma in which the relative velocities between melt and gas bubbles are negligible, we model magma flow in a one-dimensional open conduit with diffusive gas bubble growth. We calculate the ground displacements and tilts caused by spatio-temporal changes of magma pressure in the conduit. Our simulations show that magma without volatiles causes decelerated changes in volcanic inflation. Magma with gas bubble growth inflates the volcano with a constant, or accelerated, rate. Temporal changes of volcanic deformation are also affected by the magma pressure at the bottom of the conduit. When the pressure is small, the displacements and tilts increase in proportion to the 1.5th power of time. This time rate is similar to that predicted from a basic gas bubble growth model. When the pressure equals the lithostatic pressure, the effects of gas bubble growth relatively decrease and the displacements and tilts increase linearly with time.

Modelling of quasi-static adiabatic bubble formation, growth and detachment for low Bond numbers

In an effort to lessen the computational expense of bubble growth simulations without compromising its fundamental shape characteristics, an analytical model is developed. It is substantiated using validated numerical results simulating quasi-static adiabatic bubble growth for Bond numbers less than 0.07 in which its characteristic length is the radius of the cavity from which the bubble is issuing. The model's ability to predict shape and size evolution for bubble formations is shown to predict the growth and detachment volume to be in the range from 0.05% for a 0.00137 Bond number to 3% for a 0.06032 Bond number.The model builds upon a recent numerical study which showed that the shape evolution of a quasi-static bubble formation may be idealised as a spherical segment atop a cylindrical neck for low Bond number applications. By incorporating this geometry, the present work's proposed model accounts for bubble shape transformation throughout the bubble growth cycle by including a necking phenomenon in which the bulk of the bubble rises due to an elongating base as it prepares to detach. This is accomplished by introducing: (1) a volume condition which geometrically relates the neck height with the bubble's spherical segment at detachment; (2) a force instability criterion signalling the onset of detachment which relates the size of the bubble to its Bond number and cavity radius; and (3) a neck evolution growth curve. The analytical model ties these relations together with the use of the characteristics of the proposed geometry generating a full description of quasi-static adiabatic bubble growth and detachment for low Bond number formations. The resulting predicted bubble growth characteristics, such as profile, volume, centre of gravity, truncated sphericity and aspect ratio, are presented and discussed with respect to a validated numerical treatment of the problem.

Experimental and numerical studies on bubble dynamics in nonpressurized foaming systems

This work explores the influence of rheological properties on polymer foam development in nonpressurized systems. To understand the complex contributions of rheology on the mechanism of bubble growth during different stages of foam processing, visualization studies were conducted by using a polymer‐foaming microscopy setup. The evolving cellular structure during foaming was analyzed for its bubble surface density, bubble size, total bubble projected area, and bubble size distribution. Morphological analysis was used to determine the rheological processing window in terms of shear viscosity, elastic modulus, melt strength and strain‐hardening, intended for the production of foams with greater foam expansion, increased bubble density and reduced bubble size. A bubble growth model and simulation scheme was also developed to describe the bubble growth phenomena that occurred in nonpressurized foaming systems. Using thermophysical and rheological properties of polymer/gas mixtures, the growth profiles for bubbles were predicted and compared to experimentally observed data. It was verified that the viscous bubble growth model was capable of depicting the growth behaviors of bubbles under various processing conditions. Furthermore, the effects of thermophysical and rheological parameters on the bubble growth dynamics were demonstrated by a series of sensitivity studies. POLYM. ENG. SCI., 54:1947–1959, 2014. © 2013 Society of Plastics Engineers

A compound thermodynamic model for transient bubble growth in microscale

Based upon the classical bubble dynamics, a new compound thermodynamics model coupling with heat transfer mechanism for the bubble growth and collapse during explosive boiling under pulsed heating is presented from a hydrodynamic and thermodynamic perspective. The highlight in this analysis is that the relationship between the variation of the vapor temperature and the vapor pressure inside of a bubble satisfies the conditions of a polytropic process. With a selected compound polytropic index (CPI) and a determined vapor temperature variation for each different stage during the entire bubble evolution under pulsed heating, the bubble behavior can be described and predicted quite well when compared to the experimental observations for pulse ranges.For the first time, this model reveals the vapor pressure and temperature variation occurring during the bubble evolution and provides an in-depth understanding and insight into the bubble dynamics under pulsed heating. The thermodynamics analysis provides a first order solution to guide future thermal bubble MEMS design.

Bubble generation in oscillating heat pipe

In the previous models of oscillating heat pipe (OHP), only the evaporation and condensation in the liquid–vapor interfaces were considered as phase changes. Thus, only a simple sinusoidal oscillating motion of liquid slugs and vapor plugs was produced due to the small driving force. From our visualization results, the motions of liquid slugs and vapor plugs are very random and very different from the previous models. We observe that the bubble generation and growth phenomena are one of the reasons of the random motion of liquid slugs and vapor plugs. This means that those bubble generation and growth phenomena provide the large driving force. The velocity of liquid–vapor interfaces is accelerated suddenly when the TS (tube-size) bubbles, bubbles with the diameter equals to inner diameter of tube, are generated. The visualization results show that the TS bubbles will generate if the velocity of liquid slug is lower equal 0.2 m/s. The small bubble generation, before TS bubbles are generated, is ignored because they do not provide the large driving force. The model of TS bubble generation and growth is made by considering those visualization results to describe those phenomena.

Synchronization of two bubble trains in a viscous fluid: Experiment and numerical simulation

We investigate the interactions of two trains of bubbles, ejected by nozzles immersed in a viscous fluid, due only to the solution's circulation. The air fluxes (Q(1),Q(2)) are controlled independently, and we constructed parameter spaces of the periodicity of the attractors. We have observed complex behavior and many modes of phase synchronization that depend on these airflows as well as on the height (H) of the solution above the tops of the nozzles. Such synchronizations are shown in details in the parameter space (Q(1),Q(2)) and also in the (Q(1),H) space. We also observed that the coupling strength between the two trains of bubbles increases when the solution height increases. The experimental results were reasonably explained by numerical simulations of a model combining a simple bubble growth model for each bubble train and a coupling term between them, which was assumed symmetrical and proportional to the growth velocities.

DIMENSIONAL DEPENDENCE OF THE HYDRODYNAMICS OF CORE-COLLAPSE SUPERNOVAE

The multidimensional character of the hydrodynamics in core-collapse supernova (CCSN) cores is a key facilitator of explosions. Unfortunately, much of this work has necessarily been performed assuming axisymmetry and it remains unclear whether or not this compromises those results. In this work, we present analyses of simplified two- and three-dimensional CCSN models with the goal of comparing the multidimensional hydrodynamics in setups that differ only in dimension. Not surprisingly, we find many differences between 2D and 3D models. While some differences are subtle and perhaps not crucial to understanding the explosion mechanism, others are quite dramatic and make interpreting 2D CCSN models problematic. In particular, we find that imposing axisymmetry artificially produces excess power at the largest spatial scales, power that has been deemed critical in the success of previous explosion models and has been attributed solely to the standing accretion shock instability. Nevertheless, our 3D models, which have an order of magnitude less power on large scales compared to 2D models, explode earlier. Since we see explosions earlier in 3D than in 2D, the vigorous sloshing associated with the large scale power in 2D models is either not critical in any dimension or the explosion mechanism operates differently in 2D and 3D. Possibly related to the earlier explosions in 3D, we find that about 25% of the accreted material spends more time in the gain region in 3D than in 2D, being exposed to more integrated heating and reaching higher peak entropies, an effect we associate with the differing characters of turbulence in 2D and 3D. Finally, we discuss a simple model for the runaway growth of buoyant bubbles that is able to quantitatively account for the growth of the shock radius and predicts a critical luminosity relation.

Bubble Lift-Off Diameter Prediction with New Bubble Growth Model

Based on the equation that the heat conduction from the bubble underlying layer is balanced with the heat convection of bubble cooling and the latent heat of evaporation, with the help of zhao model a new bubble growth model for the forced convective subcooled boiling in the engine cooling system is developed with the consideration of the thermal boundary layer impact on the bubble growth. And an experiment is designed to verify the theoretical model. Then the Klausner model for predicting the bubble lift-off diameters is amended with the built bubble growth model. In addition, the bubble lift-off diameters at the different velocity are predicted, which is verified with the experimental data.

Model for growth of bubbles during proofing of viscoelastic dough

A model for growth of bubbles during proofing of bread dough is presented for infinitely dilute as well as finite volume fraction gas dispersions. The model accounts for diffusion of CO2, produced by fermentation, along with the resistance to bubble expansion by the viscoelasticity of the dough as described by Oldroyd-B constitutive equation. The model predicts evolution of bubble size, bubble growth rate and the profiles of CO2 concentration and normal stresses in the vicinity of the gas bubble. The growth of bubbles exhibits a lag time followed by an exponential growth phase. The lag time is found to be smaller for larger dough relaxation time, larger surface tension, smaller dough viscosity, higher temperature, higher gas volume fraction and higher CO2 production rate. The bubble growth rate is larger for smaller bubbles resulting in crossover of bubble size at longer times.

Approximate Explicit Nonlinear Receding Horizon Control for Decompression of Divers

This paper is based on a comprehensive dynamic mathematical model (Copernicus) of vascular bubble formation and growth during and after decompression from a dive. The model is founded on the statistical correlation between measurable venous gas emboli (VGE) and risk of severe decompression sickness (DCS) where VGE has been shown to be a reliable and sensitive predictor of decompression stress. By using the Copernicus model the diving decompression problem is formulated as a nonlinear optimal control problem, where the objective is to minimize the total ascend time subject to constraints on the maximum bubbles volume in the central venous pool. A recent study reveals that the optimal solution can be obtained by solving the optimization problem with equality constraints. Inspired by which, a simpler approach using barrier function is proposed in this paper, through which we achieve a more efficient and robust numerical implementation. To reduce the complexity of the nonlinear optimization problem this paper also studies the decompression profile parameterization and its effect. Furthermore, by applying multi-parametric nonlinear programming technique, an approximate explicit solution to the nonlinear optimization problem is obtained, which makes the practical implementation on a typical low-cost diving computer possible.

Lead-Free Solder Joint Void Evolution During Multiple Subsequent High-Temperature Reflows

Entrapment of volatiles during lead-free solder joint formation results in the creation of voids which may have a negative impact on the joint's mechanical and/or electrical performances. According to IPC-J-STD-001E and IPC-A-610E specifications, the post-surface-mount-technology cumulative voiding criterion in lead-free solder joints is less than 25% for a second-level interconnect. For solder joints that experience multiple high-temperature reflow processes after completion of assembly, however, it is important to understand and predict how these voids will interact and evolve during subsequent high-temperature exposures. In this paper, both in situ 2-D X-ray imaging and 3-D X-ray tomography were used to study the growth kinetics of solder joint voids during multiple reflow cycles. The results demonstrate that voids grow and move during each reflow cycle. The growth kinetics has been shown to follow a diffusion-controlled mechanism based on a model for out-gassing bubble growth in a supersaturated molten solder liquid.

On the mechanism of porosity formation during welding of titanium alloys

The mechanism of porosity formation during the fusion welding of titanium and its alloys is studied. Porosity formed during the electron beam welding of titanium is characterized using high-resolution X-ray tomography, residual gas analysis and metallographic sectioning; the results confirm that porosity formation is associated with evolution of gas, especially hydrogen. A model for hydrogen diffusion-controlled bubble growth is proposed, to aid in the interpretation of these findings. To investigate further the effect of hydrogen on porosity formation, hydrogen charging is used to achieve different hydrogen levels prior to welding. The results confirm that porosity can be suppressed even at every high hydrogen levels, when welding is carried out with optimized welding parameters and perfect joint alignment; on the other hand, porosity is exacerbated when a small beam offset is employed. This is because any beam offset alters the size of the liquid zone at the melting front, where the joint edges first become melted. It is proposed that the thickness of the liquid film at the melting front is crucial for bubble nucleation and bubble survival in the weld pool; bubbles can escape through the keyhole by breaking through this liquid film, when it is too thin. This challenges the common assumption of bubble escape by flotation to the weld pool surface. Thus the nucleation rate in the liquid zone at the melting front determines the likelihood of porosity occurring. This suggests that the beam offset is likely to be one factor influencing porosity formation in these circumstances. The paper provides fundamental insights into the mechanism of porosity formation during the welding of titanium alloys and guidance to aid in its elimination.

CO2 variability in mid-ocean ridge basalts from syn-emplacement degassing: Constraints on eruption dynamics

Basaltic glasses produced during mid-ocean ridge (MOR) eruptions display a wide range of dissolved CO2 concentrations with respect to equilibrium saturation at their eruption depth. This variability is thought to reflect the dynamics of magma ascent and emplacement, with rapid ascent and depressurization leading to supersaturated conditions and slower ascent resulting in equilibrium saturation. In this study we examine a suite of samples from the 2005–06 eruption of the East Pacific Rise (EPR), collected at ~200m intervals along two well-defined flow pathways within a single seafloor eruption to examine syn-emplacement degassing, using measurements of vesicularity, crystallinity, volatile contents, and helium. This new data set is unique because flow pathways of individual MOR lava flows have rarely been mapped and have never been systematically sampled. Here we show that a large range in dissolved CO2 concentrations exists within this single eruption that nearly spans that observed in tens of flows measured along this portion of the EPR crest. This lava flow experienced equilibrium degassing of dissolved CO2 from supersaturated conditions at the vent (and persisting over the first 750m of flow) toward equilibrium with seafloor pressures along a ~2.5km-long flow path. This was accompanied by an increase in vesicularity and characteristic bubble radius along the flow, indicating the importance of bubble growth by diffusion for degassing. Nearly constant total helium concentrations (melt+vesicles) indicate that no bubbles were lost during emplacement. With a model of diffusion-controlled bubble growth constrained by observations from the samples along the largest flow lobe of the eruption, we calculate minimum ascent rates of 0.15m/s, and – for this portion of the eruption – an eruption duration of ~30h and flow rates from 0.02 to 0.12m/s. This study represents the first quantitative assessment of eruption dynamics using volatiles in a mid-ocean ridge lava flow.

Numerical resolution of a mono-disperse model of bubble growth in magmas

Growth of gas bubbles in magmas may be modeled by a system of differential equations that account for the evolution of bubble radius and internal pressure and that are coupled with an advection–diffusion equation defining the gas flux going from magma to bubble. This system of equations is characterized by two relaxation parameters linked to the viscosity of the magma and to the diffusivity of the dissolved gas, respectively. Here, we propose a numerical scheme preserving, by construction, the total mass of water of the system. We also study the asymptotic behavior of the system of equations by letting the relaxation parameters vary from 0 to ∞, and show the numerical convergence of the solutions obtained by means of the general numerical scheme to the simplified asymptotic limits. Finally, we validate and compare our numerical results with those obtained in experiments.

Modeling of helium bubble nucleation and growth in neutron irradiated boron doped RAFM steels

Reduced activation ferritic/martensitic (RAFM) steels are promising candidates for structural materials in future fusion technology. In addition to other irradiation defects, the transmuted helium is believed to strongly influence material hardening and embrittlement behavior. A phenomenological model based on kinetic rate equations is developed to describe homogeneous nucleation and growth of helium bubbles in neutron irradiated RAFM steels. The model is adapted to different 10B doped EUROFER97 based heats, which already had been studied in past irradiation experiments. Simulations yield bubble size distributions, whereby effects of helium generation rate, surface energy, helium sinks and helium density are investigated. Peak bubble diameters under different conditions are compared to preliminary microstructural results on irradiated specimens. Helium induced hardening was calculated by applying the Dispersed Barrier Hardening model to simulated cluster size distributions. Quantitative microstructural investigations of unirradiated and irradiated specimens will be used to support and verify the model.

Visual observation and numerical studies of N2 vs. CO2 foaming behavior in core‐back foam injection molding

AbstractWe investigated, by visual observation and numerical calculations, the foaming behavior of polypropylene within a foam injection mold cavity with the environmentally benign physical blowing agents nitrogen (N2) and carbon dioxide (CO2). An 85‐ton core‐back injection‐molding machine with temperature and pressure monitoring systems as well as a high‐pressure view cell was used for the investigation. The experiments showed a prominent difference in bubble nucleation and growth between N2 and CO2 injection foaming. Even when the weight concentration of N2 dissolved in polymer was one‐third that of CO2, N2 injection foaming provided a bubble number density that was 30 times larger and a bubble size that was one‐third smaller compared to CO2 injection foaming. Classical bubble nucleation and growth models developed for batch foaming were employed to analyze these experimental results. The models reasonably explained the differences in injection foaming behavior between N2 and CO2. It was clearly demonstrated by both experiments and numerical calculations that N2 provides a higher number of bubbles with a smaller bubble size in foam injection molding compared to CO2 as a result of the lower solubility of N2 in the polymer and the larger degree of super‐saturation. POLYM. ENG. SCI., 2011. ©2011 Society of Plastics Engineers