Bubble Growth Model Research Articles

Polymeric Foaming Simulation for Extrusion Processes

A numerical simulation for polymeric foaming extrusion processes was conducted. Combining classical nucleation rate and bubble growth models with a non-Newtonian fluid model of a flow, a simultaneous bubble nucleation and growth behavior in a flow field was simulated. Simulation results were compared with the experimental data obtained by visual observations at a foaming extruder, where a polypropylene resin was physically foamed. The effects of physical parameters in foaming model on bubble size and number density calculation were intensively examined by sensitivity analysis.

Inferring volcanic degassing processes from vesicle size distributions

Both power law and exponential vesicle size distributions (VSDs) have been observed in many different types of volcanic rocks. We present results of computer simulations and laboratory analogue experiments which reproduce these findings and show that the distributions can be interpreted as the product of continuous bubble nucleation resulting from non‐equilibrium degassing. This ongoing nucleation causes the bubbles to evolve through an exponential size distribution into a power law size distribution as nucleation and growth progress. These findings may explain the apparent contradiction between present models of bubble growth in magmas, which predict that degassing in explosive eruptions is a non‐equilibrium process, and models of conduit flow, which assume perfect equilibrium degassing. The process of continuous nucleation is the mechanism whereby the volcanic system maintains near‐equilibrium in the case of rapid depressurization and slow volatile diffusion.

Degassing of water-solved gases in thin pipings during fast pressure drops

A one-dimensional model is presented, which describes the transient two-phase flow in thin pipes during fast pressure drops and degassing by use of Eulerian and Lagrangian systems. The reduction in dimension is obtained by introduction of a geometry model for bubbly and slug flow regimes. The complete model includes the transient two-phase flow, bubble formation and bubble growth. The flow model predicts rising velocities of bubbles and plugs in arbitrary inclined highly accurate pipes. The mass transfer (diffusion) of the dissolved phase is calculated by the bubble growth model. The quality of the model was examined by simulation of experimental series, whereby water was depressurised from the saturation pressure of the dissolved gas mixture (air), by variation of saturation pressure, pressure gradient and pipe geometry. The results of numerical simulation fit the experimental data well.

Bubble growth in rhyolitic melt

We report experimental data of bubble growth in natural rhyolitic melt with 1.4–2.0 wt% initial total H 2O at 0.1 MPa and 500–600°C. Growth of many bubbles is monitored in every experiment. Bubble growth rate increases with temperature and initial total H 2O. At a given temperature and initial total H 2O, bubble growth rate also increases slightly with bubble size. The average growth rate for bubbles growing in an infinite rhyolitic melt at a bubble radius of 25 μm is ∼0.038 μm/s at 600°C and 2.0 wt% H 2O t,i, ∼0.0064 μm/s at 550°C and 2.0 wt% H 2O t,i, ∼0.00034 μm/s at 500°C and 2.0 wt% H 2O t,i, and ∼0.007 μm/s at 600°C and 1.45 wt% H 2O t,i. Our data show that bubble growth is controlled by both viscous flow and diffusion, with viscous flow playing a dominant role at the initial stage. We used the numerical bubble growth model of Proussevitch and Sahagian [J. Geophys. Res. 103 (1998) 18223–18251] to calculate bubble growth rate. Using recently assessed solubility, diffusivity, and viscosity models, the bubble growth model can reproduce our experimental data to within a factor of about 2. Adjusting the viscosity model by a factor of up to 5 (within the stated 2 σ uncertainty of a factor of 8.3) generates almost perfect fits to our data. (Adjusting other parameters within uncertainty does not help significantly.) Other recent bubble growth data, except those at 0.1 MPa and 375–460°C with 5.0 wt% initial H 2O, are also consistent with the calculation. This is the first experimental verification of a bubble growth model in a silicate melt. We suggest that the verified model can be applied with confidence to quantify bubble growth during volcanic eruptions.

Recent developments in modeling gas-phase catalyzed olefin polymerization fluidized-bed reactors: The effect of bubble size variation on the reactor's performance

In the present study recent developments in modeling gas-phase catalyzed olefin polymerization fluidized-bed reactors (FBR) are critically reviewed. A new FBR model is developed to account for the effect of varying bubble size with the bed height on the reactor dynamics and the molecular properties of the polymer product. A comprehensive kinetic model for ethylene copolymerization in the presence of a multisite Ziegler–Natta catalyst is considered to describe the molecular weight developments in the FBR. The bubble-growth model developed in the present study is subsequently compared with two well-known FBR models, namely, the well-mixed and the constant bubble size model. The effect of important reactor parameters such as superficial gas velocity, maximum stable bubble size, mean particle size, catalyst injection rate, monomer/comonomer feed ratio and temperature of the feed stream, on the dynamic and steady-state behavior of the FBR is investigated. It is shown that the maximum stable bubble size, a critical parameter in the constant bubble size model, turns out to be less important when the bubble size is allowed to vary with the bed height. For typical industrial operating conditions, the constant bubble size model consistently overpredicts the emulsion phase temperature and monomer conversion, while the well-mixed model underestimates them. On the other hand, the bubble-growth model shows an intermediate behavior since it represents a more realistic description of the gas phase in the bed.

Experimental constraints on degassing of magma: isothermal bubble growth during continuous decompression from high pressure

Numerical models predict that rapid ascent of hydrous magma can lead to supersaturation of dissolved volatile constituents, possibly leading to explosive eruption. We have performed controlled decompression experiments to investigate the ascent rates required to maintain bubble–melt equilibrium. High-silica rhyolitic melts were saturated with water at 200 MPa and 825°C, decompressed to lower pressures at constant rates of 0.025, 0.25, 0.5, and 1.0 MPa s−1, and then rapidly quenched isobarically. Other samples were saturated with water over the pressure range investigated to determine equilibrium water solubility in order to quantify degassing efficiency during decompression. At a decompression rate of 0.025 MPa s−1, melt–vapor equilibrium was maintained over the entire pressure range examined: 200 to 17.5 MPa. A single bubble nucleation event occurred in response to decompression, and quenched bubble sizes can be modeled by a equilibrium bubble growth model that takes into account the number density of bubbles. At decompression rates of 0.25, 0.5, and 1.0 MPa s−1, rhyolitic melts could not degas in equilibrium when pressure decreased from 200 MPa to 140 MPa, and water supersaturation (ΔP) in the melt reached up to 60 MPa, with higher values at faster decompression rates. Further pressure release resulted in near equilibrium degassing and ΔP dropped significantly. In each case, ΔP decreased when bubbles exceeded 10 vol.%. A single, heterogeneous bubble nucleation event occurred in each experiment when ΔP<20 MPa; no other bubbles nucleated despite ΔP reaching 60 MPa, which is probably too low to trigger homogeneous nucleation. Compared to estimates for magma decompression rates during lava dome eruptions, our results indicate that magmas can degas efficiently throughout their ascent to the surface. In explosive eruptions, decompression rates may exceed those of this study and hence melts may become supersaturated with water. Such fast decompressions are expected, however, only when magma is highly vesicular, which would aid approach to equilibrium degassing.

高分子発泡成形における気泡形成シミュレーション （１） 均質核生成系の物理発泡シミュレーションと実験

The effect of processing variables, such as concentration of dissolved gas, release rate of pressure and operating temperature, on bubble growth dynamics, nucleation rate and final bubble size was examined. In the polyethylene foaming process, fundamental experiments showed that bubble size increased when the release rate of pressure decreased, whereas bubble size decreased when the operating temperature decreased. Furthermore, a higher concentration of dissolved gas resulted in smaller bubble size. These results were confirmed by numerical simulations using Shafi's nucleation and bubble growth models. Both simulation and experimental results elucidated that release rate of pressure, concentration of dissolved gas and operating temperature are the principal factors which control bubble size and foam density in the physical foaming process.

高分子発泡成形における気泡形成シミュレーション （２） 均質・不均質混合核生成系の物理発泡シミュレーション

Numerical simulations were carried out on a polyethylene/N2 physical foaming process where homogeneous and heterogeneous nucleations occurred simultaneously, using Shafi's nucleation and bubble growth models to observe the effect of nucleating agent and processing conditions on the size and the number density of bubbles. The interaction between homogeneously-nucleated and heterogeneously-nucleated bubbles could be observed when the pressure was gradually released.

Dynamics and energetics of bubble growth in magmas: Analytical formulation and numerical modeling

We have developed a model of diffusive and decompressive growth of a bubble in a finite region of melt which accounts for the energetics of volatile degassing and melt deformation as well as the interactions between magmatic system parameters such as viscosity, volatile concentration, and diffusivity. On the basis of our formulation we constructed a numerical model of bubble growth in volcanic systems. We conducted a parametric study in which a saturated magma is instantaneously decompressed to one bar and the sensitivity of the system to variations in various parameters is examined. Variations of each of seven parameters over practical ranges of magmatic conditions can change bubble growth rates by 2–4 orders of magnitude. Our numerical formulation allows determination of the relative importance of each parameter controlling bubble growth for a given or evolving set of magmatic conditions. An analysis of the modeling results reveals that the commonly invoked parabolic law for bubble growth dynamics R∼t1/2 is not applicable to magma degassing at low pressures or high water oversaturation but that a logarithmic relationship R∼log(t) is more appropriate during active bubble growth under certain conditions. A second aspect of our study involved a constant decompression bubble growth model in which an initially saturated magma was subjected to a constant rate of decompression. Model results for degassing of initially water‐saturated rhyolitic magma with a constant decompression rate show that oversaturation at the vent depends on the initial depth of magma ascent. On the basis of decompression history, explosive eruptions of silicic magmas are expected for magmas rising from chambers deeper than 2 km for ascent rates &gt;1–5 m s−1.

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.

Near-Wall Microlayer Evaporation Analysis and Experimental Study of Nucleate Pool Boiling on Inclined Surfaces

Boiling heat transfer from inclined surfaces is examined and an analytical model of bubble growth and nucleate boiling is presented. The model predicts the average heat flux during nucleate boiling by considering alternating near-wall liquid and vapor periods. It expresses the heat flux in terms of the bubble departure diameter, frequency and duration of contact with the heating surface. Experiments were conducted over a wide range of upward and downward-facing surface orientations and the results were compared to model predictions. More active microlayer agitation and mixing along the surface as well as more frequent bubble sweeps along the heating surface provide the key reasons for more effective heat transfer with downward facing surfaces as compared to upward facing cases. Additional aspects of the role of surface inclination on boiling dynamics are quantified and discussed.

The flow and heat transfer in the wedge-shaped liquid film formed during the growth of a vapour bubble

After providing simple equations that should govern the behaviour of a wedge-shaped liquid film sandwiched between the vapour in a growing bubble and the heated wall, we discuss some questions that may be important for modelling heat transfer in nucleate boiling. Considered are the curvature of the film surface and the liquid flow in the film, for example. We further examine the consequences of some simplifications adopted in models from the literature. Our analysis indicates that the bubble interface is concave-convex in the region of the three-phase-line (TPL). The convex curvature is caused by strong evaporation at the tip of the liquid wedge under no slip condition of the liquid on the wall surface. The ultra-thin layer of this wedge adhering to the wall is practically motionless and hence incapable of supplying the quantity of liquid to the film tip necessary for evaporation to occur rapidly at a fixed TPL. Based on this fact, we are led to the conclusion that, under common boiling conditions, the TPL and the film interface are not motionless, as was assumed in very recent models of bubble growth. These models are shown in the present paper to be inconsistent. Namely, the liquid flow orthogonal to the heated surface is ignored within the hydrodynamics, but this flow is indeed included into the energy balance of the models.

Generalized Model for Bubble Size and Frequency in Gas-Fluidized Beds

A generalized bubble-growth model on mean bubble size and frequency has been proposed in bubbling gas-fluidized beds. Correlations on volumetric bubble flow rate, splitting frequency of a bubble, and equilibrium bubble diameter have been considered in this model. The equilibrium bubble diameter was shown to increase linearly with the ratio of volumetric bubble flux to the splitting frequency of a bubble. The proposed model was in good agreement with the extensive data reported on mean bubble size and frequency.

The effect of the mean particle size on the dynamic behaviour of catalyzed olefin polymerization fluidized bed reactors

Recent developments in modelling gas-phase catalyzed olefin fluidized bed polymerization reactors are critically examined in order to quantify the effect of the mean particle size on the dynamic behaviour of the FBR. A comprehensive population balance model is developed to investigate the effect of particle size distribution in gas-phase fluidized bed olefin polymerization reactors. To assess the impact of mean particle size on the dynamic behaviour of an FBR, a new bubble-growth model is proposed. Simulation results show that model predictions are strongly influenced by the choice of the mean particle diameter, since its value dramatically affects the main fluidization parameters (e.g., solids terminal velocity, maximum stable bubble size, minimum fluidization velocity and the heat and mass transfer coefficients) in the bed.

Analysis of diffusion-induced bubble growth in viscoelastic liquids

Transport models of diffusion-induced bubble growth in viscoelastic liquids are developed and evaluated. A rigorous model is formulated that can be used to describe bubble growth or collapse in a non-linear viscoelastic fluid, and takes into account convective and diffusive mass transport as well as surface tension and inertial effects. Predictions for bubble growth dynamics demonstrating the importance of fluid elasticity are presented. These predictions indicate that for diffusion-induced bubble growth in viscoelastic liquids, the lower bound for growth rate is given by growth in a Newtonian fluid and the upper bound by diffusion-controlled growth. The influence of non-linear fluid rheology on bubble growth dynamics is examined and found to be relatively minor in comparison to fluid elasticity. It is shown how previously published models employing various approximations can be derived from the rigorous model. Comparisons of predicted bubble growth dynamics from the rigorous and approximate models are used to establish the ranges of applicability for two commonly-used approximations. These comparisons indicate that models using a thin boundary layer approximation have a rather limited range of applicability. An analysis of published experimental bubble growth data is also carried out using appropriate transport models.

Polymer trace devolatilization: I. Foaming experiments and model development

AbstractVolatile species are removed from polymers by devolatilization. Foaming experiments using corn syrup and low‐density polyethylene were carried out to investigate effects of volatile content, operating vacuum level and stripping agent in devolatilization. When foaming and rupture occur, a limiting foam volume growth was observed for a polymer regardless of the volatile content and the addition of stripper bubbles. A mathematical model was also developed to study the effect of stripper bubbles in polymer devolatilization. Based on the cell model for bubble growth, a bimodal model for foam growth is developed as the first step to describe the initial formation and growth of the volatile and stripper bubbles in polymer during devolatilization. When foaming and breakup occur at a critical foam volume expansion, a film model for mass transfer is used to model the second step of devolatilization. A numerical factorial study shows that the devolatilization section should be designed and operated to provide the screw channel sufficient free volume and residence time for the desired foam expansion.

Dynamics of gas‐driven eruptions: Experimental simulations using CO2‐H2O‐polymer system

We report exploratory experiments simulating gas‐driven eruptions using the CO2‐H2O system at room temperature as an analog of natural eruptive systems. The experimental apparatus consists of a test cell and a large tank. Initially, up to 1.0 wt% of CO2 is dissolved in liquid water under a pressure of up to 735 kPa in the test cell. The experiment is initiated by suddenly reducing the pressure of the test cell to a typical tank pressure of 10 kPa. The following are the main results: (1) The style of the process depends on the decompression ratio. There is a threshold decompression ratio above which rapid eruption occurs. (2) During rapid eruption, there is always fragmentation at the liquid‐vapor interface. Fragmentation may also occur in the flow interior. (3) Initially, the top of the erupting column ascends at a constant acceleration (instead of constant velocity). (4) Average bubble radius grows as t2/3. (5) When viscosity is 20 times that of pure water or greater, a static foam may be stable after expansion to 97% vesicularity. The experiments provide several insights into natural gas‐driven eruptions, including (1) the interplay between bubble growth and ascent of the erupting column must be considered for realistic modeling of bubble growth during gas‐driven eruptions, (2) buoyant rise of the bubbly magma is not necessary during an explosive volcanic eruption, and (3) CO2‐driven limnic eruptions can be explosive. The violence increases with the initial CO2 content dissolved in water.

Dynamics of vapour bubbles in nucleate boiling

This paper considers the behaviour of a vapour bubble formed at a nucleation site on a heated horizontal wall. This bubble is modeled as a spherical segment which is separated from the wall by a microlayer of intervening liquid. The liquid is presumed to be at rest at great distances from the bubble. In order to avoid unwarranted assumptions about forces acting on the bubble which are specific to all known models of bubble growth and detachment, we derive equations that govern bubble behaviour in a rigorous way from the variational equation that describes mechanical energy conservation for the whole system, which includes both the bubble and the liquid. The variational equation leads to a set of two mutually independent strongly nonlinear equations which govern bubble expansion and the motion of its centre of mass. Because these equations contain an extra unknown variable (the bubble vapour pressure), a supplementary equation that defines bubble vapour temperature must be formulated with allowance made for heat transfer to the bubble both from the bulk of the surrounding liquid and through the microlayer. The most important conclusion of this paper consists in the fact that surface tension effects result in an effective force that tends to transform the bubble into a sphere, thereby facilitating bubble detachment. This conclusion absolutely nullifies the generally, however erroneously, held belief that this effective force presses the bubble to the wall. By way of example, we consider the evolution of bubbles whose growth is thermally controlled. Our analysis provides quite a natural explanation for a number of repeatedly observed phenomena, such as the influence of gravity and surface tension on bubble growth rate and the dependence of bubble detachment size on thermophysical parameters.

Nucleate pool boiling of pure liquids and binary mixtures: Part II — Analytical model for boiling heat transfer of binary mixtures on smooth tubes and comparison of analytical models for both pure liquids and binary mixtures with experimental data

A combined physical model of bubble growth is proposed along with a corresponding bubble growth model for binary mixtures on smooth tubes. Using the general model of Wang et al.[1] and the bubble growth model for binary mixtures, an analytical model for nucleate pool boiling heat transfer of binary mixtures on smooth tubes is developed.

Nucleate pool boiling of pure liquids and binary mixtures: Part I — Analytical model for boiling heat transfer of pure liquids on smooth tubes

A mechanism is proposed for nucleate pool boiling heat transfer along with a general model for both pure liquids and binary mixtures. A combined physical model of bubble growth is also proposed along with a corresponding bubble growth model for pure liquids on smooth tubes. Using the general model and the bubble growth model for pure liquids, an analytical model for nucleate pool boiling heat transfer of pure liquids on smooth tubes is developed.