Dynamics of foam growth: Bubble growth in a limited amount of liquid

A mathematical model for the growth of a spherical gas bubble in a limited amount of a viscous liquid is presented. The growth of the bubble is assumed to be controlled by both momentum and mass transfer. The simplified approach suggested, by assuming a parabolic concentration profile for the volatile component in the liquid, results in approximated analytical expressions for the final parameters of the process. The numerical results of the model can be used to predict the increase in the bubble size as well as the decrease of the solute concentration in the liquid and the decrease of the cell density with time. The model is able to deal with real processes, such as polymer melt devolatilization and the production of polymeric foams, where many bubbles grow simultaneously. Polym. Eng. Sci. 44:1900–1906, 2004. © 2004 Society of Plastics Engineers.     

DYNAMIC SURFACE EFFECT ON THE BOILING OF MIXTURES

It is possible that the critical boiling heat flux of mixtures exceeds that of the components considerably. This effect has been attributed to the slowing down in bubble growth rate caused by the exhaustion of the volatile component near the vapor-liquid interface (the F effect) and the retardation of bubble coalescence due to surface tension gradient caused by the evaporation of component of lower surface tension (the M effect). The growth and coalescence of vapor bubbles are dynamic processes. The surface tension of a fast stretching surface of a binary mixture is higher than the static value because the component of lower surface tension cannot diffuse to the adsorbed layer promptly. This dynamic effect is represented quantitatively by a Y function.

The Y values for various binary systems are plotted versus concentration along with the F and M values in comparison with the critical heat flux values. The resemblance between these curves shows that F, M and Y effects are all partial reasons for the increasing of the critical heat flux. They are interrelated and further studies are desirable. The agreement between variation of the tray efficiencies of a fractionation distillation column and that of the Y values shows another example of that the dynamic surface effect is an important factor in many heat and mass transfer operations     

DYNAMIC SURFACE EFFECT ON THE BOILING OF MIXTURES

It is possible that the critical boiling heat flux of mixtures exceeds that of the components considerably. This effect has been attributed to the slowing down in bubble growth rate caused by the exhaustion of the volatile component near the vapor-liquid interface (the F effect) and the retardation of bubble coalescence due to surface tension gradient caused by the evaporation of component of lower surface tension (the M effect). The growth and coalescence of vapor bubbles are dynamic processes. The surface tension of a fast stretching surface of a binary mixture is higher than the static value because the component of lower surface tension cannot diffuse to the adsorbed layer promptly. This dynamic effect is represented quantitatively by a Y function.

The Y values for various binary systems are plotted versus concentration along with the F and M values in comparison with the critical heat flux values. The resemblance between these curves shows that F, M and Y effects are all partial reasons for the increasing of the critical heat flux. They are interrelated and further studies are desirable. The agreement between variation of the tray efficiencies of a fractionation distillation column and that of the Y values shows another example of that the dynamic surface effect is an important factor in many heat and mass transfer operations     

Analysis of non-explosive bubble growth within a superheated liquid droplet suspended in an immiscible liquid

Bubble growth within a volatile droplet (liquid 1) at its superheat limit suspended in an immiscible nonvolatile field liquid (liquid 2) is analysed by solving the coupled energy and momentum equations for the temperature fields in liquids 1 and 2. A numerical solution is presented for a two-phase droplet modelled as a vapour bubble growing from the centre of liquid 1. It is shown that when the properties of liquids 1 and 2 are appreciably different, the bubble growth rate can experience a significant increase or decrease when the thermal boundary layer extends into liquid 2. The present calculations are also compared with available data and the agreement is reasonable.     

Experimental and numerical studies on the effects of pressure release rate on number density of bubbles and bubble growth in a polymeric foaming process

A simulation of simultaneous bubble nucleation and growth was performed for a batch physical foaming process of polypropylene (PP)/CO₂ system under finite pressure release rate. In the batch physical foaming process, CO₂ gas is dissolved in a polymer matrix under pressure. Then, the dissolved CO₂ in the polymer matrix becomes supersaturated when the pressure is released. A certain degree of supersaturation produces CO₂ bubbles in the polymer matrix. Bubbles are expanded by diffusion of the dissolved CO₂ into the bubbles. The pressure release rate is one of the control factors determining number density of bubbles and bubble growth rate.To study the effect of pressure release rate on foaming, this paper developed a simple kinetic model for the creation and expansion of bubbles based on the model of Flumerfelt's group, established in 1996 [Shafi, M.A., Lee, J.G., Flumerfelt, R.W., 1996. Prediction of cellular structure in free expansion polymer foam processing. Polymer Engineering and Science 36, 1950-1959]. It was revised according to the kinetic experimental data on the creation and expansion of bubbles under a finite pressure release rate. The model involved a bubble nucleation rate equation for bubble creation and a set of bubble growth rate equations for bubble expansion. The calculated results of the number density of bubbles and bubble growth rate agreed well with experimental results. The number density of bubbles increased with an increase in the pressure release rate. Simulation results indicated that the maximum bubble nucleation rate is determined by the balance between the pressure release rate and the consumption rate of the physical foaming agent by the growing bubbles. The bubble growth rate also increased with an increase in the pressure release rate. Viscosity-controlled and diffusion-controlled periods exist between the bubble nucleation and coalescence periods.     

A study of bubble nucleation in a mixture of molten polymer and volatile liquid in a shear flow field

Bubble nucleation in a mixture of volatile liquid and polymer melt under shear flow conditions was investigated, using a light scattering technique. In the study, a mixture of polystyrene and trichlorofluoromethane was extruded through a slit die having glass windows and bubble nucleation in the flow channel was observed optically. A He-Ne laser was used to illuminate the nucleating and growing bubbles. The light flux scattered by the growing bubbles at a fixed angle was detected by a photomultiplier with the aid of a high-voltage power supply. The bubble nucleating site in the flow channel was located using a computer controlled tracking system, which was designed to move the entire optical system automatically in the three dimensional space, and also had the ability to follow the software control command and cooperate with the data acquisition system. When the site of bubble nucleation was located, the coordinates of this site in the flow channel and the experimental conditions were automatically recorded on a floppy diskette by entering a software command. The pressure profile along the flow channel was measured by pressure transducers, with the aid of a microprocessor-based pressure reading system. It has been found that the site of bubble nucleation varies with the position in the direction perpendicular to the flow direction, which is attributed to the nonuniform velocity and stress distributions in the slit flow channel. The present investigation suggests that bubble nucleation can be induced either by flow and/or shear stress; specifically, flow-induced bubble nucleation is the dominant mechanism at positions near the center of the die opening, and shear-induced bubble nucleation is the dominant mechanism at positions near the die wall. It should be mentioned that the bubble near the die wall may also be generated by cavitation brought about by the surface roughness of the wall and also by thermal fluctuations due to the heat transfer between the metal (die wall) and the mixture of polymer and volatile component. The present study indicates that bubble nucleation in a shear flow field can occur at an unsaturated condition. This is in contrast to bubble nucleation under static conditions, where supersaturation is necessary.     

乙烷池内核态沸腾气泡脱离直径

针对0.15、0.2、0.3 MPa 3个压力进行乙烷池内核态沸腾可视化实验研究,实验测量的热通量范围是14.27～81.22 kW·m^-2。高速摄像机采集得到竖直铜棒的光滑上表面的乙烷气泡的脱离图像,利用图像处理软件获得气泡脱离直径,并分析了Jacob数（Ja）与气泡脱离直径的变化关系。实验所测直径与引用广泛的6个关联式进行比较,Kim和Kim（2006）模型的预测效果较好,绝对平均偏差均在30%以内,但在0.15 MPa工况下,50%的预测值偏差为30%～40%。Kim和Kim（2006）模型提出Bo^1/2和Ja呈幂函数的关系。在对比基础上,用乙烷的气泡脱离直径数据拟合得到的新关联式与实验数据偏差在±30%以内。另外,选取文献中甲烷的气泡脱离直径与新关联式进行对比,在4种压力下的预测值偏差几乎均在±30%以内（只有一个预测值在±30%以外）。新关联式对甲烷和乙烷的工况具有良好的预测效果,但是由于拟合数据所用的Ja较小,在使用范围上具有一定的局限性。     

Controlled electrochemical gas bubble release from electrodes entirely and partially covered with hydrophobic materials

This paper deals with an experimental study on millimetre-size electrochemically evolved hydrogen bubbles. A method to generate gas bubbles controlled in number, size at detachment and place on a flat electrode is reported. Partially wetted composite islands are implemented on a polished metal substrate. As long as the island size is lower than a limit depending on its wettability, only one bubble spreads on the island and its size at detachment is controlled by the island perimeter. The composite, a metal–polytetrafluoroethylene (Ni–PTFE), is obtained by an electrochemical co-deposition process. On the contrary to predictions of available models for co-deposition, at current densities beyond Ni²⁺ limiting current density, the mass ratio of PTFE in the deposit strongly increases. A mechanism is proposed to describe co-deposition when hydrogen bubbles are co-evolved. The observation of gas evolution on fully hydrophobic electrodes highlights the fact that bubbles growth rate on such electrodes differs from growth rates when bubble growth is controlled by mass transport of dissolved gas. The more a bubble grows by coalescence the more its foot expands on the electrode the bigger its size at detachment. This triple line creeping mechanism explains why, when attached bubbles coalesce many times before detaching, their size at detachment increases with current density.     

The formation and growth of bubbles at a submerged orifice

A model has been developed to describe the formation of single bubbles at a submerged orifice. The model is based on a modified Rayleigh equation for bubble growth and describes the effect of gas momentum by assuming that the flow field inside the growing bubble is in the form of a circulating toroidal vortex. The equations describing the bubbling system are solved numerically using an explicit finite-difference technique. The model shows that bubble growth is characterised by an initial outward movement of the base of the bubble along the plate floor followed by an inward movement back towards the orifice which leads to a severing of the bubble from the orifice and termination of the growth cycle.Computed bubble growth rates, formation times and chamber pressure fluctuations are shown to be in good agreement with available experimental data for a wide range of system pressure (0–1.37 MN/m²) and computed bubble shapes are similar to those observed experimentally.     

液体蒸汽向单气泡中传质过程的数值分析

<正>当气体通过液相介质时,由于气液界面和气相主体间存在液体蒸汽的浓度差,使液体迅速蒸发,可以大大强化对流传热及沸腾传热.此外,在鼓泡设备中,这种溶剂汽化现象亦可能对气含率及传质带来很大影响.因此,定量地分析液体蒸汽向气泡中传质的过程是很有意义的.此过程为具人体积变化的自由边界问题.对于核状沸腾过程的自由边界问题已有很多研究,其气泡长大的控制步骤为兴高采烈相可能提供的气化潜热速率,而由于气泡中质量传递控制的移动边界问题尚未见有研究.本文提出了液体溶剂向单气泡中传质的数学模型,获得了数值解.并且对各种影响因素进行了讨论.     

双气泡间的聚并时间测量及其影响因素

聚并时间是研究气泡聚并行为过程中的一个重要参数。在消除气泡生长速度对聚并时间影响的条件下,利用自行开发的光学测试技术对双气泡间的聚并时间进行了精确的测量。考察了气泡尺寸、靠近速度、有机溶质的扩散、电解质和表面活性剂的加入对气泡聚并行为的影响。实验结果表明,双气泡间的聚并时间随着气泡直径的增加、气泡老化时间的延长、靠近速度的减小和有机溶质浓度的增大而增加。     

Prediction of cellular structure in free expansion of viscoelastic media

A systematic model is presented for a free expansion polymer foaming process that includes simultaneous nucleation and bubble growth. An influence volume approach, which couples nucleation and bubble growth, is used to account for the limited supply of dissolved gas. The melt rheology is described using the Larson viscoelastic model. The initial conditions are obtained at the upper bound of critical cluster size under conditions of elastic deformation. The resulting set of equations are solved using a combination of numerical techniques. A parametric study is conducted to examine the effects of key process variables on bubble growth, nucleation, and final bubble size distribution. It shows that the factors influencing nucleation and growth affect the ultimate bubble sizes and their distribution. The Gibbs number, a dimensionless measure of the barrier to overcome for nucleation, has the strongest impact on the cellular structure of the foam. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 67:1353–1368, 1998     

Prediction of foam growth and its nucleation in free and limited expansion

The influence volume approach (IVA) is often utilized for modeling the mass transfer process dictating bubble growth dynamics in physical foaming. However, the assumed concentration profile in the IVA method is only valid when the changes in dissolved gas concentration are small (less than 5%). In addition, the validity of the IVA method is difficult to justify in chemical foaming applications because of the difficulties involved in defining the dissolved gas concentration profile.In the present work, we define two distinct stages of bubble growth for physical foaming. These two stages are termed as free and limited expansion and are controlled by the bubble nucleation rate. Bubble nucleation is assumed to occur only in the free expansion stage. In this stage, the bubble pressure drops substantially from an initially high pressure in the supersaturated state while the dissolved gas concentration changes very little. The second stage of our two-stage mass transfer model is termed the limited expansion stage and accounts for bubble growth in the late stages of foam evolution, when the pressure changes become small. However, in the limited stage of bubble growth the dissolved gas concentration drops significantly, as the available dissolved gas is depleted. To summarize our two-stage mass transfer model of foam expansion, the pressure difference between the bubble phase and the liquid phase is the primary mechanism for driving mass transfer in the early (free) stages of foam growth and the concentration difference is the driver for bubble growth in the late (limited) stages of growth. The first stage can be regarded as the nucleation stage and it is relatively short; while the second stage can be regarded as the bubble growth stage and is much longer. Most of the bubble volume expansion takes place in the second stage.The concentration gradient at the bubble edge, which is often ignored in other models, is analyzed in detail in this paper. The details of our novel mass transfer model are also presented.     

A numerical study of flow boiling in a microchannel using the local front reconstruction method

The rapid advances in performance and miniaturization of electronic devices require a cooling technology that can remove the produced heat at a high rate with small temperature variations, as is obtained in flow boiling. To obtain insight in flow boiling, we performed numerical simulations in a 200 μm square microchannel using the local front reconstruction method. Besides validation with literature results, a parametric study shows an increasing heat removal rate and bubble growth rate with increasing wall temperature, liquid mass density, and liquid heat capacity and decreasing inlet velocity indicating the importance of phase change compared to convective transport. Finally, the heat transfer in the liquid film is studied using a Nusselt number defined with the film thickness, which is comparable to Nusselt number for falling films on hot surfaces. It is observed that convective effects are more pronounced at the bubble rear compared to the bubble front.     

Effects of probe numbers and arrangement on the measurement of charge distributions around a rising bubble in a two-dimensional fluidized bed

The charge density distribution around a rising bubble in a two-dimensional fluidized bed was measured using eight induction probes flush with the outside wall of the column and connected to charge amplifiers for recording induced charge signals as bubbles passed by. The charge distribution surrounding the bubble was reconstructed using an iterative linear back projection (LBP) algorithm by assuming that the bubble is symmetrical and the charge around the bubble remains constant as the bubble rises. The results showed that the emulsion phase far from the bubble in a two-dimensional fluidized bed of glass beads was charged negatively, and the charge density decreased gradually toward the bubble-dense phase interface, with a nearly zero charge density inside the bubble. The effects of the number of probes are also investigated. The reconstructed results using both simulated charge distribution and experimental data for different numbers of probes of the same probe diameter of 3 mm showed that increasing the number of probes can improve the reconstruction resolution. The addition of one or two more probes outside the bubble path also increased the reconstruction resolution in the region outside the bubble.     

Numerical simulation of bubble growth in viscoelastic fluid with diffusion of dissolved foaming agent

Viscoelastic simulations of bubble growth in polypropylene (PP) physical foaming were performed. A multimode Phan‐Thien Tanner (PTT) model was used to analyze the dynamic growth behavior of spherically symmetric bubbles with the diffusion of a foaming agent (CO₂). Changes in the dissolved foaming agent concentration in the polymer and in the strain of the polymer melt surrounding the bubbles were simulated with the Lagrangian FEM method. The simulation technique was validated by comparison with the bubble growth data, which were experimentally obtained from visual observations of the PP/CO₂ batch foaming system. The simulation results demonstrated that the strain‐hardening characteristic of polymer does not strongly affect the bubble growth rate. The linear viscoelastic characteristic is more influential, and the relaxation mode around 0.01 s is the most important factor in determining the bubble growth rate during the early stage of foaming. A multivariate analysis for the simulation results was also carried out. This clarified that bubble nucleus population density, surrounding pressure, initial dissolved foaming agent concentration, and diffusion coefficient are more important factors than the viscoelastic characteristics. POLYM. ENG. SCI., 45:1277–1287, 2005. © 2005 Society of Plastics Engineers     

The Model of Volatile Hydrocarbons Removal from Their Emulsions in the Flotation Process

Abstract

A mathematical model for the removal of volatile hydrocarbons from their O/W-type emulsions is presented. Two simultaneous processes are discussed: the mass transport of the dissolved hydrocarbon molecules from water into the bubble as a consequence of evaporation, and the interception of hydrocarbon droplets by a rising bubble based on a hydrodynamic model. A mesitylene (1,3,5-trimethylbenzene) results are used to test a mathematical model for both processes. Fairly good agreement is obtained.     

Theory of convective drop evaporation in direct contact with an immiscible liquid

Equations were derived for calculating the quasi-steady-state connective evaporation of a rising volatile drop in an immiscible liquid with bubble nucleation inside the drop. The solution of the energy equation was based on a potential flow model and concentric spheres configuration. The heat transfer rate was obtained in terms of superheat, degree of nonequilibrium, instantaneous diameters of the bubble and the drop, properties of the dispersed and continuous phase and the Peclet number. The solution compared well with available experimental results and other theories.     

Numerical simulation of two-phase flow with bubble break-up and coalescence coupled with population balance modeling

A computational fluid dynamic (CFD) model was developed with an improved source term based on previous work by Hagesaether et al. [1] for bubble break up and bubble coalescence to carry out numerical prediction of number density of different bubble class in turbulent dispersed flow. The numerical prediction was based on two fluid models, using the Eulerian–Eulerian approach where the liquid phase was treated as a continuum and the gas phase (bubbles) was considered as a dispersed phase. Bubble–bubble interactions, such as breakage due to turbulence and coalescence due to the combined effect of turbulence and laminar shear were considered. The result shows that the radial distributions of number densities of lower bubble classes are more than its higher counterpart. The result also shows that the Sauter mean diameter increases with the increase of height up to 1 m and then become steady. Simulated results are found to be in good agreement with the experimental data.     

Numerical simulation of polypropylene foaming process assisted by carbon dioxide: Bubble growth dynamics and stability

A mathematical model was established to simulate the bubble growth process during foaming of polypropylene (PP) by carbon dioxide, taking into account of a wide range of physical and rheological properties (solubility, diffusivity, surface tension, long-chain branching, zero shear viscosity, relaxation time, strain hardening), as well as processing conditions. By employing the Considère construction the possibility of growth instability and bubble rupture at later stage of bubble growth was predicted. The simulation revealed that the improvement of foamability of polypropylene by introducing long-chain branching was due to the well-defined viscoelastic characteristics of the melt. Rheological factors that impede bubble growth are beneficial in stabilizing the bubble growth. Stability during bubble growth is further facilitated by moderate strain hardening characteristics and elastics properties of the polymers. The diffusivity and solubility characteristics also have profound impact on the bubble growth stability, while the influence of the surface tension is negligible.