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     

Novel phenomenological discrete bubble model of freely bubbling dense gas–solid fluidized beds: Application to two‐dimensional beds

A phenomenological discrete bubble model has been developed for freely bubbling dense gas–solid fluidized beds and validated for a pseudo‐two‐dimensional fluidized bed. In this model, bubbles are treated as distinct elements and their trajectories are tracked by integrating Newton's equation of motion. The effect of bubble–bubble interactions was taken into account via a modification of the bubble velocity. The emulsion phase velocity was obtained as a superposition of the motion induced by individual bubbles, taking into account bubble–bubble interaction. This novel model predicts the bubble size evolution and the pattern of emulsion phase circulation satisfactorily. Moreover, the effects of the superficial gas velocity, bubble–bubble interactions, initial bubble diameter, and the bed aspect ratio have been carefully investigated. The simulation results indicate that bubble–bubble interactions have profound influence on both the bubble and emulsion phase characteristics. Furthermore, this novel model may become a valuable tool in the design and optimization of fluidized‐bed reactors. © 2012 American Institute of Chemical Engineers AIChE J, 2012     

On the transition stage of bubble formation on the orifice of a submerged vertical nozzle

The transition stage (self‐accelerating necking process) generally takes place before a bubble collapses. However, the mechanism is still not well understood. In this article, seven existing experiments dealing with bubble formation on the orifice of a submerged vertical nozzle are examined by solving the Young–Laplace equation. Multiple solution modes are found. Bubble in solution mode 2 has a neck and thus taller than mode 1 at the same volume. The present numerical result along with an experiment from Longuet‐Higgins et al. [J. Fluid Mech. 230, 365–390 (1991)] evidences an isometric transition from solution modes 1 to 2. This might account for the self‐accelerating necking process before a bubble collapses. Surprisingly, all of the seven existing experiments agree excellently with the bubble shapes from the Young–Laplace equation without the dynamic effect even when the bubble growth rate in the experiment is 2.25 times as large as the critical value. The gas flow rate (the dynamic effect) seems to play a role only after the transition stage. © 2011 Canadian Society for Chemical Engineering     

Hydrodynamics and simulation of air–water homogeneous bubble column under elevated pressure

A gas–liquid Eulerian computational fluid dynamics (CFD) model coupled with a population balance equation (PBE) was presented to investigate hydrodynamics of an air–water bubble column (1.8 m in height and 0.1 m in inner diameter) under elevated pressure in terms of pressure drop, gas holdup, mean bubble size, and bubble surface area. The CFD-PBE model was modified with three pressure correction factors to predict both the total gas holdup and the mean bubble size in the homogeneous bubbly flow regime. The three correction factors were optimized compared to experimental data. Increasing the pressure led to increasing the density, reducing the bubble size, and increasing the gas holdup. The bubble size distribution moved toward a smaller bubble size, as the pressure increased. The modified CFD-PBE model validated with experimental data and empirical models represented well hydrodynamics of the bubble column at P = 0.1, 1.5, and 3.5 MPa.     

A General Correlation for the Rise Velocity of Single Gas Bubbles

A correlation for the terminal rising velocity of a single gas bubble in an infinite Newtonian fluid is proposed. The model is an extension to high Reynolds numbers of a model developed recently by the author (Rodrigue, 2001a). This new equation, which is explicit in velocity, is able to predict the velocity for any bubble volume in any Newtonian fluid.     

Bubble detonation as a self-sustained solitary wave with energy release

Previous experiments have shown that a bubble detonation wave is a resonant or self-sustained solitary wave in a bubble medium. Bubble detonation is modeled by a solitary wave with energy release in bubbles. The equation describing a solitary wave of small amplitude is shown to be an analog of nonlinear Boussinesq equation of the fourth order. A comparison of the solution obtained with averaged experimental pressure profiles shows that the analytical solution is suitable for describing bubble detonation waves with a finite pressure amplitude. In the model proposed, the time of action of solitary-wave compression on a separate bubble is several times the bubble oscillation period. This result agrees with experimental data and confirms the presence of a collective resonant effect in a bubble medium. Satisfactory agreement is obtained between experimental and theoretical data on the pressure profile and extent and velocity of bubble detonation waves. __________ Translated from Fizika Goreniya i Vzryva, Vol. 43, No. 6, pp. 104–111, November–December, 2007.     

文丘里管空化器内空泡动力学特性的数值模拟

研究了文丘里管空化发生器内空泡的成长、溃灭特性,根据基本的R-P空泡运动方程,考虑了液体粘性、表面张力和可压缩性等因素的影响,运用四阶Runge-Kutta法对空泡径向非线性运动方程进行求解,得到空泡径向演变过程以及溃灭压力的变化趋势.讨论了初始汽泡半径、入口压力和文丘里管的喉径比等因素对空泡演变过程的影响.结果表明,流体的可压缩性对空泡溃灭的影响最大;空化发生器结构参数以及操作参数均对空泡运动特性产生影响,从而影响空化强度.所得结果对空化流场中空化泡演变规律的研究以及水力空化发生器的设计具有指导意义.     

Bubble breakup with permanent obstruction in an asymmetric microfluidic T‐junction

Bubble breakup with permanent obstruction in an asymmetric microfluidic T‐junction is investigated experimentally. The breakup process of bubbles can be divided into three stages: squeezing, transition, and pinch‐off stages. In the squeezing stage, the thinning of the bubble neck is mainly controlled by the velocity of the fluid flowing into the T‐junction, and the increase of the liquid viscosity can promote this process. In the transition stage, the minimum width of bubble neck decreases linearly with time. In the pinch‐off stage, the effect of the velocity of the fluid flowing into the T‐junction on the thinning of the bubble neck becomes weaker, and the increase of the liquid viscosity would delay this process. The evolution of the minimum width of the bubble neck with the remaining time before the breakup can be scaled by a power–law relationship. The bubble length has little influence on the whole breakup process of bubbles. © 2014 American Institute of Chemical Engineers AIChE J, 61: 1081–1091, 2015     

Population balance modeling of polyurethane foam formation with pressure-dependent growth kernel

Polyurethane foams are widely used materials often chosen for their useful characteristics such as low thermal conductivity, ease of application, and high strength-to-weight ratios. Computational models are needed to predict the dynamics of the flow and expansion, and the resulting material properties, to improve manufacturing processes. In this paper, a model for PMDI, a water-blown polyurethane foam, is presented. By extending a kinetics-based approach by adding bubble-scale information via a population balance equation (PBE) using the quadrature method of moments, we can track bubble size distributions during foaming. We present results from a three-dimensional computational fluid dynamics model using arbitrary Lagrangian–Eulerian interface tracking implemented in finite element software. The model compares favorably with experimental data, including dynamics, bubble distributions measured by both camera and diffusion wave spectroscopy, and post-test bubble size from scanning electron microscopy and density measurements from x-ray computed tomography.     

A unified theoretical model for breakup of bubbles and droplets in turbulent flows

Pressure has a significant effect on bubble breakup, and bubbles and droplets have very different breakup behaviors. This work aimed to propose a unified breakup model for both bubbles and droplets including the effect of pressure. A mechanism analysis was made on the internal flow through the bubble/droplet neck in the breakup process, and a mathematical model was obtained based on the Young–Laplace and Bernoulli equations. The internal flow behavior strongly depended on the pressure or gas density, and based on this mechanism, a unified breakup model was proposed for both bubbles and droplets. For the first time, this unified breakup model gave good predictions of both the effect of pressure or gas density on the bubble breakup rate and the different daughter size distributions of bubbles and droplets. The effect of the mother bubble/droplet diameter, turbulent energy dissipation rate and surface tension on the breakup rate, and daughter bubble/droplet size distribution was discussed. This bubble breakup model can be further used in a population balance model (PBM) to study the effect of pressure on the bubble size distribution and in a computational fluid dynamics‐population balance model (CFD‐PBM) coupled model to study the hydrodynamic behaviors of a bubble column at elevated pressures. © 2014 American Institute of Chemical Engineers AIChE J, 61: 1391–1403, 2015     

Effect of resistance internal on hydrodynamic behaviours and bubble characteristics in a laboratory-scale bubble column

A novel resistance internal is proposed to optimize the flow field and improve the gas–liquid contact in a co-hydrogenation reactor of coal and vacuum residuum. Local gas holdup, local liquid velocity, and characteristics of the bubble were investigated in a scaled-down laboratory model. The quantitative results showed that the resistance internals could reduce the thickness of the liquid reflux layer by a percentage up to 32% and reduce the difference in the local gas holdup at cross-sections of up to 44%. The Sauter mean diameter of the bubble decreased from 20.30 to 16.00 mm, which aroused the increase in bubble surface area by a percentage of up to 71.9%. The resistance internal promoted the breakup of the bubble with multiple mechanisms and provided diversion to fluid. In this work, improvement at multiple scales was realized, and the technical support for industrial application was provided.     

基于能量守恒的鼓泡塔双尺度流体力学模型

提出了同时描述鼓泡塔宏观流动及气泡尾涡小尺度湍动的双尺度流体力学模型,其中大尺度剪切流通过经典k-ε模型描述,而小尺度尾涡湍动则由“尾涡温度”传输方程确定。通过在运动方程添加“尾涡压力”源项,构建大小双尺度流体运动的相互作用,解释鼓泡塔内含率非均匀分布的机制。模型将鼓泡塔内的能量耗散分解3种作用机制：大尺度剪切流引起湍动耗散;小尺度尾涡湍动耗散;尾涡与壁面作用的能量耗散,较好地解决了现有鼓泡塔模型能量不守恒问题。模型计算稳定性高,模拟结果与实验结果吻合良好。     

Experimental investigation on multiscale hydrodynamics in a novel gas–Liquid–Solid three phase jet-Loop reactor

Multiphase flow hydrodynamics in a novel gas–liquid–solid jet-loop reactor (JLR) were experimentally investigated at the macroscales and mesoscales. The chord length distribution was measured by an optical fiber probe and transformed for bubble size distribution through the maximum entropy method. The impacts of key operating conditions (superficial gas and liquid velocity, solid loading) on hydrodynamics at different axial and radial locations were comprehensively investigated. JLR was found to have good solid suspension ability owing to the internal circulation of bubbles and liquid flow. The gas holdup, axial liquid velocity, and bubble velocity increase with gas velocity, while liquid velocity has little influence on them. Compared with the gas–liquid JLRs, solids decrease the gas holdup and liquid circulation, reduces the bubble velocity and delays the flow development due to the enhanced interaction between bubbles and particles (Stokes number >1). This work also provides a benchmark data for computational fluid dynamics (CFD) model validation. © 2019 American Institute of Chemical Engineers AIChE J, 65: e16537, 2019     

大孔径高气速单孔气泡形成

在内径为190mm的鼓泡塔内,研究了空气-去离子水系统在大孔径高气速条件下的单孔气泡形成。考察了五个不同的孔径,分别为4、8、10、15及21mm,孔口气速范围为0.8~154.8m·s^-1。以CCD摄像记录气泡的形状及尺寸,根据气泡长径比的变化,得到气泡初始形态转变时的临界孔口气速：当孔口气速低于20m·s^-1时,孔口气泡近似于球形,长径比小于1.1;当孔口气速大于50m·s^-1时,气泡呈现椭球形,长径比大于1.5。并对气泡尺寸与孔径及孔口气速进行关联,所得关联式对孔径大于3mm、孔口气速在10~80m·s^-1范围内所形成的气泡尺寸预测效果较好。     

Local hydrodynamics modeling of a gas–liquid–solid three-phase bubble column

A three-dimensional (3D) transient model was developed to simulate the local hydrodynamics of a gas–liquid–solid three-phase bubble column using the computational fluid dynamic method, where the multiple size group model was adopted to determine the size distribution of the gas bubbles. Model simulation results, such as the local time-averaged gas holdups and axial liquid velocities, were validated by experimental measurements under varied operating conditions, e.g., superficial gas velocities and initial solid loadings at different locations in the three-phase bubble column. Furthermore, the local transient hydrodynamic characteristics, such as gas holdups, liquid velocities, and solid holdups, as well as gas bubble size distribution were predicted reasonably by the developed model for the dynamic behaviors of the three-phase bubble column. © 2007 American Institute of Chemical Engineers AIChE J, 2007     

二维鼓泡床内气液流动特性实验与数值模拟

采用高速摄像法测量了0.20 m×0.02 m×2.00 m拟二维床内气泡尺寸分布和流型等变化规律,结果表明,随着表观气速的增大,鼓泡床内依次呈现均匀鼓泡区、过渡区和湍动区3种形式,以气泡个数概率表示的气泡尺寸分布呈对数正态分布。以计算流体力学软件ANSYS CFX 10.0为平台,采用k-ε湍流模型和GRACE曳力模型对气液鼓泡床内流体动力学行为展开了数值模拟,其结果与实验值比较吻合。研究表明,从多相流理论出发的计算流体力学模拟方法可以用来预报鼓泡床内流型过渡等流体动力学特性。     

Bubble formation and dynamics in a quiescent high‐density liquid

Gas bubble formation from a submerged orifice under constant‐flow conditions in a quiescent high‐density liquid metal, lead–bismuth eutectic (LBE), at high Reynolds numbers was investigated numerically. The numerical simulation was carried out using a coupled level‐set and volume‐of‐fluid method governed by axisymmetric Navier–Stokes equations. The ratio of liquid density to gas density for the system of interest was about 15,261. The bubble formation regimes varied from quasi‐static to inertia‐dominated and the different bubbling regimes such as period‐1 and period‐2 with pairing and coalescence were described. The volume of the detached bubble was evaluated for various Weber numbers, We, at a given Bond number, Bo, with Reynolds number . It was found that at high values of the Weber number, the computed detached bubble volumes approached a 3/5 power law. The different bubbling regimes were identified quantitatively from the time evolution of the growing bubble volume at the orifice. It was shown that the growing volume of two consecutive bubbles in the period‐2 bubbling regime was not the same whereas it was the same for the period‐1 bubbling regime. The influence of grid resolution on the transition from period‐1 to period‐2 with pairing and coalescence bubbling regimes was investigated. It was observed that the transition is extremely sensitive to the grid size. The transition of period‐1 and period‐2 with pairing and coalescence is shown on a Weber–Bond numbers map. The critical value of Weber number signalling the transition from period‐1 to period‐2 with pairing and coalescence decreases with Bond number as , which is shown to be consistent with the scaling arguments. Furthermore, comparisons of the dynamics of bubble formation and bubble coalescence in LBE and water systems are discussed. It was found that in a high Reynolds number bubble formation regime, a difference exists in the transition from period‐1 to period‐2 with pairing and coalescence between the bubbles formed in water and the bubbles formed in LBE. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3996–4012, 2015     

The dissolution of a stationary spherical bubble beneath a flat plate

The dissolution of a single stationary bubble held in place by a horizontal plate is commonly observed experimentally. For several decades the standard approach to the analysis of such dissolution data has been to apply a correction factor of ln(2)=0.69 to the Epstein-Plesset equation for an isolated bubble. In this paper, the transport equations for a stationary bubble touching a plate are solved numerically for the common case where the flow field caused by the change in system volume as the bubble dissolves can be neglected. It is found that the total bubble lifetime is not well characterised by the use of the ln(2) factor. However, in most experimental situations, the initial stages of bubble dissolution are not captured. For low gas solubilities the use of a correction factor of 0.69 to the Epstein-Plesset equation is appropriate once the initial transients have dissipated. The correction factor varies from 0.69 to 0.77 across the full range of situations described in this paper. The mathematical model is validated by comparison to experimental data.     

Impact of approximating the initial bubble pressure on cell nucleation in polymeric foaming processes

According to the classical nucleation theory, the free energy barrier for bubble nucleation, and thereby the nucleation rate, are functions of the initial bubble pressure, P_bubble,0. In almost all of the previous studies that have used computer simulations to investigate polymeric foaming processes, the value of P_bubble,0 has been approximated using the saturation pressure, P_sat. This article employs the thermodynamic equilibrium condition and the Sanchez–Lacombe (SL) equation of state (EOS) to determine the value of P_bubble,0. It is shown that using P_sat to approximate P_bubble,0 may lead to significant overestimations of the nucleation rate and the final cell density. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci 104: 902–908, 2007     

声场驱动气泡增强微流体混合的数值模拟

微流体研究中，由于雷诺数较低，流体呈层流流动，流体混合主要依靠分子扩散，混合时间长，效率低，故流体混合成为亟待解决的问题。声场激振气泡可以有效促进流体混合，已经引起了广泛关注。本文模拟研究了声场作用下气泡振动对流体混合的影响，探索了微尺度流体在声场激振下的流动特性，分析了微通道高度、入口速度、气泡间距及布置方式对流体混合的影响。结果发现，微通道高度较低时，气泡振动可以更好地促进流体混合；入口速度较小时，流体在气泡附近滞留时间较长，混合较为均匀；气泡半径较大时，旋涡扰动增强，混合效率提高；两个气泡的混合效果优于单个气泡，而气泡间距对混合效率基本无影响；微通道高度较低时，气泡同侧布置和异侧布置对流体的混合效果相接近，随着微通道高度的升高，两种布置方式对混合效果的差异逐渐显现，异侧布置具有更好的混合效果。