Assessment of a subgrid-scale model for convection-dominated mass transfer for initial transient rise of a bubble

The mass transfer between a rising bubble and the surrounding liquid is mainly determined by an extremely thin layer of dissolved gas near the bubble interface. Resolving this concentration boundary layer in numerical simulations is computationally expensive and limited to low Péclet numbers. Subgrid-scale (SGS) models mitigate the resolution requirements by approximating the mass transfer near the interface. In this contribution, we validate an improved SGS model with a single-phase simulation approach, which solves only the liquid phase at a highly resolved mesh. The mass transfer during the initial transient rise of moderately deformed bubbles in the range Re = 72–569 and Sc = 10²–10⁴ is carefully validated. The single-phase approach is able to mirror the two-phase flow field. The time-dependent local and global mass transfer of both approaches agree well. The difference in the global Sherwood number is below than 2.5%. The improved SGS model predicts the mass transfer accurately and shows marginal mesh dependency.     

Theoretische Berechnung des Stofftransports in der Umgebung einer Einzelblase

Mass transfer between a spherical bubble and a surrounding fluid has been investigated theoretically. Stationary transfer and constant diameters were assumed. The local concentration around a single bubble as well as local and mean Sherwood numbers were calculated. The velocity distributions were determined in an earlier investigation by numerical solution of the Navier—Stokes equation.The mean Sherwood number for bubbles of constant shape were proved to be a function of two dimensionsless numbers: the Schmidt- number Sc and the product Re Sc of Reynolds and Schmidt-number. Curves fitting the calculated values are lying between two limiting curves. The lower curve is valid for the limiting condition Re → 0 and Sc → ∞ while the upper curve is valid for Re → ∞ and Sc → 0.For bubbles of varying shape the Sherwood number will not follow the upper boundary but will increase considerably beyond this limit. Analysis of the mechanism of mass transfer indicates that this behavior results from periodic and aperiodic deformations of the bubble.     

Transient gas–liquid mass transfer model for thin liquid films on structured solid packings

This paper describes a model for gas–liquid mass transfer through thin liquid films present on structured packings for gas–liquid operations under dispersed gas flow regime. The model has been derived for two cases: the absorption (or desorption) of a gaseous component into the liquid film and the transfer of the gaseous component through the liquid film to the packing surface where an infinitely fast reaction takes place. These cases have been solved for three bubble geometries: rectangular, cylindrical, and spherical. For Fourier numbers below 0.3, the model corresponds to Higbie’s penetration theory for both cases. The Sherwood numbers for cylindrical and spherical bubbles are 20% and 35% higher, respectively, than for rectangular bubbles. In case of absorption and Fourier numbers exceeding 3, the effect of bubble geometry becomes more pronounced. The Sherwood numbers for cylindrical and spherical bubbles now are 55% and 100% higher, respectively, than for rectangular bubbles. In case of an infinitely fast reaction at the packing surface, the Sherwood number corresponds to Whitman’s film theory (Sh=1$Sh=1$) for all bubble geometries. In this paper also practical approximations to the derived Sherwood numbers are presented. The approximations for both cases and all bubble geometries describe all the model data within an error of 4%. The application of the model has been demonstrated for three examples: (1) gas–liquid mass transfer for a structured packing; (2) gas–liquid mass transfer in a microchannel operated with annular flow; (3) gas–liquid mass transfer in a microchannel with Taylor flow.     

Approximate theoretical solution for the Sherwood number of oscillating bubbles at different Reynolds numbers

In this paper a theoretical approach to the effect of bubble oscillations on the mass transfer rates has been carried out to get a better understanding of the effect of bubble oscillations in multiphase gas–liquid contactors. The perturbation method has been used to approximate the velocity profile surrounding the bubbles while they oscillate. The shape of the oscillating bubble was modelled taking into account the effect of the liquid viscosity on bubble oscillation frequency and amplitude. The modelled shapes match the photographs of bubbles oscillating in liquids with different viscosities. As a result, new approximate theoretical models for the Sherwood number in viscous fluids at different flow regimes have been proposed. The models extend the work already available in the literature for mass transfer rates from oscillating bubbles in inviscid fluids and provides good results in predicting the Sherwood number at high and moderate Reynolds numbers, the preferred regimes in many industrial operations where, as a result of the hydrodynamics processes involving the bubbles. Their oscillations do not completely decay.     

Direct measurement of mass transfer around a single bubble by micro-PLIFI

In this paper, an original direct and non-intrusive technique using Planar Laser Induced Florescence with Inhibition (PLIFI) is proposed to quantify the local mass transfer around a single spherical bubble rising in a quiescent liquid. The new set-up tracks the mass transferred in the bubble wake for a plane perpendicular to the bubble trajectory instead of a parallel plane as in previous works, thus avoiding optical reflection problems. A spherical bubble is formed in a glass column containing fluorescent dye. A camera with a microscopic lens is placed underneath the column to record cross-sections of the transferred oxygen. A high-speed camera is located far from the column to simultaneously record the bubble position, size, shape and velocity. The dissolved gas inhibits the fluorescence so that oxygen concentration fields can be measured. From this, a calculation method is developed to determine mass transfer on the micro-scale. Experimental results are compared to the Sherwood numbers calculated from the Frössling and Higbie models used for fully contaminated and clean spherical bubbles, respectively. Results show that all experimental Sherwood numbers occur between the two models, which gives credence to the measurements. The new technique is then developed for bubble diameters ranging from 0.7 to 2 mm in six hydrodynamic conditions (1<Re<10², 10²<Sc<10⁶).     

Determination of oxygen transfer from single air bubbles to liquids by oxygen microelectrodes

For the investigation of mass transfer from gas bubbles into liquids the concentration gradient of oxygen migrating from air bubbles was measured by means of oxygen microelectrodes. For this purpose a single air bubble was fixed by a platinum wire spiral with the liquid flowing downward. Thus the ascent of the bubble in an aerated liquid was simulated. Liquid-side mass transfer coefficients determined from concentration gradients were higher than values calculated from theory. Sherwood numbers obtained from experimental results for bubbles of larger diameters were distinctly higher than those for smaller bubbles (diameter 1 mm); the difference corresponds approximately to that predicted theoretically between bubbles with mobile and those with rigid interfaces.     

Mass transfer of a rising bubble in molten glass with instantaneous oxidation-reduction reaction

The mass transfer around a rising bubble has been studied within the field of glass melting processes. Due to the large value of liquid viscosity, creeping flow was used. The rising bubble is assumed to have a clean interface with a total mobility and the exact solution of Hadamard or Rybczynski was used to define the velocity field around the bubble. The mass transfer of oxygen in the soda-lime-silica glass melt where oxidation-reduction reactions of iron oxides occur is also described.The dimensionless mass transfer coefficient, Sherwood number, was determined as a function of the Péclet number based on the terminal rise velocity of the bubble. Two different techniques have been used: the first based on the boundary layer theory and the second using a finite element method.In order to take into account the oxidation-reduction reaction in a unified framework, a modified Péclet number has been defined as a function of two dimensionless numbers. The first is strongly linked to the equilibrium constant of the chemical reaction and the second is the glass saturation, defined as the ratio of oxygen concentration in the bulk to that at the bubble surface. The Sherwood number, taking into account the chemical reactions, increases with iron content as well as with glass reduction (i.e. small saturation level).From an application point of view, the determination of a modified Péclet number is important because it is possible to use a similar expression (determined without the reaction) by replacing the classical Péclet number by the modified one proposed herewithin.     

Effects of reaction order and convection around gas-bubbles in a gas-liquid reacting system

The mass transfer of gaseous reactant into liquid for chemical reaction is significantly affected by relative flow at the interface of gas and liquid. Two extreme cases are for a bubble behaving like a solid particle due to absorbed surfactant impurities and for a freely-internally circulating bubble with a relative interfacial velocity; the present calculations indicate a ratio of mass-transfer rates of a factor up to 1·4–2·45. The factor decreases with increasing reaction rate, becoming negligible for values of K > 2000 sec^?1. It is larger for a $\text{3}\text{2}$ order reaction than for a 1st order reaction.If there is internal circulation, the relative flow changes depending on whether the bubble is alone or in a rising bubble swarm. For small reaction rates the effect of this change in the mass transfer rate has been calculated to be 7–9% at typical bubble sizes of 0·1–0·2 cm radius. The mass transfer rate for a freely-circulating bubble is about 15% larger for a $\text{1}\text{2}$ order reaction than for a 1st order reaction at steady state. Transient and time averaged values of the Sherwood number were obtained. Shrinkage of bubbles from loss of reactant was also considered.     

Mass or heat transfer from spheroidal gas bubbles rising through a stationary liquid

Mass transfer was studied for the case of a spheroidal bubble rising through a stationary liquid. A numerical code that solves the Navier–Stokes equations and the diffusion–advection equation for the concentration was used to characterize the transfer from the bubble to the surrounding liquid phase. Simulations were carried over systematically for Reynolds number ranging from 1 to 1000, Schmidt numbers from 1 to 500 and bubble aspect ratio from 1 to 3. It appears that the use of the equivalent diameter as the characteristic length is the more appropriate to describe the transfer. The effect of bubble aspect ratio on the Sherwood number has been analyzed. At first order the extension of Boussinesq expression using the equivalent diameter can be used for practical purposes. The evolution of the correction factor that compares the Sherwood number to the one of a sphere with same equivalent Peclet number is presented and described using simple correlations. The implementation of these results into Euler–Euler simulations of mass transfer is discussed. It appears that the modification of the interfacial area combined to the modification of the Sherwood number gives a significant contribution to the interfacial source term in the equation of the concentration. Note that the results can also be considered for heat transfer and used for inviscid drops.     

Measurements and computation of low mass transfer coefficients for FCC particles with ozone decomposition reaction

The mass transfer coefficients and Sherwood numbers for catalyzed fluid cracking catalyst particles were measured and computed in a two‐dimensional (2‐D) bubbling fluidized bed, with ozone decomposition reaction. The measured and computed Sherwood numbers, using 3‐ and 2‐D kinetic theory based computational fluid dynamics simulations, were of the order of 10^?6–10^?2. The low Sherwood numbers were in reasonable agreement with the literature data for small particles, at low Reynolds numbers. The computational fluid dynamics simulations showed that it is possible to compute conversions in fluidized bed reactors without using the conventional model with empirical mass transfer coefficients. © 2011 American Institute of Chemical Engineers AIChE J, 2012     

Kinetic theory based computation of PSRI riser: Part II—Computation of mass transfer coefficient with chemical reaction

The design of circulating fluidized bed systems requires the knowledge of mass transfer coefficients or Sherwood numbers. A literature review shows that these parameters in fluidized beds differ up to seven orders of magnitude.To understand the phenomena, a kinetic theory based computation was used to simulate the PSRI challenge problem I data for flow of FCC particles in a riser, with an addition of an ozone decomposition reaction. The mass transfer coefficients and the Sherwood numbers were computed using the concept of additive resistances. The Sherwood number is of the order of 4 × 10⁻³ and the mass transfer coefficient is of the order of 2 × 10⁻³ m/s, in agreement with the measured data for fluidization of small particles and the estimated values from the particle cluster diameter in part one of this paper. The Sherwood number is high near the inlet section, then decreases to a constant value with the height of the riser. The Sherwood number also varies slightly with the reaction rate constant. The conventionally computed Sherwood number measures the radial distribution of concentration caused by the fluidized bed hydrodynamics, not the diffusional resistance between the bulk and the particle surface concentration. Hence, the extremely low literature Sherwood numbers for fluidization of fine particles do not necessarily imply very poor mass transfer.     

Mass transfer without and with chemical reaction in dispersed gas-liquid two-phase flows

Using a concentric sphere model, velocity profiles around bubbles in a bubble swarm were derived for clean and for relatively contaminated gas bubbles. The theoretical Sherwood numbers were calculated, including the effect of void fraction. These mass transfer predictions were compared with experimental results for the absorption with reaction of carbon dioxide into monoethanolamine solutions. The results indicated an appreciable contamination of the bubbles by surface active material.     

边界层方程求解传热传质复合自然对流的偏差分析

引　言动量、能量及质量的边界层方程是Naveir Stokes完整方程的简化形式 ,自上世纪初提出以来 ,应用极广 .传热传质复合自然对流是自然界和工业过程中常见的现象 ,由于存在热与物质扩散两种浮力相对大小及方向的差异以及Prandtl数和Schmit数的影响 ,使该问题颇为复杂 .Bottema     

Influence of Bubble Bouncing on Mass Transfer and Chemical Reaction

The hydrodynamics of bubbly flows is dominated by bubble‐induced turbulence and bubble‐bubble interactions. Both phenomena influence the gas‐liquid mass transfer as well as the mixing of reactants. If the time scales of mass transfer and mixing are in the same order as the time scales of a parallel‐consecutive reaction, the yield and selectivity will be affected by the local hydrodynamics. An experimental setup is presented that enables the investigation of mass transfer during well‐defined and adjustable bubble collisions. The influence of CO₂ bubble collisions on mass transfer is measured and modeled with a modified Sherwood number correlation. Further visualization of the concentration field in the vicinity of O₂ bubbles by means of laser‐induced fluorescence demonstrates the dependency of mass transfer from a chemical reaction and permits the development of a first model approach.     

The influence of hydrodynamic and mass transfer entrance effects on the operation of a parallel plate electrolytic cell

Experimental measurements of mass transfer in an electrochemical flow cell of rectangular cross section with different hydrodynamic entrance and electrode lengths have been made. For fully developed flow, average Sherwood numbers under laminar conditions vary with Graetz number to a power 0·30. For turbulent flow, fully developed mass transfer conditions occur about twelve equivalent diameters along the electrode and are best represented by the Chilton-Colburn analogy which predicts Sherwood numbers varying with Reynolds number to a power of 0·8 and Schmidt number to a one-third power. For shorter electrodes Sherwood numbers can be adequately correlated by an expression with Reynolds number to a two-thirds power and dimensionless electrode length to a power of −0·2. For hydrodynamic entrance lengths of not less than eight equivalent diameters, data in the laminar region can be expressed by an emperical boundary layer type of equation which includes terms for the hydrodynamic entrance length and electrode length. In the turbulent regime substantially developed flow occurs after eight entrance lengths and correlations with fully developed flow equations are satisfactory     

Experimental analysis of bubble mode in a plate-type absorber

An experimental analysis of ammonia-water absorption was performed in a plate-type absorber. The flow of water and ammonia gas was performed in the bubble mode. The experiments were made to examine the effects of solution flow rate and gas flow rate on the performance of the absorber. It was found that the increase of solution flow rate rarely affected the mass transfer, but improved the heat transfer. As the gas flow rate increased, slugging occurred in the bubble mode and influenced the thermal boundary layer. Finally, the results were converted into dimensionless numbers to elucidate physical phenomena and plotted as Sherwood number versus Reynolds number for mass transfer performance and Nusselt number versus Reynolds number for heat transfer performance.     

Velocities of extended bubbles in inclined tubes

The velocities of extended bubbles (slug flow bubbles) have been measured in inclined circular tubes. Eötvös numbers ranged from 4.9 to 490 and Morton numbers from 2.2 × 10⁻¹¹ to 1.5 × 10⁴. The Froude number for any angle of inclination was correlated as a function of angle and the values of the Froude numbers for the horizontal and vertical orientations.     

Liquid/solid mass transfer in an air-lift loop reactor with a dispersed solid phase

In processes with immobilized cells mass transfer across the boundary layer surrounding the support often plays an important role. Relatively little is known about external mass transfer as a function of the superficial gas velocity in bioreactors such as air-lift loop reactors. In this work ion-exchange resins were used as a solid phase to determine the mass-transfer coefficient in such a reactor. Relations between the Sherwood number and the superficial gas velocity were derived and compared with relations from the literature. Relations in which the Sherwood number is a function of the energy-dissipation rate and relations in which the relative particle velocity is calculated from the rate of free fall of the particle were compared. It was shown that the Sherwood numbers that were functions of the energy-dissipation rates were higher than could be calculated on the basis of the rate of free fall. The Sherwood number obtained was used to calculate the k_l,s of carrageenan gel beads as a solid phase in an air-lift loop reactor. © 1998 SCI.     

A Silicon Microseparator based Pervaporation Process for Separation of Ethanol/Water Mixtures using a Polymer Membrane

Abstract

The selective removal of water from ethanol through pervaporation was demonstrated in a microchannel device using a commercial membrane. Photolithography and dry etching techniques were employed for fabrication of the microseparator with hydraulic diameters of 30 µm to 80 µm. Experiments conducted at 90°C and 2–3 Torr, with Reynolds Numbers ranging from 8 to 91, resulted in an average water and ethanol permeance of 1.2×10^?3 and 8×10^?5 cm³/cm² · s · cmHg respectively. A mass transfer analysis involving Sherwood correlations was used to calculate the theoretical boundary layer resistance. The comparison of overall mass transfer coefficient with the boundary layer coefficients suggests that the membrane was the dominant resistance for this system.     

Mass transfer rates from bubbles in stirred tanks operating with viscous fluids

In this paper, the role of liquid viscosity on the mass transfer rates in stirred tank reactors has been theoretically studied. Liquid viscosity affects liquid diffusivity and bubble size distribution by defining bubble stability in the flow. A population balance, taking into account the effect of liquid viscosity on the coalescence and break-up closures, has been combined with Higbie–Kolmogorov's theory to predict the effect of liquid viscosity on the mass transfer rates. Experimental results from the literature for stirred tanks operating with one single Rushton turbine have been used as comparison. Different moderately viscous aqueous solutions (glucose, glycerol and millet-jelly) have been considered. Bubble break-up depends on the critical deformation of the bubbles in the continuum phase. A correlation between the Weber critical number and the liquid viscosity has been found. Once the bubble distribution is accurately determined, the volumetric mass transfer rate in viscous solutions can be predicted theoretically.