Mass transfer in liquid-fluidized beds at low reynolds numbers

The mass transfer data measured in the liquid fluidized beds of ion exchange resins in the Reynolds number range 0.22–6.4 are presented. A comparison was made between the experimental results and the values predicted by the recent theory of Nelson and Galloway and by the modified theory proposed by Rowe. The observed change of exponent on Reynolds number is predicted by the theory of Nelson and Galloway, but this theory fails to predict the correct dependance of Sherwood number of voidage. The modified theory proposed by Rowe predicts successfully the mass transfer coefficients at very low Reynolds number and in addition it predicts also approximately the Reynolds number region in which the change of exponent on Reynolds number is observed, but it fails to predict the correct dependance of Sherwood number on voidage in this region.     

Axial dispersion coefficients in packed beds at low reynolds numbers

Peclet numbers describing axial dispersion in gas flow through packed beds of spheres were obtained using a two measurement point, pulse technique in a test section four inches inside diameter and 5.14 feet high. Three packing sizes were investigated, corresponding to tube to particle diameter ratios of 6.4, 17, and 66 In the experiments the contribution of the velocity profile to axial spreading was reduced by using thermal conductivity detectors which responded to dispersion only in the central part of the bed cross-section. In this region of a packed bed the velocity profile is relatively flat. The results point to a particle diameter effect which is more pronounced than has been previously reported. This is in accord with the diffusive mechanism of axial dispersion in a packed bed provided dispersion caused by the velocity profile does not affect the measured pulse response. In the absence of velocity profile effects, the spreading of residence times in void cells is caused primarily by the shedding of the decelerated boundary layers on the downstream side of the particles. At low velocities however, molecular diffusion predominates. Implicit in this discussion is the hypothesis that the uniformity of shape and size of packing particles has an important bearing on the manner in which the Peclet number approaches its limiting value as the gas velocity is increased.     

Rising velocity of a swarm of spherical bubbles in power law fluids at high reynolds numbers

In this work, a theoretical scheme for estimating the rise velocity of a swarm of spherical bubbles through quiescent power law liquids at high Reynolds number is developed. The inter-bubble interactions have been accounted for by the use of a cell model. The effect of the power law index and the volume fraction of the gas on the rise velocity is elucidated. Depending upon the degree of shearthinning behaviour and the gas fraction, the swarm may rise slower or faster than a single bubble. This behaviour has been explained qualitatively in terms of two competing mechanisms.     

Surface remobilization of buoyancy-driven surfactant-laden drops at low reynolds and capillary numbers

It is well known that the terminal velocity of a drop settling in a viscous fluid is impacted by surface tension gradients. These gradients can develop because of nonuniform accumulation of surfactant on the surface as a result of a number of transport mechanisms. Here, a surfactant transport model based on a sorption-limited Frumkin framework is used to describe surfactant transport in the presence of both surface convection and diffusion at low Reynolds and capillary numbers. Constants characterizing surfactant transport in the Frumkin framework are experimentally determined and used to predict aqueous drop velocities with varying surfactant concentrations and volumes. Computation is carried out by satisfying equations governing mass, momentum, and interface species conservation. Experiments demonstrate qualitative and quantitative agreement between predicted and measured drop velocities. It is shown that surface remobilization explains some observed trends in measured velocity as the drop size decreases. © 2018 American Institute of Chemical Engineers AIChE J, 65: 294–304, 2019     

Drag coefficient—reynolds number transition for gas bubbles rising steadily in viscous fluids

Recently, the author (Rodrigue, 2001) developed a generalized correlation for the motion of gas bubbles rising steadily in uncontaminated viscous Newtonian fluids of infinite extent. It is the purpose of this note to show that this model can be written in an explicit form for the drag coefficient as a function of the Reynolds and Morton numbers. This new correlation is then used to predict the position of the minimum in a graphical representation of C_D versus Re. It is shown that the model can predict quite nicely this hydrodynamic transition for viscous fluids.     

Drag and mass transfer in non-newtonian flows through multi-particle systems at low reynolds numbers

An approximate solution for the motion of an assemblage of solid spheres moving through a power-law fluid at low Reynolds numbers is obtained using HapThe theoretical predictions based on Kuwabara's zero-vorticity cell model are very similar to those based on Happel's model. An approximation technique     

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。     

Isothermal dissolution of small rising bubbles in a low viscosity liquid

The isothermal dissolution of small single rising bubbles in a low viscosity liquid is numerically and experimentally studied. We propose a tracking-interface numerical method to analyze the dissolution of single bubbles rising in an isothermal liquid bath in the limit of small Weber numbers. The key elements of the method are the use of a frame of reference moving with the bubble and the application of different meshes to solve the mechanical and mass-diffusion problems. In addition, the gas concentration in the atmosphere over the bath, determined by mass balance of species in the gas column, is shown to be an essential component of the global dissolution problem in steady regime. Comparison with small oxygen bubbles rising in water has been carried out with remarkable agreement.     

Coalescence in a steady-state rising foam

Steady-state rising foam is an ideal medium for studying foam coalescence phenomena. The coalescence process in this type of foam is characterized analytically. The spatial distribution of bubble size depends on the effect of individual coalescence events on the bubble size distribution, the rate at which they occur, and the rise rate of the bubbles. The film rupture rate is assumed to depend on the film area and the film drainage time, and is calculated using several simple models. This analysis agrees well with measurements conducted in a laboratory foam cell. However, the time-scale of the experiments indicated that film drainage was occurring predominantly in the hydrodynamic regime, and that non-equilibrium rupture mechanisms were primarily responsible for the observed coalescence.     

Coalescence and breakup of bubbles in liquids

Coalescence and breakup in a swarm of bubbles were directly observed by using the high speed cinematography. It was designed so that three dimensional motion of bubbles could be continually followed. Influence of operating conditions and liquid properties on the frequencies of bubble passing, coalescence and breakup was examined.It was seen that there was the critical distance at which the leading bubble began to exert an noticeable influence on, the following one. The distance was about 3- to 4-fold diameter of the leading bubble. Coalescence was found to take place when more than about a half of the projected area of the following bubble was overlapped with that of the leading bubble at the critical distance. On the contrary, breakup occurred in the case of the overlapping less than about a half of the projected area of the following bubble.The increment of the following bubble velocity induced by the leading one was empirically correlated with the bubble distance and the angle between the straight line joining the centers of bubbles and the vertical axis. The Reynolds numbers had no appreciable effect on the velocity increment within the range examined.     

Coalescence efficiency of bubbles in bubble columns

Bubble size distribution was modelled by employing the population balance equation (PBE). All three bubble coalescence mechanisms (turbulence, buoyancy and laminar shear) and the main bubble breakup mechanism (breakup due to turbulent eddies) were considered in the model. Local bubble size distributions at the top and bottom of the column were obtained by solving this PBE. The results were compared with the experimental data for seven independent multiphase systems (water/air, isomax diesel/air, kerosene/air and four other liquid mixture/air) at two diverse gas velocities. The experimental adjustable constant in the coalescence efficiency function was determined by fitting the population balance to the experimental bubble size distributions. An empirical correlation was proposed for the coalescence efficiency by the dimensional analysis, which includes Reynolds and Weber numbers. © 2011 Canadian Society for Chemical Engineering     

Interaction of two in-line bubbles of equal size rising in viscous liquid

The interaction of bubbles is the key to understand gas–liquid bubbling flow. Two-dimensional axis-symmetry computational fluid dynamics simulations on the interactive bubbles were performed with VOF method,which was validated by experimental work. It is testified that several different bubble interactive behaviors could be acquired under different conditions. Firstly, for large bubbles(d: 4, 6, 8, 10 mm), the trailing bubble rising velocity and aspect ratio have negative correlations with liquid viscosity and surface tension. The influences of viscosity and surface tension on leading bubble are negligible. Secondly, for smaller bubbles(d: 1, 2 mm), the results are complicated. The two bubbles tend to move together due to the attractive force by the wake and the potential repulsive force. Especially for high viscous or high surface tension liquid, the bubble pairs undergo several times acceleration and deceleration. In addition, bubble deformation plays an important role during bubble interaction which cannot be neglected.     

Mass transfer in trickle-bed reactors at low reynolds number

Mass transfer rates were determined in a 3.4 cm i.d. trickle-bed reactor in the absence of reaction by absorption measurements and in presence of reaction. Gas flow rates were varied from 0-100 l/h and liquid flow rates from 0-1.5 l/h. The catalyst particles were crushed to an average diameter of 0.054 and 0.09 cm. Mass transfer coefficients remained unaffected by change in gas flow rate but increased with liquid rate. The data from absorption measurements were evaluated with predictions based upon plug-flow and axial dispersion model. Mass transfer coefficients were found greater in case of axial dispersion model than that of plug-flow model specially at low Reynolds number (Re₁ < 1).Hydrogenation of α-methylstyrene to cumene using a Pd/Al₂O₃ catalyst was taken as a model reaction. Intrinsic kinetic studies were made in a laboratory-stirred-autoclave. Mass transfer coefficients were determined using these intrinsic kinetic data from the process kinetic measurements in trickle-bed reactor. Mass transfer coefficients under reaction conditions were found to be considerably higher than those obtained by absorption measurements.Correlations were suggested for predicting mass transfer coefficients at low Reynolds number.The gas to liquid mass transfer coefficients for lower gas and liquid flow rates were determined in a laboratory trickle-bed reactor. The effect of axial dispersion on mass transfer was considered in order to evaluate the experimental data. Three correlations were formulated to calculate the mass transfer coefficients, which included the effect of liquid loading, particle size and the properties of the reacting substances. The gas flow rate influences the gas to liquid mass transfer only in the region of low gas velocities. In the additional investigations of gas to liquid mass transfer without reaction in trickle-bed reactor, the mass transfer coefficients were determined under reaction conditions and the intrinsic kinetics was studied in a laboratory scale stirred autoclave with suspended catalyst. A few correlations are formulated for the mass transfer coefficients. A comparison with the gas-liquid mass transfer coefficient obtained by absorption measurements showed considerable deviations, which were illustrated phenomenologically.     

Modeling of unsteady flow around accelerating sphere at moderate reynolds numbers

The flow around an accelerating spherical particle of diameter ranging from 50 to 200 m?m is studied in the range of Reynolds number between 0.1 and 100. The flow around the sphere is assumed to be laminar and two-dimensional axisymmetric. The calculated drag coefficient is compared with the theoretical predictions of added mass term and Basset history term. Appropriate corrections for those two terms are proposed as function of the acceleration rate and the particle diameter.     

Flow of power law liquids through particle assemblages at intermediate reynolds numbers

The complete equations of motion and continuity have been solved numerically using the finite element method for the flow of power law liquids through assemblages of rigid spherical particles. The inter-particle interactions have been simulated using the free surface cell model. Extensive results on drag coefficients have been obtained under a wide range of physical and operating conditions (0.9999 ≥ 0.3), 1 ≥ n ≥ 0.4 and 20 ≥ Re ≥ 1. The observed dependence of drag coefficient on voidage and non-Newtonian flow behaviour index have been explained qualitatively with the aid of order of magnitude considerations. Finally, the theoretical predictions have been validated using suitable experimental data available in the literature.     

Electrostatic collection of aerosol particles on a sphere at intermediate reynolds numbers

The experimental technique, equipment and measurements of charged and neutral drop collection efficiency by a single charged sphere (Re = 15–100) are described.