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).     

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.     

Effect of fluid dispersion coefficients on particle-to-fluid mass transfer coefficients in packed beds: Correlation of sherwood numbers

Gas-phase and liquid-phase mass transfer data published in the literature are corrected for the axial fluid dispersion coefficient values proposed by WThe corrected Sherwood numbers in the range of Reynolds number from about 3 to 10,000 are correlated by Sh = 2 + 1.1 Sc^1/3Re^0.6.     

Local mass transfer measurements from a submerged jet parallel to a flat surface

The local mass transfer rates from a submerged vertical jet parallel to the vertical electrode surface were measured by a limiting current density technique. For these studies, the parameters considered are a plate Reynolds number in the range of 1965 to 136500, electrode height over the orifice of the jet (Y) which varied from 0 to 12 times the jet orifice diameter (d), and the vertical distance of the microelectrode on the electrode plate in the range of 0.7 to 12.5 cm. The system used for measuring the limiting current was the reduction of copper ions. The relationship between the local Sherwood number (Sh) and Reynolds number (Re) was found to be Sh = 0.0004(Re)^1.5 (Sc)^0.33 This relationship is valid for Y/d ≤ 6.0.     

Mass transfer from a single fluid sphere to power-law liquids at moderate Reynolds numbers

The steady-state convective inter-phase mass transfer from a single Newtonian fluid sphere (free from surfactants) to a continuous phase with power-law viscosity has been studied at moderate Reynolds and Schmidt numbers under the conditions when the resistance to mass transfer in the dispersed phase is negligible. The species continuity equation, segregated from the momentum equations of both phases, has been numerically solved using a finite difference method. The effects of the Reynolds number (Re_o), power-law index (n_o), internal to external fluid characteristic viscosity ratio (k) and Schmidt number (Sc) on the local and average Sherwood number (Sh) have been analysed over the following ranges of conditions: 5?Re_o?200, 0.6?n_o?1.6, 0.1?k?50 and 1?Sc?1000. It has been observed that irrespective of the values of the Reynolds number and of the power-law index, as the value of k increases the average Sherwood number decreases for intermediate to large values of the Peclet number. As the value of the power-law index increases, the rate of mass transfer decreases for all values of the Reynolds number and the characteristic viscosity ratio thereby suggesting that shear-thinning behaviour facilitates mass transfer, whereas shear-thickening behaviour impedes it. Based on the present numerical results, a simple predictive correlation is proposed which can be used to estimate the rate of inter-phase mass transfer of a fluid sphere sedimenting in power-law liquids.     

随机分布纤维束间纵向层流时传质的数值模拟

按层流流动理论模型,对纤维为随机分布的中空纤维膜组件的壳程,在恒定壁面传质量和恒定壁面浓度边界条件下的传质现象进行了数值模拟,并得到了不同装填率下传质Sherwood数关联式。结果表明,纵向层流时,随机分布纤维间的传质仍可分进口段和充分发展段。对于某一给定的纤维束,在传质的充分发展段,传质Sherwood数为一个定值,且较纤维束规则排列时小得多,纤维分布不均一性将导致膜组件的传质能力下降。在传质进口段,传质Sherwood数也较纤维束规则排列时要小,装填率不但对传质系数的关联式Sh=BReaScbf(de/L)的系数B值有影响,且对该式中的Reynolds数的指数a和Schmidt数的指数b值也均有影响。     

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.     

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.     

Mass transfer with chemical reaction from spherical one or two component bubbles or drops

Unsteady state mass transfer between a one or two component bubble (or drop) and the continuous phase with a chemical reaction occuring either in the continuous or in the dispersed phase is examined. The main assumption for binary bubbles is that the rate determining step is diffusion in the continuous phase. Two limiting velocity fields. Hadamard flow (Re?:1) and potential flow (Re?:1) are used in the calculations.     

An improved subgrid scale model for front-tracking based simulations of mass transfer from bubbles

Gas–liquid bubble column reactors are often used in industry because of their favorable mass transfer characteristics. The bubble mass boundary layer in these systems is generally one order of magnitude thinner than the momentum boundary. To resolve it in simulations, a subgrid scale model will account for the sharp concentration variation in the vicinity of the interface. In this work, the subgrid scale model of Aboulhasanzadeh et al., Chem Eng Sci, 2012, 75:456–467 embedded in our in-house front tracking framework, has been improved to prevent numerical mass transfer due to remeshing operations. Furthermore, two different approximations of the mass distribution in the boundary layer have been tested. The local and global predicted Sherwood number has been verified for mass transfer from bubbles in the creeping and potential flow regimes. In addition, the correct Sherwood number has been predicted for free rising bubbles at several Eötvös and Morton numbers with industrial relevant Schmidt numbers (10³–10⁵).     

Concentration polarisation in tubular membranes —a numerical approach

A theoretical investigation ofparticle deposition onto a permeable surface of a tubular membrane is presented. Themass transport mechanisms are mathematically expressed using the two-dimensional convective diffusion equation. A numerical scheme is presented to solve the two-dimensional convective diffusion equation at the steady state for the case of nonuniform permeation velocity. This equation is solved numerically using a finite difference method. The numerical prediction of mass transfer in the mass boundary requires the use of a very dense grid. The concentration profiles along the membrane surface and the mass boundary layer are predicted. The effect of the Reynolds number, the wall Reynolds number and the Schmidt number were investigated. Correlations for the concentration boundary layer thickness δ_c/D=2(z/D)^0.33(ReSc)^{− 0.33} Re_w^−0.3 (1-0.4377 Sc^−0.0018 Re_w^−0.1551), and for the Sherwood number Sh =1.230 [(D/z) ReSc] ^0.33 (1 + 0.010 Re^−0.125Sc^1.055 Re_w.^1.132) based on the predicted values of the solute concentration profiles, are proposed, in the operating condition ranges 300 < Re < 1000, 0.02 < Re_w < 0.3 and 600 < Sc < 3200.     

The drag on a single bubble accompanied by a periodic wake

The dynamics of a single spheroidal bubble accompanied by an open periodic wake were studied in water-glycerol solutions (1 < η < 37 cP) by photographic techniques. The stability of the helical vortex, the frequency of the bubble rocking and the drag coefficient were found to be closely related. These characteristics exhibit two modes of behaviour: they are very sensitive to viscosity changes at lower Reynolds numbers Re_b; whereas at higher Re_b the viscosity does not appreaciably influence the drag coefficient C_Db and the Strouhal number S_b, while the length of the helical vortex here has reached its minimum . It has been concluded threefore that, in the transition region between the oblate spheroid and spherical cap bubbles presently investigated, the induced drag plays an important role in the overall bubble dynamics.     

Coalescence of two bubbles rising in line at low reynolds numbers

Bubble coalescence phenomena have been examined at Reynolds numbers of 0·5–80 for five different classes of bubbles. The approach velocity of the following bubble (u₂) is experimentally obtained and its behaviour with respect to the leading bubble velocity (u₁) is examined. The coalescence of bubbles with Re < 7 follows the analysis of weightless solid spheres. Bubbles having toroidal wakes (Re > 7), coalesce with two additional velocities imparted due to the wake structure. Equations are developed to predict u₁/u₂ during coalescence.     

MASS TRANSFER IN LIQUID FLUIDIZED BEDS AT LOW REYNOLDS NUMBERS

Mass transfer coefficients between fluidized ion exchange resins (Dowex 2 × 8) and dilute solutions of hydrochloric acid and of benzoic were measured in the low Reynolds number range (Re < 1.0). In the case of benzoic acid, the experimental mass transfer coefficients are in agreement with the values predicted by standard correlation Sh = (0.81 ± 0.05) ε^?1 Re ^0.5 Sc ^1/3 in the range 0.1 < Re < 1, but below this value of Re the measured mass transfer coefficients are much lower. With hydrochloric acid the experimental mass transfer data are well correlated by the equation Sh = 1.1 ε^?1 Re Sc ^1/3 in the range Re < 1. The observed decrease of the mass transfer coefficients in the low range of Reynolds numbers is most probably related to the effects which are considered in Nelson and Galloway's theory and in Rowe's modification of the same theory. The theoretical results were found useful in the interpretation of experimental data. The least squares analysis of the experimental data showed that the value of parameter α, with which the best fit between experimental and theoretical results was obtained, was α = 0.62 with benzoic acid and α = 0.7 with hydrochloric acid, in good agreement with the value value proposed by Rowe and established earlier with an NaOH system. Qualitatively, the theory predicts the correct dependence of Sh on Sc as Re → 0.     

Effect of gas sparging on mass transfer in zinc electrolytes

The effect of sparging on mass transfer is reported for zinc electrolytes containing antimony and antimony-free electrolytes. Comparative results with non-sparged electrolytes show, an enhancement in mass transfer. In the sparged electrolyte, the mass transfer coefficients,K _Zn, increase with increasing current density, antimony additions, and sulphuric acid concentration. The deposition morphology is consistent with the mass transfer results. A relationship between the mass transfer coefficients for sparged and non-sparged systems is obtained. The relationship correlates satisfactorily with the data and provides a quantitative method for determining the degree of enhancement in mass transfer coefficients due to sparging. The correlation which best represents the mass transfer data for sparged zinc electrolytes is $$Sh = 105(ReSc)^{0.23} $$ whereSh, Re, andSc are the Sherwood, Reynolds, and Schmidt numbers, respectively. The correlation represents the case where sparging is applied to a gas evolving electrode, hydrogen in this case.     

Effect of the viscosity ratio on mass transfer from a fluid sphere at low to very high Peclet numbers

The effect of the viscosity ratio on mass transfer from a fluid sphere is examined in this paper. Numerical solutions of the Navier-Stokes equations off motion and the equations of mass transfer have been obtained for the unsteady state transfer from a fluid sphere moving in an unbounded fluid medium of different viscosity. The effects of the viscosity ratio and the flow on the concentration profiles were investigated for Reynolds number, viscosity ratio and Péclet number ranges of 0?Re?400, 0?κ?1000 and , respectively. The local and average Sherwood numbers are also presented graphically. The steady state results show that the average Sherwood number is increasing as Peclet number increases for a fixed viscosity ratio. However, for a fixed Peclet number, the average Sherwood number is decreasing as the viscosity ratio increases and reaches a limit value corresponding to the average Sherwood number for a solid spherical particle. From the numerical results, a predictive equation for the Sherwood number in terms of the Peclet number, the Reynolds number and the viscosity ratio is derived.     

External mass transfer in silica monolithic stationary phases

A detailed analysis of a procedure of measurement of the external mass transfer coefficients (k_f) in RPLC systems is provided. Application is described for a system consisting of a C₁₈-silica monolithic column and a methanol/water eluent (70/30, v/v). The k_f values of butylbenzene at 298 K were derived from peak profiles recorded in pulse response experiments by subtracting the contributions of the kinetic processes, i.e., axial molecular diffusion, eddy diffusion, pore and surface diffusion, from the band variance. This approach provided the Sherwood number (Sh) for a range of Reynolds number (Re) between 0.002 and 0.005 and a Schmidt number (Sc) equal to 2.7×10³. The experimental values of k_f were compared with those estimated from literature correlations, giving a relative error of ca. 11% when Pfeffer equation was used for estimating k_f. The exponents obtained for Re and Sc in Sh were 0.43 and 0.39, values comparable with those found in literature correlations, i.e., 0.33. The k_f values estimated using the reference correlations are of the same order of magnitude as the experimental k_f values. The acquisition of more experimental data is needed for deriving an improved correlation affording more accurate estimates of k_f in stationary phases of cylindrical shape, like silica monoliths.     

Viscous flow through particle assemblages at intermediate reynolds numbers — a cell model for transport in bubble swarms

The fluid mechanical behaviour of a bubble swarm was simulated using a cell model. The Navier-Stokes equations were solved numerically for the liquid flow in a uniform assemblage of circulating, spherical bubbles. Ranges of parameters studied included, Reynolds numbers, 0–1000 and porosities, 0.4–1. The numerical calculations show the effects of variations in Reynolds numbers and porosity on: surface vorticity and pressure distributions and form and friction drag coefficients. For all Reynolds numbers investigated a standing vortex ring was absent Predicted drag coefficients and Sherwood or Nusselt numbers agree with limiting analytical solutions for low and high Reynolds numbers. The theoretical results show good agreement with experimental data for porosity as a function of superficial gas velocity. Predicted and measured Sherwood and Nusselt Numbers were in substantial disagreement, making detailed comparison unwarranted The calculations should also be valid for dispersions of uniform, circulating, spherical droplets for the special case where the droplet viscosity is much less than the viscosity of the continuous fluid     

Correlation between mass transfer and operating parameters in zinc electrowinning

The effects of antimony additions, acid concentration, and current density on mass transfer and deposition morphology were examined. The mass transfer coefficients of zinc were calculated using a codeposition method with cadmium as a tracer. The experiments were carried out for vertical electrodes in a Hull cell. The results indicate that the mass transfer coefficients increase with increasing antimony additions, acid concentration, and current density. Zinc dissolution is more severe at low current density and higher antimony levels than at higher current densities and lower antimony levels. A mass transfer correlation for pure zinc electrolyte data is $$Sh = 12.47\left( {ReSc} \right)^{0.45} $$ whereSh, Re, andSc are the Sherwood, Reynolds, and Schmidt numbers, respectively. The correlation fits very well with the experimental data. A correlation for electrolytes containing antimony was also obtained and has an exponent of 0.42. The correlations cover a wide range of operating parameters and provide a fast quantitative estimation of the change in mass transfer in zinc electrowinning.