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.     

Numerical study on steady and transient mass/heat transfer involving a liquid sphere in simple shear creeping flow

A numerical method is utilized to examine the steady and transient mass/heat transfer processes that involve a neutrally buoyant liquid sphere suspended in simple shear flow at low Reynolds numbers is described. By making use of the known Stokes velocity field, the convection‐diffusion equations are solved in the three‐dimensional spherical coordinates system. For the mass transfer either outside or inside a liquid sphere, Sherwood number Sh approaches an asymptotic value for a given viscosity ratio at sufficiently high Peclet number Pe. In terms of the numerical results obtained in this work, two new correlations are derived to predict Sh at finite Pe for various viscosity ratios. © 2013 American Institute of Chemical Engineers AIChE J, 60: 343–352, 2014     

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.     

Unsteady conjugate mass transfer of a 2D deformable droplet in a modest extensional flow in across-slot

This work aims to investigate the unsteady conjugate interphase mass transfer between a stationary deformed drop and the modest extensional flow in a cross-intersected 2D channel. It is very difficult to accurately quantify the transient mass transfer rate of solute in such a geometry. Therefore, we established a mathematical model on the basic of the Stokes equation and solved it by the boundary element method, which could deal precisely with a two-phase flow system with a deformable interface; meanwhile, the convection-diffusion equation was solved by the finite difference method to calculate the unsteady conjugate interphase mass transfer. The simulation results showed that the mass transfer rate, analyzed and characterized in terms of mean concentration variation and Sherwood number Sh, was affected by capillary number Ca, Peclet number Pe, viscosity ratio λ , interior-to-exterior diffusivity ratio K, distribution coefficient m, and wall effect factor W.     

Effect of contamination on rise velocity of bubble swarms at moderate Reynolds numbers

In this work, hydrodynamics of contaminated bubble swarms is numerically investigated using the free surface cell model combined with the spherical stagnant cap model. The governing field equations are solved numerically to elucidate the effect of Reynolds number, gas holdup and degree of contamination on the hydrodynamic behavior of bubble swarms. New extensive results are reported over the range of conditions as follows: Reynolds number, Re: 1–200, bubble holdup, Φ: 0.1–0.5, and stagnant cap angle, α: 0–180°. Finally, the effects of these parameters on streamlines and vorticity contours, surface pressure and vorticity distributions and on drag coefficients are discussed in detail. Briefly, the drag coefficients decrease with the decreasing stagnant cap angle and/or the decreasing bubble hold up and/or the increasing Reynolds number; whereas the ratio of the pressure and friction drag coefficients exhibits mixed trends with respect to these parameters.     

Mass and heat transfer from or to a single sphere in simple extensional creeping flow

The first detailed numerical investigation on the mass and heat transfer both outside and inside a solid or liquid sphere immersed in a simple extensional flow for a larger range of Peclet numbers (1–100,000) is presented. By making use of the known Stokes velocity field at small Reynolds numbers, a finite difference method with the control volume formulation is adopted to solve the convection‐diffusion transport equation. Simulation results show that the transport rate, which is represented by Sherwood number, is significantly affected by Peclet number and viscosity ratio. The flow direction, no matter a uniaxial extensional flow or a biaxial extensional flow, has no effect on the total transport rate but affects the concentration distribution a lot. Some comparisons between present simulations and previous studies are made to validate each other and confirm the reliability and applicable scopes of reported correlations. A few new correlations are put forward to predict the transfer rate at finite Peclet numbers for various values of viscosity ratios. © 2011 American Institute of Chemical Engineers AIChE J, 58: 3214–3223, 2012     

Coupled Mass and Heat Transfer between a Cone and a Power‐Law Fluid

Coupled mass and heat transfer between a cone and a non‐Newtonian fluid was studied when the concentration level of the solute in the solvent is finite (finite dilution of solute approximation). Convective heat and mass transfer between a laminar flow and a stationary cone and between a rotating cone and a quiescent fluid is investigated. Solutions of both problems are found in the form of the dependencies of Sherwood number vs. Reynolds and Schmidt numbers. Coupled thermal effects during dissolution and solute concentration level effect on the rate of mass transfer are investigated. It is found that the rate of mass transfer between a cone and a non‐Newtonian fluid increases with the increase of the solute concentration level. The suggested approach is valid for high Peclet and Schmidt numbers. Isothermal and nonisothermal cases of dissolution are considered whereby the latter is described by the coupled equations of mass and heat transfer. It is shown that for positive dimensionless heat of dissolution, K > 0, thermal effects cause the increase of the mass transfer rate in comparison with the isothermal case. On the contrary, for K < 0 thermal effects cause the decrease of the mass transfer rate in comparison with the isothermal case. The latter effect becomes more pronounced with the increase of the concentration level of the solute in a solvent.     

Mass/heat transfer from a neutrally buoyant sphere in simple shear flow at finite Reynolds and Peclet numbers

Theoretical analyses of mass/heat transfer from a neutrally buoyant particle in simple shear flow indicate that mass/heat must diffuse across a region of closed streamlines of finite thickness at zero Reynolds number, whereas spiraling streamlines allow the formation of a thin mass transfer boundary layer at small but non‐zero Reynolds numbers (Subramanian and Koch, Phys Rev Lett. 2006;96:134503; Subramanian and Koch, Phys Fluids. 2006;18: 073302). This article presents the first numerical results for mass/heat transfer at finite Reynolds and Peclet numbers. The simulations indicate that fluid particles in the flow‐gradient plane spiral away from the particle for Reynolds numbers smaller than about 2.5 while they spiral toward the particle for higher Reynolds numbers. Solutions of the Navier‐Stokes equations coupled with a boundary layer analysis of mass transfer yield predictions for the rate of mass transfer at asymptotically large Peclet numbers and Reynolds numbers up to 10. Simulations of mass transfer for zero Reynolds number and finite Peclet numbers confirm Acrivos' (Acrivos, J Fluid Mech. 1971;46:233–240) prediction that the Nusselt number approaches a finite value with increasing Peclet number. Simulations at finite Reynolds numbers and Peclet numbers up to 10,000 confirm the theoretical predictions for the concentration gradient at the particle surface at angular positions away from the flow‐gradient plane. However, the wake near the flow‐gradient plane remains too large at this Peclet number to yield a quantitative agreement of the overall rate of mass transfer with the theory for asymptotically large Peclet number. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

The heat and mass transfer from a fluid sphere at large reynolds and peclet number

Analytic expressions have been derived for the steady rate of heat or mass transfer from a fluid sphere in uniform motion at large Reynolds and Peclet number. The solution is applicable to cases where both phase resistances are of the same order of magnitude, or when the resistance of one phase is negligible. The analysis has shown that if finite values of fluid viscosity and density are considered the transfer rates are considerably less than those obtained from the ideal fluid model. The maximum flux for a fluid sphere occurs near the equatorial plane and in this respect its behavior differs significantly from a solid sphere.     

Mass transfer around bubbles,drops, and particles in uniaxial and biaxial nonlinear extensional flows

Mass transfer around spherical bubbles, drops, and solid particles in uniaxial and biaxial nonlinear extensional creeping flows at large Peclet numbers is the subject of this theoretical report. The fluid mechanics problem is governed by the viscosity ratio (λ) and the nonlinear intensity of the flow (E). The flow outside such bodies reveals a different picture than the linear case (E = 0) such as separating surfaces or closed circulations. There is a range: −1.04 < E < 0 (bubbles and drops) and −0.490 < E < 0 (particles) where the mass transfer rate is lower than the linear case. Outside these ranges, the mass transfer rate is higher than the linear case and in general it increases as |E| increases. Same mass transfer rates are expected in uniaxial and biaxial flows, except in the presence of external closed circulations where the biaxial flow overcomes the uniaxial flow. © 2018 American Institute of Chemical Engineers AIChE J, 65: 398–408, 2019     

Liquid degassing using mono‐dispersed and poly‐dispersed micron droplets

Liquid degassing using mono‐dispersed and poly‐dispersed micron droplets (148.6 and 264.8 µm) falling in an inert gas was studied experimentally and analytically in a laboratory setting. The system using poly‐dispersed droplets revealed an upper degassing limit of 93% when the inert gas to liquid flow ratio was above 15 and a fog‐type nozzle was used. Linear correlations of the Sherwood number to the Peclet number were derived for both systems of medium Reynolds numbers (10–100). The correlation for the mono‐dispersed droplets agreed well with the steady state mass transfer correlations that are available in literature, especially at small Peclet number situations. © 2012 Canadian Society for Chemical Engineering     

Coupled two-phase mass transfer to spherical drops — Trace gas absorption and oxidation by a single drop

Mass transfer across gas and liquid boundary layers into the core of drops with liquid phase first order chemical reaction has been analyzed for spherical drops in the Reynolds number range of 50 < Re_g < 400. The realistic and computationally efficient simulation of this gas absorption system is applicable in a variety of engineering fields including gas-liquid mass transfer in drops and sprays. The present paper deals with the fluid mechanics and mass transfer with chemical reaction of a single drop. In computer experiments good predictive agreement has been achieved with measured data. The theoretical results were generalized to show the influence of three major system parameters: Peclet number Pe_g or Pe_l Damköhler number Da and the distribution coefficient at the gas-liquid interface, M, on mass transfer and to demonstrate the importance of coupled gas- and liquid-phase resistances to gas absorption under practical conditions.     

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     

Enhancement of mass transfer from particles by local shear-rate and correlations with application to drug dissolution

We analyze hydrodynamic enhancement of mass (or heat) release rate from small spherical particles within fluid flows from local flow shear-rate, with application to drug dissolution. Combining asymptotic theories in the high/low shear Peclet number limits in Stokes flow with 205 carefully-developed computational experiments, we develop accurate correlations for shear enhancement of Sherwood/Nusselt number (Sh/Nu) as a function of shear Peclet and Reynolds number (S^*, Re _S). The data spanned S^* from 0 to 500 and Re _S from 0 to 10. In Stokes flow our correlations are highly accurate over the entire S^* range, whereas for finite Re _S < 1 accuracy is good for S^* up to a few thousand. Shear enhancement results from highly three-dimensional spiraling flow created by particle spin. We develop a model for particle slip velocity that is inserted into the Ranz/Marshall correlation to show that shear-rate enhancement strongly dominates convection, a result important to drug dissolution.     

A reduced‐order model for chemical species transport in a tube with a constant wall concentration

A two‐dimensional advection‐diffusion model accompanied with a parabolic velocity profile of Poiseuille flow is considered for the chemical species transport in a tube with a constant wall concentration. The Reynolds decomposition technique is applied to reduce it to an equivalent one‐dimensional model for advective‐dispersive transport in a tube through which the effective advection coefficient, the dispersion coefficient, and the effective Sherwood number are developed for the problem under study. The derived and the classical Taylor models are also compared in order to find the difference between the two arrangements. The reduced‐order model for the transport equation shows that the effective advection coefficient increases, whereas the dispersion coefficient in the tube decreases as compared to the classical Taylor equation. The effective Sherwood number for the steady state form of the developed model is found to be only a function of the Peclet number, which varies in the range of 3.215 ≤ Sh ≤ 4. These results find application in design of experiments and improve our understanding of mass transfer in microfluidic devices.     

External mass transfer from/to a single sphere in a nonlinear uniaxial extensional creeping flow

This work systematically simulates the external mass transfer from/to a spherical drop and solid par-ticle suspended in a nonlinear uniaxial extensional creeping flow.The mass transfer problem is gov-erned by three dimensionless parameters:the viscosity ratio (2),the Peclet number (Pe),and the nonlinear intensity of the flow (E).The existing mass transfer theory,valid for very large Peclet num-bers only,is expanded,by numerical simulations,to include a much larger range of Peclet numbers(1 ≤ Pe ≤ 105).The simulation results show that the dimensionless mass transfer rate,expressed as the Sherwood number (Sh),agrees well with the theoretical results at the convection-dominated regime (Pe ＞ 103).Only when E ＞ 5/4,the simulated Sh for a solid sphere in the nonlinear uniaxial extensional flow is larger than theoretical results because the theory neglects the effect of the vortex formed outside the particle on the rate of mass transfer.Empirical correlations are proposed to pre-dict the influence of the dimensionless governing parameters (λ,Pe,E) on the Sherwood number (Sh).The maximum deviations of all empirical correlations are less than 15％ when compared to the numerical simulated results.     

Mass transfer around prolate spheroidal drops in an extensional flow

Mass transfer in the continuous phase around a small eccentricity prolate spheroidal drop in an axisymmetric extensional creeping flow and at large Peclet numbers was investigated theoretically. The results show that, at very short times, the total quantity of solute transferred to or from the drop represents, at O(Ca¹), mass transfer by diffusion only around a sphere. For long times, or at steady‐state, the total quantity of solute transferred is, at O(Ca¹), slightly smaller than that of a spherical drop, and it decreases with an increase of the capillary number or the viscosity ratio. © 2012 Canadian Society for Chemical Engineering     

Laminar‐forced convection mass transfer to ordered and disordered single layer arrays of spheres

Laminar forced convection mass transfer to single layers of equidistantly and nonequidistantly spaced spheres perpendicular to the flow direction is studied. Average Sherwood numbers are reported as a function of geometric configurations and flow conditions, for open frontal area fractions between 0.25 and 0.95, Schmidt numbers between 0.7 and 10, and Reynolds numbers (based on the sphere diameter and the free stream velocity) between 0.1 and 100. For equidistantly spaced arrays of spheres, a general analytical expression is proposed for the average Sherwood number as a function of the Reynolds number, Schmidt number and the open frontal area fraction, as well as asymptotic scaling rules for small and large Reynolds. For all studied Schmidt numbers, equidistant arrays exhibit decreasing average Sherwood numbers for decreasing open frontal area fractions at low Reynolds numbers. For high Reynolds numbers, the Sherwood number approaches that of a single sphere, independent of the open frontal area fraction. For equal open frontal area fractions, the Sherwood number in nonequidistant arrays is lower than in equidistant arrays for intermediate Reynolds numbers. For very low and high Reynolds numbers, nonuniformity does not influence mass transfer. © 2012 American Institute of Chemical Engineers AIChE J, 59: 1400–1408, 2013     

Creeping flow mass transfer to a single active sphere in a random spherical inactive particle cloud at high schmidt numbers

Using the solution by Tam of Navier-Stokes equations for creeping flow around an active sphere surrounded by a random cloud of inactive spheres, an asymptotic solution of the convective diffusion equation is obtained for high Schmidt numbers. The Sherwood number for the overall mass transfer coefficient to the active sphere has been analytically related to the Peclet number as

It agrees very well with the experimental mass transfer data on single active spheres for σ = 0476, Re < 10 and large Sc. This analytical result becomes invalid as σ decreases to 0.33. Pfeffer's model for the same problem has excellent agreement with the mass transfer data on single active spheres for σ = 026, Re < 10 and Sc = 1600. Pfeffer's model seems to be quite satisfactory for the usual range of void volume fractions in packed beds. The present model seems to be more accurate at higher values of void volume fractions in packed and distended beds.     

Dispersion and Mass Transfer Effects on the Performance of an Immobilized Lipase Packed Bed Reactor During the Hydrolysis of Rice Bran Oil

The performance of an immobilized packed bed reactor for the hydrolysis of rice bran oil has been investigated and can be well described by a dispersion model with an average standard deviation of 0.0388. Global mass transfer coefficients estimated using the model and experimental data ranged from 0.095‐0.482 min^?1, depending on substrate flow rates. A dimensionless mass transfer correlation between the Sherwood number and the Reynolds number was obtained as NSh = 3.96 ×NRe^2.07.