The effect of surfactant on terminal velocity of and mass transfer from a fluid sphere in a non-newtonian fluid

The effect of interfacial tension gradients on creeping flow past a fluid sphere (bubble or drop) in a non-Newtonian fluid is investigated. The drag force and the Sherwood number for contaminated fluid spheres moving in a non-Newtonian fluid are obtained by an approximate solution of equations of motion in the creeping flow regime. It has been found that surface-active agents decrease both the terminal velocity and the mass transfer rate. The influence of the flow index of the power-law on both the drag coefficient and the Sherwood number is superimposed over the influence of the interfacial tension gradient. Comparisons between the present analysis and the available experimental data in the literature show reasonable agreement for the terminal velocity and the Sherwood number.     

Characterisation of inelastic power-law fluids using falling sphere data

Two simple methods are presented for the characterization of inelastic power law fluids from falling sphere data. The methods involve the application of shear rate or shear stress correction factors which have been derived theoretically using Slattery's solution for creeping flow about spheres. Flow curves obtained using these methods are in excellent agreement with those measured on a Weissenberg rheogoniometer for 0.83 ≤ n ≤ 1.0. The experimentally determined drag coefficients are found to be in good agreement with the predictions of Slattery's creeping flow first approximation solution. The wall correction factors of Faxen and Francis appear to be valid for inelastic non-Newtonian fluids up to a diameter ratio of at least 0.08.     

Hydrodynamic drag forces on two porous spheres moving along their centerline

This paper numerically evaluates the hydrodynamic drag force exerted on two highly porous spheres moving steadily along their centerline (sphere #1 and sphere #2) through a quiescent Newtonian fluid over a Reynolds number ranging from 0.1 to 40. At creeping flow limit, the drag forces exerted on both spheres were identical. At higher Reynolds numbers the drag force on sphere #1 was higher than sphere #2, revealing the shading effects produced by sphere #1 on sphere #2. At dimensionless diameter (β, =d_f/2k^0.5, d_f and k are floc diameter and interior permeability, respectively) >20, the spheres can be regarded nonporous. At β<20, the drag forces dropped. At β<2, the drag forces approached “no-spheres” limit. An increased size ratio of two spheres (d_f1/d_f2) would increase the drag force on sphere #1 and reduce that on sphere #2. At increasing β for both spheres, the drag force on sphere #2 was increased because of the more difficult advective flow through its interior, and at the same time the drag was reduced owing to the stronger wake flow produced by the denser sphere #1. The competition between these two effects leads to complicated dependence of drag force on sphere #2 on β value. These effects were minimal when β became low. Two identical spheres could move steadily along their centerline. At higher Reynolds number, the two spheres would move closer because of the incorporation of inertia force. For spheres of different diameters, the sphere # 2 would move faster than sphere #1 regardless of their size ratio and β value. This occurrence yielded efficient coagulation when two porous spheres were moving in-line.     

Characterisation of inelastic poweriaw fluids using falling sphere data

Two simple methods are presented for the characterization of inelastic power law fluids from falling sphere data. The methods involve the application of shear rate or shear stress correction factors which have been derived theoretically using Slattery's solution for creeping flow about spheres. Flow curves obtained using these methods are in excellent agreement with those measured on a Weissenberg rheogoniometer for 0.83 > n > 1.0. The experimentally determined drag coefficients are found to be in good agreement with the predictions of Slattery's creeping flow first approximation solution. The wall correction factors of Faxen and Francis appear to be valid for inelastic non-Newtonian fluids up to a diameter ratio of at least 0.08.     

Motion of entrained particles in gas streams

Recent work on the drag experienced by an entrained particle in relative motion through a fluid is reviewed. It is shown that fluid turbulence, acceleration, particle shape and orientation, and particle-fluid mass transfer can all have a significant effect on the value of the drag force, particularly when the non-idealities cause a change in the flow pattern around the particle. Conditions under which the drag force can be predicted with certainty are still very limited, and some potentially valuable directions for future research are suggested.     

NEWTONIAN FLUID SPHERE WITH RIGID OR MOBILE INTERFACE IN A SHEAR-THINNING LIQUID: DRAG AND MASS TRANSFER

The drag force and the mass transfer rate of a Newtonian fluid sphere, having mobile or rigid interface, moving in a power law fluid, are obtained by an approximate solution of equations of motion in the creeping flow regime. It is shown that both the drag and mass transfer increase as the flow index of the external fluid decreases.

The increase of drag due to the pseudoplastic anomaly is more significant at large viscosity ratio parameter. The results obtained are in good agreement with available experimental data and with those analyses based on variational principle when the non-Newtonian flow behavior is not very pronounced.

Also, the predicted mass transfer rates are in good agreement with the trends presented in the literature. Unlike in the case of drag force, the effect of the pseudoplastic anomaly on mass transfer rate is more pronounced for low values of the viscosity ratio parameter. The analysis was extended to include the case when the surface of the sphere was immobilized by surface-active contaminants.     

Drag on two coaxial rigid spheres moving along the axis of a cylinder filled with Carreau fluid

The boundary effect on the drag on two identical, rigid spheres moving along the axis of a long cylinder filled with a Carreau fluid for Reynolds number ranges from 0.1 to 40 is investigated. The influences of the key parameters of the problem under consideration, including the separation distance between two spheres, the relaxation time constant and the power-law index of a Carreau fluid, the Reynolds number, and the ratios (radius of sphere/radius of cylinder), on the drag acting on two spheres are investigated. We show that the boundary effect for the present case is more significant than that for the corresponding Newtonian fluid. The presence of the cylinder has the effect of enhancing the convective motion in the rear part of a sphere, thereby forming wakes and a reverse flow field, and this phenomenon is enhanced by the shear-thinning nature of a fluid. If the boundary effect is insignificant, the shear-thinning nature of a fluid has the effect of reducing the deviation of the ln(drag coefficient)-ln(Reynolds number) curve from a Stokes'-law-like relation. On the other hand, if it is significant, this deviation has a local minimum as the shear-thinning nature of a fluid varies.     

NEWTONIAN FLUID SPHERE WITH RIGID OR MOBILE INTERFACE IN A SHEAR-THINNING LIQUID: DRAG AND MASS TRANSFER

The drag force and the mass transfer rate of a Newtonian fluid sphere, having mobile or rigid interface, moving in a power law fluid, are obtained by an approximate solution of equations of motion in the creeping flow regime. It is shown that both the drag and mass transfer increase as the flow index of the external fluid decreases.

The increase of drag due to the pseudoplastic anomaly is more significant at large viscosity ratio parameter. The results obtained are in good agreement with available experimental data and with those analyses based on variational principle when the non-Newtonian flow behavior is not very pronounced.

Also, the predicted mass transfer rates are in good agreement with the trends presented in the literature. Unlike in the case of drag force, the effect of the pseudoplastic anomaly on mass transfer rate is more pronounced for low values of the viscosity ratio parameter. The analysis was extended to include the case when the surface of the sphere was immobilized by surface-active contaminants.     

Effect of Drag–Reducing Polymers on the Rate of Liquid-Solid Mass Transfer in Fixed Beds of Spheres under Forced Convection Conditions

The effect of drag–reducing polymers on the rate of liquid – solid mass transfer in a packed bed reactor under forced convection conditions was studied by measuring the rate of diffusion–controlled dissolution of copper spheres in acidified chromate solutions. The variables investigated were superficial liquid velocity, sphere diameter, bed height, and polymer concentration. The mass transfer coefficient was found to increase with increasing superficial liquid velocity. Increasing both sphere diameter and bed height were found to decrease the mass transfer coefficient. Polymer addition was found to decrease the rate of mass transfer by an amount ranging from 29.2 to 56.9% depending on superficial liquid velocity and polymer concentration. Mass transfer data were correlated in absence and in the presence of drag–reducing polymer, using the following equations, respectively: J_d = 3.71Re^–0.54 and, J_d = 2.5 Re^–0.61where J_d is mass transfer J-factor and Re is the Reynolds number.     

Surface-tension-induced interfacial convection and its effect on rates of mass transfer

Surface-tension-induced interfacial convection (Marangoni phenomena) can appear as a result of mass and heat transfer, compression and dilatation of surface films or their non-Newtonian behaviour and owing to presence in the interface of electrostatic charges. In process engineering problems the mass transfer effect is usually predominant and, depending on the geometry of the system, leads to surface renewal or changes in interfacial area. The surface renewal phenomena can appear as instabilities or disturbances and their effect on mass transfer is presented for transfer to and from drops as well as across flat interfaces in stirred and laminar flow contactors. Mass transfer coefficients and drag coefficients of drops are compared under conditions of undisturbed (diffusional) transfer, cellular convection and interfacial turbulence for stable and unstable direction of transfer. The importance of gravitational instability is indicated.     

Gas‐Solid Drag Coefficient for Ordered Arrays of Monodisperse Microspheres in Slip Flow Regime

Numerous analytical and numerical correlations for the drag force of particles in packed arrays are not applicable to microspheres because of the invalidity of the no‐slip assumption at a solid wall. The slip flow through assemblages of spheres is investigated by the lattice Boltzmann method (LBM). Three periodic arrays of static and monodisperse particles, i.e., a simple cubic, a body‐centered cubic, and a face‐centered cubic array, each with a relatively wide range of solid volume fraction, are considered. The LBM is validated for the slip flow over a single unbounded sphere and the continuum flow through spheres in a simple cubic array. The LBM results agree well with the experimental and numerical data in the literature. Simulations of slip flow through the three ordered arrays of spheres are performed. The effects of solid volume fraction and slip are both quantified within the developed drag laws.     

Mass transfer from spheres submerged in Newtonian and non-Newtonian fluids

Mass transfer from spheres immersed in Newtonian and power-law fluid flows in the high Reynolds number region is discussed using a modified penetration model. By defining appropriate characteristic velocity and length, the proposed model in which a velocity gradient at the transfer surface is taken account of can be used to evaluate the rate of mass transfer from a sphere in complicated practical flow situations. The proposed model compares reasonably well with previously reported experimental data and correlations for mass transfer from a sphere in a fluid flowing at uniform approach velocity, in a fixed or fluidised bed, and in a stirred tank. Futhermore, good agreement is found between the model and the available experimental data and correlation for mass transfer from a rotating sphere.     

Analytical expression for the friction coefficient of DLCA aggregates based on extended Kirkwood–Riseman theory

We use a self-consistent field method, which we have previously validated, to calculate the translational friction coefficient of fractal aerosol particles formed by diffusion-limited cluster aggregation (DLCA). Our method involves solving the Bhatnagar–Gross–Krook model for the velocity around a sphere in the transition flow regime. The velocity and drag results are then used in an extension of Kirkwood–Riseman theory to obtain the drag on the aggregate. Our results span a range of primary sphere Knudsen numbers from 0.01 to 100 for clusters with up to N = 2000 primary spheres. Calculated friction coefficients are in good agreement with experimental data and approach the correct continuum and free molecule limits for small and large Knudsen numbers, respectively. Results show that particles exhibit more continuum-like behavior as the number of primary spheres increase, even when the primary particle is in the free molecule regime; as an illustrative example, the friction coefficient for aggregates with primary sphere Kn = 1 is approximately equal to the continuum friction coefficient for N > 500. We estimate that our calculations are within 10% of the true values of the friction coefficients for the range of Kn and N presented here. Finally, we use our results to develop an analytical expression (Equation (38)) for the friction coefficient over a wide range of aggregate and primary particle sizes.

Copyright © 2017 American Association for Aerosol Research     

A turbulence dissipation model for particle laden flow

A dissipation transport equation for the carrier phase turbulence in particle‐laden flow is derived from fundamental principles. The equation is obtained by volume averaging, which inherently includes the effects of the particle surfaces. Three additional terms appear that reveal the effect of the particles; these terms are evaluated using Stokes drag law. Two of the terms reduce to zero and only one term remains which is identified as the production of dissipation due to the particles. The dissipation equation is then applied to cases where particles generate homogeneous turbulence, and experimental data are used to evaluate the empirical coefficients. The ratio of the coefficient of the production of dissipation (due to the presence of particles) to the coefficient of the dissipation of dissipation is found to correlate well with the relative Reynolds number. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

SETTLING VELOCITIES OF TWO EQUAL-SIZED SPHERES FOLLOWING EACH OTHER IN VISCOUS FLUIDS CONTAINED IN CYLINDERS

Experimental data are presented for two equal-sized spheres falling along the axis of a cylinder. The two spheres settle with the same velocity as that of a single falling sphere as long as their separation distance is larger than a critical value. When the distance is smaller than the critical value, the two spheres fall faster than a single sphere. The drag on the two spheres is less affected when two spheres fall in a cylinder in comparison to the situation when two spheres fall in an unbounded medium. The data are correlated and shown to agree with numerical calculations.     

Laminar flow past a permeable sphere

Flow past an isolated permeable sphere has been studied. The complete Navier-Stokes equation governs the fluid motion outside the sphere, while Brinkman's extension of Darcy's Law is assumed to hold within the porous sphere. The Navier-Stokes equation is solved using a finite difference scheme. The flow within the porous sphere is solved in two different ways, each being efficient over a particular range of Reynolds number. Drag Coefficients are presented for dimensionless permeability, β, of 5, 10, 15, and 30 and for Reynolds numbers up to 50. The computed drag coefficients are within 10% of the experimental values observed by Masliyah and Polikar for 15 < β > 33, the range covered in their work. Separation was observed only for β > 10. The onset of separation is delayed considerably in porous spheres.     

Studies in the vaporization of mercury in irrigated packed beds

Mass transfer coefficients have been measured for the vaporization of mercury flowing countercurrent to air in irrigated packed beds of spheres and Raschig rings. The measured coefficients increased with gas and liquid flow rates, and were correlated in terms of gas Reynolds number and liquid rate. The mass transfer data for liquid metal irrigation were lower than published data for wetting aqueous systems, due to the non-wetting nature of liquid metals. The lower mass transfer coefficients are believed to be attributed to a lower interfacial area for the non-wetting flow of liquid metals, although direct experimental proof was not obtained. The present results are in agreement with data for zinc absorption in molten lead in packed bed (Warner, 1959) when correlated in terms of the relative velocity and total liquid holdup. The results suggest that for liquid metal irrigated beds, the total hold-up is effective in gas phase transfer processes.     

Mass transfer between a drop or gas bubble and an isotropic turbulent flow

Theoretical formulas have been set up for calculating the coefficients of mass transfer between a drop or gas bubble and an isotropic turbulent flow for various Peclet and Reynolds numbers. The mass transfer coefficients depend mainly on the characteristics of the isotropic turbulent flow (energy dissipation, turbulence length scale, turbulence time scale), on the properties of the medium, and on the particle size. A number of practical formulas for calculating the mass transfer coefficients for turbulent flow are presented. The calculated data are compared with experimental data.     

Approximate solutions of viscous incompressible flow around fluid spheres at intermediate reynoldc's numbers

Approximate solutions of viscous incompressible flow around fluid spheres were presented. Expressions for stream functions approximately satisfying equations of motion for both exterior and interior fluid were obtained using Galerkin's method with the inclusion of inertial terms for both equations. The retention of inertial terms of the interior fluid was found to have insignificant effect on the flow behavior of exterior fluid. However, the circulation velocity within the fluid sphere was found to increase with the increase of the internal Reynold's number, which may be of importance to the study of mass transfer within fluid spheres.     

Drag on chains and agglomerates of spheres in viscous newtonian and power law fluids

New experimental data on the free settling velocity of straight chains (up to twenty spheres) and planar clusters of touching spheres in Newionian and power law media are reported. The results embrace the following ranges of conditions: 0.65 ≤ n ≤ 1; Re < - 2.5 and 1.22 < m < 48.87 Pa·sⁿ. The straight chain drag measurements are in line with theoretical predictions for Newtonian fluids. The present results in power law fluids seem to suggest that it is possible to express the drag on a straight chain of spheres in terms of that on a single sphere of equal volume. Limited results with planar clusters are satisfactorily correlated using a volume equivalent sphere diameter.