Drag on a single fluid sphere translating in power-law liquids at moderate Reynolds numbers

A numerical investigation has been carried out to obtain the steady state drag coefficients and flow patterns of a single Newtonian fluid sphere sedimenting in power-law liquids. A finite difference method based simplified marker and cell (SMAC) algorithm has been implemented on a staggered grid arrangement to solve the continuity and momentum equations. For both phases, the convective terms have been discretized using the quadratic upstream interpolation for convective kinematics (QUICK) scheme, and diffusive and non-Newtonian terms with central differencing scheme. An exponential transformation has been applied in the radial direction for the continuous phase computational domain. In order to ensure the accuracy of the solver, extensive validation has been carried out by comparing the present results with the existing literature values for a few limiting cases. Further, in this study the effects of the Reynolds number (Re_o), internal to external fluid characteristic viscosity ratio (k) and power-law index (n_o) on the continuous phase flow field, pressure drag (C_dp), friction drag (C_df) and total drag (C_D) coefficients have been analyzed over the range of parameters: 5?Re_o?500, 0.1?k?50 and 0.6?n_o?1.6. Based on numerical results obtained in this work, a simple correlation has been proposed for the total drag coefficient, which can be used to predict the rate of sedimentation of a fluid sphere in power-law liquids.     

Drag on ensembles of fluid spheres translating in a power-law liquid at moderate Reynolds numbers

This work elucidates the role of power-law rheology on the sedimentation velocity of an ensemble of mono-size spherical Newtonian droplets (free from surfactants) translating in a power-law continuous phase numerically by solving the momentum equations of both phases. A simple sphere-in-sphere cell model has been used to account for inter-drop interactions. In particular, in this study, the effects of the Reynolds number (Re_o), the internal to external fluid characteristic viscosity ratio (k), the volume fraction of the dispersed phase (Φ) and the power-law index of the continuous phase (n_o) on the external flow field, pressure drag (C_dp), friction drag (C_df) and total drag (C_d) coefficients have been analyzed over wide ranges of parameters as follows: 1 ≤ Re_o ≤ 200, 0.1 ≤ k ≤ 50, 0.2 ≤ Φ ≤ 0.6 and 0.6 ≤ n_o ≤ 1.6. Based on the extensive numerical results obtained in this work, a simple predictive correlation has been proposed for the total drag coefficient, which can be used to predict the rate of sedimentation of ensembles of Newtonian fluid spheres in power-law liquids in a new application.     

Mass transfer from ensembles of fluid spheres to a power-law liquid at moderate Reynolds and Peclet numbers

The steady convective mass transfer from ensembles of mono-size Newtonian fluid spheres to power-law liquids has been studied at moderate Reynolds and Peclet numbers. The species continuity equation segregated from momentum equations has been solved numerically using a finite difference method. A simple cell model has been used to account for the modification of the flow field due to the neighbouring droplets. Extensive numerical results have been obtained which elucidate effects of the Reynolds number (Re_o), Schmidt number (Sc), power-law index (n_o), internal to external fluid characteristic viscosity ratio (k) and the volume fraction of the dispersed phase (Φ) on the rate of mass transfer. The ranges of parameters considered herein are: 1?Re_o?200, 1?Sc?10000, 0.6?n_o?1.6, 0.1?k?50 and 0.2?Φ?0.6. For shear-thinning fluids (n_o<1), the rate of mass transfer is somewhat enhanced whereas for shear-thickening fluids (n_o>1), it decreased as compared to that in Newtonian fluids (n_o=1). A simple predictive correlation has been proposed which can be used to estimate the rate of mass transfer in liquid-liquid systems in a new application involving power-law continuous phase.     

Mass transfer from a contaminated fluid sphere

The unsteady mass transfer from a contaminated fluid sphere moving in an unbounded fluid is examined numerically for unsteady‐state transfer. The effect of the interface contamination and the flow regime on the concentration profiles, inside and outside a fluid sphere, is investigated for different ranges of Reynolds number (0 < Re < 200) and Peclet number (0 < Pe < 10⁵), viscosity ratio between the dispersed phase and the continuous phase (0 < κ < 10), and the stagnant‐cap angle (0° < θ_cap < 180°). It was found that the stagnant‐cap angle significantly influences the mass transfer from the sphere to a surrounding medium. For all Peclet and Reynolds numbers and κ, the contamination reduces the mass transfer flux. The average Sherwood number increases with an increase of stagnant‐cap angle and reaches a maximum equal to the average one for a clean fluid sphere at low viscosity ratio and large Peclet numbers. A predictive equation for the Sherwood number is derived from these numerical results. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

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     

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.     

A drag correlation of fluid particles rising through stagnant liquids in vertical pipes at intermediate Reynolds numbers

A drag correlation for a fluid particle rising along the axis of a vertical pipe at low and intermediate Reynolds numbers, Re, is proposed by making use of available correlations and a numerical database accumulated by interface tracking simulations. The accuracy of the interface tracking method has been verified through comparisons between measured and predicted velocities of single drops in vertical pipes. Being similar to drag model for solid spheres proposed by Michaelides, the developed drag correlation takes into account inertial and wall effects as their linear combination. The correlation gives good estimation of the drag coefficient for fluid particles rising through stagnant liquids in vertical pipes under the conditions of 0.13?Eo?30, −10.0?log M?2.0, 0.083?Re<200, 0?κ?10.0 and λ?0.6, where Eo is the Eötvös number, M the Morton number, κ the viscosity ratio and λ the ratio of particle diameter to pipe diameter.     

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     

Effect of blockage on heat transfer from a cylinder to power law liquids

The influence of planar confining walls on the steady forced convection heat transfer from a cylinder to power-law fluids has been investigated numerically by solving the field equations using FLUENT (version 6.2). Extensive results highlighting the effects of the Reynolds number (1?Re?40), power-law index (0.2?n?1.8), Prandtl number (1?Pr?100) and the blockage ratio (β=4 and 1.6) on the average Nusselt number have been presented. For a fixed value of the blockage ratio, the heat transfer is enhanced with the increasing degree of shear-thinning behaviour of the fluid, while an opposite trend was observed in shear-thickening fluids. Due to the modifications of the flow and temperature fields close to the cylinder, the closely placed walls (i.e., decreasing value of the blockage ratio) further enhance the rate of heat transfer as the fluid behaviour changes from Newtonian to shear-thickening fluids (n>1), the opposite influence is seen with the decreasing value of the flow behaviour index (n) in shear-thinning (n<1) fluids. Finally, the functional dependence of the present numerical results on the relevant dimensionless parameters has been presented in the form of closure relationships for their easy use in a new application.     

Air pollutant absorption by single moving droplets with drag force at moderate Reynolds numbers

Mass transfer phenomena of hydrogen chloride around single water droplets at moderate initial Reynolds numbers are investigated to simulate air pollutant absorption by droplets in wet scrubbers. Of particular interest is the uptake mechanism in the droplet under the impact of deceleration. An examination of the mass transfer inside the droplet, in view of the solute transport delay from the gas-liquid interface to the droplet interior, a maximum distribution in concentration difference between the droplet surface and the internal minimum concentration is exhibited. Meanwhile, gaseous scavenging behavior is apparently characterized by the droplet. Regarding the effect of the decelerating motion, the predictions reveal that the variation of the droplet velocity due to drag force is faster than that of the uptake process. Therefore, the absorption rates of the decelerating droplet are substantially decreased when compared with that of a droplet without drag force. As a whole, increasing initial Reynolds number causes faster decay in the droplet velocity which further reduces the mass transfer rate in the aqueous phase. This suggests that the larger the initial Reynolds number, the more significant the absorption rate of the droplet affected by the drag force.     

Slip in Flow through Assemblages of Spherical Particles at Low to Moderate Reynolds Numbers

Effects of slip velocity and volume fraction of slip spheres on the momentum transfer characteristics of assemblages of slip spheres are numerically investigated. The fluid slip along the surface of the sphere is considered by Navier's linear slip model. The dimensionless governing continuity and momentum equations are solved using a semi‐implicit marker and cell method implemented on a staggered grid arrangement in spherical coordinates. The convection and viscous terms of momentum equations are discretized by means of the QUICK scheme and a second‐order central differencing scheme, respectively. The present numerical solver is benchmarked via grid independence and comparisons with the existing literature values. Results were obtained over a wide range of pertinent dimensionless numbers such as the Reynolds number, volume fraction of the dispersed phase, and dimensionless slip parameter.     

Hydrodynamics of inhomogeneous agglomerates for a fractal dimension of 2.5 at intermediate Reynolds numbers: Effect of agglomerate sizes and Reynolds numbers

In this study, the effect of agglomerate sizes for a fractal dimension (D_f) of 2.5 on the hydrodynamics at intermediate Reynolds numbers (Re) of 1–120 was assessed. The results show that a core behaves like a solid sphere that exists in the central region inside the agglomerate. In addition, increasing the agglomerate diameter represents adding an extra permeable layer outside the agglomerate. For a larger Re or a smaller agglomerate diameter, the fluid can enter and penetrate through the agglomerate more easily, and the hydrodynamic characteristics of agglomerates deviate more from those of solid spheres. The effect of diameters on the velocity and pressure profiles becomes less significant with the increase in the diameter. Based on the simulated results, the drag ratio has an approximately linear relationship with Re, and its intercept has an exponential relationship with the dimensionless agglomerate diameter. Compared with homogeneous porous spheres, the drag ratio of the agglomerate is different. The effect of diameters on the drag ratio decreases as the diameter increases. It should be noted that the effect of radially varying permeability on inhomogeneous agglomerates should not be ignored and that the effect weakens as Re increases.     

Steady incompressible fluid flow over a bundle of cylinders at moderate reynolds numbers

The complete Navier-Stokes equations describing the steady flow of incompressible Newtonian fluids normal to an array of long cylinders have been solved numerically using the finite difference method in terms of the stream function and vorticity variables. The inter-cylinder interactions have been mimicked using the well known free surface cell model. Extensive information on the detailed structure of the flow field in terms of the surface vorticity distribution, stagnation pressure, stream line and iso-vorticity line plots, as well as on the values of the integral quantities, such as pressure, friction and total drag coefficients, have been obtained under wide ranges of conditions as follows: 0.3 ≤ ? 0.99 and 0.01 ≤ Re ≤ 100. The numerical results presented herein have been validated using the appropriate theoretical and experimental results available in the literature; the match between the present predictions and the scant experimental results is good.     

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.     

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.     

Mass transfer to a drop stream from a field liquid

Mass transfer from a stream of drops falling freely in a stagnant liquid was investigated. Drop streams were produced by a dripping method and by a jet breakup method. Water and isobutanol, mutually saturated, were used as the dispersed and the continuous phases. Sodium hydroxide was transferred from isobutanol to water drops which were initially free of solute. The mass transfer resistance is on the continuous phase side. The mass transfer coefficient and terminal velocity of drop streams were measured experimentally. The experimental results show that the mass transfer coefficient in the drop stream is affected by the shielding effect of the previous drops. The experimental data have been correlated as K_t/U_t^0.5 versus interdrop distance l, a relationship describing the effect of the interdrop distance on the mass transfer coefficient in the continuous phase.     

Effect of blockage on drag and heat transfer from a single sphere and an in-line array of three spheres

The effect of blockage ratio on the steady flow and heat transfer characteristics of incompressible fluid over a sphere and an in-line array of three spheres placed at the axis of a tube has been investigated numerically. The Navier-Stokes and thermal energy equations have been solved numerically using FLUENT for the following ranges of parameters: for a single sphere, 2 ≤ β ≤ 10; 1 ≤ Re ≤ 100; for the three-sphere system, for two values of sphere-to-sphere distance, namely s = 2 and 4. All computations were carried out for two values of the Prandtl number, i.e., 0.74 and 7, corresponding to the flow of air and water respectively. Extensive results on streamline patterns, wake characteristics (angle of separation and recirculation length), drag coefficient and Nusselt number are presented to elucidate the interplay between the blockage and the Reynolds number and their influence on drag and Nusselt number.     

Mass transfer from a freely moving single sphere to the dense phase of a gas fluidized bed of inert particles

Naphthalene spheres (particle diameter 2 <d_n < 20 mm) were vaporized in beds fluidized by air at a temperature of about 65°C. The bed material consisted of inert glass beads or alumina in the size range 100 <d_p < 700 μm. Mass transfer coefficients were measured by determining weight loss with time, as a function of d_n, d_p and the fluidization velocity U. The ratio d_n/d_p has been varied from 3 to 200. An interesting conclusion might be that there is no influence of the fluidization velocity on these transfer coefficients; they only depend on the minimum fluidization velocity U_mf. The empirical correlation ϵ_mf(j_D)_mfRe_mf^m=0.105 + 1.505 (d_n/d_p)^−1.05 with m = 0.35 + 0.29 (d_n/d_p)^−0.50, as a best fit of all the results, is accurate within 15%. At d_n/d_p = 1 it links up very well with the results of Hsiung and Thodos. For large values of d_n/d_p agreement with known results of mass transfer measurements between a fluidized bed and a wall or an object is good.     

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 particle suspended in a nonlinear uniaxial extensional creeping flow.The mass transfer problem is governed by three dimensionless parameters:the viscosity ratio(λ),the Peclet number(Pe),and the nonlinear intensity of the flow(E).The existing mass transfer theory,valid for very large Peclet numbers only,is expanded,by numerical simulations,to include a much larger range of Peclet numbers(1 ≤ Pe ≤ 10