Diffusiophoresis of a colloidal sphere in nonelectrolyte gradients parallel to one or two plane walls

The quasisteady diffusiophoretic motion of a spherical particle in a fluid solution of a nonionic solute located between two infinite parallel plane walls is studied theoretically in the absence of fluid inertia and solute convection. The imposed solute concentration gradient is constant and parallel to the two plane walls, which may be either impermeable to the solute molecules or prescribed with the far-field concentration distribution. The particle-solute interaction layer at the particle surface is assumed to be thin relative to the particle radius and to the particle-wall gap widths, but the polarization effect of the diffuse solute in the thin interfacial layer caused by the strong adsorption of the solute is incorporated. The presence of the neighboring walls causes two basic effects on the particle velocity: first, the local solute concentration gradient on the particle surface is enhanced or reduced by the walls, thereby speeding up or slowing down the particle; secondly, the walls increase viscous retardation of the moving particle. To solve the continuity and momentum equations, the general solutions are constructed from the fundamental solutions in both the rectangular and the spherical coordinate systems. The boundary conditions are enforced first at the plane walls by the Fourier transforms and then on the particle surface by a collocation technique. Numerical results for the diffusiophoretic velocity of the particle relative to that under identical conditions in an unbounded fluid solution are presented for various values of the relaxation parameter of the particle as well as the relative separation distances between the particle and the two plates. For the special case of diffusiophoretic motions of a spherical particle parallel to a single plate and on the central plane of a slit, the collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the lateral walls can reduce or enhance the particle velocity, depending on the surface properties of the particle, the relative particle-wall separation distances, and the solutal boundary condition at the walls. In general, the boundary effect on diffusiophoresis is quite complicated and relatively weak in comparison with that on sedimentation.     

Diffusiophoresis of a colloidal sphere in nonelectrolyte gradients perpendicular to two plane walls

The problem of the diffusiophoretic motion of a spherical particle in a fluid solution of a nonionic solute situated at an arbitrary position between two infinite parallel plane walls is studied theoretically in the quasisteady limit of negligible Peclet and Reynolds numbers. The applied solute concentration gradient is uniform and perpendicular to the plane walls. The particle-solute interaction layer at the particle surface is assumed to be thin relative to the particle radius and to the particle-wall gap widths, but the polarization effect of the diffuse solute in the thin interfacial layer caused by the strong adsorption of the solute is incorporated. The presence of the walls causes two basic effects on the particle velocity: first, the local solute concentration gradient on the particle surface is altered by the walls, thereby speeding up or slowing down the moving particle; second, the walls enhance the viscous retardation of the particle. A boundary-collocation method is used to semianalytically solve the solutal and hydrodynamic governing equations of the system. Numerical results for the diffusiophoretic velocity of the particle relative to that under identical conditions in an unbounded fluid solution are presented for various cases. The collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The net effect of the confining walls is always to reduce the particle velocity, irrespective of the surface properties of the particle or the relative particle-wall separation distances. The boundary effect on diffusiophoresis of a particle normal to two plane walls is found to be quite significant and generally stronger than that parallel to the walls.     

Boundary effects on osmophoresis: Motion of a spherical vesicle perpendicular to two plane walls

The problem of the osmophoretic motion of a spherical vesicle situated at an arbitrary position between two infinite parallel plane walls is studied theoretically in the quasisteady limit of negligible Peclet and Reynolds numbers. The imposed solute concentration gradient is uniform and perpendicular to the plane walls. The presence of the confining walls causes two basic effects on the vesicle velocity: first, the local concentrations on both sides of the vesicle surface are altered by the walls, thereby speeding up or slowing down the vesicle; secondly, the walls enhance the viscous interaction effect on the moving vesicle. To solve the equations of conservation of mass and momentum, the general solutions are constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinates. The boundary conditions are enforced first at the plane walls by the Hankel transform and then on the vesicle surface by a collocation technique. Numerical results for the osmophoretic velocity of the vesicle relative to that under identical conditions in an unbounded solution are presented for various values of the relevant properties of the vesicle-solution system as well as the relative separation distances between the vesicle and the plane walls. The collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the neighboring walls will enhance the vesicle velocity, but its dependence on the relative vesicle-wall separation distances is not necessarily monotonic. The boundary effect on osmophoresis of a vesicle normal to two plane walls is found to be significant and stronger than that parallel to the confining walls.     

Boundary effects on osmophoresis: motion of a spherical vesicle parallel to two plane walls

A theoretical study is presented for the quasisteady osmophoretic motion of a spherical vesicle in a solution located between two infinite parallel plane walls in the limit of negligible Reynolds and Peclet numbers. The applied solute concentration gradient is uniform and parallel to the two plane walls, which may be either impermeable to the solute molecules or prescribed with the far-field concentration distribution. The presence of the neighboring walls causes two basic effects on the vesicle velocity: first, the local concentrations on both sides of the vesicle surface are altered by the walls, thereby speeding up or slowing down the vesicle; secondly, the walls enhance the viscous interaction effect on the moving vesicle. To solve the equations of conservation of mass and momentum, the general solutions are constructed from the fundamental solutions in both the rectangular and the spherical coordinate systems. The boundary conditions are enforced first at the plane walls by the Fourier transforms and then on the vesicle surface by a collocation technique. Numerical results for the osmophoretic velocity of the vesicle relative to that under identical conditions in an unbounded solution are presented for various values of the relevant properties of the vesicle as well as the relative separation distances between the vesicle and the two plates. For the special case of osmophoretic motions of a spherical vesicle parallel to a single plate and on the central plane of a slit, the collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the lateral walls can reduce or enhance the vesicle velocity, depending upon the relevant properties of the vesicle, the relative vesicle-wall separation distances, and the solutal boundary condition at the walls. In general, the boundary effect on osmophoresis is quite complicated in comparison with that on sedimentation.     

Thermophoretic Motion of a Spherical Aerosol Particle in a Cylindrical Pore

The thermophoretic motion of a spherical aerosol particle in a cylindrical pore, with the ambient temperature gradient imposed parallel with the pore direction, is investigated. Both particle and pore surfaces can have frictional and thermal slip, and discontinuity in temperature fields across both surfaces is allowed. The relevant boundary value problem is solved with a truncated-domain boundary collocation method. It is found that the thermo-osmotic flow of the surrounding fluid caused by the thermal slippage of the pore wall plays a dominant role in determining the thermophoretic motion of the particle. This thermo-osmotic flow is directed toward the hotter region and thus leads to a toward-hot-region thermophoretic motion of the particle, which is opposite to the usual toward-cold-region particle thermophoretic motion. The effect of this thermo-osmotic flow toward particle thermophoretic motion is quite different for the particle in a closed cavity situation. For the particle in a closed cavity scenario, the thermo-osmotic flow is toward the hotter region along the cavity wall, but has to circulate back in the middle of the cavity and thus helps to push the particle toward colder region.     

Diffusiophoresis of a soft spherical particle along the axis of a cylindrical microchannel

The diffusiophoresis of a soft particle is modeled theoretically by considering a soft sphere moving along the axis of a cylindrical microchannel. This geometry allows us to examine simultaneously the boundary effect and the nature of a particle on its diffusiophoretic behavior. The soft particle, which comprises a rigid core and an ion-penetrable layer, is capable of simulating biocolloids such as cells and particles covered by an artificial membrane layer. The results of numerical simulation reveal that due to its specific structure, the diffusiophoretic behavior of a soft particle is quite different from that of a rigid particle. The influence of the cylindrical microchannel on the diffusiophoretic behavior of the particle is also very different from that of other geometries considered in the literature. We show that, in addition to the effect of double-layer polarization, the effect of electrophoresis, and the electrical interaction between the coions outside the double layer and the particle, the nature of the soft particle can also influence both quantitatively and qualitatively its diffusiophoretic behavior. Several interesting results are observed, providing valuable reference for both the design of a diffusiophoresis device and the interpretation of the relevant experimental data.     

Thermocapillary motion of a fluid droplet perpendicular to two plane walls

The quasisteady problem of the thermocapillary migration of a spherical fluid droplet situated at an arbitrary position between two infinite parallel plane walls is studied theoretically in the limit of negligible Marangoni and Reynolds numbers. The applied temperature gradient is constant and perpendicular to the plane walls. The presence of the plane walls causes two basic effects on the droplet velocity: first, the local temperature gradient on the droplet surface is altered by the walls, thereby speeding up or slowing down the droplet; secondly, the walls increase viscous retardation of the moving droplet. To solve the thermal and hydrodynamic governing equations, the general solutions are constructed from the fundamental solutions in both cylindrical and spherical coordinates. The boundary conditions are enforced first at the plane walls by the Hankel transforms and then on the droplet surface by a collocation technique. Numerical results for the thermocapillary migration velocity of the droplet relative to that under identical conditions in an unbounded medium are presented for various values of the relative viscosity and thermal conductivity of the droplet as well as the relative separation distances between the droplet and the confining walls. The collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the walls always reduces the droplet velocity, irrespective of the relative transport properties of the droplet or the relative droplet-wall separation distances. The boundary effect on thermocapillary migration of a droplet normal to two plane walls, which is relatively weak in comparison with the corresponding effect on sedimentation, is found to be quite significant and generally stronger than that parallel to the plane walls.     

Investigating the dynamics of cylindrical particles in a rotating drum using multiple radioactive particle tracking

The behavior of granular flows inside rotating drums is an ongoing area of research. Only a few studies have investigated non‐spherical particles despite the fact that particle shape is known to have a significant impact on flow behavior. In addition, the experimental techniques limit the interpretation of the results of these studies. In this work, we compared the flow behavior of cylindrical and spherical particles using the multiple radioactive particle tracking technique to capture the positions and orientations of cylindrical particles simultaneously. We analyzed two important components of the transverse flow dynamics, that is, the boundary between the active and passive layers, and the velocity profile on the free surface. For the cylindrical particles, two general models are proposed to calculate the velocity profiles on the free surface and the effective particle sizes in the active and passive layers. © 2016 American Institute of Chemical Engineers AIChE J, 62: 2622–2634, 2016     

柱状颗粒沉降过程的CFD-DEM联合仿真和实验研究

针对柱状催化剂颗粒相对于球形颗粒的不同运动特性,选择不同长度直径为2 mm的5种柱状颗粒,采用CFD-DEM数值模拟仿真,研究柱状颗粒在管状容器中沉降的运动行为,并建立柱状颗粒沉降试验台,采用高速摄像拍摄的方法进行实验研究。结果表明,在不同位置释放相同直径和长度的柱状颗粒时,靠近壁面释放的颗粒会在沉降过程中向中心漂移,且比中心释放的颗粒沉降更慢,时间更长;改变柱状颗粒与水平面的夹角,在圆管中心释放颗粒,最终颗粒都会旋转至水平状态,与水平面夹角越大,底部所受阻力越大,转动持续时间随之增加;推导柱状颗粒沉降斯托克斯方程,并通过实验数据对方程中的阻力系数进行修正,将修正后的阻力系数导入用户自定义函数(UDF)计算颗粒沉降末速度,相对误差从原来使用球形颗粒阻力系数的50%下降到17%以内,模拟较为可靠。     

CHEMICALLY INDUCED MIGRATION OF PARTICLES ACROSS FLUID STREAMLINES

The velocity of a colloidal particle that moves because of a gradient of concentration of a molecular solute depends on the concentration field at the surface of the particle. Effects of macroscopic convection of the suspending fluid on two such transport phenomena, capillary-driven movement of fluid particles and diffusiophoresis of rigid particles, are considered here. In the case of fluid particles our results also apply to motion caused by a temperature gradient. If the particles are in a laminar flow with the solute gradient directed perpendicular to the direction of flow, as might arise in the boundary layer near a surface to which the particles are being deposited, the local solute concentration field around each particle is disturbed from that of pure diffusion of the solute. Using published results for these concentration disturbances in a simple-shear flow, we determine the effect of the imposed velocity gradient on the speed of the particles in the direction of the solute gradient. For both fluid and rigid particles, the correction due to macroscopic shear is 0(Pe^3/2:) where Pe is the Peclet number based on particle radius and fluid shear rate; this effect opposes the zero-shear particle velocity. A possible consequence of this result is that by increasing the shear rate in a laminar boundary layer in the hope of enhancing the rate of particle adsorption, one may actually be decreasing the rate.     

A hydrodynamic approach to particle target efficiency in the presence of diffusiophoresis

An alternate approach to estimating the diffusiophoretic target efficiency of small particles by spherical collectors is presented. The vapor diffusion flux affecting the particle motion is incorporated into the fluid velocity field using different source terms for viscous and potential flow about the sphere. The calculated target efficiencies depend on the respective source terms, in addition to the Stokes number, collector Reynolds number, and interception parameter. The calculated diffusiophoretic target efficiency decreases for an evaporating drop and increases for a condensing drop. These effects become more pronounced as the source term becomes more positive or more negative. In the latter case, the target efficiency can exceed unity for small Stokes number and strong diffusion fluxes toward the collector surface.     

Steady‐State Modeling and Simulation of an Axial‐Radial Ammonia Synthesis Reactor

A two‐dimensional model was developed for an axial‐radial ammonia synthesis reactor of the Shiraz petrochemical plant. In this model, momentum and continuity equations as well as mass and energy balance equations are solved simultaneously by orthogonal collocation on the finite element method to obtain pressure, velocity, concentration and temperature profiles in both axial and radial directions. For the catalyst particle, the effectiveness factor is calculated by solving a two‐point boundary value differential equation. The boundary conditions for the Navier‐Stokes and continuity equations are obtained by using equations representing the phenomena of gases splitting or joining in different streams and going through holes in a thin wall. The results of the mathematical model have been compared with the plant data and a good agreement is obtained.     

Boundary effects on thermophoresis of aerosol cylinders

An analytical study is presented for the thermophoretic motion of a circular cylindrical particle in a gaseous medium with a transversely imposed temperature gradient near a large plane wall parallel to its axis in the quasisteady limit of negligible Peclet and Reynolds numbers. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model with a temperature jump, a thermal slip, and a frictional slip at the particle surface. The presence of the confining wall causes two basic effects on the particle velocity: first, the local temperature gradient on the particle surface is altered by the wall, thereby speeding up or slowing down the particle; secondly, the wall enhance the viscous retardation of the moving particle. Through the use of cylindrical bipolar coordinates, the transport equations governing this problem are solved and the wall effects on the thermophoresis of the aerosol cylinder are computed for various cases. The presence of the plane wall can reduce or enhance the particle velocity, depending upon the relative thermal conductivity and surface properties of the particle, the relative particle-wall separation distance, and the direction of the applied temperature gradient. The direction of the thermophoretic motion of a cylindrical particle near a plane wall is different from that of the prescribed thermal gradient, except when it is oriented parallel or perpendicular to the wall. The effects of the plane wall on the thermophoresis of a cylinder are found to be much more significant than those for a sphere at the same separation.     

Mathematical simulation of bioseparation in an affinity packed column

Affinity chromatography (biospecific adsorption) relies on specific interactions of biological molecules such as enzymes, antigens, antibodies, and proteins. The process consists of three steps: adsorption, washing, and elution. A mathematical model including convection, diffusion, and reversible reaction is formulated to analyse the breakthrough behaviour of the solute. A moving finite element orthogonal collocation method is applied with respect to the space variables of the governing partial differential equations of the model to evaluate the breakthrough of the solute. Danckwerts' boundary conditions are considered for the column. The validity of the numerical scheme is checked by comparison with an analytical solution for a simplified model. The results obtained from model simulation show that the breakthrough time of the solute is significantly influenced by the axial dispersion coefficient, solute concentration, ligand content, reaction kinetics, particle porosity, particle size, and flow rate. Solute recovery and bed utilisation efficiencies are evaluated for different values of the above parameters.     

Micromechanics of formation and shrinkage of a closed pore in sintering by coupled grain boundary/surface diffusion

The microscopic principle of the stress-assisted sintering is that the relative velocity between two adjoining particles is proportional to the sum of the sintering force and the mechanical force transmitted by the contact. Here, we simulated sintering of four particles by coupled grain boundary diffusion and surface diffusion, in order to analyze how the sintering force varies with the evolution of particle shape, i.e., pinch-off of pore channel, formation and shrinkage of a closed pore. The shrinkage rate of the pore volume was proportional to the relative velocity of particles, then, to the sintering force. We discussed the effect of mechanical stress on sintering also.     

SOLUTE REJECTION IN THE ULTRAFILTRATION OF POLYDISPERSE ORGANICS FROM NATURAL PRODUCTS

Solute rejection in the ultrafiltration of solutions containing polydisperse solutes was modeled using a spherical solute/cylindrical capillary model, accounting for steric hindrance and wall drag effects. A power law relationship was used for the solute radius-molecular weight relationship. The three parameters in the nondimensional model are the ratio of mean pore radius/solute radius coefficient, the exponent in the solute radius versus molecular weight relationship, and the standard deviation of the logarithms of the pore diameters. Values of the parameters, obtained by fitting the model to rejection coefficient data for solutions containing dissolved organics from wood, were self-consistent and made physical sense. The model provides a useful tool for evaluating ultrafiltration membranes for specific solute fractionation applications.     

Thermophoresis of axisymmetric aerosol particles along their axes of revolution

The axisymmetric thermophoretic motion of an aerosol particle of revolution in a uniformly prescribed temperature gradient is studied theoretically. The Knudsen number is assumed to be small so that the fluid flow is described by a continuum model. A method of distribution of a set of spherical singularities along the axis of revolution within a prolate particle or on the fundamental plane within an oblate particle is used to find the general solutions for the temperature distribution and fluid velocity field. The jump/slip conditions on the particle surface are satisfied by applying a boundary‐collocation technique to these general solutions. Numerical results for the thermophoretic velocity of the particle are obtained with good convergence behavior for various cases. For the axisymmetric thermophoresis of an aerosol spheroid with no temperature jump and frictional slip at its surface, the agreement between our results and the available analytical solutions is very good. The thermophoretic velocity of a spheroid along its axis of revolution in general increases with an increase in its axial‐to‐radial aspect ratio, but there are exceptions. For most practical cases of a spheroid with a specified aspect ratio, its thermophoretic mobility is not a monotonic function of its relative jump/slip coefficients and thermal conductivity. © 2008 American Institute of Chemical Engineers AIChE J, 2009     

Solution to the homogeneous surface diffusion model for batch adsorption systems using orthogonal collocation

A solution to the homogeneous surface diffusion model has been developed and incorporated into a batch adsorption model based on external boundary layer mass transport and homogeneous diffusion. The model has been extensively tested using three experimental adsorption systems, namely, phenol on carbon, basic yellow dye on carbon and basic blue dye on silica. The effect of initial solute concentration and adsorbent mass has been studied in 23 batch experiments, which have been modelled using the collocation solution method to solve the homogeneous surface diffusion equation. The theoretical concentration decay curves show a high degree of correlation with experimental data.     

Probabilistic model of dispersed turbulent flow in channels with rough walls

Abstract

An equation for the probability density function (PDF) for particle velocity and coordinates in a gas turbulent flow is derived. The system of equations for the first and second moments of particle velocity fluctuations is obtained. Using a method similar to Grad’s method, an approximate solution of the PDF equation was found. Based on this approximate solution, the system of equations for the averaged concentration, velocity, and second moments of particle velocity fluctuations was closed. Also, using an approximate solution, the boundary conditions on the rough wall of the channel were obtained. The boundary conditions self-consistently take into account the direction of the velocity vector of particles colliding with the surface, as well as the direction of the normal to a random plane that simulates the roughness. The fundamental difference between the mechanisms of generation of random motion of particles in channels with smooth and rough walls is shown.

Copyright © 2020 American Association for Aerosol Research     

Modeling of the Ultrafiltration of a Dextran T500 Solution in a Tubular Membrane Module

Mathematical modeling of an ultrafiltration membrane separation process, based mainly on the transmembrane pressure (TMP), is undertaken in the present work. The main objective is the prediction of the permeate flux of a solution containing Dextran T500 through a cylindrical module. The proposed model incorporates the resistance‐in‐series model coupled with the equation describing the solute (Dextran T500) transport, as well as the continuity and Navier‐Stokes equations for solution flow modeling. The model equations are solved using finite‐volume numerical methods, with appropriate initial and boundary conditions. The effects of the TMP and the length of the membrane on the mean permeate flux were also investigated. The influence of the membrane dimensions (aspect ratio) on the relative dimensionless mean permeate flux, at different inlet TMPs and different solution concentrations, respectively, have also been considered. The variations of the TMP with the membrane length as well as the influence of the Peclet number on the solute surface concentration were also examined. The numerical results obtained are compared with experimental values reported in the literature, and in general, the agreement is satisfactory enough to encourage further refinement of the model.