Shear-induced Diffusion Research Articles

Modeling of transient permeate flux in cross-flow membrane filtration incorporating multiple particle transport mechanisms

Dominant mechanisms of particle transport in cross-flow membrane filtration are unified to obtain a generalized model for time-dependent permeate flux. The unified model extends an earlier model based on shear-induced diffusion and a concentrated flowing layer to include Brownian diffusion and inertial lift. It is applicable over a broad range of contaminant sizes encompassing macromolecules, colloidal and fine particles, and large particles. The combined theory predicts an unfavorable particle size, of the order of 10 −1 μm, where the net back-transport away from the membrane attains a minimum, leading to maximum cake growth. For the system simulated in this work, this implies minimum permeate fluxes in the size range of 0.01–0.1 μm, depending on the operating time. Inside-out hollow-fiber geometry is predicted to be favorable for feed suspensions with small particles and/or low concentrations which form thin resistive cakes. However, larger particles, which form thick cakes, may result in reduced surface area available for filtration due to curvature effects in inside-out membranes, making the slit or outside-in geometry more favorable for these particles. Fine particles (< 0.1 μm) are predicted to demonstrate mass-transport limited behavior. For larger particles, different combinations of fiber radius and cross-flow velocity, resulting in the same shear rate, demonstrate different permeate fluxes.

Predicting limiting flux of skim milk in crossflow microfiltration

Four models for back-transport mechanisms in crossflow microfiltration have been investigated concerning their ability to predict the limiting permeate flux for skim milk. A tubular, ceramic membrane was used to measure the limiting fluxes for a series of crossflow velocities at two temperatures. One of the models — the shear-induced diffusion model — predicts values of the limiting flux close to our experimental values both at 55 and 15°C. The best prediction of the limiting flux is obtained by the empirical relation: flux=Re · 6.94×10 −10 m/s.

A numerical model of steady-state permeate flux during cross-flow ultrafiltration

The mass transport mechanisms during cross-flow ultrafiltration (UF) are mathematically expressed using the two-dimensional convective diffusion equation where the axial diffusion term is neglected for an axial Peclet number much greater than the transverse Peclet number. A numerical scheme is presented to solve the steady-state two-dimensional convective diffusion equation for the case of known uniform permeate flux. However, in the actual cross-flow UF process, the permeate flux along the axial direction is unknown and usually decreases with axial distance. Therefore, an iterative algorithm is developed to predict the steady-state permeate flux based on the assumption that the concentration at membrane surface cannot exceed a certain limiting value. Using the numerical model with an effective diffusion coefficient, which is considered to be the sum of molecular diffusion and shear-induced hydrodynamic diffusion coefficients, the effects of particle size, feed concentration, and axial velocity on the steady-state permeate flux were investigated.

Particle Back-Transport and Permeate Flux Behavior in Crossflow Membrane Filters

Particle residence time distributions in a membrane channel are interpreted to elucidate mechanisms of particle transport and colloidal fouling in membrane filtration. A comparison of particle size distributions in the membrane feed suspensions and deposited cakes provides evidence for selective particle transport and accumulation on membranes. These data support a previously hypothesized minimum in particle back-transport from the membrane as a function of particle size. The back-transport of smaller particles is apparently due to Brownian diffusion, while larger macrocolloids are controlled by an orthokinetic mechanism such as shear-induced diffusion. In all cases, cake specific resistances measured in the dead-end mode were higher than those of the corresponding feed suspensions. Also, cake specific resistances measured under a crossflow were higher than those in the dead-end mode. Further, the specific resistance of particle deposits on membranes increased with shear rate and decreased as the initial ...

Rheology of “Wet” Polymer Brushes via Brownian Dynamics Simulation: Steady vs Oscillatory Shear

The rheology of solvent-filled polymer brushes under steady and oscillatory shear is studied by Brownian dynamics simulation. The flow field is solved self-consistently with the polymer dynamics. We find that under steady flow the brushes are shear thinning in both normal forces and shear viscosity. During oscillatory flow, extreme shear thickening occurs when the ratio of the Weissenberg number to the dimensonless flow frequency $\mathrm{Wi}/\ensuremath{\omega}\ensuremath{\approx}3$. Shear-induced diffusion is shown to create this thickening. These findings serve to explain the recent measurements made by Klein et al. [Nature 352, 143 (1991); 370, 634 (1994); Discuss Faraday Soc. 98, 173 (1994)].

The transverse shear-induced liquid and particle tracer diffusivities of a dilute suspension of spheres undergoing a simple shear flow

We study the shear-induced self-diffusion of both a liquid tracer and a tagged spherical particle along the directions perpendicular to the ambient flow in a dilute suspension of neutrally buoyant spheres undergoing a simple shearing motion in the absence of inertia and Brownian motion effects. The calculation of the liquid diffusivity requires the velocity of a fluid point under the influence of two spheres, which was determined via Lamb's series expansion; conversely, the calculation of the particle diffusivity involves the trajectories of three spheres, which were determined using far-field and near-field asymptotic expressions. The displacements of the liquid tracer and of the tagged sphere were then computed analytically when the spheres and the tracer are all far apart, and numerically for close encounters. After summing over all possible encounters, the leading terms of the lateral liquid diffusion coefficients, both within and normal to the plane of shear, were thereby found to be 0.12γac and 0.004γac, respectively, where γ is the applied shear rate, a the radius of the spheres and c their volume fraction. The analogous coefficients of the lateral particle diffusivity were found to be 0.11γac, and 0.005γac, respectively. Also, liquid and particle diffusivities in a monolayer, with the liquid tracer and all the particle centres lying on the same plane of shear, were found to be 0.067γyac, and 0.032γac, respectively, with c denoting the areal fraction occupied by the spheres on the plane.

Steady suspension flows into two-dimensional horizontal and inclined channels

In this study a model which was developed previously is used to theoretically investigate the steady flow of a particulate suspension into two-dimensional horizontal and inclined channels. The continuity equation for the fluid and the simplified two-dimensional Navier-Stokes equations are then solved together with a particle concentration equation. This latter equation is formulated by considering the balance between the particle flux due to gravity with the corresponding particle fluxes due to convection and shear-induced diffusion. The resulting coupled system of equations is solved numerically using a specialised finite-difference method. It is found, for the parameter range for flows of proppants in hydraulic fractures, that when the suspension enters the channel with a uniform velocity profile it almost instantaneously becomes parabolic. In addition, the effects of particle sedimentation are most dominant in the entrance region, but further downstream such effects are balanced as shear-induced particle diffusion becomes more important. It is also shown that the suspension flow depends critically on the choice of the parameters used, e.g. the ratio of the particle radius to the height of the channel.

Experimental measurement of shear-induced diffusion in suspensions using long time data

A method based upon Taylor dispersion theory is used to determine the shear-induced diffusion coefficient in concentrated suspensions. The experiments are performed in a cylindrical Couette device with a suspension consisting of polystyrene spheres in a density-matched solution of glycerin and water. A sequence of several hundred transit times for a single tagged sphere to complete successive orbits within the device is measured. The data are analyzed to compute the azimuthal Taylor dispersion coefficient from which the coefficient of shear-induced diffusivity is obtained. In our experiments the particle Reynolds numbers are 𝒪(10−1). The experimental results are compared to the existing measurements of the shear-induced diffusion coefficient obtained at lower particle Reynolds numbers and based upon short-time data. We find a shear-enhanced diffusion coefficient D⊥/γ̇a2=𝒪(0.1) for a volume fraction of φ≊0.25; this is comparable to existing results from previous low particle Reynolds number studies (ℛ&amp;lt;10−3).

Shear-induced dispersion in a dilute suspension of rough spheres

In the absence of Brownian motion, inertia and inter-particle forces, two smooth spheres collide in a simple shear flow in a reversible way returning to their initial streamlines. Because the minimum separation during the collision can be less than 10−4 of the radius, quite a small surface roughness can have a significant irreversible effect on the collision. We calculate the change between the initial and final streamlines caused by roughness. Repeated random collisions in a dilute suspension lead to a diffusion of the particles across the streamlines. We calculate the shear-induced diffusivity for both self-diffusion and down-gradient diffusion.

Performance and Cost Modeling of Ultrafiltration

Models describing permeate flux, rejection, and cost are coupled to evaluate the performance and cost of ultrafiltration as a function of raw-water quality. The model for permeate flux extends a previous model for colloidal fouling based on shear-induced diffusivity to include Brownian diffusion. Contaminant removal is modeled as mechanical sieving and molecular diameter is regressed against weight to describe removal of natural organic matter (NOM). Time-dependent permeate flux is considered in estimating operating times required to achieve a specified recovery. Costs are calculated as a function of particle-size distribution in the raw water. Particles with diameters on the order of 10 −1 μm display minimum diffusivities, which leads to maximum system costs with respect to particle size. Fine materials ( <0.5 μm), with high cake resistance, demonstrate pressure-independent permeate flux for conditions typical of hollow fiber ultrafiltration. In some cases, a minimum in system costs as a function of reco...

The vertical structure of concentration fluctuation statistics in plumes dispersing in the atmospheric surface layer

This paper describes a study of the vertical structure of concentration fluctuations in a neutrally buoyant plume from an elevated point source in slightly convective to moderately stable meteorological conditions at ranges of between 12.5 and 100 m for a range of source heights between 1 and 5 m. Observations were made of concentration fluctuations in a dispersing plume using a vertical array of sixteen very fast-response photoionization detectors placed at heights between 0.5 and 16 m. Vertical profiles of a number of concentration statistics were extracted, namely, mean concentration, fluctuation intensity, intermittency factor, peak-to-mean concentration ratio, mean dissipation rate of concentration variance, and various concentration time and length scales of dominant motions in the plume (e.g., integral macro-scale, in-plume mid-scale and Taylor micro-scale). The profiles revealed a similarity to corresponding crosswind profiles for a fully elevated plume, but showed greater and greater departure from the latter shapes once the plume had grown in the vertical so that its lower dege began to interact progressively more strongly with the ground. The evolution of the concentration probability density function at a fixed range, but with decreasing height from the ground, is similar to that obtained at a fixed height but with increasing distance from the source. Concentration power spectra obtained at different heights all had an extensive inertial-convective subrange spanning at least two decades in frequency, but spectra measured near the ground had a greater proportion of the total concentration variance in the lower frequencies (energetic subrange), with a correspondingly smaller proportion in the higher frequencies (inertial-convective subrange). It is believed that these effects result from the increased mean shear near the surface, and blocking by the surface. The effect of enhanced shear-induced molecular diffusion on concentration fluctuations is examined.

Forced convection and sedimentation past a flat plate

The steady laminar flow of a well-mixed suspension of monodisperse solid spheres, convected steadily past a horizontal flat plate and sedimenting under the action of gravity, is examined. It is shown that, in the limit as Re → ∞ and ∈ → 0, where Re is the bulk Reynolds number and ∈ is the ratio of the particle radius a to the characteristic length scale L, the analysis for determining the particle concentration profile has several aspects in common with that of obtaining the temperature profile in forced-convection heat transfer from a wall to a fluid stream moving at high Reynolds and Prandtl numbers. Specifically, it is found that the particle concentration remains uniform throughout the O(Re−1/2) thick Blasius boundary layer except for two O(∈2/3) thin regions on either side of the plate, where the concentration profile becomes non-uniform owing to the presence of shear-induced particle diffusion which balances the particle flux due to convection and sedimentation. The system of equations within this concentration boundary layer admits a similarity solution near the leading edge of the plate, according to which the particle concentration along the top surface of the plate increases from its value in the free stream by an amount proportional to X5/6, with X measuring the distance along the plate, and decreases in a similar fashion along the underside. But, unlike the case of gravity settling on an inclined plate in the absence of a bulk flow at infinity considered earlier (Nir &amp; Acrivos 1990), here the concentration profile remains continuous everywhere. For values of X beyond the region near the leading edge, the particle concentration profile is obtained through the numerical solution of the relevant equations. It is found that, as predicted from the similarity solution, there exists a value of X at which the particle concentration along the top side of the plate attains its maximum value ϕm and that, beyond this point, a stagnant sediment layer will form that grows steadily in time. This critical value of X is computed as a function of ϕs, the particle volume fraction in the free stream. In contrast, but again in conformity with the similarity solution, for values of X sufficiently far removed from the leading edge along the underside of the plate, a particle-free region is predicted to form adjacent to the plate. This model, with minor modifications, can be used to describe particle migration in other shear flows, as, for example, in the case of crossflow microfiltration.

Sedimentation and sediment flow in settling tanks with inclined walls

The flow of a sediment layer that forms on an inclined plate as a consequence of the steady sedimentation of spherical particles was investigated theoretically as well as experimentally. The theoretical analysis was based on the model proposed by Nir &amp; Acrivos (1990), modified to include shear-induced diffusion due to gradients in the shear stress as well as a slip velocity along the wall due to the finite size of the particles. The resulting set of partial differential equations, which is amenable to a similarity-type solution both near the leading edge as well as far downstream, was solved numerically using a finite difference scheme thereby yielding theoretical predictions for the particle concentration and velocity profiles, plus the local sediment layer thickness, all along the plate. In addition, a new experimental technique based on laser Doppler anemometry was developed and was used to measure the particle velocity profiles in the highly concentrated sediment layer as well as the corresponding slip coefficient which relates the slip velocity to the velocity gradient adjacent to a wall. The thickness profile of the sediment layer was also measured experimentally by means of video imaging. It was found that the experimental results thus obtained for the particle velocity profile and for the local sediment layer thickness were in very good agreement with the corresponding theoretical predictions especially considering that the latter did not make use of any adjustable parameters.

EFFECTS OF AXIAL PRESSURE DROP ON THE LENGTH-AVERAGED PERMEATE FLUX IN CROSSFLOW MICROFILTRATION

Abstract A hydrodynamic model is presented which predicts the dependence of the steady-state, length-averaged permeate flux on the longitudinal pressure drop during crossflow microfiltration in both flat slit and tubular channels. The integral approach described by Romero and Davis (1988) has been extended to predict the axial variation of the transmembrane pressure drop along with that of permeate flux and cake layer thickness. The mechanism of shear-induced diffusion is employed in the analysis, which is restricted to particles with diameters of approximately 0.5–20μm (Davis, 1992), but the procedures may be extended to other particle transport mechanisms. The model predictions have been compared to the corresponding values calculated using a constant transmembrane pressure drop set equal to the arithmetic mean of those at the channel entrance and exit. The simulation results show the axial pressure drop to have the most significant effect on the average, steady-state permeate flux predictions for long,...

Shear-induced particle diffusivity dyring crossflow microfiltration of Saccharomyces cerevisiae in a radial flow filtration chamber

A method for experimental observation of the diffusivity of yeast cells in a suspension undergoing shear is presented. Local filtrate flux and cake thickness were measured in a radial flow chamber, which has been developed for this purpose. From the simultaneous numerical solution of the Navier-Stokes equation as well as the material balance equation, which govern momentum and material transport in the flat-channel geometry, simulated local flux and cake thickness were obtained for the same operating conditions as in the experiments. The shear-induced diffusion coefficient was then estimated after comparing the measurements with simulated values and minimizing the differences between them.

Influence of shear-induced migration on turbulent resuspension

In this paper we propose a model which predicts the point at which particles are first ejected from the viscous sublayer of a fluid flowing over a settled layer of particles into the turbulent core. The model, which combines viscous resuspension observations and an understanding of the structure of near-wall turbulence, is expected to be valid only for fine particles where the particle Reynolds number (based on the particle diameter and friction velocity) at resuspension is small. If a settled bed with fluid on top is sheared in a plane Couette device with the bottom plate fixed at low Reynolds nunber (based on the velocity of the top plate and the width of the gap), it has been shown that the shear-induced effective particle diffusivity arising from particle interactions causes the bed to expand. This expansion occurs in a narrow transition region between the settled bed and a region devoid of particles. If this region is thin with respect to the dimensions of the viscous sublayer of the flow, then the turbulent shear stress variations in the near-wall region will be impressed on the resuspending layer. Turbulent resuspension would be expected to occur by this mechanism when the bed has expanded enough that the upward velocity at some point in the resuspending layer caused by the turbulent eddies is greater than the downward settling due to gravity. By formulating the problem in this manner, the contribution of viscous effects to the onset of turbulent resuspension may be predicted from known quantities. The dimensionless steady-state concentration profile caused by the interaction between viscous resuspension and turbulent eddies is found to be characterized by the parameter S = β +[(9/2) ψ] 2(9/2 ψ) 2( Re P +); where β + is the dimensionless magnitude of the vertical velocity of the eddies, measured previously to be β + = 0.005; ψ is the Shields parameter τ/ ΔApga, where τ is the wall shear stress, Δp is the density difference between the particles and the fluid, g is the gravitational acceleration and a is the particle radius; and Re P + is the particle Reynolds number u ∗ d P/v , where u ∗ =(τ/p) 1 2 is the friction velocity, d P is the particle diameter, v is the kinematic viscosity and p is the fluid density. The point at which the model predicts incipient turbulent resuspension to occur is given by S ≈ 5. This point is shown to lie between the Shields criterion for the onset of first motion in a settled layer and the minimum flow condition for the complete resuspension of a settled layerm suggesting that viscous effects do play an important role in incipient turbulent resuspension at low particle Reynolds numbers.

Shear-induced resuspension in a couette device

The viscous resuspension of an initially settled bed of particles resulting from the application of a laminar shear flow was studied experimentally in a narrow-gap Couette device. The measured height of the resuspended layer was found to be in excellent agreement with that predicted theoretically using a model developed by Leighton & Acrivos, in which the downward gravitational particle flux is balanced by a corresponding upward flux due to shear-induced particle diffusion.

Shear-induced particle diffusion and longitudinal velocity fluctuations in a granular-flow mixing layer

In flows of granular material, collisions between individual particles result in the movement of particles in directions transverse to the bulk motion. If the particles were distinguishable, a macroscopic overview of the transverse motions of the particles would resemble a self-diffusion of molecules as occurs in a gas. The present granular- flow study includes measurements of the self-diffusion process, and of the corresponding profiles of the average velocity and of the streamwise component of the fluctuating velocity. The experimental facility consists of a vertical channel fed by an entrance hopper that is divided by a splitter plate. Using differently-coloured but otherwise identical glass spheres to visualize the diffusion process, the flow resembles a classic mixing-layer experiment. Unlike molecular motions, the local particle movements result from shearing of the flow; hence, the diffusion experiments were performed for different shear rates by changing the sidewall conditions of the test section, and by varying the flow rate and the channel width. In addition, experiments were also conducted using different sizes of glass beads to examine the scaling of the diffusion process. A simple analysis based on the diffusion equation shows that the thickness of the mixing layer increases with the square-root of downstream distance and depends on the magnitude of the velocity fluctuations relative to the mean velocity. The results are also consistent with other studies that suggest that the diffusion coefficient is proportional to the particle diameter and the square-root of the granular temperature.

Rheology, flow instabilities, and shear-induced diffusion in polystyrene solutions

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTRheology, flow instabilities, and shear-induced diffusion in polystyrene solutionsJules J. Magda, Chang Soon Lee, Susan J. Muller, and Ronald G. LarsonCite this: Macromolecules 1993, 26, 7, 1696–1706Publication Date (Print):March 1, 1993Publication History Published online1 May 2002Published inissue 1 March 1993https://doi.org/10.1021/ma00059a032RIGHTS & PERMISSIONSArticle Views593Altmetric-Citations77LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (1 MB) Get e-Alerts Get e-Alerts

Longitudinal shear-induced diffusion of spheres in a dilute suspension

We present a calculation of the hydrodynamic self-diffusion coefficient of a tagged particle in a dilute mono-dispersed suspension of small neutrally buoyant spheres undergoing a steady simple shearing motion. The displacement of the tagged particle parallel to the longitudinal or streamwise direction resulting from a ‘collision’ with one other particle is calculated on the assumption that inertia and Brownian motion effects are negligible. Summing over different pairs leads to a logarithmically divergent integral for the diffusivity which is rendered finite by allowing for the cutoff due to the occasional presence of another pair of particles. The longitudinal shearinduced self-diffusion coefficient is thus found to be 0.267a2γ{cln c−1 + O(c)], where γ denotes the applied shear rate, a is the radius of the spheres and c their volume concentration.