Dispersion relations for the convective instability of an acidity front in Hele-Shaw cells

Autocatalytic chemical fronts of the chlorite-tetrathionate (CT) reaction become buoyantly unstable when they travel downwards in the gravity field because they imply an unfavorable density stratification of heavier products on top of lighter reactants. When such a density fingering instability occurs in extended Hele-Shaw cells, several fingers appear at onset which can be characterized by dispersion relations giving the growth rate of the perturbations as a function of their wave number. We analyze here theoretically such dispersion curves comparing the results for various models obtained by coupling Darcy's law or Brinkman's equation to either a one-variable reaction-diffusion model for the CT reaction or an eikonal equation. Our theoretical results are compared to recent experimental data.     

Modeling electroosmotic and pressure-driven flows in porous microfluidic devices: zeta potential and porosity changes near the channel walls

This work presents analytical solutions for both pressure-driven and electroosmotic flows in microchannels incorporating porous media. Solutions are based on a volume-averaged flow model using a scaling of the Navier-Stokes equations for fluid flow. The general model allows analysis of fluid flow in channels with porous regions bordering open regions and includes viscous forces, permitting consideration of porosity and zeta potential variations near channel walls. To obtain analytical solutions problems are constrained to the linearized Poisson-Boltzmann equation and a variation of Brinkman's equation [Appl. Sci. Res., Sect. A 1, 27 (1947); 1, 81 (1947)]. Cases include one continuous porous medium, two adjacent regions of different porosities, or one open channel adjacent to a porous region, and the porous material may have a different zeta potential than that of the channel walls. Solutions are described for two geometries, including flow between two parallel plates or in a cylinder. The model illustrates the relative importance of porosity and zeta potential in different regions of each channel.     

The effect of gas slip on pressure drop and deposition of submicron particles in fibrous filters

The effect of gas slip at fibers on the drag to a flow and the deposition of submicron particles in model filters with a tree-dimensional flow field has been considered. The average values of the drag force and the efficiency of diffusion collection of particles with finite sizes in a double hexagonal three-dimensional model filter taken as a standard uniform filter have been calculated as depending on the packing density of fibers and the Knudsen number. It has been shown that, in the region of the sizes of the most penetrating particles, under preset conditions, and at specified filter parameters, the obtained collection efficiency values agree with the results of calculations performed by empirical formulas for a model fan filter. Moreover, formulas derived for a planar flow taking into account the slip effect are applicable to highly porous filters.     

Simulation of nanofibrous filters produced by the electrospinning method: 2. The effect of gas slip on the pressure drop

The dependence of pressure drop in a model filter on the distance between pairs of fibers, interfiber distances in the pairs, and the orientation of the pairs of fibers relative to flow direction is calculated with allowance for the effect of gas slip along the surface of doubled nanofibers. An isolated row of doubled parallel fibers oriented normal to the flow is selected as a model filter. Flow fields in the row of the fibers and drag forces acting upon them are calculated by the Stokes equations, which are solved by the numerical method of fundamental solutions. For pairs of fibers lying in the same plane in a row, the results of the numerical calculations agree with the analytical solution.     

Effective medium approximation and deposition of colloidal particles in fibrous and granular media

Laminar flow of fluids through fibrous and granular media and deposition of colloidal particles from a liquid suspension are two fundamental phenomena encountered in many industrial applications. An Effective Medium Approximation (EMA) is used to determine the fluid flow permeability and particle capture efficiency of random arrays of cylindrical and spherical collectors. The EMA assumes a model system in which a packing element (a single fiber in the fibrous medium and a single sphere in the granular medium) is surrounded by a fluid envelope and an effective-medium beyond the envelope. It integrates the important features of both the cell models and Brinkman's model. The Stokes equation and Brinkman equation are solved for the fluid envelope and effective medium regions, respectively, to obtain the permeability and close-to-surface velocity field around the collectors. The convective diffusion equation is then solved to determine the particle deposition rate. The analytical expressions for the permeability and particle deposition rate are derived for all possible cases of random packing of uniform and non-uniform cylinders and spheres. Effects of various system properties and operating conditions on deposition of colloidal particles are investigated. The physical or chemical conditions include the properties which affect the magnitude of double layer interaction: the electrolyte concentration and surface potentials, and the property which affects the van der Waals interaction: the Hamaker constant. It was found that the effects of the above properties is much more significant when the surface interactions play more important roles in the particle deposition process, or when the height of the total interaction energy barrier is higher than 5 kBT. Particle deposition becomes virtually impossible when the height of the repulsive energy barrier increases beyond 20 kBT.     

Deposition of aerosol nanoparticles in filters composed of fibers with porous shells

The diffusion deposition of submicron aerosol particles in model filters consisting of fibers covered with permeable porous shells is studied. An ordered system of parallel cylinders arranged perpendicular to the flow is used as a model filter. The results of calculations are given for the dependences of the capture coefficient on the shell radius, the shell permeability, the packing density of the filters, the particle radius, and the flow velocity. Calculations are performed within a wide range of Peclet numbers. It is shown that the capture coefficient and the quality criterion γ of a filter increase with the diffusion mobility of particles and shell permeability, as well as that the dependence of the quality criterion on the radius of permeable shells has a maximum. It is also shown that the capture coefficients for fibers with porous shells, calculated using the cell model and the isolated row of fibers, almost coincide with one another.     

Effect of fiber orientation on fiber wetting processes

The current work incorporates a microscopic study of the effect of fiber orientation on the fiber wetting process and flow of liquid droplets along filter fibers when subjected to airflow and gravity forces. Glass filter fibers in various combinations were oriented at various angles within a plane defined by the airflow direction and were supplied with distilled water in aerosol form. The behavior and flow of the liquid collected by the fibers were observed and measured using a specially developed microscope cell, detailed in the paper. The experimental results were compared to a theoretical model developed to describe the behavior. The theory and experimental results showed good agreement. The developed theory allows an optimum angle to be determined for the internal filter fiber structure in the design of wet filters. A sensitivity analysis of the model was conducted to determine the most important parameters. This will aid design of wet filtration systems such that maximal self-cleaning can be accomplished with minimal water use.     

Deposition of Aerosol Particles on the Row of Polydisperse Fibers Due to Interception with Allowance for Slip Effect

The efficiency of deposition of aerosol particles due to interception during the gas flow through the equidistant periodic row of parallel polydisperse fibers with random fiber radius distribution over the period was studied in the approximation of small Reynolds numbers. The case of advancing viscous flow perpendicular to the row was considered. An effect of gas slip at the fiber surface was taken into account. Quality of such model filter was studied. Expressions for deposition efficiency and filter quality averaged over random ensemble of fibers were derived; they well describe the results of numerical simulation based on lognormal fiber radius distribution including high degrees of fiber polydispersity.     

Optimization of the Aerosol Filter Parameters

Theoretical approach to the optimization of the parameters of two-stage fibrous aerosol filter is reported. The approach is based on earlier developed method for the calculation of the clogging of model filter where the particle deposit is approximated by the porous shell on the fibers. The values of optimal fiber radii are determined from the condition of maximal dust capacity of a filter at given limiting pressure drop and total initial efficiency. Optimization was performed with allowance for possible variation in the prefilter thickness.     

Spreading of liquid drops over saturated porous layers

Spreading of small liquid drops over thin porous layers saturated with the same liquid is investigated from both theoretical and experimental points of view. A theory is presented that shows that spreading is governed by the same power law as in the case of spreading over a dry solid substrate. The Brinkman's equations are used to model the liquid flow inside the porous substrate. An equation of the drop spreading is deduced, which shows that both an effective lubrication and the liquid exchange between the drop and the porous substrates are equally important. The presence of these two phenomena removes the well-known singularity at the moving three-phase contact line. Matching of the drop profile in the vicinity of the three-phase contact line with the main spherical part of the drop gives the possibility to calculate the pre-exponential factor in the spreading law via permeability and effective viscosity of the liquid in the porous layer. Unfortunately, the latter dependency turns out to be very weak. Spreading of silicone oils over different microfiltration membranes is carried out. Radii of spreading on time experimental dependencies confirm the theory predictions. Experimentally found coefficients agree with theoretical estimations.     

Diffusional deposition of nanoparticles in a 3D model fiber filter

Diffusional particle deposition from a flow on fibers at low Reynolds number Re ≪ 1 is studied in a model filter consisting of equidistant rows of parallel fibers perpendicular to the flow and forming a three-dimensional structure where one hexagonal 2D-dimensional lattice is inserted into another at the right angle. It is shown that under equal low filter packing density fiber collection efficiencies calculated within 3D and 2D models are practically the same.     

Filtration application from recycled expanded polystyrene

Water-in-oil emulsion with drop size less than 100 mum is difficult to separate. Coalescence filtration is economical and effective for separation of secondary dispersions. Coalescence performance depends on flow rate, bed depth, fiber surface properties, and drop size. The amount of surface area of the fibers directly affects the efficiency. A new recycling method was investigated in the previous work in which polystyrene (PS) sub-mum fibers were electro-spun from recycled expanded polystyrene (EPS). These fibers are mixed with micro glass fibers to modify the glass fiber filter media. The filter media are tested in the separation of water droplets from an emulsion of water droplets in oil. The experimental results in this work show that adding nanofibers to conventional micron sized fibrous filter media improves the separation efficiency of the filter media but also increases the pressure drop. An optimum in the performance occurs (significant increase in efficiency with minimal increase in pressure drop) with the addition of about 4% by mass of 500 nm diameter PS nanofibers to glass fibers for the filters.     

Analysis of constant permeate flow filtration using dead-end hollow fiber membranes

Dead-end filtration of colloids using hollow fibers has been analysed theoretically and experimentally. A mathematical model for constant flux filtration using dead-end hollow fiber membranes has been developed by combining the Hagen–Poiseuille equation, the (standard) filtration equation, and cake filtration theory of Petsev et al. [D.N. Petsev, V.M. Starov, I.B. Ivanov, Concentrated dispersions of charged colloidal particles: sedimentation, ultrafiltration and diffusion, Colloid Surf. A: Physicochem. Eng. Aspects, 81 (1993) 65–81.] to describe the time dependence of the filtration behavior of hollow fiber membranes experiencing particle deposition on their surface. Instead of using traditional constitutive equations, the resistance of the cake layer formed by the deposited colloids has been directly correlated to the cake structure. This structure is determined by application of a force balance on a particle in the cake layer combined with the assumption that an electrostatically stable cake layer of mono-sized particles would be ordered in a regular packing geometry of minimum energy. The developed model has been used to identify the relationship between the filtration behavior of the hollow fiber membrane and the particle properties, fiber size, and imposed average flux. Filtration experiments using polystyrene latex particles of relatively narrow size distribution with a single dead-end hollow fiber membrane demonstrate good consistency between experimental results and model prediction. The developed model has been used to simulate the distribution of the cake resistance, transmembrane pressure, and flux along the hollow fiber membrane and used to assess the effect of fiber size, particle size, zeta potential, and the average imposed flux on the suction pressure-time profiles, flux, and cake resistance distributions. These results provide new insights into the filtration behavior of the hollow fiber membrane under constant flux conditions.     

Constitutive modeling of polymers accounting for their hyperelasticity,plasticity, creep and viscoelastic relaxation

The hyperelastic Yeoh model has been generalized to account for creep, plasticity and viscoelasticity of polymers. The general tensorial model developed is applied to several rheometric situations: the tensile test used to measure the stress-strain curve in tension, as well as the creep and recovery tests. The resulting equations are compared to the experimental results acquired in the present work for several monolithic synthetic fibers used as specimens. The comparison revealed that the proposed phenomenological rheological constitutive equation is capable of reproducing the experimental data with a uniformly valid set of physical parameters. Moreover, it was possible to accurately predict the residual plastic deformation of the fibers.     

Inertial deposition of aerosol particles from laminar flows in fibrous filters

The deposition of aerosol particles onto filter fibers under the effect of inertial forces is studied in a wide range of Stokes numbers (St) at Reynolds numbers close to unity (Re ∼ 1). Coefficients η of the capture of inertial particles with finite sizes in model filters composed of parallel rows of identical parallel fibers located normal to the direction of a flow are determined based on the numerical solution of the Navier-Stokes and particle motion equations. It is shown that, at Re < 1 and a constant particle-to-fiber radius ratio, R = r _p/a, number St uniquely characterizes capture coefficients η for particles with different densities, while, at Re ≥ 1, the capture coefficient depends on both St and Re. At constant R and St values, the larger Re the higher the capture coefficient. The influence of the structure of the model filter on pressure drop Δp and η is investigated. A nonuniform arrangement of fibers in rows is shown to increase the Δp/U ratio at lower Re values and to make the η -St dependence more pronounced than that for systems of uniformly ordered fibers. The results of calculations agree with the experimental data.     

Rheological behavior of fiber suspensions in a turbulent channel flow

Rheological behavior of fiber suspensions in a turbulent channel flow was investigated theoretically and numerically. A model of turbulent fiber suspensions was proposed to predict the orientation distribution of fibers. The fluctuating equation for the orientation distribution function (ODF) of fibers was theoretically solved using the method of characteristics. The self-governed mean equation for orientation distribution function (ODF) was derived by relating the fluctuating ODF and angular velocities-correlated terms to the gradient of mean ODF. Then the ODF of fibers was predicted by numerically solving the mean equation for ODF. Finally the shear stress and first normal stress difference of suspensions were obtained. The results, some of which agree with the available relevant experimental data, show that the orientation distribution of fibers in the vicinity of the center of the flow is relatively broad in turbulent regime, and becomes broader with the increase of Reynolds number. The shear stress of fiber suspensions increases, while the first normal stress difference decreases, from the wall to the center of the flow for varying Reynolds number.     

The Viscous Drag of a Rough Cylinder in a Liquid Flow in a Porous Medium

The general solution to Brinkman's equations in a cylindrical coordinate system is obtained. This solution is used to calculate the viscous drag of a rough cylinder (placed in a porous medium) with an arbitrary orientation of its axis in relation to the velocity of the advancing flow. A formula is derived for the viscous drag of the cylinder as a function of the concentration of particles that have deposited on it during the initial stage of filtration.