Simulation of nanofibrous filters produced by the electrospinning method: 1. Pressure drop and deposition of nanoparticles

The peculiarities of the hydrodynamic flow field and diffusion deposition of nanoparticles in filtration layers of nanofibers obtained by spraying a polymer solution in an electric field are considered. The main attention is focused on the effect of doubled nanofibers or pairs of parallel fibers that result from longitudinal splitting of charged jets on the hydrodynamic characteristics. The calculations are performed for a periodical row of doubled parallel fibers oriented normal to the flow. The flow field and the rate of nanoparticle deposition in the row are investigated as dependences on the distances between the pairs of the fibers, interfiber distances in pairs, orientation of the pairs relative to the direction of a flow, and the relations between fiber diameters in the pairs. The equations for the flow of a viscous incompressible liquid are solved under the Stokes approximation employing the method of fundamental solutions, and the stream functions, fields of velocities, and drag forces acting upon the fibers are determined. For the found flow fields, the coefficients of diffusion capture are determined by the numerical solution of the convective diffusion equation. It is established that, when fibers are drawn together in pairs to their contact in a rarefied row, the drag force decreases twofold. This result agrees with experimental data and the analytical solution for the constrained flow around pairs of similar fibers in a rarefied row.     

A model of a dust-loaded filter with asymmetric deposit of particles on fibers

Results of numerical simulation have been reported for the flow field and diffusion deposition of nanoparticles in a model dust-loaded fibrous filter, i.e., a row of parallel fibers coated with porous permeable shells shifted toward an incident flow. The flow field and point particle collection efficiency on fibers coated with the shells have been calculated by combining the Stokes, Brinkman, and convective diffusion equations. It has been shown that the pressure drops and efficiencies of nanoparticle deposition in the filters composed by fibers with coaxial and asymmetric porous shells are almost identical.     

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.     

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.     

The Effect of Long-Range Fiber Space and Orientation Correlations on the Hydrodynamic Resistance and Diffusion Deposition in Fibrous Filters

On the basis of Brinkman's equation, the problem of hydrodynamic resistance is analyzed for a row of periodically arranged parallel circular fibers placed in a uniform porous medium. The results are applied to the fan model of a filter, for which an equation aimed at determination of Brinkman's constant is derived within the framework of a self-consistent theory. The resistance force acting on a unit length of each fiber is calculated as a function of the geometrical parameters of the model. The capture coefficient is found for diffusion deposition of aerosols on fibers of the model filter. The results obtained agree with the experimental data.     

Deposition of nanoparticles in filters composed of permeable porous fibers

The diffusion deposition of nanoparticles is studied from a flow at low Reynolds numbers in model filters composed of permeable circular porous fibers. The field of particle concentration is calculated and the capture coefficient is determined for a cell, as well as the isolated row of parallel fibers within a wide range of Peclet numbers (Pe) depending on the fiber permeability. It is shown that at Pe > 1, the diffusion capture coefficient η increases with permeability, while at Pe → ∞, it tends toward the limiting value, which is equal to the gas flow rate through the porous fiber. The capture coefficients calculated from a cell model and for a row of fibers are almost equal to each other. The diffusion deposition of aerosol particles in the highest penetration range is calculated with an allowance for their finite sizes and it is shown that the radii of most penetrable particles decrease with an increase in fiber permeability.     

Viscous Drag of Periodic Row of Polydisperse Fibers with Allowance for Slip Effect

Viscous drag in the gas flowing through an equidistant periodic row of parallel polydisperse fibers with the random fiber radius distribution over period was studied using an approximation of small Reynolds numbers. The case of viscous advancing flow perpendicular to the row was considered. The effect of gas slip at the fiber surface was taken into account. Expression was found for the resistance force averaged over random ensemble of fibers, which well describes the results of numerical simulation with the use of lognormal fiber radius distribution, including the case of high fiber polydispersity. The resistance of a very dense polydisperse row was analyzed within the framework of Keller's model with allowance for slip effect.     

Gas flow in a layer of chains formed from permeable clusters of nanoparticles

The Stokes–Brinkman flow field has been calculated in a model deposit, i.e., a row of parallel chains formed from porous spherical clusters of nanoparticles and oriented perpendicularly to the gas flow direction. The force of drag to an air flow has been calculated for the row of chains taking into account their permeability and the distance to neighboring chains. The drag forces have been found for nanodendrites composing clusters with allowance for the gas slip effect. Corresponding approximation formulas have been derived. A method has been proposed for calculating the pressure drop across a highly porous deposit of clusters of aerosol nanoparticles deposited onto a filter.     

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.     

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.     

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.     

Viscous Flow Through Polydisperse Row of Parallel Fibers: Account for the Slip Effect

A general theory of the flow through the flat periodic row of parallel polydisperse fibers randomly arranged at the fiber period is elaborated. An approximate solution of the problem with account for the slip effect at low Reynolds numbers is obtained for normal incident flow. The system of equations for determining local drag forces acting on each fiber in a row, as well as the system of equations for determining local buoyancy forces, are derived. It was demonstrated that, in the approximation obtained, the buoyancy forces are equal to zero, provided that bilateral symmetry is present in the fiber arrangement. The behavior of local forces as a function of row parameters is considered using bimodal row as an example. The form of flow function is established for the velocity field near the arbitrary fiber.     

Penetration of nanoparticles through screen-type diffusion batteries

The deposition of aerosol nanoparticles from Stokes flows in screen-type diffusion batteries designed for the determination of diffusion coefficients for suspended nanoparticles is considered. Average fiber collection efficiencies η calculated for screens consisting of two perpendicularly contacting rows of parallel equidistant straight fibers agree with the experimental data obtained for woven screens within the Peclet number range Pe = 0.15−1000. It is shown that, for dense screens, the η ∼ Pe^−2/3 power dependence is valid at Pe > 10. For rarefied screens, this dependence is fulfilled down to Pe ∼ 0.1. At Pe ≪ 1, the integral flow of particles advancing on the fibers of the first row in the screen and the fiber collection efficiency, η of an isolated row tends to a geometrical limit, which is equal to the ratio of the distance between the axes of the fibers to the fiber diameter.     

The Two-Component Fiber Row: Viscous Drag and Particle Deposition in the Regime with Interception

Theoretical expressions for a mean resistance force of a random two-component fiber row and a mean deposition efficiency of aerosol particles in this row in the regime with interception were derived. The effect of gas slip near the fibers was considered. It was shown that the derived formulas well describe the results of numerical simulation, including the modeling of the dense rows with wide fiber size distribution. Comparison with the experimental data was made.     

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.     

Stokes flow and deposition of aerosol nanoparticles in model filters composed of elliptic fibers

The calculation is implemented for the fiber collection efficiencies due to diffusion of nanoparticles in model filters, i.e., separate rows of fibers with an elliptic cross section located normal to the flow at different orientations of the ellipse axes with respect to the flow. The Stokes flow field in the system of the fibers is found by the method of fundamental solutions. The concentration field of Brownian particles and the efficiency of their deposition onto the fibers are determined from the numerical solution of the equation for the convective diffusion. The dependence of the capture coefficient on the Peclet number for elliptic fibers is shown to have the form η = APe^−m, where exponent m changes from 2/3 to 3/4 at the parallel and normal orientation of the major axes of the ellipses with respect to the flow, respectively. It is shown that, from the viewpoint of aerosol nanoparticle capture, the best filters are those in which the fibers have a maximum midsection at the same cross-sectional area.     

Stokes flow in periodic systems of parallel cylinders with porous permeable shells

The problem solved in this study concerns steady-state flow of a viscous incompressible liquid at low Reynolds numbers in a model filter consisting of parallel cylinders with porous permeable shells. Both a separate row and lattices (square and hexagonal) of cylinders directed perpendicularly to the flow are considered. The flow field outside and inside the porous shell is found from the solutions to the Stokes and Brinkman equations. The drag force and the filtration efficiency are determined both as functions of the ratio between the cylinder diameter and the distance between the axes of adjacent cylinders and as functions of the thickness and permeability of the shells. The cell model is shown to be applicable for describing the flow field in a hexagonal lattice of cylinders with porous shells within a wide range of packing densities. Original Russian Text ? V.A. Kirsh, 2006, published in Kolloidnyi Zhurnal, 2006, Vol. 68, No. 2, pp. 198–206.     

Mathematical modeling of gas–liquid membrane contactors using random distribution of fibers

This work presents a mathematical model for gas absorption in microporous hollow fiber membrane contactors by using a random distribution of fibers. The chemical absorption of carbon dioxide into aqueous amine solutions and sulfur dioxide into water were simulated by this model. The nonlinear mathematical expressions of the component material balance for the liquid, membrane, and gas were solved simultaneously by using a numerical method. The results from the model were compared with four sets of different experimental data in the literature. In addition, the contactors were modeled based on the assumption of regular arrangement of fibers in the shell side by using Happel's free surface as well as plug flow models. The plug flow model was employed to compare the various available equations in the literature for the shell side mass transfer coefficient. The results indicate that the channeling of gas in the shell side decreases the efficiency of contactor significantly. It was found that the random distribution of fibers is a suitable method to simulate the commercial modules. The results also indicate that, the regular Happel's free surface model and the plug flow model are more suitable for handmade modules. The influence of shell side channeling on the contactor performance were investigated in different fiber packing densities, and in various gas and liquid flow rates.     

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.     

Dead-end ultrafiltration in hollow fiber modules: Module design and process simulation

Ultrafiltration in a hollow-fiber module operating with outside-in and dead-end flow at a constant flow rate was simulated using a model that takes into account the longitudinal pressure drops inside the fibers and within the fiber bundle. The model considers both the filtration phase during which the membrane is fouled by the formation of a filter cake and the backwash phase in which it is cleaned, so as to predict the net rate of production of the module during an operating cycle.The results show that there is a combination of packing density and fiber diameter that gives a maximum net flow rate. Furthermore, this model allows the influence of operating conditions and feed properties on the module performance to be estimated. This can be used to determine how operating parameters must be modified when there is a change in the feed properties.