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.     

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.     

Deposition of Aerosol Nanoparticles in Fibrous Filters

The deposition of aerosol nanoparticles on model fibrous filters of different porosity at small Reynolds numbers was considered. The efficiency of particle deposition on a fiber was determined by the numerical solution of the convective diffusion equation in the ordered systems of parallel fibers located perpendicularly to a flow. The calculation results agree with the known experiments on the model filters over the Peclet number (Pe) range from 0.05 to 1000. It was shown that the Natanson–Stechkina formula for the coefficient of capture obtained in the limit of thin diffusion boundary layer at Pe 1 is valid within a wide range of the Peclet numbers up to Pe 1.     

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.     

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.     

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.     

Nanoaerosol Filtration with Fine-Fiber Polydisperse Filters

The self-consistent theory of nanoaerosol filtration has been considered at small Peclet numbers. It has been shown that, at Pe < 1, it is necessary to consider the deposition of particles simultaneously on thePe < entire ensemble of fibers. Only at can the filtration efficiency be found from the collection efficiencies calculated separately for each fiber. The self-consistent theory has been used to estimate the efficiencies of filtration with polydisperse fibrous filters. The penetrations of particles through filters with different variances of the functions of fiber length distribution over fiber radii have been compared at a constant packing density or a constant total length of all fibers. It has been shown that, at a constant packing density, a rise in the variance leads to a decrease in the filtration efficiency, while, at a constant total fiber length, the efficiency is almost independent of the width of the distribution function.     

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.     

Diffusion Deposition of Finite Size Particles on Fibrous Filters at Intermediate Knudsen Numbers

Diffusion deposition of aerosol particles from the flow on model filter made of ultrathin fibers with allowance for the intrinsic size of particles was considered in the range of Knudsen numbers Kn = 0.1–10. The problem was solved in the approximation of diffuse boundary layer. It was shown that the value of capture coefficient at small Kn seems to be larger and, at larger Kn, lower than the sum of individual capture coefficients due to the diffusion and interception. The calculated value of the radius of the most penetrable particles at Kn 1 agrees with experimental data.     

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.     

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.     

The Effect of van der Waals Forces on the Deposition of Highly Dispersed Aerosol Particles on Ultrafine Fibers

The effect of van der Waals forces on the collection of highly dispersed aerosol particles with ultrafine fiber filters was studied theoretically. The capture coefficient was found from the numerical solution of the equation of convective diffusion with the account of the particle size, the effect of van der Waals forces acting between a particle and a fiber, and the gas slip effect at the surface of ultrafine fibers. It was shown that allowance being made for van der Waals forces markedly affects the capture coefficient within the maximal particle penetration range and that the radius of the most penetrating particles decreases with the rising effect of these forces.     

The Deposition of Aerosol Submicron Particles on Ultrafine Fiber Filters

The diffusion deposition of submicron aerosol particles of a finite size on a model filter composed of parallel ultrafine fibers with a radius comparable with the mean free path of air molecules was considered. The diffusion capture coefficient with allowance made for particle interception _DR is found by the numerical solution of the elliptic equation of steady-state convective diffusion in the wide ranges of interception parameter R, Peclet (Pe) and Knudsen (Kn) numbers at small Reynolds numbers. It was shown that, at small Kn numbers, the _DR value exceeds the sum of capture coefficients due to specific deposition mechanisms, interception and diffusion, = _R + _D , whereas, at Kn > 1, _DR . Within the range of intermediate Pe, Kn, and R numbers, the radius of the most penetrating particles is higher than the fiber radius.     

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.     

Survival and relaxation time, pore size distribution moments, and viscous permeability in random unidirectional fiber structures

Computer simulation results are presented for the mean survival time, principal relaxation time, mean pore size, and mean square pore size, for random porous structures consisting of parallel nonoverlapping or partially overlapping fibers. The numerical procedure is based on a discrete step-by-step random walk mechanism simulating the Brownian diffusion trajectories of molecules in the porous media. Numerical results on the viscous permeability of these structures are computed with a method based on electrical conduction principles and compared to a variational bound derived from the mean survival time. The results show that nonoverlapping fiber structures exhibit lower values of the dimensionless mean survival time, principal relaxation time, mean pore size, and mean square pore size than randomly overlapping fiber structures of the same porosity, while partially overlapping fiber structures show behavior intermediate to those of the two extreme cases. The mean square pore size (second moment of the pore size distribution) is found to be a very good predictor of the mean survival time for non-, partially, and randomly overlapping fiber structures. Dimensionless groups representing the deviation of variational bounds from our simulation results vary in practically the same range as the corresponding values reported earlier for beds of spherical particles. A universal scaling expression of the literature relating the mean survival time to structural properties [S. Torquato and C. L. Y. Yeong, J. Chem. Phys. 106, 8814 (1997)] agrees very well with our results for all examined fiber structures, thus validated for the first time for porous media formed by partially overlapping particles. The permeability behavior of partially overlapping fiber structures resembles that of nonoverlapping fiber structures for flow parallel to the fibers, but not for transverse flow, where percolation phenomena prevail. The permeability results for beds of unidirectional partially overlapping fibers of moderate and low hard-core porosity compliment successfully earlier numerical data on the permeability of similar structures originating from high-porosity beds of nonoverlapping fibers.     

Deposition of charged aerosol nanoparticles in diffusion batteries

The deposition of weakly charged aerosol nanoparticles onto fibers in a diffusion battery designed to measure the diffusion coefficients of particles is considered. The fiber collection efficiancies as functions of particle size and charge are determined by the numerical solution of the equation of convective diffusion in a system of parallel uncharged fibers located normal to a flow. It is shown of effect of the single charge of nanoparticles produced by a differential mobility analyzer on their deposition is negligible and may be ignored when calibrating diffusion batteries.     

Diffusional Deposition of Heavy Submicron Aerosol Particles on Fibrous Filters

The deposition of Brownian submicron aerosol particles of high density in fibrous filters with allowance for interception effect, influence of gravitational, van der Waals forces, and the gas slip on the surface of ultrafine fibers was considered. Based on the numerical solution of the equation of convective diffusion in the field of external forces, the capture coefficient was calculated as a function of particle size and density, the angle between the vectors of gravity force, and the face flow velocity. It was shown that, for descending flow, the radius of most penetrating dense particles appeared to be noticeably smaller than for the ascending flow.__________Translated from Kolloidnyi Zhurnal, Vol. 67, No. 3, 2005, pp. 352–356.Original Russian Text Copyright © 2005 by Kirsh.     

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.     

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.     

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.