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

The Stokes flow field and aerosol particle deposition from flows in model filters, i.e., separate layers of granules with square and hexagonal structures, have been calculated taking into account the effect of gas slip at granule surface. Approximating formulas have been derived for granule drag forces to a flow. The efficiencies of diffusion collection of particles have been calculated in a wide range of Peclet numbers with allowance for a finite particle size and the existence of a Knudsen boundary layer, the layer thickness being comparable with the particle sizes. The applicability of the cell model to the calculation of granular filters has been discussed.     

Nanoparticle deposition in granular filters at Reynolds numbers higher than unity

The influence of the inertia of a viscous incompressible liquid flow on the viscous drag and diffusion deposition of particles in model granular filters at Reynolds numbers higher than unity, Re > 1, has been considered. The granule drag forces and particle-collection efficiencies in isolated layers with square and hexagonal packings of granules have been calculated. The influence on each other of approaching monolayers of granules on pressure drop and nanoparticle deposition has been studied. It has been shown that, at Re > 1, the collection efficiency dramatically increases due to the effect of interception.     

Diffusion deposition of aerosol nanoparticles in model granular filters

The diffusion deposition of point aerosol particles from a flow in model granular (grained) filters, i.e., separate layers composed of parallel chains of spherical granules, has been studied at small Reynolds numbers. Numerical solution of the Stokes and convective diffusion equations has been employed to determine the drag forces and granule collection efficiencies as depending on the Peclet diffusion number in a range Pe = 0.02–2 × 10⁴ and the ratio between the granule diameter and the distance between chain axes. Layers of closed chains with square and hexagonal packings have been considered. Approximation formulas have been derived for calculation of nanoparticle penetration in model granular filters.     

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.     

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.     

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.     

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.     

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.     

Inertial Deposition of Aerosol Particles on Fiberous Filters

The inertial deposition of aerosol particles on model filters at low and intermediate Stokes numbers and low Reynolds numbers was studied. It was shown that the efficiency of the inertial deposition of submicron particles is markedly affected by retarded van der Waals forces and the gas slip in the vicinity of thin fibers.     

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.     

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.     

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.     

用直接数值模拟对移动颗粒层除尘的研究及实验对比

利用气固两相流数值模拟计算模型，分别采用不同粒径的移动颗粒层过滤除尘器，对不同粒径粉尘颗粒的碰撞次数进行统计，并对移动床除尘中过滤介质尺寸与粉尘粒径尺寸之间的相互选择性进行了初步研究。模拟计算了在同一风速下碰撞次数与粉尘粒径以及移动层颗粒径之间的关系。计算统计的结果与实验结果对比发现，二者存在定性上的一致。结果表明，在移动床过滤除尘器中不同粒径的过滤层对不同粒径尘粒具有明显的选择性。     

Efficiency of aerosol particle collection with filtering materials containing dispersed inclusions

During the flow around spherical inclusions in a homogeneous porous medium, the gas flow field is calculated using the Brinkman equation. The expression is derived for the effective permeability of the porous medium containing the rarefied ensemble of randomly arranged dispersed spherical particles of arbitrary sizes. It is shown that the Darcy approximation is applicable to the large-size inclusions. The influence of dispersed inclusions of spheroidal shape on both the pressure drop and the efficiency of aerosol particle collection with a filter is studied in the Darcy approximation. It is shown that dispersed inclusions can both increase and decrease the quality criterion of a filter depending on the values of parameters.     

Numerical Analysis of 3‐D Flow Past a Stationary Sphere with Slip Condition at Low and Moderate Reynolds Numbers

Computations are performed to determine the steady 3‐D viscous fluid flow forces acting on the stationary spherical suspended particle at low and moderate Reynolds numbers in the range of 0.1≤Re≤200. A slip is supposed on the boundary so that the slip velocity becomes proportional to the shear stress. This model possesses a single parameter to account for the slip coefficient λ (Pa.s/m), which is made dimensionless and is called Trostel number (Tr=λ a/μ). Decreasing slip, increases drag in all Reynolds limits, but slip has smaller effects on drag coefficient at lower Reynolds number regimes. Increasing slip at known Reynolds number causes to delay of flow separation and inflect point creation in velocity profiles. At full slip conditions, shear drag coefficient will be zero and radial drag coefficient reaches to its maximum values. Flow around of sphere at full‐slip condition is not equal to potential flow around a sphere. Present numerical results corresponding to full slip (Tr→0) are in complete accord with certain results of flow around of inviscid bubbles, and the results corresponding to no‐slip (Tr→∞) have excellent agreement with the results predicted by the no‐slip boundary condition.     

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.     

Deposition of latex particles on alumina fibers

The deposition of sub-micron latex particles during flow through beds of fine alumina fibers has been studied as a function of pH and ionic strength in the vicinity of the fiber isoelectric point (i.e.p.). Conditions were chosen so that the predominant capture mechanism was diffusion. Results have been compared with the “classical” theory of convective diffusion to cylinders and with modern theory, which takes into account colloidal and hydrodynamic interactions between particles and collector.Initial fiber collection efficiencies, determined at the i.e.p. where the fibers bear no surface charge, are considerably lower than those predicted by classical theory and are insensitive to ionic strength. This lack of dependence on ionic strength suggests that neither constant potential nor constant charge conditions are maintained during particle-fiber encounters and that some intermediate condition is more appropriate. Particle capture results obtained at pH values above and below the fiber i.e.p. agree qualitatively with the predictions of the Spielman and Friedlander “surface reaction” model, although this is not strictly valid below the i.e.p. where the fibers are positively charged and there is no repulsion barrier. Under these conditions a significant enhancement of capture efficiency is observed at low ionic strength, as a result of double layer attraction. Above the i.e.p., where the fibers and particles are both negatively charged, double layer repulsion causes a large reduction in capture efficiency, although the predicted is even larger.The saturation coverage of the fibers by deposited particles was found to decrease strongly as the ionic strength was decreased, indicating the importance of lateral interactions between particles.     

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.     

Modeling of submicrometer aerosol penetration through sintered granular membrane filters

We present a deep-bed aerosol filtration model that can be used to estimate the efficiency of sintered granular membrane filters in the region of the most penetrating particle size. In this region the capture of submicrometer aerosols, much smaller than the filter pore size, takes place mainly via Brownian diffusion and direct interception acting in synergy. By modeling the disordered sintered grain packing of such filters as a simple cubic lattice, and mapping the corresponding 3D connected pore volume onto a discrete cylindrical pore network, the efficiency of a granular filter can be estimated, using new analytical results for the efficiency of cylindrical pores. This model for aerosol penetration in sintered granular filters includes flow slip and the kinetics of particle capture by the pore surface. With a unique choice for two parameters, namely the structural tortuosity and effective kinetic coefficient of particle adsorption, this semiempirical model can account for the experimental efficiency of a new class of "high-efficiency particulate air" ceramic membrane filters as a function of particle size over a wide range of filter thickness and texture (pore size and porosity) and operating conditions (face velocity).