Modeling filter material with an orthotropic structure based on woven mesh

. A description is given of a filter material with an orthotropic structure based on woven mesh; its advantages are considered, and design options for filter elements from this material are given that are of practical interest for cleaning various liquids, including transmission ones. A model is proposed and the results of calculations of the structural and hydrodynamic properties of a filter material with an orthotropic structure based on woven mesh are presented.


Introduction
Mesh filter materials (MFMs) based on metal wire mesh are widely used for cleaning various liquids and gases. MFMs have a number of advantages over other filter materials (FMs), due to the combination of high strength and permeability; stability of the porous structure, which excludes the migration of FM particles into the cleaned medium; high heat resistance; and the ability to multiple regeneration [1,2]. To produce MFM with high performance, it is reasonable to use woven metal mesh (WMM) with square cells. A significant disadvantage of such MFM is the low dirt capacity due to the fact that the method of surface filtering is implemented instead of a deep one in their practical use. On this account, filter element (FE) from MFM must often be replaced or regenerated [3]. At the same time, the regeneration process is difficult. It is impossible to completely remove all impurity particles stuck in the cells. Their failure to pass when using FE after regeneration is not guaranteed and the cells themselves are subject to deformation. However, using simple design and technological solutions, it is possible to to create new FMs applying MFMs, operating in the deep filtering mode. It is possible with their help to realize multi-stage purification of liquids and gases. Such MFMs can be almost completely cleaned of trapped particles of contaminants during the regeneration process. Such FMs can be produced by simple packing (laying by stack) of mesh with square cells, when the warp and weft wires are located one above the other, respectively [4,5]. The image of the structure of such a material is shown in Figure 1. In this case the structure can be called orthotropic, since it is obvious that in two mutually perpendicular directions, both the pores (namely, the fineness of cleaning) and the throughput (permeability coefficient) are significantly different. Two basic FE designs with an orthotropic structure based on WMMs are shown in Figure 2 and they reveal how the filtering ability of the laying is realized, shown in Figure  1,b.  It is obvious that, in comparison with traditional filtering of the medium to be cleaned through the cells of the woven meshes, the use of the orthotropy of their structure makes it possible to reduce the fineness of cleaning and facilitate the regeneration process: when developing a filter element, the contour of the elementary filter cell becomes open, and the contaminants on the filter material are retained only by adhesion forces.
In accordance with the foregoing, a prediction of the properties of such materials at the design stage is of great interest. The objective of this paper is to develop a model for calculating the properties of filter materials with an orthotropic structure based on woven metal meshes.

Experimental results and discussion
When developing the model, we will use an approach based on the building unit cells. Figure 4 shows a model representation of a fragment of the material under consideration based on WMMs with square cells. The dimensions of the presented cell are characterized by the diameter of the wire d and the step of weaving the wire in the grid l. According to reference [6], the structural properties of FM include porosity, specific surface area, and pore sizes. Porosity P is determined by the ratio of the pore volume V п to the total volume of the porous body V [6].
For the considered unit cell The specific surface S spec. is equal to the ratio of the surface area of the frame of the unit cell S surf to its volume V. As follows from Figure   ( The pore sizes are determined by the maximum circle diameters inscribed in the cross sections of the faces of the unit cell and will be characterized by values d пx in the direction of the ОX axis and d пy in the direction of the OY axis (3) The hydrodynamic properties of the material are determined by the permeability coefficients at various filtration modes. The calculation of the viscosity coefficient k μ and inertial permeability coefficient k  is carried out by analogy with the calculation of the hydrodynamic properties of the cellular material considered in reference [7]. For this purpose, we use the well-known binominal equation of filtration that describes the fluid flow in a porous medium: where рpressure drop on a layer of porous material; Н 0filtering layer thickness; fluid density; μdynamic viscosity of fluid; vfiltration rate. Let us consider the fluid flow in a unit cell along the OX axis. In this case, we distinguish two regions of fluid passage through the unit cell ( Figure 6), namely: Ithe region lying at the input and output of the fluid flow from the unit cell, and IIthe region located in the center of the unit cell.
To determine the values of k μx and k x , it is necessary to calculate a number of auxiliary quantities. The unit cell height h and its cross-sectional area S 0 are equal to d h 2  , and dl S x 2 0  , respectively.
Cross-sectional areas of region I -S 1x and region II -S 2x : Cross-section perimeters of region I -N 1x and region II -N 2x : The lengths of region Il 1x and region IIl 2x are defined as From the continuity of the flow , we find the fluid velocities v 1x , and v 2x , respectively in regions I and II 4 / The hydraulic radii of the cross-sections of region I -R 1x and region II -R 2x are equal to Reynolds criteria Rе 1x , Rе 2x for flow in regions I and II are equal to 2 2 (10) The pressure loss on viscous friction within the unit cell is determined by the Darcy-Weisbach formula: