S. I. Vasin

Hydrodynamic permeability of membranes built up by particles covered by porous shells: Cell models

A review is presented on an application of a cell method for investigations of hydrodynamic permeability of porous/dispersed media and membranes. Based on the cell method, a hydrodynamic permeability is calculated of a porous layer/membrane built up by solid particles with a porous shell and non-porous impermeable interior. Four known boundary conditions on the outer cell boundary are considered and compared: Happels, Kuvabaras, Kvashnins and Cunninghams (usually referred to as Mehta-Morses condition). For description of a flow inside the porous shell Brinkmans equations are used. A flow around an isolated spherical particle with a porous shell is considered and a number of limiting cases are shown. These are compared with the corresponding results obtained earlier.

Hydrodynamic permeability of aggregates of porous particles with an impermeable core.

A hydrodynamic permeability of membranes built up by porous cylindrical or spherical particles with impermeable core is investigated. Different versions of a cell method are used to calculate the hydrodynamic permeability of the membranes. Four known boundary conditions, namely, Happels, Kuwabaras, Kvashnins and Cunningham/Mehta-Morses, are considered on the outer surface of the cell. Comparison of the resulting hydrodynamic permeability is undertaken. A possible jump of a shear stress at the fluid-membrane interface, its impact on the hydrodynamic permeability is also investigated. New results related to the calculated hydrodynamic permeability and the theoretical values of Kozeny constant are reported. Both transversal and normal flows of liquid with respect to the cylindrical fibers that compose the membrane are studied. The deduced theoretical results can be applied for the investigation of the hydrodynamic permeability of colloidal cake layers on the membrane surface, the hydrodynamic permeability of woven materials.

Cell models for flows in concentrated media composed of rigid impenetrable cylinders covered with a porous layer

AbsractThe hydrodynamic permeability of a membrane simulated by a set of identical impenetrable cylinders covered with a porous layer is calculated by the Happel-Brenner cell method. Both transverse and longitudinal flows of filtering liquid with respect to the cylindrical fibers that compose the membrane are studied. Boundary conditions on the cell surface that correspond to the Happel, Kuwabara, Kvashnin, and Cunningham models are considered. Brinkman equations are used to describe the flow of liquid in the porous layer. Results that correspond to previously published data are obtained for the limiting cases. Theoretical values of Kozeny constants are calculated. The models proposed can be used to describe the processes of reverse osmosis, as well as nano- and ultrafiltration.

Permeability of complex porous media

AbsractHydrodynamic permeability of membrane composed of a set of porous spherical particles with rigid impenetrable cores is calculated. The cell method proposed by Happel and Brenner is used in calculations. All known boundary conditions on the cell surface, such as the Happel, Kuwabara, Kvashnin, and Cunningham (Mehta-Morse) models, are considered. The flow of liquid is described by the Brinkman equations. The problem of flow around a single spherical particle covered with porous layer by the uniform flow of viscous incompressible liquid is solved. Theoretical and empirical results are compared. Different limiting cases for which the derived formulas lead to results known from published literature are considered.

Liquid Flow inside a Cylindrical Capillary with Walls Covered with a Porous Layer (Gel)

Viscous incompressible liquid flow in a long cylindrical capillary, the internal surface of which is covered with a permeable porous layer, is studied within the frameworks of three mathematical models. In the first model, the liquid flow in the porous layer is described by the Brinkman equation; according to the second one, the presence of the porous layer is taken into account using the Navier slip boundary conditions; and, in the third model, the Navier condition is imposed on the porous layer-liquid interface, with the flow inside the porous layer being excluded. The theoretical predictions are compared with the experimental data that one of us has obtained for liquid flow rates in porous capillaries. The validity and appropriateness of the application of the proposed models are discussed.

Permeability of medium composed of cylindrical fibers with fractal porous adlayer

The flow of viscous liquid in the porous medium formed by cylindrical fibers coated with a fractal porous adlayer is considered. Based on the Happel-Brenner cell method, the hydrodynamic permeability of a medium is calculated using the Brinkman equations. The analysis is performed for boundary conditions on cell surfaces of four types corresponding to the Happel, Kuwabara, Kvashnin, and Cunningham models. Different (transversal, longitudinal, and random) orientations of fibers with respect to liquid flow are considered.

Flow of viscous liquid in porous model medium with fractal structure

The flow of viscous liquid in a porous model medium with a fractal structure is considered. The hydrodynamic permeability of a medium is calculated according to the Happel and Brenner cell method using Brinkman equations. An analysis is carried out for boundary conditions on the cell surfaces of four types corresponding to the Happel, Kuwabara, Kvashnin, and Cunningham models. Situations that correspond to the flow of viscous liquid in a porous medium formed by fractal aggregates and the uniform flow of viscous liquid around the single composite particle are analyzed.

Hydrodynamic permeability of biporous membrane

This paper concerns the hydrodynamic permeability of biporous medium built up by porous cylindrical particles located in another porous medium by using cell model technique. It is continuation of the previous work of authors where biporous membrane was built up by porous spherical particles embedded in accompanying porous medium. Four known boundary conditions, namely, Happel’s, Kuwabara’s, Kvashnin’s and Cunningham/Mehta-Morse’s, are considered on the outer surface of the cell. The variation of hydrodynamic permeability of biporous medium (membrane) with viscosity ratio, Brinkman constants, and solid fraction are presented and discussed graphically. Comparison of the resulting hydrodynamic permeability is undertaken. Some previous results for dimensionless hydrodynamic permeability have been verified.

Uniform liquid flow around porous spherical capsule

The problem of a liquid flow that is uniform at infinity around a spherical porous capsule is solved. The flow in a porous layer is described by the Brinkman equation assuming that the viscosity of the Brinkman medium differs from the viscosity of the liquid flowing around. The tangential stress jump condition is imposed on the porous medium-liquid interface. Velocity and pressure distributions are determined and the hydrodynamic force applied to the capsule is calculated.

Cell model of biporous medium (membrane)

The hydrodynamic permeability of biporous medium (membrane) modeled by the set of porous particles located in the porous medium with other rheological properties is calculated using the cell method. All known boundary conditions on the cell surface, i.e., the Happel, Kuwabara, Kvashnin, and Cunningham boundary conditions, are considered. Significant limiting cases that lead to new or already known results are studied.