КОЛЛОИДНЫЙ ЖУРНАЛ, 2013, том 75, № 4, с. 524-533
УДК 532.546.2
HYDRODYNAMIC PERMEABILITY OF BIPOROUS MEMBRANE
© 2013 г. Pramod Kumar \kdav1, Ashish Tiwari2, Satya Deo3, Manoj Kumar \kdav1, Anatoly Filippov4*, Sergey V&sin4, Elena Sherysheva5
1 Department of Mathematics, National Institute of Technology Patna
Patna-800005 (Bihar), India 2Department of Mathematics, Birla Institute of Technology and Science Pilani-333031, (Rajasthan), India 3Department of Mathematics, University of Allahabad Allahabad-211002 (U.P.), India 4Department of Higher Mathematics, Gubkin Russian State University of Oil and Gas Leninskii pr. 65-1, Moscow, 119991 Russia 5Frumkin Institute of Physical Chemistry and Electrochemistry RAS Leninskii pr. 31, Moscow, 119991 Russia Поступила в редакцию 12.09.2012 г.
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.
DOI: 10.7868/S0023291213040186
INTRODUCTION
Flow through porous media has been a topic of longstanding interest for researchers due to its numerous applications in bio-mechanics, physical sciences and chemical engineering etc. For effective use of a porous medium in the above areas, the structure of porous layer should be viewed from all angles e.g. it is not necessary that the particles always have a smooth homogeneous surface but also have a rough surface or a surface covered by porous shell. For the medium of high porosity, the sum suggested by Brinkman [1] is more suitable for describing the flow through the porous medium. He evaluated the viscous force exerted by a flowing fluid on a dense swarm of particles by modifying Darcy's equation for porous medium, which is commonly known as Brinkman equation. In many technical and technological problems that arise during the study of the permeability of aqueous conglomerates composed primarily of one size particles, it is important to determine the regularities of its variations upon the addition of a certain amount of particles with quite different characteristic sizes. These problems include the determination of the permeabil-
* Corresponding author. E-mail address: a.filippov@mtu-net.ru.
ity of sugar syrups on the growth of large crystals and the formation of clusters [2], finding the permeability of forming ion-exchange membranes during the variations in their structural composition, the determination of the permeability of liquid concrete upon the addition of large gravel, etc.
The problem of flow through a swarm of particles become complex, if we consider the solution of the flow field over the entire swarm by taking exact positions of particles. In order to avoid the above complication, it is sufficient to obtain the analytical expression by considering the effects of the neighboring particles on the flow field around a single particle of the swarm, which can be used to develop relatively simple and reliable models for heat and mass transfer. This has lead to the development ofparticle-in-cell models.
Uchida [3] proposed a cell model for a sedimenting swarm of particles, considering spherical particle surrounded by a fluid envelope with cubic outer boundary. Happel [4, 5] proposed cell models in which the particle and outer envelope, both are spherical/cylindrical. He solved the problem when the inner sphere/cylinder is solid with respective boundary conditions on the cell surface. The Happel model assumes uniform velocity condition and no tangential stress at
the cell surface. The merit of this formulation is that, it leads to an axially symmetric flow that has a simple analytical solution in closed form, and thus can be used for heat and mass transfer calculations. Analytical solutions of particle-in-cell models discussed above are always practically useful to many industrial problems, but the solutions of creeping flow for the above models have not been found in case of complex geometry. Kuwabara [6] proposed again a cell model in which he used the nil vorticity condition on the cell surface to investigate the flow through swarm of spherical/cylindrical particles. However, Kuwabara formulation requires a small exchange of mechanical energy with the environment. The mechanical power given by the sphere to the fluid is not all consumed by viscous dissipation in the fluid layer. Apart from this, Kvashnin [7] and Mehta-Morse [8] gave their respective boundary conditions for the outer cell surface. Kvashnin [7] proposed the condition that the tangential component of velocity reaches a minimum at the cell surface with respect to radial distance, signifying the symmetry on the cell. However, Mehta-Morse [8] used Cunningham's [9] approach by assuming the tangential velocity as a component of the fluid velocity, signifying the homogeneity of the flow on the cell boundary. The importance of the Mehta-Morse [8] boundary condition is that since we are interested in the flow behavior on a large scale, we shall average the flow variables on the small scale over a cell volume to obtain large scale behavior.
A Cartesian-tensor solution of the Brinkman equation was investigated by Qin and Kaloni [10] and they also evaluated the drag force on a porous sphere in an unbounded medium. Flow through beds of porous particles was studied by Davis and Stone [11] and they evaluated the overall bed permeability of swarm by using cell model. Vasin and Filippov [12], Filippov et al. [13] evaluated the hydrodynamic permeability of membrane of porous spherical particles using Mehta-Morse condition on the cell surface. Recently, Vasin et al. [14, 15] compared all four cell models to evaluate the permeability of membrane of porous spherical particles with a permeable shell and discussed the effect of different parameters on the hydrodynamic permeability of the membrane for all the four above mentioned boundary conditions. Deo and Yadav [16] studied the problem of Stokes flow through a swarm of porous circular cylinder-in-cell enclosing an impermeable core with Kuwabara's and Happel's boundary conditions. The stream function for a slow viscous flow through an array of porous cylindrical particles with Happel's boundary condition was considered in [17]. The flow patterns along with the drag force exerted to each porous cylindrical particle in a cell were evaluated.
The hydrodynamic permeability of membranes built up by spherical particles covered by porous shells was discussed by Yadav et al. [18]. Deo et al. [19] studied hydrodynamic permeability of membranes built up by porous cylindrical or spherical particles with impermeable core using cell model technique. They used
different versions of a cell method to calculate the hydrodynamic permeability of the membranes and utilized the boundary condition of tangential stress jump at the interface between porous shell and clear liquid. They studied also both transversal and normal flows of liquid with respect to the cylindrical fibers that compose the membrane. The hydrodynamic permeability of biporous medium (membrane) modeled by the set of porous spherical particles located in the porous medium with other rheological properties is calculated using the cell method by Vasin et al. [20]. The motivations of mentioned papers and especially last article [20] lead us to discuss the present problem for porous cylindrical particles located not in clear liquid but in the porous medium, which includes some earlier known results.
Therefore this paper concerns the hydrodynamic permeability of biporous medium (membrane) built up by porous cylindrical particles embedded in another porous medium by using cell model technique. It is assumed that the biporous medium composed two types ofparticles — conventionally small (as referred to their diameter) cylindrical or spherical particles and large cylindrical particles, which are parallel each other. So, in comparison to large particle, the concentration of small particles is higher and characteristic size is lower. The medium formed by the particles of the fine fraction is assumed to be continuous as it makes the large particles submerge into the porous medium of small cylindrical or spherical particles. Such kind of medium can be constructed, for example, when homogeneous porous pattern (ion-exchange membrane material) is reinforced by the set of identical parallel porous fibers. Four known boundary conditions, namely, Happel's, Kuwabara's, Kvashnin's and Cun-ningham/Mehta—Morse's, are considered on the outer surface of the cell. The variation of hydrodynamic permeability of biporous medium (membrane) with viscosity ratio m, Brinkman constants si, i = 1, 2, and solid volume fraction y are presented and discussed graphically. Some previous results for dimensionless hydrodynamic permeability have been verified. The problem of flow in media with low and high concentrations of large particles leads precisely to this physical and mathematical formulation. For the medium of high concentrations of large particles, small cylindrical particles merely fill the space between large cylindrical or spherical particles, thus forming a highly porous medium. The only distinction between two aforementioned situations is in the difference of the relative concentrations of large and small particles in a conglomerate.
MATHEMATICAL FORMU
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