Kozeny Constant Research Articles

Asymptotic analysis of electrohydrodynamic flow through a swarm of porous cylindrical particles

The present article reveals the study of an electrohydrodynamic flow through a membrane composed of a swarm of porous layered cylindrical particles adopting a heat transfer approach. The configuration of the proposed theoretical model is segregated into two regions in which the region proximate to the solid core of the cylindrical particle is a porous region. However, a region surrounded by a porous region is a non-porous (clear fluid) region. The thermal equations are employed under steady-state conditions to establish the temperature distribution when heat conduction prevails over heat convection. The Brinkman and Stokes equations regulate fluid flow through a swarm of porous layered cylindrical particles in porous and non-porous regions, respectively. With the purpose of addressing an electric field in the fluid flow process through a swarm of porous layered cylindrical particles to understand the role of a Hartmann electric number, the momentum equation and the charge density are coupled and nonlinear. The nonlinear second-order differential equation governs the momentum equation and regulates fluid flow through a swarm of porous cylindrical particles. The solutions of the energy equations for both regions are analytically obtained. The asymptotic expansions of velocities for porous and non-porous regions have been derived using the perturbation technique for the small and large values of the nonlinearity parameter α. The effects of various parameters like Hartmann electric number, Grashof number, radiation parameter, viscosity ratio parameter, and porosity of the porous material on the hydrodynamical permeability, Kozeny constant of the membrane, and temperature are analyzed graphically. A noteworthy observation is that a rising Hartmann electric number, the ratio of electric force to the viscous force, enhances the velocity, which is relatively more significant for higher permeability and hence enhances the membrane permeability; however, decay in Kozeny constant is reported with a rising Hartmann electric number. Significant velocity and membrane permeability growth are described with a rising Grashof number, a ratio of thermal buoyancy and viscous forces. The observations from the present study hold promise for advancing our understanding of critical physical and biological applications, including wastewater treatment filtration processes, petroleum reservoir rocks, and blood flow through smooth muscle cells.

Creeping flow of non-Newtonian fluid through membrane of porous cylindrical particles: A particle-in-cell approach

The present study is an attempt to deal with hydrodynamic and thermal aspects of the incompressible Carreau fluid flow past a membrane consisting of uniformly distributed aggregates of porous cylindrical particles enclosing a solid core which aims to provide a comprehensive study of the impact of non-Newtonian nature of Carreau fluid in the filtration process through membranes. The non-Newtonian characteristic of Carreau fluid is adopted to describe the mechanism of the pseudoplastic flow through membranes. The layout of the fluid flow pattern is separated into two distinct areas in which the area adjacent to the solid core of the cylindrical particle is considered as porous. However, the region surrounding the porous cylindrical particle is taken as non-porous (clear fluid region). The Brinkman equation governs the porous region, whereas the non-porous region is regulated by the Stokes equation. The nonlinear governing equations of the Carreau fluid flow in the different regions are solved using an asymptotic series expansion in terms of the small parameters, such as Weissenberg number ( We ≪ 1 ) and a non-dimensional parameter ( S ≪ 1 ), for the higher permeability of the porous material. For large permeability, the expression of velocity is derived, and the same has been used to compute the hydrodynamic permeability, Kozeny constant, and temperature profile. The numerical scheme (NDSolve in Mathematica) is used to solve the singularly perturbed boundary value problems in the case of small permeability of the porous medium [i.e., ( S ≫ 1 )]. The graphical analysis illustrating the outcomes of the effects of varying control parameters such as the power-law index, viscosity ratio parameter, permeability of the porous medium, Weissenberg number, and Nusselt number on the membrane permeability, Kozeny constant and temperature profile are discussed comprehensively and validated with previously published works on the Newtonian fluid in the limiting cases. The notable determination of the present study is that the Carreau fluid parameters, such as the Weissenberg number, power-law index, and viscosity ratio parameter, have a significant impact on the velocity, and hence, the membrane permeability, Kozeny constant, and temperature profile. The results showed a significant increase in the flow velocity and hydrodynamic permeability as the dominance of elastic forces over viscous forces increased in the case of high permeability ( S ≪ 1 ). The velocity gets a slight reduction for lower permeability of the porous material ( S ≫ 1 ); however, the hydrodynamic permeability behaves similar to the higher permeability of the porous material. The findings of the proposed work may be instrumented in analyzing various processes, including wastewater treatment filtration processes, and blood flow through smooth muscle cells. The proposed work, however, requires experimental verification.

Parallel and perpendicular flows of a couple stress fluid past a solid cylinder in cell model: Slip condition

The axisymmetric steady flow of a couple stress fluid between two concentric cylinders with a slip effect is investigated with the help of the cell model technique. Here, the inner cylinder is rigid, and the outer cylinder is fictitious. The tangential slip, vanishing of normal velocity, and zero couple stress conditions are applied on the inner cylindrical surface. In addition, zero shear stress (Happel's model), continuity of normal velocity component, and zero couple stress conditions are used on the outer cylindrical surface. We consider two flow problems: the first is the parallel flow, and the second is the perpendicular flow to the cylinder in the cell model. Also, we have discussed the random case. For all the cases, the Kozeny constant is calculated. We described some special cases and compared them with well-known results. The effects of slip and couple stress parameters on the Kozeny constant with fixed value of couple stress viscosity parameter are presented graphically. The influence of the couple stress viscosity parameter on the Kozeny constant with fixed values of couple stress, and slip parameters for parallel flow are expressed graphically. The numerical values for the Kozeny constant for different values of fractional void volume are tabulated. We also obtained the results of the consistent couple stress theory as a special case.

Numerical Simulation and Experimental Studies of Gas Pressure Infiltration Al-356/SiC Composites

In this study, the filling process, solidification parameters, temperature distribution, and residual stress distribution of gas pressure-infiltrated Al-356/SiC composites were investigated through simulation and experiment. In addition, a series of orthogonal tests was also carried out to precisely demonstrate the preheating temperature, infiltration temperature, and infiltration pressure. After a thorough analysis, the orthogonal tests revealed that the optimal process parameters are as follows: the SiC preheating temperature is 550 °C, the infiltration temperature of the Al-356 alloy is 620 °C, and the infiltration pressure is 8 MPa. The simulation results revealed that pressure had a sharp decrease of ~87% during filling, and the critical pressure was ~0.12 MPa. The velocity decreased with the increase in the filling time, and the average velocity was ~2.60 ms−1. Feasible analysis suggested that critical pressure is ~0.11 MPa and average velocity is ~4.20 ms−1; this difference is attributed to apparent velocity and the Kozeny constant. In the solidification process, shrinkage porosity appeared in the centers of the composites, which is evident with scanning electron microscopy. Moreover, the stress concentration of 171.3 MPa appeared in the composite region connected with the runner, which is the cause of the nucleation of the crack. However, based on the optimum orthogonal parameters and simulative results, the stress concentration was reduced, and crack-free and porosity-free composites were achieved.

Prediction of permeability of five-harness satin fabric by a modified Kozeny constant determined from experiments

Permeability is a critical parameter not only in flow simulation analysis but also in liquid composite molding process. When a liquid resin is infused into a dry preform, the impregnation is mainly characterized by the permeability. The permeability of a dry preform can be obtained through theoretical and experimental methods. In the theoretical estimation of permeability, the effects of fiber arrangement as well as fabric type and form for various types of preforms are not sufficiently reflected in the calculation. Thus, there is a gap between the theoretical and experimental permeability. Recently, experimental determination has been gaining considerable attention as a mean to obtain accurate permeability values; however, it requires a number of trials. In this study, the permeability of the Hexforce G0926 5HS (five-harness satin) carbon fabric preform is estimated using representative theoretical prediction models, the Gebart and Kozeny–Carman equations. In addition to the Kozeny–Carman permeability (using the Kozeny constant values from literature), the Kozeny constant obtained through experiments was used to obtain a modified Kozeny–Carman permeability. All three calculated permeabilities were compared and verified with the fabric manufacturer’s reference value. The results showed that the modified Kozeny–Carman permeability using the experimentally determined Kozeny constant was closest to the reference value at 57% fiber volume fraction. Further, the predicted permeability was compared with other experimental permeability values from literature over the 40%–65% range of fiber volume fraction. We found that the modified Kozeny–Carman permeability once again came closest to the literature values. Finally, an optimized fitting equation was proposed to replace the Kozeny–Carman equation for predicting the permeability of Hexforce G0926 5HS carbon fabric over the 40%–65% fiber volume fraction range.

Creeping flow of Jeffrey fluid through a swarm of porous cylindrical particles: Brinkman–Forchheimer model

The majority of the previous studies analyzed the flow of fluids with constant viscosity through membranes composed of porous cylindrical particles using the particle-in-cell approach with the Brinkman equation governing the flow through porous media. However, a slight variation in temperature affects the viscosity of the fluids and hence affects the filtration process of fluids through membranes. The motivation of this problem came from the fact that viscosity is concentration dependent due to presence of impurities and contaminants in the fluids and hence can be taken as function of position or temperature. The present work is a theoretical attempt to investigate the impact of temperature-dependent viscosity on the creeping flow of Jeffrey fluid through membrane consisting of the aggregates of the porous cylindrical particles. The flow pattern of the Jeffrey fluid is taken along the axial direction of the cylindrical particles, and the cell model approach is utilized to formulate the governing equations driven by a constant pressure gradient. The flow regime is divided into two-layer form, one is inside the porous cylindrical particle enclosing a solid core, which is governed by the Brinkman–Forchheimer equation, and another one is outside of the porous cylindrical particle, which is governed by the Stokes equation. Being a nonlinear equation, an analytical solution of the Brinkman–Forchheimer equation is intractable. To overcome this difficulty, the regular and singular perturbation methods have been employed to solve the Brinkman–Forchheimer equation under the assumption of temperature-dependent viscosity for small and large permeability of the porous medium, respectively; however, an analytical approach is utilized to solve the Stokes equation. The analytical expressions for velocity in different regions, hydrodynamic permeability of the membrane, and Kozeny constant are derived. The impact of various control parameters such as viscosity parameter, Forchheimer number, permeability of the porous medium, and Jeffrey fluid parameter on the above quantities are discussed and validated with previously published works on the Newtonian fluid in the limiting cases. The present work is in good agreement with the previously published work on Newtonian fluid under constant viscosity assumptions where the porous media flow was governed by the Brinkman equation. The remarkable observation of the present study is that higher viscosity and Jeffrey fluid parameters lead to enhanced velocity profile and hence the hydrodynamical permeability of the membranes. However, a decay in the Kozeny constant is observed with the increasing viscosity and Jeffrey fluid parameters. The coating of porous layer can be attributed to adsorption of polymers on the solid particles and further makes the present model to be more relevant in understanding the membrane filtration process.

Permeability variation analysis using the superficial diameter correlation with porosity change

Permeability characterization is a major factor for ensuring more environment-friendly operations and economically viable industrial applications related to carbon capture sequestration, hydrocarbon recovery, nuclear waste disposal, and remediation in groundwater. Regardless, the permeability variation caused by changes in formation stress is simply defined as the power-law function of porosity. An alternative formula can be presented using the Kozeny–Carman equation based on hydraulic diameter and tortuosity. However, the hydraulic tortuosity and the Kozeny constant cannot be precisely measured because of the extremely complex and microscale pores. Accordingly, this study considers the Kozeny–Carman equation for presenting the other definable variables and more general correlations for performing permeability variation analyses. Herein, the effective tortuosity and effective and superficial diameters of porous media are deduced adopting the conventional viscous flow theory. Subsequently, the Kozeny–Carman equation is improved by replacing the immeasurable variables with the effective variables. The correlations of all the key geometric variables with permeability variation are investigated via pore-scale simulations based on two types of 20-series porous medium models with a wide range of porosity (13.4%–47.4%) and permeability (0.0073 –18.3 Darcy). Herein, several impressive functional aspects of the superficial diameter were discovered with porosity changes, such as quadratic functional correlations, parallel shifts for each flow path, and less sensitive variations in low porosity ranges. Consequently, this study proved that permeability variations can be more precisely and generally estimated using the quadratic correlations of the superficial diameter with porosity changes.

Permeability of carbon refractory materials used in a blast furnace hearth

Permeability of experimental and standard micropore carbon materials, before and after the heat treatment at 1550 °C, was investigated using gas permeability measurements (GP) and mercury intrusion porosimetry (MIP). Permeability of both materials increased after the annealing. Experimental micropore carbon material was less permeable than standard one, both before and after the heat treatment. Measured gas permeabilities were compared with permeabilities determined based on MIP measurements using two different models: Hagen−Poiseuille and Carman−Kozeny equation. Using the inverse problem approach, for the first time Carman−Kozeny constant was determined for micropore carbon materials. Carman−Kozeny model gave more accurate results (i.e. close to gas permeability measurements) than Hagen−Poiseuille model. Performed investigations showed that mercury intrusion porosimetry might be alternative method to investigate infiltration resistance of low permeable materials.

Flow past composite cylindrical shell of porous layer with a liquid core: magnetic effect

The creeping flow of a magnetic fluid perpendicular to a porous cylindrical shell is investigated, employing the unit cell model. The viscous fluid is assumed to be flowing in three zones divided as fluid, annular porous, and cavity regions, respectively. We apply a uniform magnetic field in a transverse direction of flow and then emphasize the influence of the Hartmann layers which are developed in their vicinity. Modified Stokes and modified Brinkman’s equations are employed in the liquid and porous regions, respectively. Happel and Kuwabara cell models are used as the interface conditions at the cell surface, and at the fluid–porous interface, continuity of velocity components, continuity of normal stresses, and stress jump condition for tangential stresses are applied. An expression for Kozeny constant for the cylindrical shell is presented. Representation of Kozeny constant under the influences of the pertinent parameters such as Hartmann numbers, stress jump coefficient, fractional void volume, viscosity ratios, permeability, and separation is displayed through graphs and a table. The results are compared with the cases which do not involve magnetic effect. They reveal the strong impact of Hartmann’s numbers on the resisting force experienced on the cylindrical shell. The results agree well with previous available works.

Magnetohydrodynamic creeping flow around a weakly permeable spherical particle in cell models

The present paper studies the impact of applied uniform transverse magnetic field on the flow of incompressible conducting fluid around a weakly permeable spherical particle bounded by a spherical container. Analytical solution of the problem is obtained using Happel and Kuwabara cell models. The concerned flow is parted in two regions, bounded fluid region and internal porous region, to be governed by Stokes and Darcy’s law respectively. At the interface between the fluid and the permeable region, the boundary conditions used are continuity of normal component of velocity, Saffman’s boundary condition and continuity of pressure. For the cell surface, Happel and Kuwabara models together with continuity in radial component of the velocity has been used. Expressions for drag force, hydrodynamic permeability and Kozeny constant acting on the spherical particle under magnetic effect are presented. Representation of hydrodynamic permeability for varying permeability parameters, particle volume fraction, slip parameter and Hartmann numbers are represented graphically. Also, the magnitude of Kozeny constant for weakly permeable and semipermeable sphere under a magnetic effect has been presented. In limiting cases many important results are obtained.

Establishment of the Permeability Model for Soft Solid Sludge Conditioned with Flocculants.

Permeability plays a decisive role in the dewatering process and reflects the difficulty of filtration, especially for soft solid material such as sludge. In this paper, the physicochemical properties and dewatering performance of sludge conditioned with different kinds of flocculants were investigated. Results showed that the flocculant could change the sludge microstructure such as floc morphology, specific surface area, and fractal dimension. Compared with filtration pressure, flocculants had a greater influence on sludge permeability which was a significant negative correlation with filtration pressure and was a positive correlation with flocculant dosage. In order to describe the fact that fluid flows through the porous voids for soft solid sludge, the improved Kozeny constant was corrected. Research showed that permeability was more significant in the dewatering process for the sludge conditioned with inorganic flocculants than that with organic flocculants. The Kozeny constant was not only relevant with suspension nature but also with filtration pressure. The range of the improved Kozeny constant was reasonably determined based on flocculant type, concentration, and filtration pressure, which was of great help to project applications. For raw sludge, the improved Kozeny constant was 958 times than that of the original value, and it decreased significantly for conditioned sludge.

Impact of magnetic field on flow past cylindrical shell using cell model

The present paper concerns about finding the impact of applied transverse magnetic field on parallel cylindrical shell of magneto viscous fluid by unit cell model. The considered flow is divided into three regions, bounded fluid region, porous region and inner cavity region, where the flow in the bounded and cavity regions is governed by Stokes equation and flow in the annular porous region is governed by Brinkman’s equation in the presence of magnetic field. The boundary conditions used at the fluid–porous interface are continuity of velocity components and stress jump condition for tangential stresses together with Happel and Kuwabara boundary conditions. Expression for volumetric flow rate in the presence of transverse magnetic field is calculated, and limiting cases leads to some well-known results. The effect of Kozeny constant versus fractional void volume for varying permeability, Hartmann numbers, viscosity ratio, separation parameter and stress jump coefficient is tabulated and represented by graphs. In the limits of the motion of porous cylinder and impermeable cylinder in the cell, the numerical values of the Kozeny constant are in good agreement with the available values in the literature.

Modeling Based on Kozeny Model of Air Permeability of Nonwoven -Punched Original and Treated

An analysis is made of application of Kozeny approach for estimating air permeability of nonwoven needle-punched fabrics based on fibres with 0.33 and 1.7 tex linear densities and a blend of these fibres as well as of treated materials based on polyester and bicomponent fibres. The inverse problem, which boils down to calculation of the Kozeny constant using experimentally derived permeability coefficient of fabrics and materials is solved. The Kozeny constant depends on the composition of the fabrics and materials and diminishes with transition from fabrics made with fibres of different linear densities to treated materials.

An Equivalent Spherical Particle System to Describe Characteristics of Flow in a Dense Packing of Non-spherical Particles

Effective characteristics, such as the effective particle diameter and hydraulic radius, are usually used to determine the pressure loss associated with single-phase flow through an isotropic, dense non-spherical granular packing. These quantities can be considered as those of an equivalent spherical particle system. This paper reviews different methods to select effective particle diameters taking into account the effect of particle shape. A new method is proposed to determine an equivalent spherical particle system with both the hydraulic radius and the pressure loss equivalent to those in the actual non-spherical particle packing. This equivalent system can be used to determine the characteristics of flow in the actual system. The analyses provide theoretical justification for adoption of proper effective particle diameters and porosity as well as the dependency of Ergun–Kozeny constant on the shape factor of particles. It is also shown theoretically that the critical Reynolds number for Darcian flow depends on the porosity and the shape factor of particles.

Establishment and application of fractal capillary tube bundle model of porous media

In view of the problem of statistical regression constant in the model of capillary tube bundles in the porous media, a capillary bundle percolation model with fractal geometry was reconstructed. The function expressions of the fractal coefficient and Kozeny constant were deduced. The relationship between the macroscopic fractal properties of porous media and the fractal dimension and the micro pore parameters were obtained. Results show: Fractal coefficient is a function of fractal dimension, maximum pore radius and minimum pore radius; The macroscopic physical properties of porous media are a function of the fractal dimension and the radius of the capillary (the maximum capillary radius and the minimum capillary radius). The expression does not contain any empirical or experimental constants. In the fractal capillary percolation model, the relationship between the three kinds of surface volume, skeleton volume and pore volume are the same as the traditional equal diameter straight capillary bundle model. The Kozeny constant can be accurately described by the function expression of the z-h coefficient, which is used for correcting the difference between real and ideal porous media model.

Permeability from Microscopy: Review of a Dream

The Kozeny–Carman and Timur-type equations connecting porosity and permeability contain rock-textural constants such as tortuosity and specific surface area. Sometimes these are combined in single factors as Kozeny constant or flow zone index. The partial differential equations of flow in triple-porosity rocks contain transfer factors, interporosity flow shape factors between different kinds of pores, as well as their individual storativities. Without knowing these constants, no meaningful permeability prediction or flow simulation is possible. The paper reviews the main ideas of how to find such rock-textural properties directly from the microscopic image.

Parallel and perpendicular flows of a micropolar fluid between slip cylinder and coaxial fictitious cylindrical shell in cell models

The quasisteady two-dimensional micropolar fluid flow between two coaxial cylinders is considered. The inner cylinder is assumed to be solid while the outer one is fictitious. Happel's cell model has been applied. Linear slip boundary conditions on the surface of the solid cylindrical particle are used for both velocity and microrotation. The modified Happels boundary conditions are applied at the fictitious cylindrical surface. Two flow problems are considered: the flow parallel to cylinders and the flow perpendicular to cylinders. Analytical expressions for Kozeny constant are obtained. The numerical values for the Kozeny constant are tabulated and represented graphically for various values of the physical parameters of the problem.

Transport properties of electrospun fibrous membranes with controlled anisotropy

In-plane gas permeability measurements were performed on electrospun polyacrilonitrile fibrous mats. Samples were cast onto a drum rotating at various speeds in order to create materials with different degrees of fiber alignment. Permeability measurements were made in both parallel and perpendicular directions and large increases were seen for flow in the fiber direction as expected. Because these measurements were performed in-plane it was necessary to mechanically compress the samples during measurement. Tests were performed at several compressed thicknesses, but it was found that the samples did not compress uniformly so it was not possible to determine a unique Carman–Kozeny constant for the material. A methodology was developed by which the Carman–Kozeny constant was determined for each compression that produced a linear trend against compressed porosity. It was proposed that the true Carman–Kozeny constant for the material can be determined by extrapolating this trend to the value of the uncompressed porosity, and good agreement with the literature values was obtained.

Determination of Kozeny Constant Based on Porosity and Pore to Throat Size Ratio in Porous Medium with Rectangular Rods

Kozeny-Carman permeability equation is an important relation for the determination of permeability in porous media. In this study, the permeabilities of porous media that contains rectangular rods are determined, numerically. The applicability of Kozeny-Carman equation for the periodic porous media is investigated and the effects of porosity and pore to throat size ratio on Kozeny constant are studied. The continuity and Navier-Stokes equations are solved to determine the velocity and pressure fields in the voids between the rods. Based on the obtained flow field, the permeability values for different porosities from 0.2 to 0.9 and pore to throat size ratio values from 1.63 to 7.46 are computed. Then Kozeny constants for different porous media with various porosity and pore to throat size ratios are obtained and a relationship between Kozeny constant, porosity and pore to throat size ratio is constructed. The study reveals that the pore to throat size ratio is an important geometrical parameter that should be taken into account for deriving a correlation for permeability. The suggestion of a fixed value for Kozeny constant makes the application of Kozeny-Carman permeability equation too narrow for a very specific porous medium. However, it is possible to apply the Kozeny-Carman permeability equation for wide ranges of porous media with different geometrical parameters (various porosity, hydraulic diameter, particle size and aspect ratio) if Kozeny constant is a function of two parameters as porosity and pore to throat size ratios.

Nanoscale infiltration behaviour and through-thickness permeability of carbon nanotube buckypapers

Carbon nanotube thin films or ‘buckypapers’ show potential for various applications including electrodes for energy devices, nanoscale filtration devices and composite materials. This paper reports on the study of through-thickness permeability of different buckypaper materials. The infiltration behaviours of different liquids into four types of buckypaper were investigated. Infiltration of the liquids into buckypaper was found to follow Darcy’s law, except in the case of epoxy resin solution permeation into SWNT buckypaper. The results revealed that the permeability of SWNT buckypaper was of the order of 10−19 m2, which is about two orders of magnitude lower than the 10−17 m2 permeability for the MWNT buckypaper. The factors of wider pores, higher porosity and less surface area appear to contribute to a higher permeability, which is consistent with Darcy’s law and the Kozeny–Carman model. The Kozeny constants of buckypapers correlated well with the tortuosity of their flow paths and nanoscale pore size. The polarity of working fluids did not show an impact on the permeability. Solutions with molecular size near the size of the nanopores in the buckypaper led to lower permeability due to the occurrence of pore blockage. In addition, a threshold pressure existed for liquid to infiltrate into nanoscale pores in buckypapers, which does not exist in fibre reinforcement preforms.