Kozeny Equation Research Articles

Modeling of transient permeate flux decline during crossflow microfiltration of non-alcoholic beer with consideration of particle size distribution

► Modeling of transient flux in crossflow MF of rough beer was performed. ► Predict the cake layer composition with different particle sizes at each time step. ► Effect of particle sizes on membrane fouling and cake layer formation studied. ► Better agreement of model with data considering particle size distribution (PSD). Crossflow microfiltration of non-alcoholic beer is investigated numerically and it has been verified by experimental data. Due to the presence of particles with different sizes in feed suspension, a modified combination of three mechanisms of particle back-diffusion is developed to predict particle deposition and cake layer buildup during the process. The simulation results show that smaller particles (about 1 μm) are the main contributor to the cake layer due to a minimum in back transport and are the main reason of the flux decline. On the other hand, larger particles ( a p > 20 μm) are swept away along the membrane during the filtration process and move toward the membrane exit due to the crossflow. Therefore, the cake enriches with finer particles as the permeate flux decreases. The specific cake resistance was determined experimentally and Carman–Kozeny equation fails to provide a complete theoretical perspective of specific resistance of irregular, non-uniformly sized particles that formed the cake layer. The results of numerical simulation are in good agreement with observed data.

Evaluation of through-thickness permeability and the capillary effect in vacuum assisted liquid molding process

Through-thickness penetration under vacuum assistance is crucial for resin film infusion (RFI) and vacuum assistant resin transfer molding (VARTM) process. In this paper, values of the through-thickness unsaturated permeability (TTUP) and capillary pressure (Pc) are estimated based on the infiltration velocity in preforms of carbon fiber fabric and glass fiber fabric, respectively, measured by a specially designed apparatus. It reveals that, for the through-thickness permeation, the Pc values generally decrease with increasing fiber content. Relatively accurate TTUP can be obtained by counting Pc into the permeation dynamics. If Pc is neglected, liquids with good-wettability, such as silicone oil, tend to result in larger TTUPs. The corrected TTUPs show good agreement according to Carman–Kozeny, Gutowski modified Carman–Kozeny equation, and Gebart model, respectively. The resultant permeability resistance parameters of the preforms indicate that the penetration in carbon fabric bed is slower than in glass fabric bed. However, for fiber volume fraction more than 60%, the corrected TTUPs show no significant difference for all the preforms.

Single- and two-phase flow in microfluidic porous media analogs based on Voronoi tessellation

The objective of this study was to create a microfluidic model of complex porous media for studying single and multiphase flows. Most experimental porous media models consist of periodic geometries that lend themselves to comparison with well-developed theoretical predictions. However, many real porous media such as geological formations and biological tissues contain a degree of randomness and complexity at certain length scales that is not adequately represented in periodic geometries. To design an experimental tool to study these complex geometries, we created microfluidic models of random homogeneous and heterogeneous networks based on Voronoi tessellations. These networks consisted of approximately 600 grains separated by a highly connected network of channels with an overall porosity of 0.11-0.20. We found that introducing heterogeneities in the form of large cavities within the network changed the permeability in a way that cannot be predicted by the classical porosity-permeability relationship known as the Kozeny equation. The values of permeability found in experiments were in excellent agreement with those calculated from three-dimensional lattice Boltzmann simulations. In two-phase flow experiments of oil displacement with water we found that the wettability of channel walls determined the pattern of water invasion, while the network topology determined the residual oil saturation. The presence of cavities increased the microscopic sweeping efficiency in water-oil displacement. These results suggest that complex network topologies lead to fluid flow behavior that is difficult to predict based solely on porosity. The novelty of this approach is a unique geometry generation algorithm coupled with microfabrication techniques to produce pore scale models of stochastic homogeneous and heterogeneous porous media. The ability to perform and visualize multiphase flow experiments within these geometries will be useful in measuring the mechanism(s) of displacement within micro- and nanoscale pores.

Numerical simulation of full-penetration laser beam welding of thick aluminium plates with inductive support

A three-dimensional laminar steady-state numerical model was developed to investigate the influence of an alternating current (ac) magnetic field during high-power full-penetration laser welding on the weld pool dynamics and weld cross section of a 20 mm thick aluminium plate in flat position. Three-dimensional heat transfer, fluid dynamics including phase transition and electromagnetic field partial differential equations were solved iteratively with the commercial finite element software COMSOL Multiphysics using temperature-dependent material properties up to evaporation temperature. Thermocapillary convection at the weld pool surfaces, natural convection and latent heat of solid–liquid phase transition were taken into account in this model. Solidification was modelled by the Carman–Kozeny equation for porous media morphology. The ac magnet was mounted on the root side of the weld specimen. The magnetic field was aligned perpendicular to the welding direction. The flow pattern in the melt and thus also the temperature distribution were significantly changed by the application of oscillating magnetic fields. It was shown that the application of an ac magnetic field to laser beam welding allows for a prevention of the gravity drop-out. The simulation results are in good qualitative agreement with the experimental observations.

Prediction of permeability of regular scaffolds for skeletal tissue engineering: A combined computational and experimental study

Scaffold permeability is a key parameter combining geometrical features such as pore shape, size and interconnectivity, porosity and specific surface area. It can influence the success of bone tissue engineering scaffolds, by affecting oxygen and nutrient transport, cell seeding efficiency, in vitro three-dimensional (3D) cell culture and, ultimately, the amount of bone formation. An accurate and efficient prediction of scaffold permeability would be highly useful as part of a scaffold design process. The aim of this study was (i) to determine the accuracy of computational fluid dynamics (CFD) models for prediction of the permeability coefficient of three different regular Ti6Al4V scaffolds (each having a different porosity) by comparison with experimentally measured values and (ii) to verify the validity of the semi-empirical Kozeny equation to calculate the permeability analytically. To do so, five CFD geometrical models per scaffold porosity were built, based on different geometrical inputs: either based on the scaffold’s computer-aided design (CAD) or derived from 3D microfocus X-ray computed tomography (micro-CT) data of the additive manufactured (AM) scaffolds. For the latter the influence of the spatial image resolution and the image analysis algorithm used to determine the scaffold’s architectural features on the predicted permeability was analysed. CFD models based on high-resolution micro-CT images could predict the permeability coefficients of the studied scaffolds: depending on scaffold porosity and image analysis algorithm, relative differences between measured and predicted permeability values were between 2% and 27%. Finally, the analytical Kozeny equation was found to be valid. A linear correlation between the ratio Φ3/Ss2 and the permeability coefficient k was found for the predicted (by means of CFD) as well as measured values (relative difference of 16.4% between respective Kozeny coefficients), thus resulting in accurate and efficient calculation of the permeability of regular AM scaffolds.

Permeability prediction in chalks

The velocity of elastic waves is the primary datum available for acquiring information about subsurface characteristics such as lithology and porosity. Cheap and quick (spatial coverage, ease of measurement) information of permeability can be achieved, if sonic velocity is used for permeability prediction, so we have investigated the use of velocity data to predict permeability. The compressional velocity from wireline logs and core plugs of the chalk reservoir in the South Arne field, North Sea, has been used for this study. We compared various methods of permeability prediction from velocities. The relationships between permeability and porosity from core data were first examined using Kozeny's equation. The data were analyzed for any correlations to the specific surface of the grain, Sg, and to the hydraulic property defined as the flow zone indicator (FZI). These two methods use two different approaches to enhance permeability prediction from Kozeny's equation. The FZI is based on a concept of a tortuous flow path in a granular bed. The Sg concept considers the pore space that is exposed to fluid flow and models permeability resulting from effective flow parallel to pressure drop. The porosity-permeability relationships were replaced by relationships between velocity of elastic waves and permeability using laboratory data, and the relationships were then applied to well-log data. We found that the permeability prediction in chalk and possibly other sediments with large surface areas could be improved significantly using the effective specific surface as the fluid-flow concept. The FZI unit is appropriate for highly permeable sedimentary rocks such as sandstones and limestones that have small surface areas.

Evaluation of different methods for the prediction of saturated hydraulic conductivity in tilled and untilled soils

The larger the bulk density of the soil, the smaller the saturated hydraulic conductivity (Ks), however, the relationship between Ks and dry bulk density for tilled and untilled conditions is different. Ks is lower in tilled soil than in untilled soil with the same texture at the same bulk density. The purpose of this study was to compare different models for the prediction of Ks for two soil textures under both tilled and non-tilled conditions. We compred two models based on the non-similar media concept (NSMC-0, NSMC-1), a model based on the similar media concept (SMC) and a model based on the Kozeny equation and Poiseuille law for prediction of Ks (KC-1 and KC-2). This study was conducted at two areas with loam and silty clay loam soils under tilled and untilled conditions. It is concluded that the SMC model is not able to predict Ks under either tilled or untilled conditions. Further, the NSMC-0 model, along with an equation to estimate the shape factor, was able to predict Ks versus dry bulk density for tilled soils. According to our study, under untilled conditions, the KC-1 and NSMC-1 models, and under tilled conditions, the NSMC-1 and NSMC-0 models, predicted Ks accurately. It is concluded that the NSMC models together with the optimized Kozeny–Carman models could reliably be used to predict Ks in different soil textures.

Microstructural effects on the permeability of periodic fibrous porous media

An analytical–numerical approach is presented for computing the macroscopic permeability of fibrous porous media taking into account their microstructure. A finite element (FE) based model for viscous, incompressible flow through a regular array of cylinders/fibers is employed for predicting the permeability associated with this type of media. High resolution data, obtained from our simulations, are utilized for validating the commonly used semi-analytical models of drag relations from which the permeability is often derived. The effect of porosity, or volume fraction, on the macroscopic permeability is studied. Also microstructure parameters like particle shape, orientation and unit cell staggered angle are varied. The results are compared with the Carman–Kozeny (CK) equation and the Kozeny factor (often assumed to be constant) dependence on the microstructural parameters is reported and used as an attempt to predict a closed form relation for the permeability in a variety of structures, shapes and wide range of porosities.

Two-Region Extended Archie's Law Model for Soil Air Permeability and Gas Diffusivity

The air permeability (k a ) and soil gas diffusion coefficients (D p ) are controlling factors for gas transport and fate in variably saturated soils. We developed a unified model for k a and D based on the classical Archie's law, extended by: (i) allowing for two-region gas transport behavior for structured soils, with the natural field moisture condition (set at -100 cm H 2 O matric potential [pF 2]) as the reference (spliced) point between the large-pore (drained pore diameter ≥30 μm at pF ≤ 2) and the small-pore (subsequently drained pores 2) regions, and (ii) including a percolation threshold, set as 10% of the total porosity for structureless porous media or 10% of the porosity in the large-pore region for structured soils. The resulting extended Archie's law with reference point (EXAR) models for k a and D were fitted to the measured data. For both structureless and structured porous media, Archie's saturation exponent (n) was higher for D p than for k a , indicating higher water blockage effects on gas diffusion. For structured soils, the saturation exponent for the large-pore region (n 1 ) was lower than for the small-pore region (n 2 ). Generally, n 1 values of ∼1 for k a and 2 for D p and n 2 values of 4/3 for k a and 7/3 for D described the data well. Two reference-point expressions for k a at pF 2 were also developed and tested together with existing models for D at pF 2 against independent data across soil types. The best-performing reference-point models were a k a model based on the classical Kozeny equation and the Moldrup D p model.

Permeability of JSC-1A: A lunar soil simulant

The permeability of lunar soil simulant, JSC-1A, is measured over a range of bulk densities from 1550 to 2000kgm−3. The corresponding viscous flow permeability is 1×10−12m2 to 6.1×10−12m2 for this bulk density range. Implications of these values on the contamination of regolith by rockets, on barrier/enhancement to bulk flow of ice, and on cratering are discussed. Although the particle size and shape distribution of the JSC-1A are extremely wide, the permeability measurements agree surprisingly well with the Carman–Kozeny equation. The results provide evidence that the Carman–Kozeny model could be applicable to other naturally occurring soils if effective soil properties are considered.

Use of fluorescent microorganisms to perform in vivo and in situ local characterization of microbial deposits

This study presents an original method for in situ characterisation of the structure and organization of a microbial deposit accumulated on a membrane surface during filtration operations. The strategy is based on coupling macroscopic measurements (filtration results) and local observations at the microscopic level. This approach has been performed thanks to the use of self-fluorescent microorganisms (modified microorganisms) and the design of a specific filtration cell allowing a direct microscopic visualisation of the deposit to be performed without any treatment which could damage or modify the structure. Indeed the use of living fluorescent cells has allowed a non-destructive, in situ and in vivo, study of the deposits. As an example of application, the influence of microorganism size and morphology on the filtration performances and the deposit characteristics were examined in the case of dead-end microfiltration of model suspensions (yeasts and bacteria) obtained under controlled conditions. For these compact deposits the upper part of the deposits, corresponding to a 30 μm thickness slice, could be precisely analysed at the microscopic level and for yeasts/bacteria mixed deposit, the local microorganisms organization in the slice could be analysed. For pure yeasts deposit, the porosity obtained by image analysis, 12%, was in agreement with the calculated value by Carman–Kozeny equation using permeate flow measurements.

Compression-dependent permeability measurement for random soft porous media and its implications to lift generation

In the past decade, foundations have been laid for understanding the lift generation in a soft porous medium under rapid compaction (Feng and Weinbaum, 2000. Journal of Fluid Mechanics 422, 282–317; Wu et al., 2005b. Journal of Fluid Mechanics 542, 281–304; Wu et al., 2004a. Physical Review Letters 93(19), 194501; Barabadi et al., 2009. Journal of Heat Transfer 131(10), 101006-1–101006-12; Al-Chidiac et al., 2009. Journal of Porous Media 12(11), 1019–1035). One of the key parameters that affects the lift generation is the variation of the Darcy permeability as a function of compression. This critical problem is investigated in the current study using a novel experimental setup, namely a permeameter. Three different, soft, synthetic, fibrous, porous materials were chosen for the study. Their microstructures were characterized using a scanning electron microscope and a stereomicroscope. Their porosities were precisely measured using a water displacement method. By carefully controlling the air flow through the materials contained in a long Plexiglas tube of the permeameter, one obtained consistent results for the Darcy permeability of the tested material as a function of its porosity. Fluffing the porous materials caused disturbance of their microstructures thus variations in the permeability, especially in the high porosity range. The experimental data was curve-fitted and compared to established expressions. It showed that the Nogai Expression (Nogai and Ihara, 1978. Journal of Textile Machinery Society of Japan 31(12), T166–T170) provided the best fit for the change of permeability as a function of compression for the fibrous materials studied herein. The Carman–Kozeny equation, however, is only applicable for one of the fibrous materials. This finding is consistent with the theoretical predictions by Barabadi et al. (2009).

Porous polymer nanoparticle layers

Multilayers of colloidal polymer particles provide micro- or mesopores and can be used as a separation media similar to micro- and ultrafiltration membranes. Materials derived from polymer particles are of interest for making novel functional membranes, because they combine mesoporosity, high specific surface area and the possibility to apply functional groups on the outer and inner surface of the separation layer. The aim of the current work was to build up mesoporous multilayers of polymeric nanoparticles for testing its structural properties and certain separation performances. For these purposes different latices with average particle sizes between 20 and 80 nm were synthesised by emulsion polymerisation and the latices were further cleaned by dialysis and ion-exchange to remove surfactants. Some latices were further surface modified with a sulphonated reagent to produce highly charged particles. The latex particles were deposited on micro- and ultrafiltration membranes by a dead-end filtration process and the hydraulic cake resistances of the very thin nanoparticle layers (filter cake) were determined in filtration experiments. The Carman–Kozeny equation was used to derive the porosity of the layers. It was shown that the rejection of dextrane increased with decreasing average diameter of applied particles and the particle layers had molecular weight cut offs in the range of ultrafiltration membranes. The calculations using the data from filtration experiments show that the porosity of a layer obtained by the filtration process increases when smaller particles are applied and these calculations were confirmed by rejection experiments. The results on the cake resistances for different latices indicate that the surface properties of the nanoparticles applied strongly affect the cake properties and the mechanical stability of such a layer formed by filtration.

Characterization of CO2 storage properties using core analysis techniques and thin section data

Abstract Carbon dioxide capture and sequestration in saline aquifers is considered one of the most important technologies for short and medium term climate change mitigation. There are many challenges for wide scale deployment of sequestration, including developing a sufficiently accurate fundamental understanding of the processes which govern the movement of supercritical CO2 in saline aquifers. A method for testing our understanding of the underlying physics is to conduct numerical simulations intended to replicate a multiphase flow experiment in which CO2 is injected into a brine saturated rock core. This paper focuses on methods to integrate pore scale rock properties into permeability models of cores subjected to such experiments. We use thin sections to measure pore scale features of rocks, and test correlations that relate permeability to porosity using the Carman–Kozeny equation. Incorporation of these pore scale features into sub-core scale maps of permeability is being used to history match core-scale multi-phase flow experiments.

On the generation of lift forces in random soft porous media

In this paper, we examine the generation of pressure and lift forces in a random soft fibrous media layer that is confined between two planar surfaces, an infinite horizontal lower boundary and a horizontal inclined upper boundary, in the lubrication limit where the characteristic thickness of the fibre layerH≪Lthe length of the inclined surface. The model for the fibre layer is a Brinkman equation and the Darcy permeabilityKpis described by the widely used Carman–Kozeny equation for random porous media. Two cases are considered: (a) an inclined upper boundary which slides freely on top of a stationary fibre layer which is firmly attached to the lower boundary and (b) an inclined stationary upper boundary with an attached fibre layer in which the horizontal lower boundary slides freely in its own plane beneath it. Superficially, the problems appear equivalent to the classical problem for a slider bearing where the solutions for the pressure distribution and lift force are independent of which boundary is moving. In this problem there is an optimum compression ratiok=h1/h2= 2.2, whereh1andh2are the heights at the leading and trailing edges, for maximum lift force. However, this symmetry is lost if the intervening space is filled with a soft porous fibrous material since the Brinkman equation is not invariant under a transformation of coordinates in which the inherently unsteady problem in case (a) is transformed to a steady reference frame in which the inclined upper boundary is stationary and the horizontal boundary with the adhered fibre layer moves below it. Although in the steady reference frame case (a) now appears to resemble case (b), the solutions are strikingly different and depend critically on the value of the dimensionless fibre interaction layer thickness$\alpha\,{=}\, H/\sqrt{K_p}$. For α ≪ 1 the solutions for both cases approach the classical solution for a slider bearing. For α ≫ 1 we show, using asymptotic analysis that the solutions diverge dramatically. In case (a) the pressure and lift force increase as α2and asymptotically approach a limiting behaviour for large values of α, first predicted in Feng and Weinbaum (J. Fluid Mech., vol. 422, 2000, p. 288), while in case (b) the pressure and lift force decay as α−2since the inclined upper boundary is screened by the fibre layer and the amount of fluid dragged through the fluid gap decreases as α increases and vanishes for α ≫ 1. The solution in case (a), where the inclined upper boundary moves, is of particular interest since it reveals the potential to generate enormous lift forces using commercially available inexpensive soft porous materials provided the lateral leakage at the edge of the planform can be eliminated through the use of a channel with impermeable sidewalls as first proposed in the work by Wu, Andreopolous and Weinbaum (Phys. Rev. Lett., vol. 93, 2004, p. 194501). The behaviour is illustrated for both a toboggan sliding in such a channel and a larger planform that might be useful in commercial transportation.

Validity of the permeability Carman–Kozeny equation: A volume averaging approach

Abstract A volume averaging approach is used to estimate the porous media permeability. Contrary to traditional methods that rely on solving the Navier–Stokes equations for laminar flow, this approach has the advantage that it does not require the specification of some physical conditions and parameters (pressure drop and viscosity). Numerical results on synthetic models of porous media showed that (i) the local porous medium configuration has an important effect on the permeability value, and (ii) the Carman–Kozeny equation cannot describe the permeability behavior as a function of porosity and characteristic lengths. In turn, our results indicate that simple empirical equations, commonly used in practice, are unable to describe the permeability functionalities over a broad range of porous media configurations.

An approach to defining tortuosity and cementation factor in carbonate reservoir rocks

Tortuosity and cementation factor are two critical parameters that significantly affect estimates of reservoir properties. Tortuosity factor can be used to estimate permeability using the Carman–Kozeny equation and is an important parameter for formation resistivity factor calculation using a modified version of Archie's formula. It is also used to predict water saturation of reservoir rocks. Tortuosity as an input parameter in Biot's equation can be used to estimate velocity dispersion. In this work, based on the Generalized Archie Equation Curve Fitting (GAECF), tortuosity and cementation factor are determined for selected intervals in a carbonate reservoir. Formation resistivity factor (FRF) analysis was carried out on both core samples and on well log data. The analyses were carried out separately on porous and recognized tight intervals. The intervals selected were fracture-free and include both matrix porosity and somewhat vuggy porosity. FRF analysis on well log data indicates that the carbonate intervals examined follow the GAECF well. However, unreasonable values for the tortuosity parameter, ‘ a’, and cementation factor, ‘ m’, were obtained. To overcome this, the average minimum apparent tortuosity value was calculated for these intervals. Approximate values for tortuosity factor from both core samples and well log data lie close to each other (for porous intervals, zones S1, S7 and S11). This indicates that the results obtained from FRF analysis on well log data can be effectively and reliably used as an alternative to FRF studies on core samples.

Determination of cake porosity using image analysis in a coagulation–microfiltration system

Porosity of a cake layer formed through particle deposition plays an important role in the filtration performance of low-pressure membranes. An experimental analytical method for the direct measurement of cake porosity in a coagulation–microfiltration was developed using a CLSM and image analysis technique. A series of aggregates suspensions with different sizes and fractal dimensions were prepared with 0.1 μm of fluorescent polystyrene latex beads. Microfiltration of each suspension was carried out to form a cake layer whose porosity was measured using CLSM with an optical section of 2 μm and an image analysis program. To validate this method, measured porosities were compared with those calculated from specific cake resistances using the Carman–Kozeny equation. The two sets of porosities were in good agreement, confirmed statistically by a t-test. Based on this analysis technique, the effects of size and fractal dimension of aggregates on cake porosity and compressibility were investigated under various operating pressures. For low fractal dimensions the cake porosity was barely dependent on the fractal dimension of aggregates, whereas for high fractal dimensions it decreased with increasing fractal dimension.

Effect of adsorbed water upon permeabilities of pressure-filtered ceramic nanopowder compacts

The permeabilities of nanocrystalline 3 mol% yttria-stabilized zirconia (3Y-TZP), silica, and boehmite powder compacts are measured during their formation by constant rate pressure filtration. The Carman–Kozeny equation with no account for the effect of adsorbed water often overestimates by a factor of 2 or more the measured permeabilities, with increasing deviation with decreasing permeability. A permeability equation from the literature and one derived here, both taking into account the effect of adsorbed water, show significant improvement over the classic Carman–Kozeny equation for predicting measured permeabilities. The equation derived here allows straightforward predictions to be made of how permeability will change as the critical point of drying (when shrinkage stops) is approached.

Modeling the Relationship between Soil Bulk Density and the Hydraulic Conductivity Function

A model is presented that could quantify and predict the effect of an increase in soil bulk density on the hydraulic conductivity function (HCF). Two approaches are formulated to predict the saturated hydraulic conductivity of compacted soils. The first approach is a general expression based on the Kozeny equation that requires only information on the bulk density. The second approach exploits information contained in the water retention curve (WRC). This approach, which relies on Assouline's model for soil HCFs, also provides a basis for a proposed expression to predict the unsaturated HCF of compacted soils. It is shown that a relationship between η, the power parameter in the expression for the HCF, and ε, the coefficient of variation of the WRC, is applicable to a large number of both compacted and uncompacted soils. The model was verified against measured HCFs for soils at different bulk density values. Both the saturated and unsaturated hydraulic conductivities were well reproduced by the suggested expressions.