The rheology of pseudoplastic fluids in porous media using network modeling

This paper considers the rheology of pseudoplastic (shear thinning) fluids in porous media. The central problem studied is the relationship between the viscometric behavior of the polymer solution and its observed behavior in the porous matrix. In the past, a number of macroscopic approaches have been applied, usually based on capillary bundle models of the porous medium. These simplified models have been used along with constitutive equations describing the fluid behavior (usually of power law type) to establish semiempirical macroscopic equations describing the flow of non-Newtonian fluids in porous media. This procedure has been reasonably successful in correlating experimental results on the flow of polymer solutions through both consolidated and unconsolidated porous materials. However, it does not allow an interpretation of polymer flow in porous media in terms of the flows on a microscopic scale; nor does it allow us to predict changes in macroscopic behavior resulting from variations at a microscopic level in the characteristics of the porous medium such as pore size distribution. In this work, we use a network approach to the modeling of non-Newtonian rheology, in order to understand some of the more detailed features of polymjer flow in porous media. This approach provides a mathematical bridge between the behavior of the non-Newtonian fluid in a single capillary and the macroscopic behavior as deduced from the pressure drop-flow rate relation across the whole network model. It demonstrates the importance of flow redistribution within the elements of the capillary network as the overall pressure gradient varies. As an example of a pseudoplastic fluid in a porous medium, we consider the flow of xanthan biopolymer. This polymer is important as a displacing fluid viscosifier in enhanced oil recovery applications and, for that reason, a considerable amount of experimental data has been published on the flow of xanthan solutions in various porous media.     

Modeling Disjoining Pressures in Submicrometer Liquid-Filled Cylindrical Geometries

This work develops models for calculating the disjoining pressures of a cylindrical fluid "plug", specifically in submicrometer cylindrical pores. This modeling produces closed-form, cylindrical-pore disjoining pressures for London/van der Waals and solute/pore-wall adsorption interactions, which are the slit-pore models with the characteristic pore size replaced by the radius and multiplied by 6, resulting in a 48-fold or more increase in magnitude. In addition, this work contains a numerical solution for electrostatic interactions. The result of the numerical solution was a 9-fold increase in the modeled disjoining pressure compared to that in the slit-pore model. The cylindrical models may apply to the chemical coating of the interior walls of cylindrical pores or to the thermodynamics within droplets after the breakup of a fluid coating a surface. However, the application used as the base case in this paper is the extension of transport and thermodynamic laws for porous media, previously developed with capillary pressure models, to fully saturated porous media with submicrometer-sized pores. As such, the models could apply to mass transport in ultrafiltration, nanofiltration, and reverse-osmosis membranes. Copyright 2001 Academic Press.     

Electrophoresis of a charged colloidal particle in porous media: boundary effect of a solid plane

Electrokinetic treatments such as the electrophoretic technique have been applied successfully to various soil remediation and contaminant removal situations. To understand further the fundamental features involved, the electrophoretic motion of a charged particle in porous media is investigated theoretically in this study, focusing on the boundary effect of a nearby solid plane toward which the particle moves perpendicularly. The porous medium is modeled as a Brinkman fluid with a characteristic screening length (λ(-1)) that can be obtained directly from the experimental data. General electrokinetic equations are used to describe the system and are solved with a pseudospectral method based on Chebyshev polynomials. We found that the particle motion is deterred by the boundary effect in general. The closer the particle is to the boundary, the more severe this effect is. Up to a 90% reduction in particle mobility is observed in some situations. This indicates that a drastic overestimation (10-fold!) of the overall transport rate of particles may occur for large-scale in situ operations in porous media, such as soil remediation utilizing large planar electrodes, should a portable analytical formula valid for bulk systems only be used. Correction factors for various situations in porous media are presented as convenient charts with which to aid engineers and researchers in the field of environmental engineering, for instance, as a realistic estimation of the actual transport rate obtainable. In addition, the results of present study can be applied to biomedical engineering and drug delivery as well because polymer gels and skin barriers both have a porous essence.     

Dusty-gas model. Allowance for surface forces

The dusty-gas model has been extended to the case of nanoporous media, in which the action of surface forces must be taken into account. A basic set of transport equations underlying the model has been derived proceeding from a set of kinetic equations for an ordinary gas and dust particles. In the kinetic equations, the interaction between the gas and dust particles is represented as a sum of a long-range (analog of surface forces) and short-range components. The contribution of the long-range component has been taken into account in the self-consistent approximation, while the short-range component has been considered in the standard manner. Allowance for the surface forces has been shown to result in a substantial modification of the equation for gas transport through porous bodies, with this modification being most pronounced at nonuniform temperatures.     

Gas mixture flow in nanoporous media in the presence of surface forces. The dusty-gas model

The dusty-gas model has been generalized to the case of gas mixture flow in nanoporous media under the conditions of the action of surface forces. A basic set of transport equations has been derived proceeding from kinetic equations for a gas mixture and dust particles. To take into account the surface forces, the interaction between a gas and dust particles has been represented as a sum of a long-range potential, which reflects the surface forces, and a short-range potential, which describes gas molecule scattering on the surface of pore walls. The contribution of the long-range component has been taken into account in the self-consistent approximation, while the short-range component has been considered in the standard manner. The surface forces have been shown to have a substantial effect on the transfer of mixed gases through porous bodies; in particular, it becomes possible to separate mixture components due to different potentials of the interaction of their molecules with pore surface.     

Fluid dynamics in capillary and chip electrochromatography

This review is concerned with the phenomenological fluid dynamics in capillary and chip electrochromatography (EC) using high-surface-area random porous media as stationary phases. Specifically, the pore space morphology of packed beds and monoliths is analyzed with respect to the nonuniformity of local and macroscopic EOF, as well as the achievable separation efficiency. It is first pointed out that the pore-level velocity profile of EOF through packed beds and monoliths is generally nonuniform. This contrasts with the plug-like EOF profile in a single homogeneous channel and is caused by a nonuniform distribution of the local electrical field strength in porous media due to the continuously converging and diverging pores. Wall effects of geometrical and electrokinetic nature form another origin for EOF nonuniformities in packed beds which are caused by packing hard particles against a hard wall with different zeta potential. The influence of the resulting, systematic porosity fluctuations close to the confining wall over a distance of a few particle diameters becomes aggravated at low column-to-particle diameter ratio. Due to the hierarchical structure of the pore space in packed beds and silica-based monoliths which are characterized by discrete intraparticle (intraskeleton) mesoporous and interparticle (interskeleton) macroporous spatial domains, charge-selective transport prevails within the porous particles and the monolith skeleton under most general conditions. It forms the basis for electrical field-induced concentration polarization (CP). Simultaneously, a finite and -- depending on morphology -- often significant perfusive EOF is realized in these hierarchically structured materials. The data collected in this review show that the existence of CP and its relative intensity compared to perfusive EOF form fundamental ingredients which tune the fluid dynamics in EC employing monoliths and packed beds as stationary phases. This addresses the (electro)hydrodynamics, associated hydrodynamic dispersion, as well as the migration and retention of charged analytes.     

Transition from creeping via viscous-inertial to turbulent flow in fixed beds

This review is concerned with the analysis of flow regimes in porous media, in particular, in fixed beds of spherical particles used as reactors in engineering applications, or as separation units in liquid chromatography. A transition from creeping via viscous-inertial to turbulent flow is discussed based on macro-scale transport behaviour with respect to the pressure drop-flow rate dependence, in particular, the deviation from Darcy's law, as well as direct microscopic data which reflect concomitant changes in the pore-level hydrodynamics. In contrast to the flow behaviour in straight pipes, the transition from laminar to turbulent flow in fixed particulate beds is not sharp, but proceeds gradually through a viscous-inertial flow regime. The onset of this steady, nonlinear regime and increasing role of inertial forces is macroscopically manifested in the failure of Darcy's law to describe flow through fixed beds at higher Reynolds numbers. While the physical reasons for this failure still are not completely understood, it is not caused by turbulence which occurs at Reynolds numbers about two orders of magnitude above those for which a deviation from Darcy's law is observed. Microscopic analysis shows that this steady, nonlinear flow regime is characterized by the development of an inertial core in the pore-level profile, i.e., at increasing Reynolds number velocity profiles in individual pores become flatter towards the center of the pores, while the velocity gradient increases close to the solid-liquid interface. Further, regions with local backflow and stationary eddies are demonstrated for the laminar flow regime in fixed beds. The onset of local fluctuations (end of laminar regime) is observed at superficial Reynolds numbers on the order of 100. Complementary analysis of hydrodynamic dispersion suggests that this unsteady flow accelerates lateral equilibration between different velocities in fixed beds which, in turn, reduces spreading in the longitudial (macroscopic flow) direction.     

Diffusive flux of nanoparticles through chemically modified alumina membranes

Surface chemistry plays an important role in determining flux through porous media such as in the environment. In this paper diffusive flux of nanoparticles through alkylsilane modified porous alumina is measured as a model for understanding transport in porous media of differing surface chemistries. Experiments are performed as a function of particle size, pore diameter, attached hydrocarbon chain length and chain terminus, and solvent. Particle fluxes are monitored by the change in absorbance of the solution in the receiving side of a diffusion cell. In general, flux increases when the membranes are modified with alkylsilanes compared to untreated membranes, which is attributed to the hydrophobic nature of the porous membranes and differences in wettability. We find that flux decreases, in both hexane and aqueous solutions, when the hydrocarbon chain lining the interior pore wall increases in length. The rate and selectivity of transport across these membranes is related to the partition coefficient (K(p)) and the diffusion coefficient (D) of the permeating species. By conducting experiments as a function of initial particle concentration, we find that K(p)D increases with increasing particle size, is greater in alkylsilane-modified pores, and larger in hexane solution than water. The impact of the alkylsilane terminus (-CH(3), -Br, -NH(2), -COOH) on permeation in water is also examined. In water, the highest K(p)D is observed when the membranes are modified with carboxylic acid terminated silanes and lowest with amine terminated silanes as a result of electrostatic effects during translocation.     

Mesoscopic simulations of phase distribution effects on the effective thermal conductivity of microgranular porous media

This paper analyzes the phase distribution effects on the effective thermal conductivity (ETC) of multi-phase microgranular porous media using mesoscopic statistics based numerical methods. A multi-parameter random generation-growth method, quartet structure generation set (QSGS), is developed for replicating microstructures of multi-phase granular porous media based on the macroscopic statistical information, such as the volume fractions and the phase interactions. The phase distribution characteristics and the interphase connections are controlled by adjusting the related parameters. Then the energy transport equations through porous media are solved by a lattice Boltzmann method developed by us with multi-phase conjugate heat transfer considered. The results indicate that a smaller average particle size could lead to a larger effective thermal conductivity of two-phase porous media for a certain porosity. For the anisotropic media, if the larger directional growth probability is along the direction of temperature gradient, the effective thermal conductivity in the parallel direction is enhanced as a result, and that in the vertical direction will be weakened. For multi-phase porous media, the degree of phase conglomeration is determined by the phase interactions. A larger liquid-liquid interaction leads to a higher degree of liquid phase conglomeration and therefore a larger effective thermal conductivity of the porous media.     

Chemico-electromechanical coupling in microporous media

We determine the macroscopic transport properties of isotropic microporous media by volume-averaging the local Nernst-Planck and Navier-Stokes equations in nonisothermal conditions. In such media, the excess of charge that counterbalances the charge deficiency of the surface of the minerals is partitioned between the Gouy-Chapman layer and the Stern layer. The Stern layer of sorbed counterions is attached to the solid phase, while the Gouy-Chapman diffuse layer is assumed to have a thickness comparable to the size of the pores. Rather than using Poisson-Boltzmann distributions to describe the ionic concentrations in the pore space of the medium, we rely on Donnan distributions obtained by equating the chemical potentials of the water molecules and ions between a reservoir of ions and the pore space of the medium. The macroscopic Maxwell equations and the macroscopic linear constitutive transport equations are derived in the vicinity of equilibrium, assuming that the porous material is deformable. In the vicinity of thermodynamic equilibrium, the cross-coupling phenomena of the macroscopic constitutive equations of transport follow Onsager reciprocity. In addition, all the material properties entering the constitutive equations depend only on two textural properties, the permeability and the electrical formation factor.     

Resolving the coupled effects of hydrodynamics and DLVO forces on colloid attachment in porous media

Transport of colloidal particles in porous media is governed by the rate at which the colloids strike and stick to collector surfaces. Classic filtration theory has considered the influence of system hydrodynamics on determining the rate at which colloids strike collector surfaces, but has neglected the influence of hydrodynamic forces in the calculation of the collision efficiency. Computational simulations based on the sphere-in-cell model were conducted that considered the influence of hydrodynamic and Derjaguin-Landau-Verwey-Overbeek (DLVO) forces on colloid attachment to collectors of various shape and size. Our analysis indicated that hydrodynamic and DLVO forces and collector shape and size significantly influenced the colloid collision efficiency. Colloid attachment was only possible on regions of the collector where the torque from hydrodynamic shear acting on colloids adjacent to collector surfaces was less than the adhesive (DLVO) torque that resists detachment. The fraction of the collector surface area on which attachment was possible increased with solution ionic strength, collector size, and decreasing flow velocity. Simulations demonstrated that quantitative evaluation of colloid transport through porous media will require nontraditional approaches that account for hydrodynamic and DLVO forces as well as collector shape and size.     

Protein adsorption in porous adsorbent particles: A multiscale modeling study on inner radial humps in the concentration profiles of adsorbed protein induced by nonuniform ligand density distributions

Recently published results determined from molecular dynamics (MD) modeling and simulation studies have shown that the spatial distribution of the density of immobilized charged ligands in ion‐exchange porous adsorbent particles is most likely nonuniform and the adsorbent particles also exhibit local nonelectroneutrality. In this work, the functional forms of the nonuniform spatial distributions of the density of the immobilized ligands in four different porous adsorbent media that were determined by MD studies were employed in a macroscopic continuum model describing the transport and adsorption of a single protein in the porous particles of the four different adsorbent media. The results clearly show that inner radial humps in the concentration profiles of the adsorbed protein can occur when the spatial distribution of the density of the immobilized ligands in the porous adsorbent particles is nonuniform and also has local maxima or minima along the radial direction in the particle. The results also indicate that the rate at which the equilibrium condition is approached depends significantly on the functional form of the spatial distribution of the density of the immobilized ligands. When adsorption equilibrium has been reached, the concentration profile of the adsorbed protein exhibits the shape of the spatial distribution of the density of the immobilized ligands. The results suggest that the technique of confocal scanning laser microscopy could be used to measure the concentration profile of an adsorbed protein at equilibrium and this measurement could provide the spatial distribution of the density of the immobilized ligands, and such measurements could also be used for quality control of the adsorbent medium. The results in this work have also implications in the modeling, design, analysis, and quality control of systems involving biocatalysis. Furthermore, the results clearly indicate that it is very important to study the dynamic behavior of an adsorption system having a nonuniform spatial distribution in the density of the immobilized charged ligands and where (i) both monovalent and multivalent interactions between the single charged adsorbate and the immobilized charged ligands occur and (ii) the values of the pH and ionic strength are such that the electrophoretic effects are active.     

微孔中简单流体粘度的分子动力学模拟及关联模型

用分子动力学模拟计算了微孔介质中流体氩在不同温度、不同密度和不同孔径下的剪切粘度.并根据Chapman-Enskog关于硬球流体传递性质的理论以及Heyes的关于Lennard-Jones流体粘度的表达式,提出了两个描述微孔介质中流体粘度的模型,该模型可以计算微孔中流体氩在不同状态下的粘度值.通过与计算机模拟值的比较,证明这两个微孔流体粘度模型是可用的.     

Flow of electrolyte through porous piezoelectric medium: macroscopic equations

The aim of this contribution is to elaborate a general framework for modelling flows of two-ionic species electrolytes through porous piezoelectric media.By using the method of two-scale asymptotic expansions, the macroscopic phenomenological equations describing electrokinetics of such a structure are derived and the formulae for the effective mechanical and nonmechanical coefficients are given. Natural jump conditions are assumed on the interfaces between the piezoelectric skeleton and conductive fluid.     

Flow-through particles for the high-performance liquid chromatographic separation of biomolecules: perfusion chromatography

This paper reports a new technique for reducing resistance to stagnant mobile phase mass transfer without sacrificing high adsorbent capacity or necessitating extremely high pressure operation. The technique involves the flow of liquid through a porous chromatographic particle, and has thus been termed "perfusion chromatography". This is accomplished with 6000-8000 A pores which transect the particle. Data from electron microscopy, column efficiency, frontal analysis and theoretical modelling all suggest that mobile phase will flow through these large pores. In this manner, solutes enter the interior of the particles through a combination of convective and diffusional transport, with convection dominating for Peclet numbers greater than one. The implications of flow through particles on bandspreading, resolution and dynamic loading capacity are examined. It is shown that the rate of solute transport is strongly coupled to mobile phase velocity such that bandspreading, resolution of proteins and dynamic loading capacity are unaffected by increases in mobile phase velocity up to several thousand centimeters per hour. The surface area of this very large-pore diameter material is enhanced by using a network of smaller, 500-1500 A interconnecting pores between the throughpores. Scanning electron micrographs show that the pore network is continuous and that no point in the matrix is more than 5000-10,000 A from a through-pore. As a consequence, diffusional path lengths are minimized and the large porous particles take on the transport characteristics of much smaller particles but with a fraction of the pressure drop. Capacity and resolution studies show that these materials bind and separate an amount of protein equivalent to that of conventional high-performance liquid chromatography as well as low performance agarose-based media at greater than 10-100 times higher mobile phase velocity with no loss in resolution.     

Effect of depletion interactions on transport of colloidal particles in porous media

The influence of depletion interactions on the transport of micrometer-sized, negatively charged polystyrene latex particles through porous media was studied by analysis of particle breakthrough curves as a response to short-pulse particle injections to the inlet of a packed column of glass beads. The column outlet latex particle concentration profiles and the total amount of particles exiting the column were determined as a function of the concentration of small, silica nanoparticles in the solution and the bulk flow rate. Because of similar charges, the silica particles do not adsorb to either the latex particles or glass beads and thus induce an attractive depletion force between the latex particles and glass bead collectors. The total column outlet latex particle amount was calculated by integrating the measured breakthrough concentration curve and compared to the known amount of injected particles at the column inlet. It was found that the particle recovery was a decreasing function of the silica nanoparticle concentration and the carrier fluid residence time, and an increasing function of the velocity in the bed. In addition, removing the silica nanoparticles from the flowing solution caused a second outlet peak to appear, suggesting that some of the polystyrene particles were captured in secondary energy wells. The experimental data were interpreted using the predicted potential energy profile between a single particle and a glass bead, which was assumed to consist of electrostatic, van der Waals, and depletion components. The results indicate that secondary energy wells significantly affect particle transport behavior through porous media.