Membrane Transport Model Research Articles

Numerical modeling of a long-term in situ chemical osmosis experiment in the Pierre Shale, South Dakota

We have numerically modeled evolving fluid pressures and concentrations from a nine-year in situ osmosis experiment in the Pierre Shale, South Dakota. These data were obtained and recently interpreted by one of us (C.E.N.) as indicating a potentially significant role for chemical osmosis in media like the Pierre Shale. That analysis considered only the final pressure differentials among boreholes that were assumed to represent osmotic equilibrium. For this study, the system evolution was modeled using a recently developed transient model for membrane transport. The model simulates hydraulically and chemically driven fluid and solute transport. The results yield an estimate of the thickness of the water film between the clay platelets b of 40 Å, which corresponds to an osmotic efficiency σ of 0.21 for the ambient pore water salinity of 3.5 g/l TDS. These values largely confirm the results of the earlier equilibrium analysis. However, the new model analysis provides additional constraints suggesting that intrinsic permeability k = 1.4 × 10 −19 m 2, specific storage S s = 1.7 × 10 −5 m −1, and diffusion coefficient D* = 6 × 10 −11 m 2/s. The k value is larger than certain independent estimates which range from 10 −21 to 10 −20; it may indicate opening of microcracks during the experiments. The fact that the complex transient pressure and concentration behavior for the individual wells could be reproduced quite accurately, and the inferred parameter values appear to be realistic for the Pierre Shale, suggests that the new model is a useful tool for modeling transient coupled flows in groundwater systems.

A General Channel Model Accounts for Channel, Carrier, Countertransport and Cotransport Kinetics

In this work we propose a unifying model of mediated membrane transport, based upon the idea that the integral membrane proteins involved in these processes operate via complex channel mechanisms. In the first part, we briefly review literature about the structural aspects of membrane transporters. We conclude that there is a substantial amount of evidence suggesting that most membrane proteins performing transport are embodied with channel-like structures that may constitute the translocation paths. This includes cases where the phenomenological transport kinetics do not correspond to the classical channel behavior. In the second part of this article we introduce the general channel model of mediated transport and employ it to derive specific examples, like simple one- or two-ligand channels, water-ligand channels, simple carriers, co- and counter-transport systems and more complex water-ligand carriers. We show that, for the most part, these particular cases can be obtained by the application of the techniques of diagram reduction to the full model. The necessary conditions for diagram reduction reflect physical properties of the protein and its surroundings.

Transport Phenomena in Polymer Electrolyte Membranes: II. Binary Friction Membrane Model

The insight gained from the analysis conducted in Part I (see the preceding article) is used in the development of a general transport model for water and protons in perfluorosulfonic acid membranes based on the binary friction model. As a tool for investigating the unknown parameters in the general membrane transport model, a simplified conductivity model is derived to represent conditions found in alternating current (ac) impedance conductivity measurements. This binary friction conductivity model (BFCM) is applied to 1100 equivalent weight (EW) Nafion, and compared to other established membrane models. It is shown to provide a more consistent fit to the data over the entire range of water contents and at different temperatures. The subset of transport coefficients in the BFCM is the same as in the general binary friction membrane model (BFM2), and thus with additional data on water transport, the BFM2 model and all its required parameters can be fully specified. The paper discusses possible experimental investigations and fundamental simulations to determine the model parameters required to apply the general BFM2 to predict coupled proton and water transport in PEM fuel cells.

Modeling of mass transport of aqueous solutions of multi-solute organics through reverse osmosis membranes in case of solute–membrane affinity: Part 1. Model development and simulation

A two-dimensional mathematical model is developed for transport of multi-solute liquid (aqueous) solutions through reverse osmosis membranes in which strong affinity may exist between the membrane polymer and the solutes. The final objective of this research study is to predict the performance of reverse osmosis (RO) or nanofiltration (NF) membranes in the case of multi-solute aqueous systems in the presence of strong solute–membrane attraction. Modeling of membrane transport in this case is complex because of the interactions between solute, solvent and membrane. The model, which assumes a micro-porous structure for the membrane, is an extension of a single-solute model based on the preferential sorption—capillary flow mechanism, and takes detailed solute–solvent–membrane interactions into account. The model is believed to be the first to have been able to describe the anomalous behavior of such systems for multi-component aqueous solutions. The developed model is used to simulate the performance of cellulose acetate membrane in aqueous toluene–benzene systems by varying the operating pressure and mole fraction of toluene/benzene system. Generally, the separation of both solutes will decrease as the operating pressure is increased, with toluene having higher separations than benzene. The permeation flux also decreases with increasing operating pressure. However, the permeation flux of mixture increases as solutes concentration in feed solution is increased. The solutes separation values are higher at higher solutes concentrations of the feed.

A pore diffusion transport model for forecasting the performance of membrane processes

Membrane processes in water treatment often experience permeate flux decline due to fouling caused by natural organic matter (NOM). A membrane transport model incorporating surface and internal pore fouling was developed for prediction and simulation of permeate flux and concentrations of permeate and concentrate. This model considered resistances due to membrane material, concentration polarization, gel formation, and internal pore fouling. Contact angle measurements employed for evaluating hydrophobicity and membrane fouling potential showed that the polyethersulfone HFK-131 ultrafiltration membranes were more hydrophobic, and therefore, more susceptible to organic fouling than the cross-linked aromatic polyamide NF-70 nanofiltration membranes. Membrane filtration tests were conducted with groundwater containing NOM under various operation conditions to evaluate permeate flux decline patterns due to organic fouling, as well as solute concentration profiles in the permeate and concentrate streams. The predictive capability of the proposed model was excellent as evidenced from comparison of experimental results and model simulations. Model simulation and sensitivity studies provided a well-founded theory based on the dependence of transport resistance components and membrane performance on various input parameters. These parameters included the mass-transfer coefficient, solution dynamic viscosity, membrane resistance, and resistance per unit gel-layer thickness.

Modeling of a spiral-wound module and estimation of model parameters using numerical techniques

A model for a spiral-wound reverse osmosis system using the three-parameter membrane transport model by Spiegler-Kedem is presented. The pressure drops in the permeate channel, feed channel and also the variation of the mass transfer coefficient along the feed channel were taken into account. An analytical solution was not possible due to the large number of nonlinear model equations; therefore, a computer solution utilizing finite differences was employed. The data generated by the simulation of the proposed model clearly indicate that neglecting the variation in the mass transfer coefficient and pressure drop along the flow channels can lead to errors in permeate concentration, though the effect on permeate flow rate may not be significant. The significance of the reflection coefficient in the membrane transport model was also investigated. A method for estimation of the model parameters is also presented; previously reported experimental data were analyzed. Using this parameter-estimation program, a correlation for the mass transfer coefficient in the feed channel is proposed and compared with the correlation available in the literature.

Recent Methodologies for the Estimation of N-Octanol / Water Partition Coefficients and their Use in the Prediction of Membrane Transport Properties of Drugs

The lipophilicity of drug molecules (represented as the logarithm of the n-octanol/water partition coefficient) often strongly correlates with their pharmacological and toxic activities. It is therefore, not surprising that there is considerable interest in developing mathematical models capable to accurately predict their value for new drug candidates. In this review, current major approaches for estimating partition coefficients are described and some of their advantages and disadvantages are discussed. Recent uses of these partition coefficient algorithms in the development of membrane transport models are also discussed.

Introduction of Lyo- and Cryoprotective Carbohydrates into Mammalian Cells through Swelling-Activated Channels

As sugars and related compounds are increasingly being used as potential cryo- and lyoprotective agents for preservation of rare and valuable mammalian cells and tissues, introduction of these compounds, especially saccharide, into a living cell is one of the key technology. In the present study, volume changes of human T-lymphocytes (Jurkat line) exposed to hypotonic carbohydrate-substituted solutions of different composition and osmolality were studied by videomicroscopy. The complex volumetric data were analyzed with a membrane transport model that allowed the estimation of the hydraulic conductivity and volume-dependent solute permeabilities. We found that under hypotonic stress of 100 mOsm, the membrane permeability to monomeric carbohydrates inereased dramatically (apparently due to channel activation caused by extensive cell swelling), whereas oligosaccharide permeability remained very poor. The size-selectivity of the swelling-activated sugar permeation was confirmed by direct chromatographic measurements of intracellular sugars. The results of this study are of interest for biotechnology.

Intracellular Delivery of Carbohydrates into Mammalian Cells through Swelling-activated Pathways

Volume changes of human T-lymphocytes (Jurkat line) exposed to hypotonic carbohydrate-substituted solutions of different composition and osmolality were studied by videomicroscopy. In 200 mOsm media the cells first swelled within 1-2 min and then underwent regulatory volume decrease (RVD) to their original isotonic volume within 10-15 min. RVD also occurred in strongly hypotonic 100 mOsm solutions of di- and trisaccharides (trehalose, sucrose, raffinose). In contrast to oligosaccharide media, 100 mOsm solutions of monomeric carbohydrates (glucose, galactose, inositol and sorbitol) inhibited RVD. The complex volumetric data were analyzed with a membrane transport model that allowed the estimation of the hydraulic conductivity and volume-dependent solute permeabilities. We found that under slightly hypotonic stress (200 mOsm) the cell membrane was impermeable to all carbohydrates studied here. Upon osmolality decrease to 100 mOsm, the membrane permeability to monomeric carbohydrates increased dramatically (apparently due to channel activation caused by extensive cell swelling), whereas oligosaccharide permeability remained very poor. The size-selectivity of the swelling-activated sugar permeation was confirmed by direct chromatographic measurements of intracellular sugars. The results of this study are of interest for biotechnology, where sugars and related compounds are increasingly being used as potential cryo- and lyoprotective agents for preservation of rare and valuable mammalian cells and tissues.

Effect of concentration polarisation and osmotic pressure on flux in organic solvent nanofiltration

The separation of molecules present in organic solvents by nanofiltration has potential application in several industries, and organic solvent stable nanofiltration (NF) membranes have recently become available. There is a rapidly growing body of information available on the processes controlling solvent fluxes and solute rejections in solvent nanofiltration. However, previous work has mainly been carried out with dilute solutions (<1 wt.% solute in solvent), whereas in actual applications, solutes will be more concentrated (>5 wt.%) and phenomena such as concentration polarisation and osmotic pressure may contribute to the solvent flux, as in aqueous systems. In order to improve our understanding of organic solvent nanofiltration phenomena, experiments were performed in a cross-flow rig in which NF was carried out in a continuous mode. Solutions of different concentrations (up to 20 wt.%) of tetraoctylammonium bromide and docosane in toluene were used. Description of the experimental data, including prediction of solute rejection, was performed using the solution diffusion model for membrane transport and the film theory for liquid mass transfer effects. The results show that the organic systems cannot always be described by a simple osmotic pressure model. The flux through the membrane is affected by the cross-flow velocity, indicating that concentration polarisation induces mass transfer limitations. The fit between the model and the experimental data is markedly improved by allowing the activities of the solution components to vary, indicating that these systems are non-ideal.

Modeling of a radial flow hollow fiber module and estimation of model parameters using numerical techniques

A numerical solution is developed for modeling as well as predicting the performance of a radial flow hollow fiber reverse osmosis (HFRO) module. The three-parameter Spiegler–Kedem (S–K) model is used for describing the mass transport across the membrane. The solution proposed considers pressure drops in both permeate and bulk streams and includes concentration polarization. The numerical solution uses finite difference method. Two limiting cases of the system of equations are used to arrive at analytical solutions and check the validity of the numerical solution. The simulation of the above three-parameter membrane model is compared with a two-parameter membrane model and an analytical solution available in literature and the significance of the third membrane parameter, the reflection coefficient, is investigated in detail. It is found that when the reflection coefficient is decreased from its maximum value of one, the permeate flow rates are higher and the permeate concentration differs substantially from that reported by the two parameter membrane model of Sekino [J. Membr. Sci. 85 (1993) 241]. Under the simulation conditions investigated (1> σ>0.9), the maximum difference in the permeate flow rates between the two can be as large as 14% while the differences in the permeate concentrations can be as large as 4431%. The mass transfer coefficient and the membrane model parameters are determined by using an optimization technique—the Simplex Search. The data for phenol separation obtained on a laboratory scale B9 HFRO experimental setup and the experimental data available in literature are used for the estimation of the model parameters as well as the mass transfer coefficient in the B9 module. These estimated parameters are then used to predict and compare performance at some other operating conditions. The theoretically predicted and the experimental values for both present and previous studies are found to be in good agreement. The analysis of the phenol separation data clearly shows the two parameter membrane transport model used in previous studies may not be sufficient for accurate design and analysis of a radial flow hollow fiber system for many solute-membrane systems. Finally, the mass transfer correlation for the shell side of the hollow fiber module is also developed and compared with the correlation available in literature.

A GENERAL MODEL FOR MULTICOMPONENT TRANSPORT IN NONPOROUS MEMBRANES BASED ON MAXWELL-STEFAN FORMULATION

A “general” model of membrane transport was formulated using the mechanistic Maxwell-Stefan approach and generalized driving forces, which included the contribution from the various internal and external driving forces. Transient models of dialysis and pervaporation were developed that used exactly the same general model to describe the transport through the membrane. In this model, the bulk solution/polymer equilibria were described by a modified Flory-Huggins model, and the concentration dependence of ternary Maxwell-Stefan diffusivities was described by a natural extension of the binary Vignes relationship to a multicomponent system. A notable advantage of the general model lies in the fact that the Maxwell-Stefan diffusivities retain their physical significance irrespective of the number of components present. This offers the opportunity of recovering many of the model parameters from relatively simple binary experiments. The results obtained indicate that the general model is capable of describing the transient dialysis and pervaporation of the {ethanol-water}/silicone rubber system with an identical set of concentration dependent equilibrium and diffusive parameters. The general model provides a solid framework for the theoretical description of diverse processes employing a nonporous polymer as the selective separation barrier.

The Determination of Multiple Steady States in Two Families of Biological Systems

The capacity of computational multiple steady states in two biological systems are determined by the Deficiency One Algorithm and the Subnetwork Analysis. One is a bacterial glycolysis model involving the generation of ATP, and the other one is an active membrane transport model, which is performed by pump proteins coupled to a source of metabolic energy. Mass action kinetics, is assumed and both models consist of eight coupled non-linear equations. A set of rate constants and two corresponding steady states are computed. The phenomena of bistability and hysteresis are discussed. The bifurcation of multiple steady states is also displayed. A signature of multiplicity is derived, which can be applied to mechanism identifications if steady state concentrations for some species are measured. The capacity of steady state multiplicity is extended to their families of reaction networks.

Analysis of modified surface force pore flow model with concentration polarization and comparison with Spiegler–Kedem model in reverse osmosis systems

The Modified Surface Force Pore Flow (MD-SF-PF) model is used for predicting the performance of the sodium chloride–water and sodium sulphate–water reverse osmosis systems. Unlike the previous analysis available in literature on this model, the present work takes concentration polarization into account explicitly. The required mass transfer coefficient is estimated by relating it to the feed flow rate through two additional parameters. The model equations being non-linear are solved using the orthogonal collocation technique. The model solution code is validated by comparing results with those available in literature. Simulation studies indicate observed rejection versus flux has a similar trend to that predicted by the Spiegler–Kedem model. From some of the experimental observations of Murthy [Studies on membrane transport models, Ph.D. thesis, I.I.T. Delhi, July 1996], model solution and mass transfer parameters are estimated through a combination of the Downhill Simplex method and Monte Carlo search. Simulated results using above parameters are found to be in very good agreement with the remaining experimental observations. Prediction of the membrane performance and mass transfer coefficient are also found to be very close to those estimated through the Spiegler–Kedem model. The membrane specific parameters such as the membrane thickness and pore radius are calculated from experimental data for the two different systems on a similar cellulose acetate membrane. The closeness of these parameter values again shows the validity of MD-SF-PF model.

The deduction of fine structural details of asymmetric nanofiltration membranes using theoretical models

Asymmetric cellulose acetate nanofiltration (NF) membranes were prepared at different shear rate between 152.0 and 506.67 s −1 to investigate the effect of casting shear rate on the fine structural details of the membrane and membrane performance by using sodium chloride solution. The experimental data is modeled based on the pore flow, solution–diffusion mechanisms and the extended Nernst–Plank equation. The Spiegler–Kedem membrane transport model was used to evaluate the membrane parameters such as reflection coefficient, and solute permeability, P s. The fine structural details of the nanofiltration membrane were evaluated in terms of effective pore radius r p, effective charge density X d, ratio of effective membrane thickness to membrane porosity. The measurement was conducted using steric-hindrance pore (SHP) model. The effective charge density X d, was determined using Teorell–Meyer–Sievers (TMS) model. The modeling results show that, the obtained values were in the range of the commercial available NF membranes.

Voltammetry of the phase transfer of polypeptide protamines across polarized liquid/liquid interfaces.

Using natural polypeptide protamines as the illustrating example, we observed the phase transfer of biological polyions at polarized interfaces between two immiscible electrolyte solutions (ITIES) for the first time by micropipet voltammetry. Analysis of the limiting currents indicates that each protamine carries 20 positive charges as expected from the number of arginine residues in the molecule. Also, it was shown that the shape of the voltammograms is limited by the interfacial transfer of protamines. Coupling of the transfer reactions with interfacial adsorption/desorption of protamines was also observed. ITIES will serve as a basis of amperometric sensors for biological macromolecules and also as a model for understanding potential-driven membrane transport of these molecules.

Temperature effects on sieving characteristics of thin-film composite nanofiltration membranes: pore size distributions and transport parameters

Crossflow filtration experiments were performed to measure transport of water and hydrophilic neutral organic solutes spanning a range of molecular sizes across two commercial thin-film composite nanofiltration (NF) membranes in the temperature range 5–41 °C. Non-viscous contributions to activation energies of pure water permeation across these polymeric membranes were calculated to be 3.9 and 6.4 kJ mol −1. Analysis of solute rejection using a phenomenological model of membrane transport revealed that sizes of pores that contributed to rejection followed a lognormal distribution at any given temperature. Additionally, increasing temperature increased mean pore radii and the molecular weight cutoff suggesting changes in the structure and morphology of the polymer matrix comprising the membrane barrier layer. Consistent with the free volume theory of activated gas transport, activation energies of neutral solute permeability in aqueous systems also increased with Stokes radius and molecular weight indicating their hindered diffusion in membrane pores. All activation energies for pore diffusion calculated in this study were greater than just the viscous contribution to bulk diffusion demonstrating hindered transport across the nanofiltration membranes. Finally, similar to gas transport across zeolites and rubbers, the activation energy and the Arrhenius pre-exponential factor for hindered diffusion coefficients increased with solute size and were highly correlated with each other.

Multicomponent transport across nonporous polymeric membranes

The theoretical description of multicomponent transport across nonporous polymeric membranes was investigated using two alternative frameworks: the phenomenological approach of irreversible thermodynamics and the mechanistic Stefan—Maxwell formulation. The transport models developed account for potential equilibrium and/or kinetic coupling of fluxes and the contribution of diffusion induced non-selective flow within the polymer. The models were validated against transient dialysis and pervaporation data for the {ethanol-water}/silicone rubber system. A critical assessment was obtained by recovering the model parameters from the dialysis data and using the same parameters to predict the transient pervaporation performance. An empirical modification of the Flory—Huggins model was developed to describe the sorption of small polar solutes into hydrophobic membranes, which provided a physically realistic description of the sorption equilibria for the {ethanol-water}/silicone rubber system. The irreversible thermodynamics approach was used to develop transient models of dialysis and pervaporation. Average phenomenological diffusion coefficients recovered from dialysis data can give a good qualitative prediction of pervaporation performance. However, a quantitative prediction requires the explicit inclusion of the concentration dependence of the diffusivities, which is best achieved within the mechanistic Stefan—Maxwell formulation. A generic model of membrane transport was formulated using the mechanistic Stephan—Maxwell approach and generalised driving forces, which included the contribution from the various internal and external driving forces. The results obtained indicate that the generic model is capable of describing the transient dialysis and pervaporation of the {ethanol-water}/silicone rubber system with an identical set of concentration dependent equilibrium and diffusive parameters.

MULTIPHYSICS FINITE ELEMENT MODEL OF SMALL SOLUTE TRANSPORT IN HOLLOW FIBER HEMOFILTER MEMBRANES

Previous mathematical models of hemofilter solute transport have usually employed lumped compartmental representations and uniform membrane processes, and have required a number of assumptions for analytical efficiency. These models often do not address regional variations in membrane transport, convection-diffusion interaction, and region-dependent processes such as hack-filtration. These influences are of particular importance in therapies such as high-volume hemofiltration. In order to overcome these limitations, we developed a finite element model of hollow fiber membrane transport that incorporates both momentum and mass transport. The geometry is based on the specifications of the AN69S hollow fiber hemofilter (Gambro), and modeled as a 2-D axisymmetric region with distributed blood, membrane and dialysis phases. Navier-Stokes equations were used for momentum transport in blood and dialysis phases, Brinkman equations for porous flow for membrane momentum transport, and the convection-diffusion equation for mass transport in all phases. Anisotropic solute diffusivity and Brinkman permeability coefficients were used to simulate membrane channel transport. Boundary conditions were established for pressure, velocity and concentration at membrane boundaries. The model was solved with FEMLAB (Comsol, Inc.). The current model supports transport of small solutes in aqueous medium. Model parameters can be adjusted to simulate various hemofilter operating conditions. Model performance parallels that of experimental data for comparable hemofilters.

Predictive membrane transport model for nanofiltration processes in water treatment

AbstractA membrane transport model was developed for prediction and simulation of membrane filtration (nanofiltration) dynamics with reference to permeate flux. It incorporates important phenomenological aspects of membrane transport, such as concentration polarization and gel layer formation, and illustrates the concentration of solutes as foulants in the mass‐transfer boundary layer on the membrane surface. Membrane filtration tests using tannic acid as a model organic compound were designed for investigating permeate fluxes, as well as solute concentration profiles for permeates and concentrates. Membrane performance experiments were conducted under various operation conditions by varying several parameters including solute concentrations, transmembrane pressures, and reject flow rates. The tests showed that the NF‐45 membrane composed of polypiperazine amide was more susceptible to organic fouling by tannic acid than the NF‐70 membrane made of cross‐linked aromatic polyamide. These observations were supported by surface‐potential measurements that demonstrated higher negative surface charges and greater hydrophilicity for the NF‐70 membrane in the presence of tannic acid. The predictive capability of the membrane transport model was evaluated using the results from membrane filtration tests. Model sensitivity studies were conducted to obtain information on effects of various input parameters pertaining to operating conditions and fluid‐dynamic regimes.