A novel approach for modeling concentration polarization in crossflow membrane filtration based on the equivalence of osmotic pressure model and filtration theory

A theoretical model for prediction of permeate flux during crossflow membrane filtration of rigid hard spherical solute particles is developed. The model utilizes the equivalence of the hydrodynamic and thermodynamic principles governing the equilibrium in a concentration polarization layer. A combination of the two approaches yields an analytical expression for the permeate flux. The model predicts the local variation of permeate flux in a filtration channel, as well as provides a simple expression for the channel-averaged flux. A criterion for the formation of a filter cake is presented and is used to predict the downstream position in the filtration channel where cake layer build-up initiates. The predictions of permeate flux using the model compare remarkably well with a detailed numerical solution of the convective diffusion equation coupled with the osmotic pressure model. Based on the model, a novel graphical technique for prediction of the local permeate flux in a crossflow filtration channel has also been presented.     

Prediction of permeate flux during osmotic pressure-controlled electric field-enhanced cross-flow ultrafiltration

Electric field-enhanced cross-flow ultrafiltration has been carried out to separate protein, bovine serum albumin, from aqueous solution using a 30,000 molecular weight cutoff membrane. A theoretical model is developed to predict permeate flux under a laminar flow regime including the effects of external d.c. electric field and suction through the membrane for osmotic pressure-controlled ultrafiltration. The governing equations of the concentration profile in the developing mass transfer boundary layer in a rectangular channel are solved using a similarity solution method. The effect of d.c. electric field on the variation of membrane surface concentration and permeate flux along the length of the channel is quantified using this model. The expression of Sherwood number relation for estimation of mass transfer coefficient is derived. The analysis revealed that there is a significant effect of electric field on the mass transfer coefficient. A detailed parametric study has been carried out to observe the effect of feed concentration, electric field, cross-flow velocity, and pressure on the permeate flux. For 1 kg/m3 BSA solution, by applying a d.c. electric field of 1000 V/m, the permeate flux increases from 42 to 98 L/m2 h compared to that with zero electric field. The experimental results are successfully compared with the model predicted results.     

Generalized integral and similarity solutions of the concentration profiles for osmotic pressure controlled ultrafiltration

A mathematical model describing the concentration polarization phenomenon during osmotic pressure controlled ultrafiltration is presented. Generalized integral and similarity solutions of the concentration profile in the mass transfer boundary layer are obtained. The parameters governing the shape of the concentration profile vary with time in case of a batch cell and axial distance in a cross flow cell. The model is used to predict the permeate flux and the solute rejection simultaneously during unstirred batch cell and cross flow UF. The results obtained by integral and similarity solutions are compared with the results of detailed numerical solution of the governing equations for both the systems. The predictions of permeate flux from the generalized integral method are also compared with some approximate solutions in order to assess the limitations of the various approximations. UF experiments were performed with Dextran (T-20) in cross flow system and with PEG-6000 and Dextran (T-40 and T-20) in unstirred batch cell. Predictions of the model are in remarkably good agreement with detailed simulation as well as experimental results. Moreover, the integral solution can also account for the variation of diffusivity with solute concentration. Comparisons show that (a) while the generalized integral method is much simpler than the detailed numerical solutions, it is much more general and accurate than other analytical and semi-analytical solutions, and, (b) the proposed solution predicts the osmotic pressure controlled flux decline accurately over a wide range of operating conditions. The expression for gel layer governed UF (constant membrane surface concentration) is found to be an asymptotic case of the present solution.     

A mass transfer model for the prediction of rejection and flux during ultrafiltration of PEG-6000

A mass transfer model in case of ultrafiltration is proposed in the present study which is capable of predicting the permeate volumetric flux and rejection at different pressure, concentration and stirrer speed. The model is based on the steady state mass balance over the boundary layer, coupled with the results from irreversible thermodynamics. It first predicts the membrane surface and permeate concentrations — which are then utilized to calculate rejection. Permeate flux is then predicted using the result obtained from filtration theory. The model utilizes four parameters, namely, solvent permeability, solute permeability, reflection coefficient and specific cake resistance. These parameters along with the known values of the operating conditions and solution properties enable one to predict the flux as a function of time and rejection. The computed results are found to be in good agreement with the previously published data of Bhattacharjee and Bhattacharya during ultrafiltration of PEG-6000 by cellulose acetate membrane.     

Effect of ethanol on ultrafiltration of bovine albumin solutions with organic membranes

This paper investigates the ultrafiltration of albumin-ethanol solutions on polysulfone hollow fiber membranes with 30 kDa cut-off. The aim is to identify the mechanisms responsible for the observed permeate flux reduction in presence of ethanol. The variations of permeate flux with transmembrane pressure and wall shear rate fit the usual pattern of flux limitation by concentration polarization. Thus, although ethanol significantly increases the permeate viscosity, the data show that the flux decrease is not a direct consequence of the viscosity increase but rather due to reduced albumin diffusivity which decreases the back transport to the bulk solution. The specific resistance of the albumin layer on the membrane was found to be unaffected by the presence of ethanol. However the fouling potential of our solutions was found to be significantly increased by the addition of ethanol. Thus the observed flux reduction due to ethanol seems to be explained by a combination of a thicker polarization layer caused by reduced back transport and increased membrane fouling. A 10% increase in filtrated volume can be obtained by imposing periodic retrofiltrations which decrease fouling.     

Concentration Polarization of Interacting Solute Particles in Cross-Flow Membrane Filtration

A theoretical approach for predicting the influence of interparticle interactions on concentration polarization and the ensuing permeate flux decline during cross-flow membrane filtration of charged solute particles is presented. The Ornstein-Zernike integral equation is solved using appropriate closures corresponding to hard-spherical and long-range solute-solute interactions to predict the radial distribution function of the solute particles in a concentrated solution (dispersion). Two properties of the solution, namely the osmotic pressure and the diffusion coefficient, are determined on the basis of the radial distribution function at different solute concentrations. Incorporation of the concentration dependence of these two properties in the concentration polarization model comprising the convective-diffusion equation and the osmotic-pressure governed permeate flux equation leads to the coupled prediction of the solute concentration profile and the local permeate flux. The approach leads to a direct quantitative incorporation of solute-solute interactions in the framework of a standard theory of concentration polarization. The developed model is used to study the effects of ionic strength and electrostatic potential on the variations of solute diffusivity and osmotic pressure. Finally, the combined influence of these two properties on the permeate flux decline behavior during cross-flow membrane filtration of charged solute particles is predicted. Copyright 1999 Academic Press.     

Effect of electric field during gel-layer controlled ultrafiltration of synthetic and fruit juice

Electric field enhanced ultrafiltration of pectin–sucrose mixture (synthetic juice) and mosambi (Citrus sinensis (L.) Osbeck) fruit juice using 50,000 (MWCO) polyerthersulfon membrane is studied in a cross-flow cell. Pectin, completely rejected by the membrane, forms a gel type layer over the membrane surface. Under the application of an external dc electric field across the membrane, gel-layer formation is restricted leading to an enhancement of permeate flux. During ultrafiltration of synthetic juice, application of dc electric field (800 V/m) increases the permeate flux to almost threefold compared to that with zero electric field. A theoretical model based on integral method assuming suitable concentration profile in the boundary layer is developed. The proposed model is used to predict the permeate flux in gel-layer governed electric field enhanced ultrafiltration. Predictions of the model are successfully compared with the experimental results under a wide range of operating conditions. Experiments with fruit juice also demonstrated significant increase in flux with the application of a suitable electric field.     

Microfiltration of amphoteric surfactant using ceramic membranes

The microfiltration of commercially available amphoteric surfactant using ceramic membranes has been investigated. Various combinations operating conditions such as pH, electrolyte and surfactant concentrations were employed. Zeta potential and adsorption isotherms were obtained for the components of membrane surfactant system as functions of pH using surfactant or indifferent electrolyte (KCl). The shift in the membrane isoelectric point induced by the surfactant is linked to the carboxylic groups present on the surfactant which are believed to play a dominant role in the net surface charge of the membrane. A minimum in the permeate flux was found at the pH corresponding to the isoelectric point of the zwitterionic surfactant. This behaviour is ascribed to the interactions occurring between the surfactant–surfactant molecules and the surfactant–membrane. The higher fluxes obtained at low pH as compared to high pH arise from different fouling mechanisms and ionic strengths. Lower fluxes were found when inorganic electrolytes were used in conjunction with surfactant. However, as the valency of the salt increases, flux behaviour of the zwitterionic surfactant (close to isoelectric point) does not vary whilst the cationic and anionic state of the surfactants are much more affected. Interactions between surfactant molecules as a result of the charge screening effects by the larger valence ions are encouraged. The permeate flux declines with an increasing surfactant concentration even though some concentrations fall under the critical micelle concentration (c.m.c.). This is attributed to concentration polarisation in which the accumulated surfactant concentration at the membrane surface could form a stable viscous phase which is resistant to permeate flow in the secondary layer next to the membrane surface. This paper demonstrates the role interactions such as surfactant–surfactant and surfactant–membrane play in influencing the filterability of surfactant solutions using ceramic membranes.     

Numerical model of non-isothermal pervaporation in a rectangular channel

A numerical model of non-isothermal pervaporation was developed to investigate the development of the velocity, concentration and temperature fields in rectangular membrane module geometry. The model consists of the coupled Navier–Stokes equations to describe the flow field, the energy equation for the temperature field, and the species convection-diffusion equations for the concentration fields of the solution species. The coupled nonlinear transport equations were solved simultaneously for the velocity, temperature and concentration fields via a finite element approach. Simulation test cases for trichloroethylene/water, ethanol/water and iso-propanol/water pervaporation, under laminar flow conditions, revealed temperature drop axially along the module and orthogonal to the membrane surface. The nonlinear character of the concentration and temperature boundary-layers are most significant near the membrane surface. Estimation of the mass transfer coefficient assuming isothermal assumption conditions can significantly deviate from the non-isothermal predictions. For laminar conditions, predictions of the feed-side mass transfer coefficient converged to predictions from the classical Lévêque solution as the feed temperature approached the permeate temperature.     

Effect of colloids on salt transport in crossflow nanofiltration

The role of colloid deposition on the performance of a salt-rejecting NF membrane was evaluated by modeling salt transport using a two-layer transport model, which quantified the relative contributions of advection and diffusion in the cake and the membrane layers, and the effects of flux on the membrane sieving coefficient. The model was able to accurately describe how the measured permeate concentration, rejection, osmotic pressure, and flux decline varied with time. The two-layer model confirmed that the Peclet number in the cake layer was about an order of magnitude higher than that in the membrane layer, leading to significant concentration polarization at the membrane surface, as shown by others. However, the cake layer also increased overall resistance, which resulted in flux decline during constant pressure operation. Flux decline caused an increase in the actual sieving coefficient, leading to higher solute flux, lower observed rejection, and thus lower the bulk concentration. These coupled phenomena tended to mitigate the increase in concentration polarization caused by the cake. Therefore, as predicted by the model and verified by experiment, the osmotic pressure does not increase monotonically as the cake grows, and in fact can decrease when the cake layer is thick and the flux decline is significant. In our experimental system, the pressure drop across the cake layer, which was proportional to the cake thickness, was significant under the conditions studied. The effects of cake-enhanced osmotic pressure analyzed here are lower than those observed in previous studies, possibly because the transport model employed explicitly accounts for the effect of flux decline due to cake growth on the membrane sieving coefficient, and possibly because we used a somewhat different methodology to estimate cake porosity.     

Contribution of convection,diffusion and migration to electrolyte transport through nanofiltration membranes

Transport mechanisms through nanofiltration membranes are investigated in terms of contribution of convection, diffusion and migration to electrolyte transport. A Donnan steric pore model, based on the application of the extended Nernst-Planck equation and the assumption of a Donnan equilibrium at both membrane-solution interfaces, is used. The study is focused on the transport of symmetrical electrolytes (with symmetric or asymmetric diffusion coefficients). The influence of effective membrane charge density, permeate volume flux, pore radius and effective membrane thickness to porosity ratio on the contribution of the different transport mechanisms is investigated. Convection appears to be the dominant mechanism involved in electrolyte transport at low membrane charge and/or high permeate volume flux and effective membrane thickness to porosity ratio. Transport is mainly governed by diffusion when the membrane is strongly charged, particularly at low permeate volume flux and effective membrane thickness to porosity ratio. Electromigration is likely to be the dominant mechanism involved in electrolyte transport only if the diffusion coefficient of coions is greater than that of counterions.     

Theoretical analysis of the influence of the axial variation of the transmembrane pressure in cross-flow filtration of rigid spheres

A model of the axial and the radial transmembrane pressure drop in a cylindrical cross-flow filtration module was developed by performing a hydrodynamic analysis of the fluid flow based on the momentum and the continuity equations. Use of this expression for the transmembrane pressure drop together with the resistance model and the concept of shear induced diffusion of the particles at the membrane surface resulted in an expression of the permeate flux. The predictions of the transmembrane pressure drop, the permeate flux and the particles near the membrane surface are discussed for cases with and without the formation of a stagnant layer. The importance of the cylindrical membrane fiber dimensions on the permeate flux is also discussed.     

Simulation of Non-Linear Flow Density in Transition State from the Mathematical Model of the Diffusion of Metal Anions Through a Flat Sheet Supported Liquid Membrane

A mathematical model is developed for the carrier facilitated transport of metal ions through a flat sheet support liquid membrane (FSSLM) in transition state from Fick's second law. From this model, and from Fick's first law, the flow density is derived as a non-linear concentration gradient. Both expressions, concentration and flow density, depend on the thickness of the membrane and on time. Since the rate constant plays an important role in the model, it is considered as the parameter that controls the system and an equation for it is obtained. This equation explains the velocity of the co-transport process. The proposed model takes into account the species co-transported together with the metal ions. An equation for the number of moles of this species is obtained as a function of the metal species. The concentration gradient of this species explains the behaviour of pH in the feed phase during the process. The model is tested against experimental data corresponding to the transport of metal anions in acidic solution and it is shown that the co-transport process is reproduced with high accuracy.     

Modelling of pressure-driven membrane separation of electrolytes.: `High temperature' approximation

A model of pressure-driven membrane process of electrolyte separation is presented. The electric field potential assumed as being known, exact solution for permeate composition is readily obtained. All species are assumed to have the same convection velocity. Local electroneutrality condition is not used. The electric potential has been taken into account under high temperature approximation, thus reducing the problem to algebraic equation in exp(Ψ), where Ψ is dimensionless flow potential, and making it possible to calculate concentrations of ions in permeate. Negative retention is shown to be possible for one-component electrolyte solution. For electrolyte mixtures, concentration of ion with high charge is shown to “govern” the membrane selectivity in respect to low-charge ions. Results obtained are in qualitative accordance with the earlier experimental data on membrane separation of reaction mixtures in homogeneous catalysis.     

Effect of proton concentration on membrane potential across a weak amphoteric polymer membrane

An amphoteric membrane consists of both positively and negatively fixed charge groups chemically bound to the polymer chains. If the external solution is changed from alkali to acid, it is possible to obtain an experimental result in which the membrane potential changes from positive to negative through the isoelectric point. It was characterized by examining the relationship between membrane potential and proton concentration (pH) obtained from both experimental and theoretical considerations. The Nernst-Planck flux equation and the Donnan equilibrium theory were also solved for a four-component system combined with the dissociation constant, in order to discuss the pH dependence of membrane potential in a weak amphoteric membrane by comparing the experimental results with the calculated results. It was proven that the calculated results substantially deviated from the theoretical results despite a similar tendency. Such a deviation was caused by the fact that the original theory disregarded the activity coefficient and the ionic mobility, which were dependent on the fixed charge concentration in a membrane. The original theoretical model was modified by adding the effect of a fixed charge group to the activity coefficient and ionic mobility. The calculated results using the modified model explained well the experimental results if the parameter called charge effectiveness, phi, was introduced into the equations. Introduction of phi into the prediction of membrane potential was already done by Kobatake et al. in a system of a strong polyelectrolyte monopolar membrane/salt aqueous solution. In this study, it was proved that phi can also be introduced into a weak amphoteric polymer membrane/salt aqueous solution system. Finally it was also concluded that the Donnan equilibrium and the Nernst-Planck flux equation were still applicable for examining the transport phenomena for the system of a weak amphoteric charged membrane and electrolyte solutions at various pH.     

Hollow-fiber membrane adsorber: Mathematical model

Two opposite design strategies for ultrafiltration/microfiltration filters: (1) reduction of concentration polarization and particle deposition to increase permeate velocity and (2) utilization of particle deposition on membrane surface to produce an additional (to permeate) volume of clarified water, are analyzed. It is shown that the first strategy is always associated with additional expenditures in power or other material resources, making it not enough cost-effective to be competitive with non-membrane filtration processes in some water treatment applications. At the same time, the second strategy does not require additional power expenditures and provides high water recovery and cost-effectiveness. The mathematical model describing the performance of hollow-fiber membrane adsorber, which represents a second-strategy filter, is studied. A general form of the particle-deposition equation is introduced, and its terms are analyzed. As a result, its linearized form, looking like a linear equation of reversible adsorption, is chosen. A numerical solution to the system of governing equations is obtained and used to assess the accuracy of approximate solutions. A new approximate solution allowing one to evaluate the adsorber particle retentions with an acceptable accuracy is suggested.     

Hydrophobic modification of poly(phthalazinone ether sulfone ketone) hollow fiber membrane for vacuum membrane distillation

New hydrophobic poly(phthalazinone ether sulfone ketone) (PPESK) hollow fiber composite membranes coated with silicone rubber and with sol–gel polytrifluoropropylsiloxane were obtained by surface-coated modification method. The effects of coating time, coating temperature and the concentration of silicone rubber solution on the vacuum membrane distillation (VMD) properties of silicone rubber coated membranes were investigated. It was found that high water permeate flux could be gotten in low temperature and low concentration of silicone rubber solution. When the coating temperature is 60 °C, the coating time is 9 h and the concentration of silicone rubber solution is 5 g L⁻¹ the water permeate flux of the silicone rubber coated membrane is 3.5 L m⁻² h⁻¹. The prepolymerization time influence the performance of polytrifluoropropylsiloxane coated membranes, and higher prepolymerization time decrease the water permeate flux of the membrane. The water permeate flux and the salt rejection was 3.7 L m⁻² h⁻¹ and 94.6%, respectively in 30 min prepolymerization period. The VMD performances of two composite membranes during long-term operation were studied, and the results indicated that the VMD performances of two composite membranes are quite stable. The salt rejection of silicone rubber coated membrane decreased from 99 to 95% and the water permeate flux fluctuated between 2.0 and 2.5 L m⁻² h⁻¹. The salt rejection of polytrifluoropropylsiloxane coated membrane decreased from 98 to 94% and the water permeate flux fluctuated in 1 L m⁻² h⁻¹ range.     

A constant flux based mathematical model for predicting permeate flux decline in constant pressure protein ultrafiltration

This paper discusses a novel approach for predicting permeate flux decline in constant pressure ultrafiltration of protein solutions. A constant pressure process is assumed to be made up of a large number of small, sequential, constant flux ultrafiltration steps: the flux decreasing due to fouling and other related factors at the end of each step. The advantage of this approach is that constant flux ultrafiltration is easier to study, characterize, and model than constant pressure ultrafiltration. Consequently model parameters can be obtained in reliable and reproducible manner. Constant pressure ultrafiltration is dynamic in nature since both the magnitude of osmotic back-pressure and the extent of membrane fouling decrease as the permeate flux decreases with time. The proposed model takes into consideration the interplay between permeate flux, concentration polarization, and membrane fouling. The model demonstrates that the initial rapid flux decline is due to a combination of concentration polarization and membrane fouling while during the remaining part of the process, the effect of concentration polarization becomes negligible. The model also shows that concentration polarization affects the initial flux decline only at higher transmembrane pressures. This model which was validated using experimental data is conceptually simpler than other available models and easy to use. In addition to its value as a predictive tool it would particularly be useful for deciding appropriate start-up conditions in ultrafiltration processes.     

Pulsed-electric-field crossflow ultrafiltration of bovine serum albumin

A conventional crossflow ultrafiltration (CUF) apparatus was modified by the inclusion of electrodes which permitted a pulsed electric field to be produced across the ultrafiltration membrane (PEF-UF process). Using this apparatus, a discontinuous electrophoretic velocity was imposed upon the proteins being concentrated, opposing their convective movement toward the CUF membrane. This resulted in a lower concentration of rejected solute protein in the fluid boundary layer adjacent to the high-pressure side of the membrane and, hence, in a lower solute-related filtration resistance than in the case of conventional ultrafiltration (zero electric field). Studies of the PEF-UF process with bovine serum albumin (BSA) in the range of 0.5–5% w/v demonstrated a 25–40% decrease in the solute-related resistance to the permeate flux compared to the case of a zero electric field. Accordingly, higher permeate fluxes and, therefore, higher rates of concentration of the protein solution were obtained than for conventional crossflow ultrafiltration. When the electric field was reimposed following a period of operation under conventional CUF conditions, the permeate flux could be restored to nearly the same higher value observed initially for the PEF-UF process.     

Modeling of flux decline during crossflow ultrafiltration of colloidal suspensions

Mass transfer during crossflow ultrafiltration is mathematically expressed using the two-dimensional convective–diffusion equation. Numerical simulations showed that mass transfer in crossflow filtration quickly reaches a steady-state for constant boundary conditions. Hence, the unsteady nature of the permeate flux decline must be caused by changes in the hydraulic boundary condition at the membrane surface due to cake formation during filtration. A step-wise pseudo steady-state model was developed to predict the flux decline due to concentration polarization during crossflow ultrafiltration. An iterative algorithm was employed to predict the amount of flux decline for each finite time interval until the true steady-state permeate flux is established. For model verification, crossflow filtration of monodisperse polystyrene latex suspensions ranging from 0.064 to 2.16 μm in diameter was studied under constant transmembrane pressure mode. Besides the crossflow filtration tests, dead-end filtration tests were also carried out to independently determine a model parameter, the specific cake resistance. Another model parameter, the effective diffusion coefficient, is defined as the sum of molecular and shear-induced hydrodynamic diffusion coefficients. The step-wise pseudo steady-state model predictions are in good agreement with experimental results of flux decline during crossflow ultrafiltration of colloidal suspensions. Experimental variations in particle size, feed concentration, and crossflow velocity were also effectively modeled.