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.     

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.     

Unsteady state flux response: a method to determine the nature of the solute and gel layer in membrane filtration

The unsteady-state permeate flux response to a step change in transmembrane pressure is shown to result in unique flux–pressure profiles for the three types of solutes common in membrane ultrafiltration (UF): (a) solutes which exert an osmotic pressure but do not form a ‘gel’; (b) solutes which do not exert an osmotic pressure but form a ‘gel’ and (c) solutes which exert an osmotic pressure and also form a ‘gel’. It is also shown that for stirred cell UF, changes in the bulk feed solution properties (concentration, volume) are negligible on the time scale needed to attain a stable permeate flux. Unsteady-state permeate flux measurements could therefore be made at short filtration times so that the results would not be masked by changes in bulk properties.     

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.     

A unified model for flux prediction during batch cell ultrafiltration

An analysis of the flux decline encountered during ultrafiltration (UF) in a batch cell is presented by including the combined influence of the osmotic pressure and the gel layer. A predictive model for the flux decline in unstirred and stirred batch cell UF processes is developed by unifying the osmotic pressure and gel layer models. UF experiments were performed in a batch cell with polymeric solutes (PEG, dextran and PVA) and a protein (BSA), ranging widely in molecular weights and physico-chemical properties, under various operating conditions (pressure, solution pH, and stirrer speed). The present unified model predictions match closely with the experimental flux behaviour for all cases, while individual osmotic pressure and gel layer models are found to be inadequate.     

Separation of aromatic alcohols using micellar-enhanced ultrafiltration and recovery of surfactant

The removals of single aromatic alcohols, including para nitro phenol (PNP), meta nitro phenol (MNP), phenol (P), catechol (CC), beta napthol (BN) and ortho chloro phenol (OCP) from aqueous solution have been studied using micellar-enhanced ultrafiltration (MEUF). Cetyl (hexadecyl) pyridinium chloride (CPC) has been taken as the cationic surfactant. An organic polyamide membrane of molecular weight cut-off 1000 is used in the MEUF experiments. Experiments are conducted using unstirred batch cell and a continuous cross flow cell. The effects of surfactant-to-solute concentration ratio in the feed, transmembrane pressure drop and cross flow rate on the permeate flux and observed retention of each solute have been studied in detail. The retention of solutes without using surfactant varies from 3 to 15% only at a typical feed solute concentration of 0.09 kg/m³. However, under the same operating pressure (345 kPa), retention increases to about 66–98% depending on the nature of solute at the end of 30 min of experiment in the batch cell using surfactant micelles (10 kg/m³). The maximum retention of solute is obtained at surfactant-to-solute concentration ratio of 110. Free surfactant molecules present in the permeate and retentate are then recovered by a two-step chemical treatment process. In the first step, the surfactant is precipitated by potassium iodide and in the second step, the surfactant is recovered from the precipitate by the addition of cupric chloride. Optimum consumptions of potassium iodide and cupric chloride are also obtained experimentally.     

Hydrodynamic resistance of concentration polarization boundary layers in ultrafiltration

The influence of concentration polarization on the permeate flux in the ultrafiltration of aqueous Dextran T70 solutions can be described by (i) the osmotic pressure model and (ii) the boundary layer resistance model. In the latter model the hydrodynamic resistance of the non-gelled boundary layer is computed using permeability data of the Dextran molecules obtained by sedimentation experiments. It is shown both in theory and experiment that the two models are equivalent.     

Prediction of mass transfer coefficient with suction for turbulent flow in cross flow ultrafiltration

Sherwood number relations for the prediction of the mass transfer coefficient for developing concentration boundary layer have been obtained for turbulent flow regime from first principles. The common flow modules, namely, rectangular channel, tubular and radial cross flow are considered. The relationships developed include the effect of suction through the membrane. Relevant relations for the estimation of mass transfer coefficient for cross flow ultrafiltration are formulated. The proposed Sherwood relations are used in conjunction with the osmotic pressure model to predict the permeate flux in osmotic pressure governed ultrafiltration. The simulated results are compared with the experimental data obtained from the literature. A detailed parametric study has been performed to observe the effects of the operating conditions on the filtration performance in terms of the permeate quantity and quality.     

On the prediction of permeate flux for nanofiltration of concentrated aqueous solutions with thin-film composite polyamide membranes

The nanofiltration of binary aqueous solutions of glucose, sucrose and sodium sulfate was investigated using thin-film composite polyamide membranes with different molecular weight cut-off's. The NF experiments, in total recycle mode, were performed in a plate-and-frame module Lab 20 (AlfaLaval), at 22 °C and with a flowrate of 8.2 L/min, using the membranes NF90, NF200 and NF270 from FilmTec (Dow Chemical), for transmembrane pressures between 1 and 6 MPa and with aqueous solutions with osmotic pressures of between 0.5 and 3.0 MPa. The permeate flux was predicted by the osmotic pressure model, using the membrane hydraulic resistance and the solution viscosity inside the membrane pores, and computing the concentration polarization with recourse to a mass-transfer correlation specific for the plate-and-frame module used. The flux predictions, using the pure water viscosity, agree reasonably with the experimental data only for low transmembrane pressures and with the most diluted solutions. For higher transmembrane pressures and for higher solute concentration the predicted fluxes can be as far as 2.5, 4.1 and 9.6 times higher than the experimental one, for the aqueous solutions of Na₂SO₄, glucose and sucrose, respectively. These deviations are strongly reduced when the pure water viscosity is replaced by the solution viscosity adjacent to the membrane. In this case, the maximum deviation between predictions and experiments occurs also for higher transmembrane pressures and for higher solute concentration, but the maximum ratio between predicted values and the experiments were reduced now to 1.8, 2.1 and 2.9, for the aqueous solutions of Na₂SO₄, glucose and sucrose, respectively. Even using the solution viscosity adjacent to the membrane, and for the systems investigated, the osmotic pressure model must used with caution for design purposes because it may over predict the permeate flux by a factor of about 2 when the solute concentration is high.     

Influence of protein–protein interactions on bulk mass transport during ultrafiltration

Bulk mass transfer limitations can have a significant effect on the flux and selectivity during membrane ultrafiltration. Most previous studies of these phenomena have employed the simple stagnant film analysis, but this model is unable to account for the effects of solute–solute interactions on mass transport. We have developed a generalized framework for multicomponent mass transfer that includes both thermodynamic and hydrodynamic (frictional) interactions. Thermodynamic (virial) coefficients were evaluated from osmotic pressure data for albumin (BSA) and immunoglobulins (IgG), while hydrodynamic interaction parameters were determined from filtrate flux data obtained in a stirred cell using fully retentive membranes. The protein concentration profiles in the bulk solution were evaluated by numerical solution of the governing continuity equations incorporating the multicomponent diffusive flux. This model was used to analyze flux and protein transmission data obtained for the filtration of BSA and IgG mixtures through partially permeable membranes. The model accurately predicted the large reduction in flux and BSA transmission upon addition of IgG. These effects were due to the coupling between BSA and IgG mass transfer caused by protein–protein interactions.     

Micellar enhanced ultrafiltration of phenolic derivatives from their mixtures

Micellar enhanced ultrafiltration (MEUF) of different phenolic derivatives including meta-nitrophenol (MNP), catechol (CC), para-nitrophenol (PNP), and beta-napthol (BN) in their binary mixture has been studied. A 1:1 ratio of the mixture of (i) MNP with CC and (ii) PNP with BN is taken for the MEUF experiments using a cationic surfactant, namely, cetyl(hexadecyl)pyridinium chloride (CPC). An organic polyamide membrane with molecular weight cutoff of 1000 is used. Experiments are conducted using an unstirred batch cell and a continuous cross-flow cell. The effects of various operating conditions, e.g., concentrations of surfactant and solute in the feed, transmembrane pressure drop, and cross-flow rate (for cross-flow experiments) on the permeate flux and the observed retention of each solute have been studied in detail. The retention of solutes without using the surfactant varies from 3 to 15% only at a typical feed solute concentration of 0.09 kg/m3, whereas, under the same operating pressure (345 kPa), retention at the end of the experiment increases to about 66 to 99.8% depending on the nature of solute in the batch cell using surfactant micelles (10 kg/m3). Retention of solutes is less in the case of the two-component feed solution compared to the single-component feed solution. An increase in flux to the range of 9 to 16% is realized in cross-flow experiments compared to batch cell flux after one hour of operation.     

Flux decline during nanofiltration of naturally-occurring dissolved organic matter: effects of osmotic pressure,membrane permeability,and cake formation

Nanofiltration of naturally-occurring dissolved organic matter (NOM) by an aromatic polyamide membrane was measured in a crossflow bench-scale test cell and modeled using a semi-empirical osmotic pressure/cake formation model. Our objective was to examine flux decline due to NOM fouling while explicitly accounting for flux decline due to osmotic effects and changes in membrane permeability. This approach allowed quantification of the effect of ionic composition on specific NOM cake resistance, and yielded insight into flux decline due to enhanced NaCl rejection by the NOM deposit. In the absence of NOM, increasing NaCl concentration reduced salt rejection and decreased membrane permeability. Flux decline was modeled by accounting for changes in osmotic pressure with time, and by employing an effective permeability. The addition of calcium significantly reduced rejection of sodium and feed conductivity, and thus mitigated flux decline. Increasing pH from 4 (near membrane pI) to 10 increased the effective permeability but also increased NaCl rejection, which resulted in greater flux decline. The presence of NOM caused greater flux decline resulting from a combination of NOM cake resistance and increased rejection of NaCl by negatively charged NOM functional groups. Increasing NaCl concentration had little effect on the mass of NOM deposited, but significantly increased the specific resistance of the NOM cake. The effect of ionic strength on specific resistance correlated with a reduction in NOM size, estimated by separate UF permeation experiments and size exclusion chromatography analysis of UF permeate. Therefore, increased specific cake resistance is consistent with a more compact, less porous cake. Flux decline by NOM solutions showed a maximum at pH 7, where salt rejection was also a maximum. Binding of calcium reduced the ability of NOM to enhance NaCl rejection, and likely increased NOM cake resistance. Flux decline caused by NOM fouling in the presence of calcium was only significantly different than that caused by NOM in a solution of NaCl at the same ionic strength when the calcium concentration corresponded to saturation of NOM binding sites.     

The dynamics of polarisation in unstirred and stirred ultrafiltration

Polarisation of a retentive UF membrane has been studied for three types of solute, a colloid (silica sol), a protein (albumin) and a branched chain polymer (Dextran), with and without stirring. For unstirred conditions the data have been analysed by modified constantpressure filtration theory, which gives specific resistances, a, of the solutes consistent with their sizes and conformation. The simple Carman—Kozeny relationship approximates a for the colloid and the protein. For all three solutes a increased with pressure and concentration. Times to steady-state flux for stirred conditions ranged from 10 to 50 seconds, with longest times for the lowest concentrations and the largest solute. The amount of solute in the polarised layer was estimated from the measured cake (gel) resistances and the known specific resistances. Layer thicknesses ranged up to to ? 5 μm for the protein, ? 6μm for the Dextran and ? 20 μm for the silica sol. Slight deviations of flux—time profiles from the filtration model are explained by membrane—solute interactions, such as irreversible pore plugging and reversible pore obstruction.     

Modeling and analytical simulation of rotating disk ultrafiltration module

An unsteady state mass transfer model has been developed for rotating disk ultrafiltration module. Starting from the basic physics of the system, analytical expression of back transport flux generated due to rotation-induced shear field is determined, which is subsequently incorporated in the fundamental material balance equation. In order to get an analytical solution of governing partial differential equation via Laplace transformation, pseudo steady state consideration is imposed both on permeate as well as back transport flux. Once the analytical form of concentration field is obtained using the expression permeate flux, membrane surface concentration are evaluated using polymer solution theory and irreversible thermodynamics. Finally an iterative scheme is designed to simulate the permeate flux and membrane surface concentration under specified set of operating parameters. The prediction from this model is found to be in good agreement with experimental data obtained from PEG-6000/water system using cellulose acetate membrane of 5000 Da molecular weight cut-off.     

Rejection of phosphates by a ZrO2 ultrafiltration membrane

The potential of using ultrafiltration for separation of salt solutions has been explored. Solutions of phosphates were filtered through commercially available ZrO₂ ultrafiltration membranes, with a cut-off value of 15 kD. In the experiments, effects of cross flow, permeate flux, pH and ionic strength of the solution on rejection were the main items of interest. The process is modelled using the Maxwell-Stefan equations for mass transfer, accounting for the three different driving forces that govern the process (gradients in electrical potential, pressure and concentrations). The rejections observed for the phosphate ions were surprisingly high (up to 80%) considering the cut-off value of the membrane used. They were also strongly influenced by the ionic strength of the solution, indicating that electrical effects are important. The rejection curves are well described by the Maxwell-Stefan model, in which the charge of the membrane was assumed to be dependent upon solute concentration according to a Freundlich isotherm. The model is also able to describe the effect of concentration polarisation in the liquid boundary layer in front of the membrane.     

Modelling of the pore size distribution of ultrafiltration membranes

A dilute aqueous solution of polydisperse neutral dextrans was used to determine the sieving properties (flux and rejection) of porous polyacrylonitrile membranes. Gel ermeation chromatography was used to measure the solute mole and concentration in the permeate. From these data, rejection coefficients were calculated as a function of solute molecular size. A mathematical model was then developed to relate the flux and solute rejection to pore size distribution and the total number of pores, based upon the assumption that solute rejection was the result of purely geometric considerations. As a first approximation, a solute molecule was considered either too large to enter a membrane pore, or if it entered, its concentration in the permeate from that pore, as well as the solvent flux through the pore, were not affected. This model also considered the effects of steric hindrance and hydrodynamic lag on the convection of solute through a membrane. The shape and sharpness of pore size distributions were found to be useful in comparisons of ultrafiltration membranes.     

A multisolute osmotic virial equation for solutions of interest in biology

The osmotic virial equation was used to predict osmolalities of solutions of interest in biology. The second osmotic virial coefficients, Bi, account for the interactions between identical solute molecules. For multisolute solutions, the second osmotic virial cross coefficient, Bij, describes the interaction between two different solutes. We propose to use as a mixing rule for the cross coefficient the arithmetic average of the second osmotic virial coefficients of the pure species, so that only binary solution measurements are required for multisolute solution predictions. Single-solute data were fit to obtain the osmotic virial coefficients of the pure species. Using those coefficients with the proposed mixing rule, predictions were made of ternary solution osmolality, without any fitting parameters. This method is shown to make reasonably accurate predictions for three very different ternary aqueous solutions: (i) glycerol + dimethyl sulfoxide + water, (ii) hemoglobin + an ideal, dilute solute + water, and (iii) bovine serum albumin + ovalbumin + water.     

Nanofiltration studies of larger organic microsolutes in methanol solutions

Multistep organic solvent-based pharmaceutical syntheses of larger organic microsolutes having molecular weights (MW) in the range of 300–1000 generally require athermal separation processes because the active molecules and the intermediates are thermally labile. To that end, nanofiltration (NF) of methanol solutions of three selected solutes, safranin O (MW 351), brilliant blue R (MW 826) and vitamin B₁₂ (MW 1355) has been studied in a batch stirred cell for a dilute solution of each individual solute at 3034 kPa (440 psig). The solvent-resistant membranes investigated and their manufacturer-specified molecular weight cut-offs (MWCO) are MPF-44 (250), MPF-50 (700) and MPF-60 (400). During an initial transient period, the solvent flux decreased with time and the solute rejection increased with time for every membrane reaching a steady state after about 12 h. This behavior resulting from membrane compaction and pore size reduction was partially reversible. Additional studies using higher feed solute concentrations (1 and 3 wt.%) show considerable reduction in solvent flux and increase in solute rejection; the effect appears to be far more than that due to an increase in osmotic pressure and possible reasons for such a behavior have been suggested. The observed solute rejection values are generally significantly lower than the manufacturer-specified MWCO values. Additional studies varying the feed solution pressure through the membrane MPF-60 indicate that the variation of the percent rejection of solutes safranin O and brilliant blue R with the solvent flux tends to follow the relation suggested by the Finely Porous Model.     

The deformation of dextran molecules. Causes and consequences in ultrafiltration

The transfer of dextran T70 solutions through a skinned polysulfone hollow fiber membrane was studied with and without applied pressure. The molecular weight distributions of dextran in the feed and in the permeate were obtained by high pressure liquid chromatography. Two different phenomena appear to play important roles with regard to solute transfer. One is related to the shear stress imposed by the flow at the pore entrances, i.e. to permeate flux, and the other is related to the influence of solute concentration on the expansion of the macromolecular chains. These phenomena explain the observed variations with operating conditions of the overall rejection coefficient.