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.     

In situ characterization by SAXS of concentration polarization layers during cross-flow ultrafiltration of Laponite dispersions

The structural organization inside the concentration polarization layer during cross-flow membrane separation process of Laponite colloidal dispersions has been characterized for the first time by in situ time-resolved small-angle X-ray scattering (SAXS). Thanks to the development of new "SAXS cross-flow filtration cells", concentration profiles have been measured as a function of the distance z from the membrane surface with 50 μm accuracy and linked to the permeation flux, cross-flow, and transmembrane pressure registered simultaneously. Different rheological behaviors (thixotropic gel with a yield stress or shear thinning sol) have been explored by controlling the mutual interactions between the particles as a result on the addition of peptizer. The structural reversibility of the concentration polarization layer has been demonstrated being in agreement with permeation flux measurements. These observations were related to structure of the dispersions under flow and their osmotic pressure.     

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.     

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.     

Tubular ultrafiltration flux prediction for oil-in-water emulsions: analysis of series resistances

Using the resistance-in-series (RIS) approach to permeate flux modeling, a general relationship between permeate flux, transmembrane pressure, cross-flow velocity, and feed kinematic viscosity was developed for the tubular ultrafiltration (UF) of synthetic oil-in-water emulsions. The fouling layer resistance, R_f, was 63% of the total membrane resistance, R_m′; however, concentration polarization was the predominant factor controlling resistance in the tubular UF system. An explicit form of the resistance index, Φ, was postulated based on the observed interactions between Φ, cross-flow velocity and feed kinematic viscosity and the RIS model was modified to further describe the interactions between permeate flux and operational parameters. The modified model adequately predicted flux–pressure data over the range of experimental variables examined in this study. Additionally, a set point operating pressure was determined as a function of cross-flow velocity and feed viscosity to achieve a balance between polarization and total membrane resistance.     

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.     

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.     

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.     

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.     

Modeling of the permeate flux decline during MF and UF cross-flow filtration of soy sauce lees

Cross-flow ultrafiltration and microfiltration have been used to recover refined soy sauce from soy sauce lees for over 25 years. The precise mechanism which dominated the permeate flux during batch cross-flow filtration has not been clarified. In the present study, we proposed a modified analytical method incorporated with the concept of deadend filtration to determine the initial flux of cross-flow filtration and carried out the permeate recycle and batch cross-flow filtration experiments using soy sauce lees. We used UF and MF flat membrane (0.006 m² polysulfone) module under different transmembrane pressures (TMP) and cross-flow velocities. The modified analysis provided an accurate prediction of permeate flux during the filtration of soy sauce lees, because this model can consider the change in J₀ at initial stage of filtration which was caused by the pore constriction and plugging inside membrane, and these changes may not proceed when the cake was formed on the membrane surface. Mean specific resistance of the cake increased with TMP due to the compaction of the cake and decreased with cross-flow velocity due to the change of deposited particle size, but less depended on the membrane in the present study. These results indicate that the value of J₀ determined by modified method was relevant to exclude the effects of the initial membrane fouling by pore constriction due to protein adsorption and plugging with small particles. The modified analytical method for the cake filtration developed in the present study was considered to be capable of selecting an appropriate operating conditions for many cross-flow filtration systems with UF, MF membranes.     

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.     

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.     

Studies on secondary layer formation and its characterization during cross-flow filtration of microbial cells

The formation of membrane sublayers during cross-flow filtration was studied with a standardized E. coli suspension both in a tubular and a flat channel module with different membrane materials. The height of the layers was calculated for different experimental conditions. Transmembrane pressure, cross-flow velocity, compressibility of the suspended particles, properties of the suspension, particle size and concentration were all found to have a significant effect on the formation of membrane sublayers. A decrease of the layer thickness and corresponding filtration resistance with increasing channel length was observed due to the longitudal transmembrane pressure gradient. The filtration resistance of the layer is found to be the dominant factor determining the flux rate.     

In situ monitoring techniques for concentration polarization and fouling phenomena in membrane filtration

Membrane fouling and subsequent permeate flux decline are inevitably associated with pressure-driven membrane processes. Despite the myriad of studies on membrane fouling and related phenomena--concentration polarization, cake formation and pore plugging--the fundamental mechanisms and processes involved are still not fully understood. A key to breakthroughs in understanding of fouling phenomena is the development of novel, non-invasive, in situ quantification of physico-chemical processes occurring during membrane filtration. State-of-the-art in situ monitoring techniques for concentration polarization, cake formation and fouling phenomena in pressure-driven membrane filtration are critically reviewed in this paper. The review addresses the physical principles and applications of the techniques as well as their strengths and deficiencies. Emphasis is given to techniques relevant to fouling phenomena where particles and solutes accumulate on the membrane surface such that pore plugging is negligible. The relevance of the techniques to specific processes and mechanisms involved in membrane fouling is also elaborated and discussed.     

Flux decline during cross flow membrane filtration of electrolytic solution in presence of charged nano-colloids: a simple electrokinetic model

An electrokinetic transport based approach for quantification of reversible flux decline due to the concentration polarization of an electrolyte solution in presence of charged colloids is presented. The model envisions the electrolyte transport across a charged cake or gel layer as transport of ions through charged cylindrical capillaries. This model is coupled with the standard theory of concentration polarization during cross flow membrane filtration. The analysis is carried out entirely in terms of generalized, non-dimensional variables. A dimensionless group termed as the scaled gel layer resistance evolves from the analysis, which accounts for the electrical properties of the charged nano-colloids and the electrolyte solution. A parametric study is performed to elucidate the coupled influence of mass transfer, membrane resistance, gel resistance, and electrical properties of the gel-electrolyte polarized layer. The effects of these parameters are examined on the filtration performance through the model equations.     

Modeling of the Permeate Flux during Microfiltration of BSA-Adsorbed Microspheres in a Stirred Cell

A study on the variation of the permeate flux was performed in a stirred cell charged with microspheres, to investigate the effects of the stirrer speeds (300, 400, and 600 rpm) and the BSA concentration (0.1, 0.2, 0.4, and 0.8 g/L) under constant pressure. The permeate flux increased over time before the saturation point, but it began to decrease after that point. An increase of the BSA concentration and the stirrer speed resulted in the rapid increase of the permeate flux. This is contrary to the observation of the conventional filtration experiments using a stirred cell. A resistance-in-series model was employed for the modeling of the permeate flux. The cake resistance (R(c), induced by the concentration polarization of microspheres) and the fouling resistance (R(f), induced by the adsorption of BSA inside the membrane pore) must be considered simultaneously for the modeling. These modeling results were in good agreement with the experimental data. These can be applied to the special system considering both R(c) and R(f) as well as the general filtration systems using a stirred cell. Copyright 2000 Academic Press.     

Resistance-in-series analysis in cross-flow ultrafiltration of fermentation broths of Bacillus subtilis culture

Quantitative analysis of various resistances that lead to flux decline during cross-flow ultrafiltration (UF) of the fermentation broth of Bacillus subtilis ATCC (American Type Culture Collection) 21332 culture was studied. Polyethersulfone membrane with a molecular weight cut-off (MWCO) of 100 kDa was used. Prior to cross-flow UF, the broth was treated by acid precipitation (pH 4.0) and centrifugation, and the precipitate was re-dissolved in NaOH solution. Experiments were performed at a feed pH of 7.0, a feed surfactin concentration of 1.48 g L⁻¹, and a cross-flow velocity of 0.32 m s⁻¹ but at different transmembrane pressures (ΔP, 20–100 kPa). The resistance-in-series model was used to analyze the flux behavior, which involves the resistances of membrane itself and cake as well as those due to adsorption and solute concentration polarization. It was shown that the resistance due to solute concentration polarization and of membrane dominated under the conditions examined. The resistances due to cake formation and solute adsorption were comparable, and their sum contributed below 20% of the overall resistance.     

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.     

Modeling of Fouling of Crossflow Microfiltration Membranes

Abstract

Steady-state and transient models are reviewed for predicting flux decline for crossflow microfiltration under conditions in which both external cake buildup and internal membrane fouling are contributing factors. Experimental work is not covered in the scope of this review, although reference is made to a few recent studies which have compared experimental measurements with theory. The steady-state cake thickness and permeate flux are governed by the concentration polarization layer adjacent to the cake of rejected particles which forms on the membrane surface. Depending on the characteristic particle size and the tangential shear rate, Brownian diffusion, shear-induced diffusion, or inertial lift is considered to be the dominant mechanism for particle back-transport in the polarization layer. For typical shear rates, Brownian diffusion is important for submicron particles, inertial lift is important for particles larger than approximately ten microns, and shear-induced diffusion is dominant for intermediate-sized particles. For short times, it is shown that the transient flux decline due to cake buildup is closely approximated by deadend batch filtration theory, independent of the tangential shear rate. For long times, however, the steady or quasi-steady flux increases with shear rate, because the tangential flow sweeps particles toward the filter exit and reduces cake buildup.