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.     

Free-solvent model shows osmotic pressure is the dominant factor in limiting flux during protein ultrafiltration

In protein ultrafiltration (UF), the limiting flux phenomenon has been generally considered a consequence of the presence of membrane fouling or the perceived formation of a cake/gel layer that develops at high operating pressures. Subsequently, numerous theoretical models on gel/cake physics have been made to address how these factors can result in limiting flux. In a paradigm shift, the present article reestablishes the significance of osmotic pressure by examining its contribution to limiting flux in the framework of the recently developed free solvent osmotic pressure model. The resulting free-solvent-based flux model (FSB) uses the Kedem–Katchalsky model, film theory and the free solvent representation for osmotic pressure in its development. Single protein tangential-flow diafiltration experiments (30 kDa MWCO CRC membranes) were also conducted using ovalbumin (OVA, 45 kDa), bovine serum albumin (BSA, 69 kDa), and immuno-gamma globulin (IgG, 155 kDa) in moderate NaCl buffered solutions at pH 4.5, 5.4, 7 and 7.4. The membrane was preconditioned to minimize membrane fouling development during the experimental procedure. The pressure was randomly selected and flux and sieving were determined. The experimental results clearly demonstrated that the limiting flux phenomenon is not dominated by membrane fouling and the FSB model theoretically illustrates that osmotic pressure is the primary factor in limiting flux during UF. The FSB model provides excellent agreement with the experimental results while producing realistic protein wall concentrations. In addition, the pH dependence of the limiting flux is shown to correlate to the pH dependency of the specific protein diffusion coefficient.     

The effect of imposed flux on biofouling in reverse osmosis: Role of concentration polarisation and biofilm enhanced osmotic pressure phenomena

This paper describes a systematic study of biofouling in reverse osmosis process using model bacteria of Pseudomonas fluorescens and employing a sodium chloride tracer response technique for fouling characterization. It was found that the growth of biofilm at constant flux following initial bacteria colonization of the membrane surface increased with imposed flux. The rationale was that biofilm growth was nutrient dependent, where the nutrient availability at the membrane wall was controlled by the magnitude of concentration polarization, which is driven by flux. The salt tracer response showed that the biofouling comprised a hydraulic resistance and induced an enhanced osmotic pressure phenomenon; known as the biofilm enhanced osmotic pressure (BEOP) effect [M. Herzberg, M. Elimelech, Biofouling of reverse osmosis membranes: role of biofilm-enhanced osmotic pressure, Journal of Membrane Science 295 (2007) 11–20], due to hindered back diffusion of solutes through the tortuous path of the heterogeneous structure of the biofilm. For the conditions studied, the contribution of BEOP to transmembrane pressure increase was greater than the hydraulic resistance.     

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.     

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.     

Physical properties of sodium carboxymethyl cellulose molecules adsorbed on a polyacrylonitrile ultrafiltration membrane

Ultrafiltration experiments showed that the graphical relationship between flux and pressure was a straight line through the origin, provided that the wall shear rate of the bulk fluid was higher than a certain critical value or the pressure was below a critical value. A higher critical shear rate corresponded to a higher critical pressure. For these conditions the total hydrodynamic resistance was only slightly greater than the resistance of a clean membrane for pure water. This additional resistance is attributed to a (mono-) molecular layer of macromolecules which is adsorbed on the membrane in the absence of both a concentration polarization layer and a conventional gel layer.At steady state ultrafiltration conditions, an increase of the flux was obtained after replacing the bulk solution by distilled water at constant experimental conditions, which is attributed to the removal of the concentration polarization layer whereas a mono-molecular layer of macromolecules remained adsorbed on the membrane. For these conditions the flux vs. pressure relationship showed a qualitatively similar behaviour as for ultrafiltration conditions.At a constant shear rate the flux vs. pressure relationship was a straight line through the origin for pressures below the critical pressure, the value of which increased with the shear rate. This linear relationship was reversible, showing no hysteresis. However, if the pressure was higher than its critical value, the flux vs. pressure relationship was no longer a straight line as a consequence of the occurrence of an additional hydrodynamic resistance which did not disappear entirely upon lowering the pressure below its critical value. For the explanation of these phenomena it is assumed that freely moveable parts of the adsorbed macromolecules can block the entrance region of the pores in the membrane if the pressure is beyond its critical value.On the other hand, for pressures below the critical pressure or shear rates beyond the critical shear rate, the pores of the membrane are deblocked. This blocking and deblocking of pores by parts of adsorbed macromolecules apparently takes place in a partly reversible way.     

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.     

Ideal limiting fluxes in ultrafiltration: comparison of various theoretical relationships

The ultrafiltration of macromolecules is characterised by a limiting flux at high transmembrane pressures. There is also some evidence that at high pressures and low crossflow velocities the flux decreases slightly with increasing pressure. It is confirmed from a theoretical viewpoint that this can only be caused by a decrease in the average mass-transfer coefficient due to concentration increases in the boundary layer. At the practical level, we propose an expression which, for a given system, enables the ideal flux to be estimated a priori as a function of the transmembrane pressure. The ideal flux is defined as that flux which would occur in the absence of fouling and gelation. The model includes the influence of both osmotic pressure and the variation in viscosity due to concentration polarisation. Thus for predictive purposes knowledge of osmotic pressure and viscosity as a function of concentration is required. The only membrane parameter that has to be experimentally determined is the membrane permeability. In the absence of adsorption (which is the ideal case) this is the permeability to the pure solvent. The model has been tested against Jonsson's data for the ultrafiltration of dextran solutions. The results are most encouraging.     

LiCl-NaCl-KCl-H2O溶液体系渗透压的分子动力学模拟

盐湖卤水中含有大量的锂、 钾资源, 发展卤水的渗透压预测模型和方法, 对于资源的综合利用具有重要的指导意义. 本文采用分子动力学(MD)方法结合水溶液渗透压模拟(OPAS)技术, 研究了LiCl-NaCl-KCl-H₂O体系的渗透压模拟方法, 计算了298.15 K下体系多种组成和浓度的渗透压, 并将计算结果与实验值和Pitzer模型计算值进行了对比. 结果显示, 模拟计算值与实验值和Pitzer模型计算值渗透压的趋势基本一致, 包括同种溶液不同浓度和同浓度不同溶液之间的渗透压关系. 该方法可以应用于多元混合物体系对渗透压进行定性和半定量的计算.     

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.     

Concentration and temperature polarization effects during osmotic membrane distillation

Pineapple juice is one of the popular fruit juice due to its pleasant aroma and flavor. Concentration of clarified pineapple juice was carried out by osmotic membrane distillation in a plate and frame membrane module. Concentration and temperature polarization effects are found to have significant role on flux reduction during osmotic membrane distillation process. The contribution of these polarization effects on reduction of the driving force (in turn the flux) at various process conditions such as osmotic agent concentration (2–10 mol/kg (1 molality = 1 mol/kg)), flow rate (25–100 ml/min) of feed and osmotic agent are studied. Concentration polarization has more significant effect on flux reduction when compared to temperature polarization. The experimental fluxes were in good agreement with theoretical fluxes when calculated by considering both concentration and temperature polarization effects. The pineapple juice was concentrated up to a total soluble solids content of 62°Brix at ambient temperature.     

Rotating reverse osmosis: a dynamic model for flux and rejection.

Reverse osmosis (RO) is a compact process for the removal of ionic and organic pollutants from contaminated water. However, flux decline and rejection deterioration due to concentration polarization and membrane fouling hinders the application of RO technology. In this study, a rotating cylindrical RO membrane is theoretically investigated as a novel method to reduce polarization and fouling. A dynamic model based on RO membrane transport incorporating concentration polarization is used to predict the performance of rotating RO system. Operating parameters such as rotational speed and transmembrane pressure play an important role in determining the flux and rejection in rotating RO. For a given geometry, a rotational speed sufficient to generate Taylor vortices in the annulus is essential to maintain high flux as well as high rejection. The flux and rejection were calculated for wide range of operating pressures and rotational speeds.     

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.     

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.     

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.     

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.     

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 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.