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.     

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.     

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.     

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.     

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.     

Prediction of dynamic permeate flux during cross-flow ultrafiltration of polyethylene glycol using concentration polarization-gel layer model

A tubular ultrafiltration model which couples the formation of a cake layer on the membrane surface and the presence of a polarized layer above the cake has been developed, which contains a single constant and the cake layer resistance to be evaluated from experiments. In the model, the tangential flow of feed material is assumed to induce a shearing effect on the cake layer resulting in the re-entrainment the particles into the bulk stream. The validity of the model over a range of cross-flow velocity, transmembrane pressure (TMP) and solute concentration was confirmed using experimental permeate fluxes obtained from the ultrafiltration of polyethylene glycol. Excellent prediction is observed for solute concentrations above some critical value at which a well developed cake layer is believed to have been formed. For concentrations below this value, the model under predicted the steady-state permeate fluxes. By ignoring the presence of the polarized layer, the model always over predict the dynamic fluxes.     

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 model for the prediction of the thermal degradation and ignition of wood under constant and variable heat flux

The ignition of combustible materials is an important aspect of the processes taking place in an unwanted fire. In this work, an experimental and theoretical study of the ignition process of wood has been carried out. Experiments of both spontaneous and piloted ignition have been performed. Constant and decreasing variable heat fluxes have been tested. A mathematical model has been used to predict the time to ignition of wood for the different operating conditions used. The solution of the model provides the temperature at each point of the solid, the local solid conversion and the time to ignition of the material. In general, a good agreement between experimental and theoretical results is obtained.     

An improved model for mass transfer during the formation of polymeric membranes by the immersion-precipitation process

An analysis is presented, which describes the isothermal ternary diffusion process encountered in the formation of a cellulose acetate polymeric membrane by a direct immersion-precipitation of polymeric solutions in a nonsolvent bath. A material coordinate was employed to derive the mass transfer equations for the membrane solution and the convective mass transfer in the coagulation bath was taken into account by solving the hydrodynamic boundary layer equations. Diffusion coefficients were measured and used to deduce ternary phenomenological coefficients. The computed results are found to agree with the experimental precipitation time and membrane morphologies observed in scanning electron photomicrographs. © 1994 John Wiley & Sons, Inc.     

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.     

A concentration polarization model for the ultrafiltration of nonionic surfactants

A theoretical model has been developed that describes ultrafiltration of nonionic surfactants. The model takes into account the fact that surfactants start to aggregate and form micelles at the critical micelle concentration. The model can be used to predict the performance of the membrane if the transport properties inside and at the membrane surface as well as the surfactant association behavior, are known. Three hydrophilic ultrafiltration membranes, made of regenerated cellulose, were used in the investigation. The cut-offs of the membranes were 10,000, 20,000, and 30,000 Da. The surfactant used in the investigation was the nonionic surfactant Triton X-100. The influence of the concentration of surfactant, transmembrane pressure and pure water flux were studied theoretically and experimentally. From the results presented in this work it can be concluded that the calculated values are in good agreement with experimental data.     

Dimensional analysis of steady state flux for microfiltration and ultrafiltration membranes

Dimensional analysis of the mass, length and time shows that the steady state flux observed for microfiltration or ultrafiltration through inorganic composite membrane can be expressed using two dimensionless numbers. The shear stress number N_S compares the shear stress against the membrane wall to the driving pressure, while the resistance number N_f compares the convective cross-flow transport to the drived transport through a layer, whose resistance is the sum of all the resistances induced by the different processes which limit the mass transport. Experimental data obtained in ultrafiltration of hydrocarbon emulsions and microfiltration of methanogenic bacteria suspensions and secondary treated wastewater were recalculated in terms of these dimensionless groups. Straight lines were plotted whose slope depends solely on the suspension and the membrane and not on the solute concentration. A negative slope and a positive intersection with the N_S axis means that a cake layer or a polarization layer can be completely eliminated at a critical cross-flow velocity; this was the case for an inorganic particles suspension and for the methanogenic suspension. A straight line of negative slope followed by a plateau means that an irreversible fouling is superimposed to the reversible phenomenon; this was observed for a secondary treated wastewater. A positive slope means that fouling predominates; this was observed with hydrocarbon emulsions.     

Fouling during the cross-flow ultrafiltration of proteins: a mass-transfer model

A model was designed to predict the effect of pH and ionic strength on fouling in the cross-flow ultrafiltration (UF)of protein solutions. Ilias and Govind’s numerical approach for concentration polarisation was combined with the Stokes–Einstein generalised equation and Bowen and Jenner’s osmotic pressure model. Coagulation was predicted when the mass-transport equation diverged to concentrations higher than that of the maximum osmotic pressure. This occurred at concentrations exceeding that of the maximum diffusivity, when diffusion became too weak to resist drag forces. In this case, one or more monolayers of protein deposited. The model was experimentally tested with 1 g/l BSA and Amicon H1P30-20 modules, for a range of pressures, ionic strengths and pH.     

The relationship between membrane surface pore characteristics and flux for ultrafiltration membranes

Surface porosities of Amicon XM100A and XM300 membranes have been measured by electron microscopy and found to be less than 1 per cent. From the measured pore size distributions it is deduced that 50 per cent of the solvent flow is through 20 to 25 per cent of the pores.The conventional model for concentration polarisation in ultrafiltration (UF), which assumes a homogeneously permeable membrane surface, has been modified to account for regions of differing permeability. An effective free area correction factor (≤ 1.0) has been introduced to allow for the effect of membrane surface properties on gel-polarised UF flux.Ultrafiltration experiments with protein solutions and membranes with a range of water fluxes confirm that gel-polarised UF flux is dependent on membrane permeability and surface properties. Effective free area correction factors vary from about 0.4 to 1.0 with values < 1.0 obtained for membranes with water fluxes typically < 150 1/m² hr at 100 kPaSupport for the effective free area concept in UF is provided by an analogy between a gel-polarised UF membrane and a composite reverse osmosis membrane. In both cases the magnitude of the upper ‘controlling’ resistance may be influenced by the pore size and spacing of the lower supporting structure.     

Quantitative description of the rejection of polymeric catalysts by ultrafiltration membranes

The rejection capability of ultrafiltration membranes for polymeric catalysts like enzymes was measured under well defined conditions. A simple model has been developed to describe the course of retention during continuous operation. Even at very low concentrations of the polymer, a significant influence of concentration polarization on the retention could be found. The course of retention could be described numerically simply by using distribution parameters for the pore size of the membrane and the particle size of the polymeric catalyst.     

Study of the competitive isotherm model and the mass transfer kinetics for a BET binary system

The competitive adsorption behavior of the binary mixture of phenetole (ethoxy-benzene) and propyl benzoate in a reversed-phase system was investigated. The adsorption equilibrium data of the single-component systems were acquired by frontal analysis. The same data for binary mixtures were acquired by the perturbation method. For both compounds, the single-component isotherm data fit best to the multilayer BET model. The experimental overloaded band profiles are in excellent agreement with the profiles calculated with either the general rate model or the modified transport-dispersive models. The competitive adsorption data were modeled using the ideal adsorbed solution (IAS) theory. The numerical values of the coefficients were derived by fitting the retention times of the perturbation pulses to those calculated using the IAS theory compiled with the coherence conditions. Finally, the elution profiles of binary mixtures were recorded. They compared very well with those calculated. As a characteristic feature of this case, an unusual retainment effect of the chromatographic band of the more retained component by the less retained one was observed. The combination of the General Rate Model and the adsorption isotherm model allowed an accurate prediction of the band profiles.     

Analysis of the slip effect on the permeate flux in membrane ultrafiltration

A method for predicting the mass transfer coefficient as well as the limiting permeate flux in membrane ultrafiltration has been found, based upon the boundary-layer theory which takes into account the slip velocity on the membrane surface. The theory presupposes the existence of a slip flow on a porous membrane surface, especially for the limiting permeate-flux operations. Further, the slip velocity increases with the size of the pores of the membrane, with feed velocity and also with feed concentration. The theory also showed that the permeate flux increases with the increase of the slip velocity. A considerable improvement in theoretical prediction of the permeate flux is expected if the slip flow effect is taken into consideration.     

A simplified mass transfer analysis for multicomponent condensation

This paper considers the problem of calculating the condensation fluxes N_i for n-component vapour mixtures, taking account of diffusional interactions in the vapour phase. A simplified method for the flux calculation is suggested requiring the assumption that the matrix of multicomponent mass transfer coefficients [W], defined by the n-1 dimensional matrix relation (N) = [W](Δx), is constant along the diffusion path. This assumption allows the calculation of the steady-state fluxes N_i in an explicit manner without iterations. For conditions of equimolar diffusion, the simplified method developed here coincides with the linearized theory development of Toor and Stewart and Prober. For other cases involving diffusion through an inert gas, for example, the method suggested here is simpler than the Toor-Stewart-Prober approach. The accuracy of the simplified method is demonstrated with a few typical examples involving mass transfer in a ternary gas phase. Though condensation in the presence of an inert gas has been treated explicitly here, the developed simplified method should find application in other mass transfer processes such as distillation, absorption and extraction.