Extended Nernst Planck Equation Research Articles

Facilitated transport of monovalent salt mixture through ultrafiltration Na-Mordenite membrane: Numerical investigations of electric and dielectric contributions

In this paper, a knowledge model is used to describe the rejection of ionic solutions: pure monovalent salt–water solutions and four halide ions–water mixture. This model is based on a description of the transport within the pores via the extended Nernst–Planck equation. Electric and dielectric contributions on the separation selectivity are investigated from experimental results. The first version of the numerical model, based on Donnan Steric Pore Model, cannot be used to simulate the experimental curves obtained with the mixture of halide monovalent salts. Then, the introduction of ion energy hydrations in the model to describe the dielectric contribution of each ion, which are proportional to the polarizability of ions, is necessary to adapt the model. This result confirms the contribution of Van der Waals forces in the rejection process observed.

On the salt rejection properties of nanofiltration polyamide membranes formed by interfacial polymerization

Nanofiltration polyamide membranes obtained by interfacial polymerization possess skin layers containing amine and carboxylic acid groups that are distributed in an inhomogeneous fashion, leading to a bipolar fixed charge distribution (i.e. the sign of the surface charge density changes over the skin layer thickness). In this work we have modified the standard NF model (based on steric/Donnan exclusion and the extended Nernst–Planck equation) so as to account for the spatial variations of the fixed charge inside pores. The retention performances of NF membranes having bipolar charge distributions that capture the main electrostatic features of polyamide membranes have been investigated by considering 1-2, 2-1 and 1-1 electrolytes. In the range of volume fluxes usually obtained with NF polymer membranes, calculations show that the theoretical retention sequence of polyamide membranes for the above-mentioned electrolytes is 1-2 > 2-1 > 1-1, in agreement with experimental data available in the literature. This retention sequence has been shown to be specific of membranes with bipolar charge distributions since both homogeneously charged membranes and membranes with unipolar charge distributions (i.e. the concentration of charged surface groups can vary over the skin layer thickness but the sign of surface groups remains the same) would be more permeable to asymmetric electrolytes having divalent counterions than to symmetric mono-monovalent electrolytes. Conclusions drawn in this work are also likely to benefit the design of new NF membranes with targeted distribution of ionizable surface groups for desalination and water treatment.

Modelling of isothermal coupled moisture–ion transport in cementitious materials

A numerical model has been developed to predict isothermal coupled moisture–ion transport in cementitious materials. Ionic transport is described by the extended Nernst–Planck equation, which accounts for advection of the liquid phase. Moisture transport includes Fickian water-vapour relative diffusion and Darcian liquid-phase movement. The ion effect on liquid/vapour water equilibrium is taken into account. The variations of the transport properties vs the degree of saturation are described by integral functions or analytical formulas. A Freundlich's type description along with instantaneous Friedel's salt formation is considered for chloride binding at equilibrium. A kinetic equation is added in the cases of non-instantaneous binding processes. The governing equations, as well as the methods of assessment of the material properties required as input data, are described in the paper. Moreover, examples of application of the model in lab conditions are provided, which highlight its capability of predicting moisture/ionic concentration profiles even in complex configurations.

Evaluation of the Effective Charge Density on Low Pressure Nanofiltration with the Separation Characteristics of Monovalent and Divalent Solutes in the Production of Drinking Water

The electric charge on a membrane was investigated by analyzing the experimental rejection of various monovalent and divalent ionic solutes. The characteristics of the separation of ionic solutes using various nanofiltration membranes were obtained from an experimental nanofiltration set-up, with a surface area of 40 cm 2 under the operational pressures between 0.25-0.3 MPa. The state of the membrane electric charge was observed using separation coefficients, i.e., the permeation ratio of monovalent to divalent ions. To confirm the state of the membrane charge observed via the separation coefficient, a calculation using the extended Nernst-Planck equation, coupled with the Donnan equilibrium, assuming different electric charge states of the membrane, was compared with the experimental rejection of ionic solutes. The examination of the characteristics of separation using three types of nanofiltration mem branes showed that one of the membranes carried a negative/positive double charge density inside, while other two membranes carried either a positive or negative charge density.

A transport model considering charge adsorption inside pores to describe salts rejection by nanofiltration membranes

In the present work, rejection of mineral salts by nanofiltration membranes is modeled by a new knowledge model considering a charge distribution along the membrane pores. The model is classically based upon the coupling between the extended Nernst–Planck equation of transport within the pores with the Donnan, steric and dielectric interfacial exclusion. In literature, this kind of model is currently being used to describe salt rejections and it requires assessment of the membrane charge density and the dielectric constant of the solution inside the pores or the dielectric constant of the membrane material. Experimental estimation of these parameters, especially the membrane charge density, is difficult. For this reason, these parameters are usually adjusted so as to fit experimental rejections. The novelty of the proposed approach lies with the membrane charge density, which is not considered constant along the pores. Indeed, in this work, the membrane charge density was introduced in the model by means of adsorption isotherms, which were determined beforehand from the streaming potential measurements. In this case, only the dielectric constant inside pores had to be adjusted. The good agreement found between experimental and simulated rejection curves with coherent dielectric constants proves that the model theory is consistent. It has also proved that the estimated membrane charge density was, clearly, not overestimated as it was previously reported and made it possible to foresee a fully predictive model.

Effect of highly concentrated salt on retention of organic solutes by nanofiltration polymeric membranes

During desalination of feed with highly concentrated salt by nanofiltration (NF), predictive modeling was difficult due to the effect of salt on retention of organic solutes. Consequently, a better understanding of salt effect on membrane and organic solutes was required. In this study, four well-known commercially available NF polymeric membranes, NF270, NF-, Desal-5 DL and Nanomax50, were analyzed by a model based on an extended Nernst–Planck equation, using highly concentrated glucose and sodium chloride (NaCl) solutions. The results showed that with increasing salt concentration, the solute-to-pore size ratio ( λ i) decreased while the ratio of effective membrane thickness to porosity (Δ x/ ɛ) increased, indicating that the effect of salt may include decreasing solutes size, increasing membrane pore size, and increasing effective membrane thickness. Moreover, such salt effect appeared to be independent of membrane and solute types, and the correction model could well predict the retention of charged solutes at high salt concentration because electrostatic repulsion effect between charged solutes and membranes was completely screened by the salt ions. Meanwhile, several hypotheses such as membrane swelling, hydration layer thinning and particle collision were provided to explain the change of model parameters by highly concentrated salt.

Infl uence of an inhomogeneous membrane charge density on the rejection of electrolytes by nanofiltration membranes

The salt rejection of membranes with inhomogeneous fixed charge densities including both symmetric and asymmetric distributions has been investigated theoretically. The rejection rates for the different fixed charge distributions, have been calculated by using a standard nanofiltration (NF) model based on the extended Nernst-Planck equation, the local electroneutrality condition and a modified Donnan equation including both steric and electrostatic effects and have been compared with the rejection rate obtained for a homogeneous distribution with the same average fixed charge concentration. The effects of the ion diffusion coefficients and the pore radius on the membrane rejection have also been investigated for the various fixed charge distributions. It has been shown that inhomogeneous fixed charge distributions may significantly affect the rejection properties of NF membranes. The value of the fixed charge concentration at the pore inlet plays a major role and high charge densities at the pore inlet tend to increase the salt rejection. Nonetheless, it has also been shown that strong axial gradients of the fixed charge concentration can affect significantly the ion transport within pores.

Coupled model of extended Nernst–Planck equation and film theory in nanofiltration for xylo-oligosaccharide syrup

This work presents an observed retention ( R obs ) model for xylo-oligosaccharides (XOs) syrup in nanofiltration (NF) process. A commercial-purchased HDS-12-2540 NF membrane was operated in batch recycling model. The volume flux ( J v ), trans-membrane pressure (Δ P), solute concentrations in feed solution ( C b ) and permeate solution ( C p ) were collected and the experimental R obs was obtained. On the basis of the extended Nernst–Planck equation and film theory, a theoretical R obs model was established. The model parameters including solute permeability ( P s ) and reflection coefficient ( σ) for arabinose, xylose and xylobiose were estimated by curving fitting using genetic algorithm (GA) method. It was found that the reflection coefficient ( σ) for arabinose (0.895) was higher than that for xylose (0.776), which indicated that the membrane retention to arabinose was stronger. Membrane structural parameters, such as ratio of solute radius to pore radius ( λ) and ratio of effective membrane porosity to membrane thickness ( A k /Δ x), were also estimated using steric-hindrance pore (SHP) model.

Modeling the electric transport of sulfuric and phosphoric acids through anion-exchange membranes

The electric transport of the H 2SO 4 and H 3PO 4 mixture through two types of anion-exchange membranes (strong base: ACM; weak base: AAV) was investigated. The H 2SO 4 removal efficiency from the mixture and the H 3PO 4 retention efficiency were calculated. It was found that the effectiveness of the separation process is diminished by the association of phosphoric anions with the fixed charges which reduces the effective charge concentration. The transport of acids was quantitatively described by the model based on the extended Nernst–Planck equation and the Donnan equilibria. It satisfactorily describes the electro-membrane process except its last stage when H 2SO 4 concentration at the cathode side of a membrane is of a low value. When the association equilibrium between H 2PO 4 − and the fixed charges of the ACM membrane is introduced into the model, the concentration changes of H 3PO 4 could be much better described. Assuming the ideal Donnan equilibrium to occur, the transport of the both acids cannot be described by the same parameter related to the membrane porosity, tortuosity and the ion diffusivities. Regarding H 3PO 4, different ionic equilibria were taken into account in order to show their effects on the values of optimal model parameters.

Treatment of landfill leachates by nanofiltration

Landfill leachate contains high concentrations of organic matter, color, heavy metals and toxic substances. This study presents the feasibility of a commercial nanofiltration membrane (NF-300) in the removal of pollutants from a landfill leachate generated from the Treatment Stabilization and Disposal Facility in Gujarat state of India. Two different leachate samples (Leachates A and B) were collected from the downstream side of closed landfill cells A and B. The average quality of the leachate was 67 719 mg/L COD, 217 mg/L ammonical nitrogen, 22 418 mg/L BOD, 3847 mg/L chlorides and 909 mg/L sulphate. The operating variables studied were applied pressure (4–20 atm), feed flowrate (5–15 L/min) and pH (2, 4, 5.5 and 6.7). It was observed that the solute rejection ( R O) increased with increase in feed pressure and decreased with increase in feed concentration at constant feed flowrate. In the present study, the rejection of cations followed the sequence: R O (Cr 3+) > R O (Ni 2+) > R O (Zn 2+) > R O (Cu 2+) > R O (Cd 2+) for leachates A and B. The order of solute rejection sequence is inversely proportional to the diffusion coefficients. The rejection of sulphate ions by the NF-300 membrane was 83 and 85%, while the rejection of chlorides was 62 and 65% for leachates A and B, respectively. The NF-300 membrane was characterized by using the combined-film theory-Spiegler–Kedem (CFSK) model based on irreversible thermodynamics and the ion transport model based on the extended Nernst–Planck equation. The membrane transport parameters were estimated using the Levenberg–Marquadt method. The estimated parameters were used to predict the membrane performance and the predicted values are in good agreement with the experimental results.

Nanofiltration process for separating Cr(III) from acid solutions: Experimental and modelling analysis

This study aims to analyse the nanofiltration process at laboratorial scale, for separating Cr(III) from acid solutions. Thus, several experiments were tested and the results interpreted with a suitable mathematical model. The rejection of trivalent chromium salts under acidic conditions using the nanofiltration membrane Desal 5 DK was investigated. The experimental data were described by a Donnan steric partitioning pore and dielectric exclusion model which was based on the extended Nernst–Planck equation and incorporates also the concentration polarization effect in the membrane/feed solution interface. Such approach enabled to estimate the charge membrane density which varies from 1.5 × 10 − 3 to 0.1 mol m − 3 , and the pore dielectric constant nearly 50. The predictions obtained with the mathematical model are in good agreement with the experimental data. The parametric sensitivity analysis showed that Cr(III) concentration profile across the membrane is particularly sensitive to initial concentrations of chromium and chloride. The pore dielectric constant also has a significant effect on the observed rejections of counter-ions (Cr(III) and H +) and co-ion (Cl −). The model used may be helpful to optimize NF process applied to the treatment of metal salt solutions under acidic conditions.

Rejection behavior of nickel ions from synthetic wastewater containing Na 2SO 4, NiSO 4, MgCl 2 and CaCl 2 salts by nanofiltration and characterization of the membrane

Nanofiltration (NF) has attracted increasing attention over the recent years due to the development of new applications. This paper presents the separation of heavy metal ions such as nickel from synthetic wastewater of nickel electroless plating industry. The wastewater from nickel electroless plating industry contains various metals such as nickel, sodium, magnesium and zinc. In order to ascertain the effect of ion rejection in a mixed salt system, the rejection behaviour of nickel ion in synthetic wastewater of Na 2SO 4, NiSO 4, MgCl 2 and CaCl 2 is tested by using NF300. The maximum observed solute rejection of nickel ion is 97.96% and 96.87% for an initial feed concentration of 5 and 10 ppm, respectively. The NF300 membrane is characterized by using combined film theory–Spiegler–Kedem (CFSK) model based on irreversible thermodynamics and ion transport model based on the extended Nernst–Planck equation. Boundary layer thicknesses, enrichment factors, concentration polarization modulus as well as membrane transport parameters are estimated by using the Levenberg–Marquadt method. The estimated parameters are used to predict the membrane performance and found that the predicted values are in good agreement with the experimental results.

Transport of salt mixtures through nanofiltration membranes: Numerical identification of electric and dielectric contributions

In this paper, a knowledge model is used to describe the rejection of ionic components in various proportions obtained by mixing two salts (NaCl and Na 2SO 4 or NaCl and MgCl 2). This model is based on a description of the transport within the pores via the extended Nernst–Planck equation and the steric, electric and dielectric exclusion phenomena at the membrane/solution interfaces. Electric and dielectric contributions on the separation selectivity are investigated from experimental results by estimating, respectively, the membrane charge density X d and the solvation energy through the dielectric constant within the pores ɛ p . A new way of numerical identification is presented in this paper, which consists in identifying simultaneously this couple of parameters ( ɛ p , X d ) from mixture experiments. This method appears to be particularly convenient, as it does not require additional assumption to decouple their influences on rejection. The obtained values are discussed with respect to the mixture type and its composition. It was found that the values of both ɛ p and X d are physically consistent and show monotonous behaviors with the proportions of ions. In particular, ɛ p decreases linearly when the proportion of divalent ions increases.

Separation of phosphoric acid from an industrial rinsing water by means of nanofiltration

Separation of phosphoric acid from industrial rinsing water containing dissolved aluminium was evaluated with four commercial flat sheet nanofiltration membranes. The MPF34 (Koch) and Desal-5 DK (GE Osmonics) membranes effectively retained aluminium. The aluminium removal efficiency of the NF membranes was always more than 98% for Desal-5 DK (GE Osmonics) and MPF34 (Koch). The phosphoric acid recovered in the permeate stream was 77% for Desal-5 DK and 56% for MPF34. It is observed from the study that the membranes can successfully be applied for the separation of phosphoric acid from industrial rinsing wastewaters. Membrane stability tests for the Desal-5 DK membrane showed a low decreasing of the aluminium rejection and a significant increasing of permeate flux. The extended Nernst–Planck equation, coupled with steric hindrance, the Donnan equilibrium, dielectric exclusion and chemical speciation, explained the experimental results

Modeling of batch and semi-batch membrane filtration processes

A mathematical frame for modeling batch and semi-batch membrane filtration processes is provided. The approach followed in this work uses the feed concentrations as a basis for the calculations, rather than the concentration factor. A practical computational algorithm is proposed. Our method hands separately the design equations describing the engineering aspects of batch and semi-batch systems and the models of mass transfer through the membrane. Thus, different methods can be applied to compute the permeate flux and rejection without having to modify the general framework. In particular, we present an empirical approach to characterize the membrane separation behavior based on a minimal number of experiments. Moreover, we consider irreversible thermodynamics models and a transport model based on the extended Nernst–Planck equations. Finally, various batch and semi-batch nanofiltration operations are carried out with an organic/electrolyte binary test solution to validate the proposed algorithms.

Modelling the separation by nanofiltration of a multi-ionic solution relevant to an industrial process

The current work focuses on the application of nanofiltration (NF) to the isolation of a pharmaceutical product, clavulanate (CA −), from clarified fermentation broths, which show a complex composition with five main identified ions (K +, Cl −, NH 4 +, SO 4 2− and CA −). Our aim is to predict the rejection rates of these five ions, with the NF membrane Desal-DK, which may influence the separation of CA − and play a role in the whole downstream process. Concentration polarization of this multi-ionic solution was addressed both through the application of the extended Nernst–Planck equations in the film layer, assuming limiting values for the boundary layer thickness, and by applying a new methodology which dispenses this arbitrary assumption. The latter approach was chosen to calculate the intrinsic rejection rates. The transport through the membrane pores was modelled using the Steric, Electric, Dielectric and Exclusion (SEDE) model in order to predict the intrinsic rejections of the five ions in solution. The membrane volume charge density was assessed from tangential streaming potential measurements for the system membrane/five ions solution. With a single fitted parameter, ɛ p (the dielectric constant of the solution inside pores) and considering that the ion selectivity is governed by steric, Donnan and Born dielectric exclusion mechanisms, the predicted intrinsic rejections rates of the five ions are in good agreement with the experimental values.

Electric transport of sulfuric acid through anion-exchange membranes in aqueous solutions

The current efficiency of membrane electrolysis of sulfuric acid solution using two types of anion-exchange membranes (ACM and AAV) has been determined. The transport phenomena in the membrane surrounded by polarization layers have been described by the model based on the extended Nernst–Planck equation and the Donnan equilibria, applied locally. It has been found that in the case of ACM two fitting parameters – the concentration of fixed charges ( c ¯ m = 1.1 M ) and the pore factor ( k DV θ = 0.006) – are needed for a relatively good fitting of model to the experimental data. However, the weak-base membrane AAV should be characterized by three parameters—the maximum concentration of fixed charges which can be formed in that membrane ( c ¯ m , max = 1.2 M ), the protonization constant ( K H = 6) and k DV θ = 0.0025. For both membranes the optimal values of c ¯ m are much lower than the experimental ones. The probable reasons of that discrepancy have been discussed. The thickness of the polarization layer, l pol, necessary in the model fitting has been calculated from the experimentally determined limiting current density using the equation derived for sulfuric acid. Comparing the values of l pol determined for ACM in the H 2SO 4 and HNO 3 solutions it was possible to find that only the sulfate anions carry the electric charge in that membrane in contact with dilute solutions of H 2SO 4.

Separation of weak and strong acids by electro-electrodialysis—Experiment and theory

The experimental results of the separation of acetic acid (HA) from the sulfuric acid by the electro-electrodialysis (EED) method and the modeling of process have been presented. The Neosepta membranes CMX and ACM have been used. It has been found that the efficiency of retention of HA is high (>0.9) when the process is conducted below the limiting current density with respect to HSO 4 − or SO 4 2− anions. The observed current efficiency of the H 2SO 4 removal was rather low ( CE S = ca. 0.7, when the initial concentration of H 2SO 4 in the mixture was 1 or 2 M) which was caused by the nonideal selectivity of the anion-exchange membrane. The experimental results have been described by the model based on the extended Nernst-Planck equation and the Donnan equilibrium. Since the efficiency of the process depended mainly on the selectivity of anion-exchange membrane (ACM), the concentration of fixed charges of that membrane, c ¯ m , and the ratio of volume fraction of pores to their squared tortuosity, V p/θ 2, have been chosen as the main fitting parameters. It has been found that the fitting of the EED data depends mainly on c ¯ m , whereas in the modeling of diffusion experiment (or an EED experiment conducted at low current density) both parameters are important. The best fit has been obtained for c ¯ m = 0.5 − 0.6 M , i.e. ca. one order smaller than that determined experimentally. The obtained optimal value of V p/θ 2, equal to 0.013, is consistent with those previously obtained for other Neosepta anion-exchange membrane.

Prediction of the concentration polarization in the nanofiltration/reverse osmosis of dilute multi-ionic solutions

The aim of this work was to develop a simple and accurate model for predicting the concentration polarization index in the nanofiltration (NF)/reverse osmosis (RO) of dilute multi-ionic solutions. On the grounds of this model, the total flux of the ion i at the feed-solution/membrane interface consists of the sum of the diffusion, convection and migration fluxes, the former of which is determined by conventional mass-transfer correlations duly corrected to take into account the permeation through the membrane (suction effect). The coupling of the ionic fluxes is enforced by the electroneutrality requirement at the feed-solution/membrane interface. The model developed dispenses with the arbitrary assumption of the thickness of a film layer in the vicinity of the membrane surface. Assessing the accuracy/validity of this model with multi-ionic solutions would be rather harsh, thus the model accuracy and ranges of validity were ascertained for a benchmark case: NF/RO of single salt solutions. The model predicts approximate concentration polarization indexes of the salts A +B −, A + 2B 2− and A + 3B 3− (or A 2B − 2 and A 3+B − 3) with positive deviations lower than 10% with respect to the benchmark concentration polarization index, for ions diffusivities ratios, D 1/ D 2 (or D 2/ D 1), in the range 0.16–5.5 and ϕ ≡ J v / k c < 3 , where J v is the permeation flux and k c is the mass-transfer coefficient of the salt for vanishing mass-transfer rates at impermeable walls. The main assumption of the model – the individual mass-transfer coefficients of the ions are independent of each other – appears to hold in a broad range of conditions, for single salt solutions. The model developed was expeditely applied to predict the concentration polarization in the nanofiltration of solutions containing Na +, Cl − and a dye 3− (experimental data of Bowen and Mohammad [AIChE J. 44 (8) (1998) 1799–1812]), and its predictions are in fair agreement with the predictions of the extended Nernst–Planck equations in the film layer of the “slowest” ion.

Importance of the cross-effects in the transport through ion-exchange membranes

The attempt to reduce the number of coefficients in the transport equations of linear non-equilibrium thermodynamics (LNET) has been presented. By neglecting some of the interactions between transported species the equations relating the independent experimental transport quantities have been derived. These relations have been tested using the experimental data determined for the Nafion and Neosepta membranes in contact with various 1:1 electrolytes. Generally, better predictions of the transport quantities have been obtained for the more swollen Nafion membrane. The most reliable results for the Nafion membrane have been found assuming no counterion–coion interactions ( r 12 = 0). This assumption simplifies the LNET equations to the extended Nernst-Planck equation with the convection factor. The neglecting of other interactions can lead to satisfactory results only in some certain cases. It has been demonstrated that the Monte Carlo method can help in obtaining more reliable results.