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Membrane separation has enjoyed tremendous advances in relevant material and engineering sciences, making it the fastest growing technology in water treatment. Although membranes as a broad-spectrum physical barrier have great advantages over conventional treatment processes in a myriad of applications, the need for higher selectivity and specificity in membrane separation is rising as we move to target contaminants at trace concentrations and to recover valuable chemicals from wastewater with low energy consumption. In this review, we discuss the drivers, fundamental science, and potential enabling materials for high selectivity membranes, as well as their applications in different water treatment processes. Membrane materials and processes that show promise to achieve high selectivity for water, ions, and small molecules—as well as the mechanisms involved—are highlighted. We further identify practical needs, knowledge gaps, and technological barriers in both material development and process design for high selectivity membrane processes. Finally, we discuss research priorities in the context of existing and future water supply paradigms.

Post-treatment of brine produced by reverse osmosis (RO) is a great challenge as it often requires high energy input and works under extreme operating conditions. In this study, brine from a RO plant in El Paso, TX USA was successfully treated using a pilot-scale nanophotonics enhanced solar membrane distillation (NESMD) system. The novel NESMD reactor has a nanophotonic membrane surface area of 0.2 m2 and an internal heat recovery system to recover the latent heat released during vapor condensation. By utilizing a sweeping gas operational mode under real solar irradiation (585–827 W m−2), the NESMD realized successful desalination of RO brine with the membrane flux reaching 0.45–0.65 kg m−2 h−1 and total dissolved solids (TDS) removal greater than 99.5% without an external heat condenser. The decrease in the feed flow rate to the evaporation channel of the NESMD system led to an increase in the gained output ratio (GOR) from 0.35 to 0.62. To the best of our knowledge, this is the largest photothermal reactor utilized for the desalination of real RO brine under practical solar irradiation. Compared with conventional brine treatment processes that require high temperature or pressure, the NESMD desalinates RO brine at near-ambient temperature and pressure with free solar energy, providing a promising approach for water desalination, and RO brine post treatment.

Water matrix composition impacts water treatment performance. However, matrix composition impacts have rarely been studied for electrochemical water treatment processes, and the correlation between the composition and the treatment efficiency is lacking. This work evaluated the electrochemical reduction of nitrate (ERN) using different complex water matrices: groundwater, brackish water, and reverse osmosis (RO) concentrate/brine. The ERN was conducted using a tin (Sn) cathode because of the high selectivity towards nitrogen evolution reported for Sn electrocatalysts. The co-existence of calcium (Ca

Electro-driven technologies are viewed as a potential alternative to the current state-of-the-art technology, reverse osmosis, for the desalination of brackish waters. Capacitive deionization (CDI), based on the principle of electrosorption, has been intensively researched under the premise of being energy efficient. However, electrodialysis (ED), despite being a more mature electro-driven technology, has yet to be extensively compared to CDI in terms of energetic performance. In this study, we utilize Nernst-Planck based models for continuous flow ED and constant-current membrane capacitive deionization (MCDI) to systematically evaluate the energy consumption of the two processes. By ensuring equivalently sized ED and MCDI systems-in addition to using the same feed salinity, salt removal, water recovery, and productivity across the two technologies-energy consumption is appropriately compared. We find that ED consumes less energy (has higher energy efficiency) than MCDI for all investigated conditions. Notably, our results indicate that the performance gap between ED and MCDI is substantial for typical brackish water desalination conditions (e.g., 3 g L-1 feed salinity, 0.5 g L-1 product water, 80% water recovery, and 15 L m-2 h-1 productivity), with the energy efficiency of ED often exceeding 30% and being nearly an order of magnitude greater than MCDI. We provide further insights into the inherent limitations of each technology by comparing their respective components of energy consumption, and explain why MCDI is unable to attain the performance of ED, even with ideal and optimized operation.

Based upon an international workshop, this perspective evaluates how nano-scale pore structures and unique properties that emerge at nano- and sub-nano-size domains could improve the energy efficiency and selectivity of electroseparation or electrocatalytic processes for treating potable or waste waters. An Eisenhower matrix prioritizes the urgency or impact of addressing potential barriers or opportunities. There has been little optimization of electrochemical reactors to increase mass transport rates of pollutants to, from, and within electrode surfaces, which become important as nano-porous structures are engineered into electrodes. A “trap-and-zap” strategy is discussed wherein nanostructures (pores, sieves, and crystal facets) are employed to allow localized concentration of target pollutants relative to background solutes (i.e., localized pollutant trapping). The trapping is followed by localized production of tailored reactive oxygen species to selectively degrade the target pollutant (i.e., localized zapping). Frequently overlooked in much of the electrode-material development literature, nano-scale structures touted to be highly “reactive” towards target pollutants may also be the most susceptible to material degradation (i.e., aging) or fouling by mineral scales that form due to localized pH changes. A need exists to study localized pH and electric-field related aging or fouling mechanisms and strategies to limit or reverse adverse outcomes from aging or fouling. This perspective provides examples of the trends and identifies promising directions to advance nano-materials and engineering principles to exploit the growing need for near chemical-free, advanced oxidation/reduction or separation processes enabled through electrochemistry.

Per- and polyfluoroalkyl substances (PFAS), which are present in many waters, have detrimental impacts on human health and the environment. Reverse osmosis (RO) and nanofiltration (NF) have shown excellent PFAS separation performance in water treatment; however, these membrane systems do not destroy PFAS but produce concentrated residual streams that need to be managed. Complete destruction of PFAS in RO and NF concentrate streams is ideal, but long-term sequestration strategies are also employed. Because no single technology is adequate for all situations, a range of processes are reviewed here that hold promise as components of treatment schemes for PFAS-laden membrane system concentrates. Attention is also given to relevant concentration processes because it is beneficial to reduce concentrate volume prior to PFAS destruction or sequestration. Given the costs and challenges of managing PFAS in membrane concentrates, it is critical to evaluate both established and emerging technologies in selecting processes for immediate use and continued research.

Poly(ether sulfone) (PES) is commonly used polymer in ultrafiltration (UF) membranes which, due to its hydrophobic nature, frequently incorporates the copolymer, polyvinylpyrrolidone (PVP), to impr...

Capacitive deionization (CDI), which is based on the electrosorption of ions by porous electrodes, is an emerging technology for brackish water desalination. Understanding the key drivers of energy consumption in CDI and benchmarking CDI with reverse osmosis (RO), the current state-of-the-art for brackish and seawater desalination, is crucial to guide the future development of desalination technologies. In this study, we develop system-scale models to analyze the energy consumption and energy efficiency of CDI and RO over a wide range of material properties and operating conditions. Using our models, we explore how the energetic performance of CDI and RO compare as a function of feed salinity, water recovery, salt rejection, and average water flux, which is normalized by electrode and membrane area in CDI and RO, respectively. Our analysis shows that RO is significantly more energy efficient than CDI, particularly when targeting higher salinity feed streams and higher salt rejection values. For brackish water with a salt concentration of 2000 mg L−1, achieving 50% water recovery and 75% salt rejection, with an average water flux of 10 L m−2 h−1 using CDI requires a specific energy consumption of 0.85 kWh m−3, more than eight times that of RO (0.09 kWh m−3). Importantly, our results also indicate that current efforts to improve electrode materials can only marginally reduce the energy consumption of CDI. We conclude with a discussion highlighting other important factors, such as capital cost, electrode stability, and membrane fouling, which affect the efficacy of CDI and RO for low-salinity desalination.

The resistivity of aggregates used as fill within mechanically stabilized earth (MSE) walls has been found to be a good indicator of its potential to corrode metal reinforcements. As such it is important to accurately measure the fill resistivity. Previous studies have found that resistivity is affected by the gradation of the material, as finer particles normally have much lower resistivity than coarse particles. The resistivity is also greatly dependent on moisture content. To estimate how the variation in moisture content can affect the resistivity, and the in-service performance of MSE walls, laboratory resistivity measurements were performed on 23 materials at varying moisture contents. The results were used to develop a model to estimate the resistivity of a given material knowing its minimum resistivity, porosity, and degree of saturation greater than 50%. Model predictions can potentially be used to estimate the resistivity of in-service wall fill and to harmonize field and laboratory resistivity measurements. The model could be used in future corrosion modeling that considers the seasonal or long-term moisture contents of the fill, similar to the recently developed Mechanistic Empirical Pavement Design (MEPDG) procedure.

Membrane capacitive deionization (MCDI) is a simple and low-cost method for brackish water desalination involving reversible electrosorption using high surface area, porous electrodes paired with ion-exchange membranes. Ion-exchange membranes improve charge efficiency and salt adsorption capacity by limiting the transport of co-ions and inhibiting faradaic reactions at the electrode surface. Effective ion-exchange membranes for MCDI should have high permselectivity and low ionic resistance, but there is typically a trade-off between these two properties. In this work, we studied partially sulfonated pentablock copolymer (sPBC) as a cation-exchange coating for MCDI electrodes. sPBC ion exchange coatings of varying ion exchange capacity (IEC, 1.0, 1.5, 2.0 meq g−1) and a range of casting solvent compositions (10–60 wt% n-propanol in toluene) were prepared. Transmission electron microscopy analysis of the membranes showed a morphological change from a micellar to lamellar and then to an inverse micellar structure with increasing polarity of the casting solvent. Water uptake and salt permeability increased with increasing IEC and casting solvent polarity over the entire range of conditions tested. MCDI device studies indicated that charge efficiency and salt adsorption capacity both increased with water uptake over a range of casting solvent compositions due to morphological changes in the sPBC film. This work demonstrates an effective solution-processible ion-exchange layer for MCDI using a self-assembling block copolymer and suggests that ideal ion-exchange coatings for MCDI should have high water uptake to minimize ionic resistance while at the same time maintaining a high charge density of fixed charged groups to achieve high permselectivity.

Reliable forecasts of water demand that account for factors that drive demand are imperative to understanding future urban water needs. The effects of meteorological dynamics and sociocultu...

International audience; Electrochemical advanced oxidation processes (EAOPs) have shown excellent capabilities to the abatement of recalcitrant organic pollutants. The Fenton-based electrochemical systems have shown great performance for in-situ generation of H2O2, allowing an efficient • OH production for the mineralization of organic pollutants. These systems have been widely applied at bench scale; however, studies to scale-up to a higher technology readiness level (TRL) are lacking. One of the main scale-up challenges of the Fenton-based systems is the implementation of air diffusion electrodes (ADE) in flow-by electrochemical cells. The ADE adds additional complexity with respect to mass transfer effects due to hydraulic reactor design and ADE gas pressure control. Therefore, this work experimentally investigated residence time distribution and platinum-sheet electrode mass transfer effects due to (a) the liquid cross-flow velocity through the electrochemical cell, (b) the gas pressure of the air-diffusion electrode (ADE), and (c) the presence of mesh sheet mass transfer promoters between the electrodes. Analysis of experimental results revealed a synergistic improvement of mass transfer with the ADE gas flow and the presence of mesh promoters. Engineers could exploit this synergistic effect to design electrochemical cells with significantly lower capital cost.

This study compares the scaling behavior of membrane distillation (MD) with that of nanophotonics-enabled solar membrane distillation (NESMD). Previous research has shown that NESMD, due to its localized surface heating driven by photothermal membrane coatings, is an energy-efficient system for off-grid desalination; however, concerns remained regarding the scaling behavior of self-heating surfaces. In this work, bench-scale experiments were performed, using model brackish water, to compare the scaling propensity of NESMD with MD. The results showed NESMD to be highly resistant to scaling; a three times higher salt concentration factor (c/c0) was achieved in NESMD compared to MD without any decline in flux. Analyses of the scaling layer on NESMD membranes revealed that salt deposition was 1/4 of that observed for MD. Scaling resistance in NESMD is attributed to its lower operating temperature, which increases the solubility of common scalants and decreases salt precipitation rates. Precipitation kinetics measurements revealed an order of magnitude faster precipitation under heated conditions (62 °C, k = 8.7 × 10-2 s-1) compared to ambient temperature (22 °C, k = 7.1 × 10-3 s-1). These results demonstrate a distinct advantage of NESMD over MD for the treatment of high scaling potential water, where scaling is a barrier to high water recovery.

The service life of mechanically stabilized earth walls depends on the corrosion rate of the metallic reinforcement used in their construction. The resistivity of the backfill aggregates needs to be measured accurately to estimate realistically the corrosion rate of the reinforcement. Resistivity testing is usually performed using the traditional soil box on the portion of the aggregates that passes a No. 10 or No. 8 sieve to either select or reject the backfill. For a more reasonable characterization of the corrosivity of coarse backfills, it is desirable to use their actual gradations. To that end, several resistivity boxes that were double and quadruple the dimensions of the original box were constructed. In addition to the three standard gradations specified by the Texas Department of Transportation, over 20 backfill materials sampled from sources throughout Texas were fractionated to fines, fine sand, coarse sand, and gravel. Resistivity tests were performed separately on each of these four constituents for each backfill. The results were used to evaluate a relationship that would allow the estimation of the resistivity of any desired backfill gradations from the resistivity values of these four constituents. The proposed model looks promising since the resistivity of the backfill composed of the actual gradation can be estimated with reasonable certainty. The results of this study can potentially help highway agencies and contractors use a number of local quarries that are currently disqualified based on the resistivity values obtained from only testing materials that pass a No. 8 or No. 10 sieve.

Zero Discharge Desalination (ZDD) is a very-high-recovery hybrid desalination system, typically comprised of a primary desalter (such as reverse osmosis (RO) or nanofiltration (NF)) and electrodialysis metathesis (EDM). The EDM acts as a “kidney” by removing troublesome salts from the concentrate of the primary desalter, which allows for additional recovery of potable water. A mathematical model was developed to simulate ZDD system performance using mass balance, desalination design equations, and experimental data. Model results confirm that ZDD can achieve >97% system recovery for brackish water with (a) a feed total dissolved solids (TDS) concentration of 60% of the TDS); and (c) a silica content of

Pervaporation desalination has several advantages over competing desalination technologies, most notably an ability to select for or against volatile organic compounds and the ability to process high salinity feeds at a low transmembrane pressure. Pervaporation has not been commercialized for desalination applications because of its energy intensity. However, emerging processes such as hydraulic fracturing produce high total dissolved solids (>30–45 g L−1) byproduct streams that exceed the operational limits of traditional reverse osmosis and could be treated by pervaporation. Here, we demonstrate free-standing pervaporation membranes with excellent permeance and high salt removal based on a partially sulfonated pentablock terpolymer with the tradename Nexar™. Pervaporation membranes were easily cast from this material with desalination performances comparable or superior to commercially available membranes. We found that the polymer degree of sulfonation and casting solvent polarity had a significant impact on the membranes’ water uptake but only a modest impact on the pervaporation desalination performance. Membranes with a degree of sulfonation of 52% (2.0 meq g−1 IEC) and a casting solution composed of 50 wt% n-propanol and 50 wt% toluene achieved a water flux of 3.32 kg m−2 h−1 (permeance 135 kg m−2 h−1 bar−1) with 99.5% salt removal in pervaporation from a 32 g L−1 sodium chloride feed solution at room temperature. We demonstrated that dense, non-porous Nexar™ pervaporation membrane permeance and salt separation performance were superior to commercial pervaporation membranes and equivalent to commercial membrane distillation membranes, which have much larger pores. This study demonstrates that commercially available sulfonated pentablock terpolymers are excellent membranes for pervaporation desalination because of their ease of casting and excellent performance.

Electrocatalytic-driven processes show promise for water purification, including abating persistent organic pollutants via advanced oxidation processes. Since the early 90s, thousands of research articles have systematically explored degradation of organic pollutants in batch systems and synthetic waters. Despite these research efforts, commercial treatment units for water treatment in the field are barely reported. Translation of scientific and technological advances into the market place requires identification of niche applications. Here we consider point-of-use devices. Techno-economic analysis (TEA) of electrochemical oxidation was benchmarked against commercially-available competition in the water purification sector, specifically carbon block adsorption devices. Results identified electrode material as one of the driving capital costs. Decreasing boron-doped diamond cost by ten-fold or exploring alternative emergent materials such as Ti4O7 can enhance competitiveness of electrochemical devices in the market. Meanwhile, the first-order rate constant was identified as one of the most relevant parameters because it affects operational expenses. Strategies to enhance target pollutant selectivity in real water matrices is a research need to ensure efficient water purification with minimum electrical energy requirements. TEA showed a promising scenario for electrocatalytic devices to overcome commercially-available technologies. This initial TEA identified key challenges and research needs to overcome cost limitations and to beat traditional systems in reliability in the road map towards market implementation.

Membrane capacitive deionization (MCDI) is a low-cost technology for desalination. Typically, MCDI electrodes are fabricated using a slurry of nanoparticles in an organic solvent along with polyvinylidene fluoride (PVDF) polymeric binder. Recent studies of the environmental impact of CDI have pointed to the organic solvents used in the fabrication of CDI electrodes as key contributors to the overall environmental impact of the technology. Here, we report a scalable, aqueous processing approach to prepare MCDI electrodes using water-soluble polymer poly(vinyl alcohol) (PVA) as a binder and ion-exchange polymer. Electrodes are prepared by depositing aqueous slurry of activated carbon and PVA binder followed by coating with a thin layer of PVA-based cation- or anion-exchange polymer. When coated with ion-exchange layers, the PVA-bound electrodes exhibit salt adsorption capacities up to 14.4 mg/g and charge efficiencies up to 86.3%, higher than typically achieved for activated carbon electrodes with a hydrophobic polymer binder and ion-exchange membranes (5-13 mg/g). Furthermore, when paired with low-resistance commercial ion-exchange membranes, salt adsorption capacities exceed 18 mg/g. Our overall approach demonstrates a simple, environmentally friendly, cost-effective, and scalable method for the fabrication of high-capacity MCDI electrodes.

Reverse electrodialysis (RED) can produce electric energy from waste heat using thermolytic solutions (e.g., NH4HCO3) where waste heat is used to regenerate the high concentration (HC) and low concentration (LC) solutions. The salinity difference between the two solutions in RED is converted into electric potential across an ion exchange membrane (IEM), exploiting the liquid junction potential. Theoretical calculation of the junction potential is cumbersome because the activity coefficients and equilibrium speciation of individual ions are complicated for highly concentrated NH4HCO3 solution. We used a simplification of the Planck–Henderson equation to approximate the junction potential in thermolytic RED systems based on conductivity measurements, and this approximation was consistent with experimentally measured junction potentials. The experimental results also found that NH4HCO3 created greater junction potentials across anion exchange membranes than (NH4)2CO3 solution for a given molar concentration ratio. The junction potential was hardly affected by the magnitude of HC as long as the concentration ratio between HC and LC was maintained. Based on the experimental findings, we recommend that thermolytic RED systems be operated under neutral pH and high concentration ratio conditions (above 1:100 ratio). These findings provide information essential for designing and operating thermolytic RED systems for future study and practical application.

The objective of this research was to investigate the sensitivity of electrodialysis performance to variations in voltage application and membrane type when treating brackish water reverse osmosis (BWRO) concentrate waste, which typically exceeds multiple salt solubility limits. Synthetic BWRO concentrates from Arizona, Texas, and Florida of 7890–18,600 mg/L total dissolved solids were prepared with 6–10 mg/L of poly-phosphonate antiscalants. Experimentation was performed using a laboratory-scale electrodialyzer a nominal transfer area of 64 cm 2 per membrane. Flow, pressure, conductivity, temperature, and pH were measured continuously, and periodic process samples were analyzed for anion and cation concentrations. The three BWRO concentrates were successfully treated with stack voltage applications of 1.0–1.5 V/cell-pair with initial current densities of 200–600 A/m 2 and final salinity removal ratios up to 98%. This paper shows consistent specific energy consumption (approximately 0.03 kWh/m 3 per Volt/cell-pair applied per meq/L separated) for electrodialysis treatment for several concentrates across a range of salinity and composition. Successive electrodialysis treatment recovered more than 78% of BWRO concentrate without precipitation, corresponding to calcite and dolomite saturation ratios of 15. These results demonstrate that electrodialysis processes can effectively minimize concentrate waste from BWRO processes, with simulated system recoveries up to 95%.

The objective of this research was to investigate the sensitivity of electrodialysis performance to variations in voltage application and membranes when treating brackish water reverse osmosis concentrate waste. Synthetic BWRO concentrates from Arizona and Texas of 7890–14,800 mg/L total dissolved solids were prepared with poly-phosphonate antiscalants. Experimentation was performed using a laboratory-scale electrodialyzer with two sets of membranes (AMV-CMV and PCSA-PCSK) with a nominal transfer area of 64 cm 2 per membrane. Flow, pressure, conductivity, temperature, and pH were measured continuously, and periodic samples were analyzed for specific anion and cation concentrations. The BWRO concentrates were successfully treated with stack voltage applications of 0.5–1.5 V/cell-pair for salinity removal ratios up to 99% with current density less than 500 A/m 2 . This paper highlights that (1) the specific energy consumption was proportional to the applied voltage and equivalent concentration separated ( i.e. , approximately 0.03 kW h/m 3 per Volt/cell-pair applied per meq/L separated); (2) lower voltage applications decreased the relative separation rate of sulfate compared to chloride; and (3) water transport by electro-osmosis was independent of voltage application or resulting current densities, while it is affected by the ion exchange membranes.

The objective of this research was to investigate the sensitivity of electrodialysis performance to variations in hydraulic flow when treating brackish water reverse osmosis (BWRO) concentrate waste. A synthetic BWRO concentrate from Arizona of 7890 mg/L total dissolved solids was prepared with poly-phosphonate antiscalants, and desalinated with a laboratory-scale electrodialyzer with 10 cell-pairs and a transfer area of 64 cm 2 per membrane. Flow, pressure, conductivity, temperature, and pH were measured continuously, and periodic process samples were analyzed by ion chromatography and inductively coupled plasma-optical emission spectrometry for anion and cation concentrations, respectively. The BWRO concentrate was successfully treated with a stack voltage application of 1.0 V/cell-pair and current densities less than 280 A/m 2 for salinity removal ratios up to 99% (without precipitation). The superficial velocities were controlled in a range of 1.2 to 4.8 cm/s, which corresponded to Reynolds numbers of 10 to 40. This paper shows the polarization parameter (ranging from 2.0 to 3.6 A/m 2 per meq/L) as a function of Reynolds number and removal ratio, and, at maximum sensitivity, the polarization parameter was proportional to Reynolds number raised to the 0.132 power.

In Texas, Arizona, and New Mexico, colonias refer to unincorporated rural settlements along the U.S.–Mexico border. Colonias lack governance and public services normally provided by local government (Ward, 1999). Residents typically rely on well water or hauled water stored in above-ground containers. This study attempted to quantify and compare water-related perceptions and practices of colonia residents. No significant differences were observed between colonia residents using well water versus hauled-stored water for water quality perceptions and water use practices. Most, however, had negative perceptions of their water supply; a majority perceived daily water supplies as not potable. Significant paradoxical discrepancies between perceptions and practice were identified. This study adds to a small but growing literature on subjective dimensions of quality of life indicators for colonia residents. Additional studies are needed to quantify the type and level of health risks posed by compromised water supplies for this vulnerable population. Understanding differences in perceptions and practices associated with water sources could help to identify which subpopulations of colonia residents are in greatest need of water infrastructure or remediation.

The performance of electrodialysis is strongly dependent on the boundary layer thickness near an ion-exchange membrane. While a thinner boundary layer is known to enhance the ionic separation, the effects of statistical properties of the boundary layer thickness, such as the variation of the thickness and the correlation between the two boundary layers facing across the spacer, have not been elucidated. These effects were estimated by the Monte Carlo method incorporated into an analytical model. The analytical model simulates the binary ionic transport in four distinct regions of an electrodialysis cell pair: the bulk solution, boundary layer, ion-exchange membrane, and interface between the aqueous solution and ion-exchange membrane. The model current and potential relationships found that a greater variation or more positive correlation improves the ionic separation in the non-Ohmic regime. A bench-scale electrodialyzer was operated in a batch recycle system to develop steady-state current and potential relationships. Comparison between the model and experimental results found that the mean boundary layer thickness was tens of micrometers and the standard deviation of the thickness was similar to or greater than the mean thickness with the sheet-flow type mesh spacer.

In ion-exchange membrane systems such as electrodialysis, the overlimiting current phenomenon still remains a difficult topic to study due to the complicated solution method for the Nernst–Planck–Poisson model. In this study, the Nernst–Planck–Poisson equations were prepared to simulate the steady-state binary ionic transport in the diffusion boundary layer near an ion-exchange membrane. The system of differential equations was converted into the Painleve equation of the second kind in such a way that the converted model domain explicitly shows the transition point from the space-charge region to the electroneutral region in the diffusion boundary layer even before the differential equation is solved. Based on this property, mathematical expressions were proposed to estimate the limiting current density and the width of the space-charge region in the diffusion boundary layer near an ideally perm-selective ion-exchange membrane. The so-called Airy solution of the Painleve equation of the second kind was used to describe the ionic transport in the space-charge region. It was also found that the Airy function of the second kind with its derivative describes the behavior of the electric double layer developed from the ion-exchange membrane surface. In addition, a relatively simple numerical method, including a stability criterion, was used to solve the Painleve equation of the second kind to simulate the ionic transport in the diffusion boundary layer near an ion-exchange membrane.

In electrodialysis desalination, the boundary layer near ion-exchange membranes is the limiting region for the overall rate of ionic separation due to concentration polarization over tens of micrometers in that layer. Under high current conditions, this sharp concentration gradient, creating substantial ionic diffusion, can drive a preferential separation for certain ions depending on their concentration and diffusivity in the solution. Thus, this study tested a hypothesis that the boundary layer affects the competitive transport between di- and mono-valent cations, which is known to be governed primarily by the partitioning with cation-exchange membranes. A laboratory-scale electrodialyzer was operated at steady state with a mixture of 10mM KCl and 10mM CaCl(2) at various flow rates. Increased flows increased the relative calcium transport. A two-dimensional model was built with analytical solutions of the Nernst-Planck equation. In the model, the boundary layer thickness was considered as a random variable defined with three statistical parameters: mean, standard deviation, and correlation coefficient between the thicknesses of the two boundary layers facing across a spacer. Model simulations with the Monte Carlo method found that a greater calcium separation was achieved with a smaller mean, greater standard deviation, or more negative correlation coefficient. The model and experimental results were compared for the cationic transport number as well as the current and potential relationship. The mean boundary layer thickness was found to decrease from 40 to less than 10 μm as the superficial water velocity increased from 1.06 to 4.24 cm/s. The standard deviation was greater than the mean thickness at slower water velocities and smaller at faster water velocities.