Validity of the Boltzmann equation to describe Donnan equilibrium at the membrane–solution interface

Surface conductivity in electrokinetic systems with porous and charged interfaces: Analytical approximations and numerical results

We derive an approximate analytical representation of the conductivity for a 1D system with porous and charged layers grafted onto parallel plates. Our theory improves on prior work by developing approximate analytical expressions applicable over an arbitrary range of potentials, both large and small as compared to the thermal voltage (RTF). Further, we describe these results in a framework of simplifying nondimensional parameters, indicating the relative dominance of various physicochemical processes. We demonstrate the efficacy of our approximate expression with comparisons to numerical representations of the exact analytical conductivity. Finally, we utilize this conductivity expression, in concert with other components of the electrokinetic coupling matrix, to describe the streaming potential and electroviscous effect in systems with porous and charged layers.

A mechanistic model for electrochemical nutrient recovery systems

Electrochemical membrane technologies such as electrodialysis have been identified as key technologies to enable nutrient recovery from wastewater. However, current electrochemical models are focused on simpler solutions than wastewater and omit key outputs such as pH, or total cell potential. A combined physico-chemical and electrochemical model was developed which includes the mechanisms of competitive transport of ions, implicit inclusion of H(+) and OH(-), pH (including ionic activity and ion pairing), different factors contributing to total cell potential and a novel method for ion exchange membrane transport. The model outputs compare well with measurements from experiments and simulate secondary effects such as electrode reactions and current leakage. Results found that membrane, rather than boundary layer or bulk resistance was the major contributor to potential drop, and that apparent boundary layers were relatively thick (3 ± 1 mm). Non-ideal solution effects such as ion-pairing and ionic activity had a major impact, particularly on multi-valent Ca(2+) ions, which enhances the capability of electrodialysis to recover monovalent nutrient ions such as K(+) and NH4(+). Decreased resistivity of ion exchange membranes to specific ions (for example, in this case nitrate) could also be detected. The methods here are validated using a comparatively simple synthetic solution of five ionic components, but are able to be easily scaled for a more complex solution, and are also compatible with additional mechanisms such as precipitation, fouling, and scaling.

Understanding Ammonium Transport in Bioelectrochemical Systems towards its Recovery

We report an integrated experimental and simulation study of ammonia recovery using microbial electrolysis cells (MECs). The transport of various species during the batch-mode operation of an MEC was examined experimentally and the results were used to validate the mathematical model for such an operation. It was found that, while the generated electrical current through the system tends to acidify (or basify) the anolyte (or catholyte), their effects are buffered by a cascade of chemical groups such as the NH₃/NH₄⁺ group, leading to relatively stable pH values in both anolyte and catholyte. The transport of NH₄⁺ ions accounts for ~90% of the total current, thus quantitatively confirming that the NH₄⁺ ions serve as effective proton shuttles during MEC operations. Analysis further indicated that, because of the Donnan equilibrium at cation exchange membrane-anolyte/catholyte interfaces, the Na⁺ ion in the anolyte actually facilitates the transport of NH₄⁺ ions during the early stage of a batch cycle and they compete with the NH₄⁺ ions weakly at later time. These insights, along with a new and simple method for predicting the strength of ammonia diffusion from the catholyte toward the anolyte, will help effective design and operation of bioeletrochemical system-based ammonia recovery systems.

Resistance identification and rational process design in Capacitive Deionization

Capacitive Deionization (CDI) is an electrochemical method for water desalination employing porous carbon electrodes. To enhance the performance of CDI, identification of electronic and ionic resistances in the CDI cell is important. In this work, we outline a method to identify these resistances. We illustrate our method by calculating the resistances in a CDI cell with membranes (MCDI) and by using this knowledge to improve the cell design. To identify the resistances, we derive a full-scale MCDI model. This model is validated against experimental data and used to calculate the ionic resistances across the MCDI cell. We present a novel way to measure the electronic resistances in a CDI cell, as well as the spacer channel thickness and porosity after assembly of the MCDI cell. We identify that for inflow salt concentrations of 20 mM the resistance is mainly located in the spacer channel and the external electrical circuit, not in the electrodes. Based on these findings, we show that the carbon electrode thickness can be increased without significantly increasing the energy consumption per mol salt removed, which has the advantage that the desalination time can be lengthened significantly.

Predicting the Rejection of Major Seawater Ions by Spiral-Wound Nanofiltration Membranes

Seawater nanofiltration (SWNF) generates a softened permeate stream and a retentate stream in which the multivalent ions accumulate, offering opportunities for practical utilization of both streams. This study presents an approach to simulation of SWNF including all major seawater ions (Na(+), Cl(-), Ca(2+), Mg(2+), and SO4(2-)) based on the Nernst-Planck equation, and uses it for permeate and retentate streams composition prediction. The number of degrees of freedom in the system was reduced by assuming a very high ionic permeability for Na(+), which only weakly affected the other parameters in the system. Two alternatives were examined to analyze the importance of concentration dependence of ion permeabilities: The assumption of constant ion permeabilities resulted in a reasonable fit with experimental data. However, for the permeate composition the overall fit was significantly improved (P < 0.0001) when the permeabilities of Ca(2+) and Mg(2+) were allowed to depend on the ratio of their total concentration to Na(+). This type of dependence emphasizes the strong interaction of divalent ions with the membrane and its effect on the membrane fixed charge through screening or charge reversal. When this effect was included, model predictions closely matched the experimental results obtained, corroborating the phenomenological approach proposed in this study.

Theory of the formation of the electric double layer at the ion exchange membrane-solution interface

This work aims to extend the study of the formation of the electric double layer at the interface defined by a solution and an ion-exchange membrane on the basis of the Nernst-Planck and Poisson equations, including different values of the counter-ion diffusion coefficient and the dielectric constant in the solution and membrane phases. The network simulation method is used to obtain the time evolution of the electric potential, the displacement electric vector, the electric charge density and the ionic concentrations at the interface between a binary electrolyte solution and a cation-exchange membrane with total co-ion exclusion. The numerical results for the temporal evolution of the interfacial electric potential and the surface electric charge are compared with analytical solutions derived in the limit of the shortest times by considering the Poisson equation for a simple cationic diffusion process. The steady-state results are justified from the Gouy-Chapman theory for the diffuse double layer in the limits of similar and high bathing ionic concentrations with respect to the fixed-charge concentration inside the membrane. Interesting new physical insights arise from the interpretation of the process of the formation of the electric double layer at the ion exchange membrane-solution interface on the basis of a membrane model with total co-ion exclusion.

Energy from CO2 using capacitive electrodes - a model for energy extraction cycles

A model is presented for the process of harvesting electrical energy from CO2 emissions using capacitive cells. The principle consists of controlling the mixing process of a concentrated CO2 gas stream with a dilute CO2 gas stream (as, for example, exhaust gas and air), thereby converting part of the released mixing energy into electrical energy. The model describes the transient reactive transport of CO2 gas absorbed in water or in monoethanolamine (MEA) solutions, under the assumption of local chemical equilibrium. The model combines the selective transport of ions through ion-exchange membranes, the accumulation of charge in the porous carbon electrodes and the coupling between the ionic current and the produced electrical current and power. We demonstrate that the model can be used to calculate the energy that can be extracted by mixing concentrated and dilute CO2 containing gas streams. Our calculation results for the process using MEA solutions have various counterintuitive features, including: 1. When dynamic equilibrium is reached in the cyclical process, the electrical charge in the anode is negative both during charging and discharging; 2. Placing an anion-exchange membrane (AEM) in the system is not required, the energy per cycle is just as large with or without an AEM.

Spatiotemporal pH dynamics in concentration polarization near ion-selective membranes

We present a detailed analysis of the transient pH dynamics for a weak, buffered electrolyte subject to voltage-driven transport through an ion-selective membrane. We show that pH fronts emanate from the concentration polarization zone next to the membrane and that these propagating fronts change the pH in the system several units from its equilibrium value. The analysis is based on a 1D model using the unsteady Poisson-Nernst-Planck equations with nonequilibrium chemistry and without assumptions of electroneutrality or asymptotically thin electric double layers. Nonequilibrium chemical effects, especially for water splitting, are shown to be important for the dynamical and spatiotemporal evolution of the pH fronts. Nonetheless, the model also shows that at steady state the assumption of chemical equilibrium can still lead to good approximations of the global pH distribution. Moreover, our model shows that the transport of the hydronium ion in the extended space charge region is governed by a balance between electromigration and water self-ionization. On the basis of this observation, we present a simple model showing that the net flux of the hydronium ion is proportional to the length of the extended space charge region and the water self-ionization rate. To demonstrate these effects in practice, we have adopted the experiment of Mai et al. (Mai, J.; Miller, H.; Hatch, A. V. Spatiotemporal Mapping of Concentration Polarization Induced pH Changes at Nanoconstrictions. ACS Nano 2012, 6, 10206) as a model problem, and by including the full chemistry and transport, we show that the present model can capture the experimentally observed pH fronts. Our model can, among other things, be used to predict and engineer pH dynamics, which can be essential to the performance of membrane-based systems for biochemical separation and analysis.

Theory of ion transport with fast acid-base equilibrations in bioelectrochemical systems

Bioelectrochemical systems recover valuable components and energy in the form of hydrogen or electricity from aqueous organic streams. We derive a one-dimensional steady-state model for ion transport in a bioelectrochemical system, with the ions subject to diffusional and electrical forces. Since most of the ionic species can undergo acid-base reactions, ion transport is combined in our model with infinitely fast ion acid-base equilibrations. The model describes the current-induced ammonia evaporation and recovery at the cathode side of a bioelectrochemical system that runs on an organic stream containing ammonium ions. We identify that the rate of ammonia evaporation depends not only on the current but also on the flow rate of gas in the cathode chamber, the diffusion of ammonia from the cathode back into the anode chamber, through the ion exchange membrane placed in between, and the membrane charge density.

Thermodynamic, energy efficiency, and power density analysis of reverse electrodialysis power generation with natural salinity gradients

Reverse electrodialysis (RED) can harness the Gibbs free energy of mixing when fresh river water flows into the sea for sustainable power generation. In this study, we carry out a thermodynamic and energy efficiency analysis of RED power generation, and assess the membrane power density. First, we present a reversible thermodynamic model for RED and verify that the theoretical maximum extractable work in a reversible RED process is identical to the Gibbs free energy of mixing. Work extraction in an irreversible process with maximized power density using a constant-resistance load is then examined to assess the energy conversion efficiency and power density. With equal volumes of seawater and river water, energy conversion efficiency of ∼ 33-44% can be obtained in RED, while the rest is lost through dissipation in the internal resistance of the ion-exchange membrane stack. We show that imperfections in the selectivity of typical ion exchange membranes (namely, co-ion transport, osmosis, and electro-osmosis) can detrimentally lower efficiency by up to 26%, with co-ion leakage being the dominant effect. Further inspection of the power density profile during RED revealed inherent ineffectiveness toward the end of the process. By judicious early discontinuation of the controlled mixing process, the overall power density performance can be considerably enhanced by up to 7-fold, without significant compromise to the energy efficiency. Additionally, membrane resistance was found to be an important factor in determining the power densities attainable. Lastly, the performance of an RED stack was examined for different membrane conductivities and intermembrane distances simulating high performance membranes and stack design. By thoughtful selection of the operating parameters, an efficiency of ∼ 37% and an overall gross power density of 3.5 W/m(2) represent the maximum performance that can potentially be achieved in a seawater-river water RED system with low-resistance ion exchange membranes (0.5 Ω cm(2)) at very small spacing intervals (50 μm).