Donnan Equations Research Articles

Numerical Investigation for The Valorization of Waste from HMD Cell Using Electrodialysis

The Hydromagnetic desalination (HMD) system is a continuous process with several advantages, including a high water recovery ratio, and can be favored economically by producing several industrial byproducts instead of discharging the highly concentrated brine to the environment. In the current work, the ions concentration in the Electrodialysis (ED) technique is simulated using COMSOL Multiphysics V.5.2 software. The ED cell simulated in this paper contains two selective membranes (anion and cation) with a width of 0.25 mm each. The salt is to be taken away in the middle domain. The cell operation has been simulated to separate the sodium and chlorine ions from the HMD brine waste solution at 40 or 55ºC temperatures at different voltages and concentrations. In this two-dimensional model, the Nernst-Plank equation has been used to describe ion flux and charge transport in the electrolyte solution. Secondary current distribution theory and the electroneutrality condition have been used in the mathematical model. Finally, Donnan equations have been used to provide the exact fulfillment of boundary conditions for constant voltage mode. The simulation shows that the highest efficiency is obtained at high temperatures and voltage with the lowest feed concentration. Finally, the results have been validated using experimental data from the literature, and a satisfying agreement has been found.

Modelling of anion-exchange membrane transport properties with taking into account the change in exchange capacity and swelling when varying bathing solution concentration and pH

In this paper we take into account the variation of anion-exchange membrane swelling and effective exchange capacity with changing the concentration and/or pH of the bathing solution when simulating membrane transport characteristics (the electric conductivity, diffusion permeability and ion transport numbers). Two main structural elements of the membrane are distinguished: a microporous polyelectrolyte gel assumed homogeneous, and macropores including structural defects and voids. It is supposed that only microporous gel exerts osmotic pressure, which causes membrane swelling described in our model by the Gregor equation, which employs the mole fractions of free and bound water. Three types of ion-exchange functional groups, namely, the secondary, tertiary and quaternary amino groups are taken into account. The degree of dissociation of these groups depends on the internal solution pH, which is described by equilibriums between protonated and deprotonated forms. The Donnan equations and electroneutrality condition are used to calculate the ion concentration in the membrane. The microheterogeneous model based on the Nernst-Planck equations is used to describe the ion transport. We take into account that the ion diffusion coefficients depend on the degree of gel swelling, this dependence is described by the Mackie-Meares equation. The model parameters include the equivalent volume of dry polyelectrolyte gel, the volume fraction of macropores, the structural parameters and others.The results of calculations are compared with experimental data on the electric conductivity, diffusion permeability and ion transport numbers for two heterogeneous anion-exchange membranes MA-40 and MA-41 (Shchekinoazot, Russia) differed by the fractions of the weakly basic functional groups. A good quantitative agreement between the theory and experiment is found for both membranes using the same set of ion hydration numbers and chemical equilibrium constants for secondary and tertiary amino groups. In particular, it is shown that with increasing pH of the bathing solution, the membrane diffusion permeability passes through a maximum, which is explained by the corresponding maximum of water content. A higher water content results in larger membrane pores, which leads to higher values of ion diffusion coefficients and higher amount of sorbed electrolyte. Together with varying water content, the variation in membrane effective exchange capacity as a function of external solution pH, determines a non-trivial behavior of electric conductivity, diffusion permeability and ion transport numbers in the membrane equilibrated with NaCl solutions of different concentration and pH.

Electrochemical Impedance of Ion-Exchange Membranes in Ternary Solutions with Two Counterions

This paper aims to study the electrochemical impedance of a system consisting of an ion-exchange membrane and two diffusion boundary layers on both sides of the membrane bathed in ternary electrolyte solutions with two counterions. An approximate analytical expression has been arranged for the impedance of an ideal ion-exchange membrane system for counterions with identical charge number on the basis of the Nernst–Planck–Donnan equations under electroneutrality. The electrochemical impedance for different values of the direct electric current of ion-exchange membrane systems with univalent ions has been numerically obtained from the Nernst–Planck and Poisson equations. The results have been compared with the approximate analytical expressions in different arrangements of practical interest, including conventional electrodialysis, bi-ionic conditions, and reverse electrodialysis. Our results provide new insights into the role played by the generalized expressions of the transport numbers, electrolyte diffu...

Concentration of Sodium Hydroxide Solutions by Electrodialysis

Several commercial ion-exchange membranes (FumaTech, DuPont) were tested in the electrodialytic oncentration of NaOH solutions in the range 1–5 M. It was found that the FumaTech cation-exchange membrane showed better performance than the bilayer Nafion membranes (DuPont) because of lower diluting effect caused by the electro-osmotic flow. However, neither of the tested membranes showed high enough efficiency for a practical application. The experimental results were modeled using the extended Nernst-Planck and Donnan equations. It was found that the derived model with two fitted parameters (effective fixed-charge concentration, tortuosity-porosity factor) satisfactorily describes the investigated process in the concentration range 1–4 M.

Modelling the swelling and osmotic properties of clay soils. Part II: The physical approach

The increasing use of clays, with a high montmorillonite content in their mineralogical composition, as hydraulic and contaminant barriers for landfill and soil remediation applications needs to be supported by an adequate theoretical modelling of the mechanical behaviour and transport properties, in order to assess the expected performances in the long term. The framework of the thermodynamics of irreversible processes was adopted in a companion paper to derive phenomenological constitutive equations for a clay soil characterised by swelling and osmotic phenomena, without specifying any of the physical mechanisms that occur at the pore scale. In this paper, a physical approach is proposed in order to provide an interpretation of the phenomenological parameters, obtained from laboratory tests. The soil structure is assumed to be constituted by montmorillonite lamellae, that can be aggregated to form the so-called tactoids, which have a slit-like geometry. Chemical equilibrium is assumed to be established between the bulk electrolyte solution and the internal pore solution at the macroscopic scale, so that the hydraulic pressure and ion concentrations can be evaluated through the Donnan equations. Water and ion transport is described at the pore scale through the generalised Navier–Stokes equation and the generalised Nernst–Planck equations, respectively. Mechanical behaviour is modelled taking into account intergranular contact stresses. The approach is applied to interpret literature experimental results, showing how it can reduce the number of tests that need to be carried out and provide insight into the physical mechanisms that determine the observed phenomena.

Modeling the electric transport of HCl and H3PO4mixture through anion-exchange membranes

The electric transport of the mixture of hydrochloric and phosphoric acids through strong base (Neosepta ACM) and weak base (Selemion AAV) anion-exchange membranes was investigated. The instantaneous efficiency of HCl removal from the cathode solution, <TEX>$CE_{Cl}$</TEX>, with and without <TEX>$H_3PO_4$</TEX> was determined. It was found that <TEX>$CE_{Cl}$</TEX> was 0.8-0.9 if the number of moles of elementary charge passed through the system, <TEX>$n_F$</TEX>, did not exceed ca. 80% of the initial number of HCl moles in the cathode solution, <TEX>$n_{Cl,ca,0}$</TEX>. The retention efficiency of <TEX>$H_3PO_4$</TEX> in that range was close to one. The transport of acid mixtures was satisfactorily described by a model based on the extended Nernst-Planck and Donnan equations for <TEX>$n_F$</TEX> not exceeding <TEX>$n_{Cl,ca,0}$</TEX>. Among the tested model parameters, most important were: concentration of fixed charges, the porosity-tortuosity coefficient, and the partition coefficient of an undissociated form of <TEX>$H_3PO_4$</TEX>. For the both membranes, the obtained optimal values of fixed charge concentration, <TEX>$\bar{c}_m$</TEX>, were up to 40% lower than the literature values of <TEX>$\bar{c}_m$</TEX> obtained from the equilibrium measurements. Regarding the <TEX>$H_3PO_4$</TEX> equilibria, it was sufficient to consider <TEX>$H_3PO_4$</TEX> as a monoprotic acid.

Membrane Potential in Multi-Ionic Mixtures

The membrane potential arising through charged porous membranes separating electrolyte mixtures at identical hydrostatic pressures but different concentrations was investigated theoretically by means of the steric, electric, and dielectric exclusion (SEDE) model. Transport phenomena were described through the Nernst-Planck formalism, while ion partitioning at the membrane/solution interfaces was accounted for by means of modified Donnan equations including steric and dielectric effects. The high concentration limit of the membrane potential depends on the mixture composition and the pore size as well. A specific feature of membrane potential in multi-ionic systems is the dependence at high concentration on the effective dielectric constant of the solution confined inside pores. Indeed, the effective dielectric constant inside pores does not affect the high concentration limit of the membrane potential in the case of single salt solutions. The low concentration limit of the membrane potential is independent of the mixture composition, the effective dielectric constant inside pores, and the pore radius, but it is ruled by counterions with the highest charge number. The membrane potential measured at high salt concentration with single salt solutions and electrolyte mixtures could be used to determine the pore size and the effective dielectric constant inside pores, respectively. This may constitute an alternative way for membrane characterization with the advantage of avoiding the need for additional rejection rate measurements.

Ionic transport through ion-exchange and bipolar membranes

Problems of steady-state and non-steady-state multi-component ion transport through ion-exchange and bipolar membranes and adjacent diffusion layers are considered. Equations of material balance of electrolyte components and the Poisson equation are used as the mathematical model. An efficient numerical method providing splitting of equations of material balance of electrolyte components and the Poisson equation is proposed. The method requires no explicit consideration of interfaces and the use of Donnan equations; it provides exact fulfillment of boundary conditions both for the mode of constant current density and for the mode of constant voltage. The following results of simulating the ion transport through ion-exchange and bipolar membranes and adjacent diffusion layers are presented: the distributions of concentrations, electrical potential, charge, and partial current densities and the voltammograms of electromembrane systems.

Application of membrane techniques in a water softening process

The paper presents the results of tests carried out on selected membranes, SEPA (Osmonics Inc., USA). The efficiency of membrane water softening was evaluated on the basis of permeate flux and the physicochemical analysis of raw water and permeate. The transport-separation properties of the membranes were determined using (a) the model solutions of NaCl and MgSO 4 and (b) the natural waters of high degree of hardness. The performed tests indicate that the DS-5-DK and DS-3-SE membranes may be applied to the softening of water for drinking and household purposes. The model based on the extended Nernst—Planck and the Donnan equations has been quantitatively fitted to the experimental data. The obtained values of model parameters indicate that in the case of RO membranes a mechanism other than the Donnan exclusion is responsible for the high retention of ions. The membrane filtration modelling will be performed in the next stage of project.

Electrophysiological properties of nerve cell bodies in the sixth abdominal ganglion of the maine lobster, Homarus americanus

1. 1. Nerve cell bodies located in the rostral half of the sixth abdominal ganglion of Homarus americanus comprise a unimodal population with respect to their resting potentials. At 21°C the mean resting potential of this population is 53·3 mV and the median (and mode) is 54 mV. 2. 2. Electrophysiological studies show that the resting membrane potentials of these cells can be described by a Donnan distribution for K + and Cl −. A prediction from the Donnan equations that large anions, such as CH 3SO 4 −, are permeable through the soma membranes is supported by the results of osmometric experiments. 3. 3. Since low temperature, ouabain, NaCN and lack of oxygen do not have an effect on the resting membrane potentials of these cells, it is concluded that metabolically dependent processes such as a Na + pump are not significantly involved in the maintenance of the resting potential. 4. 4. The cells have effective resistances ranging from 0·73 to 1·13 megohms, produce delayed rectification, but no other electrogenic responses to depolarization. 5. 5. The cell bodies lack chemo-sensitivity to the following agents: acetylcholine, GABA, glycine, glutamate, DOPA, dopamine, adrenalin and serotonin. 6. 6. It is suggested that the functions of the large somas of lobster neurons are primarily nutritive and trophic, and that the lobster soma plays no significant role in nervous integration.

A re-examination of the donnan distribution as a mechanism for membrane potentials and potassium and chloride ion distributions in crab muscle fibers

1. 1. The adductor muscle in the meropodite of the walking leg of the blue crab, Callinectes sapidus, was studied with respect to the nature of the resting membrane potential of the muscle fiber and the ionic distributions that contribute to this potential. Equations describing a Donnan distribution for the crab muscle were utilized. Measurements of the changes in membrane potentials in response to alterations of the potassium, sodium and chloride contents of the external medium correlated well with predictions from the Donnan equations. 2. 2. Cable constants were determined on surface fibers of the adductor muscle. These fibers were found to have membrane properties intermediate between fast and slow crustacean muscle fibers. 3. 3. Ion analysis of whole adductor muscle showed a discrepancy between the experimental values for potassium and chloride and their values predicted from the Donnan equations. However, excellent agreement between the experimental and predicted values were obtained when the experimental values were corrected for “bound” potassium, “bound” chloride and non-solvent water. The results of experimental tests of some osmometric predictions of the Donnan equations are also described. 4. 4. The results of these electrophysiological, osmometric and ion analyses studies strongly support the hypothesis that the striated muscle fibers of the brackish-water crab, Callinectes sapidus, are characterized by a Donnan-like distribution.

The Electric Double Layer and The Donnan Equilibrium in Relation to Plant Cell Walls

space in plant tissues into "water free space" (W.F.S.) and "Donnan free space" (D.F.S.) is examined in systems which contain electrically charged surfaces separated by various distances. It is suggested that plant cell walls should be described in terms of a system of electric double layers and not by classical Donnan equations. An approximate theory is presented which resl1lts in an expression for the equivalent width of D.F.S. in terms of the external concentration but which is independent of the surface charge density.

Complex coacervation: IV. Thermodynamic calculations on four‐component systems

AbstractThe Donnan equations in complex coacervation, are derived and discussed for a simple case. Calculations are given on the influence of the mixing proportion of two poly‐electrolytes on complex coacervation.

The distribution of ions at interfaces: The concept of a surface phase

Recent theories of the distribution of ions at a charged surface have been examined, and have been compared with the practically convenient treatment based on the idea of a definite “surface phase.” In particular, the equations of the Gouy theory have been shown to be compatible with those for a “surface phase” of thickness 1 κ which the Donnan equations have been applied.