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The paper deals with relationships between the individual transmembrane fluxes of binary electrolyte solution components and the experimentally measurable quantities describing rates of transfer processes, namely, the electric current, the transmembrane volume flow and the rates of concentration changes in the solutions adjacent to the membrane. Also, we collected and rigorously defined the kinetic coefficients describing the membrane selective and electrokinetic properties. A set of useful relationships between these coefficients is derived. An important specificity of the proposed analysis is that it does not use the Irreversible Thermodynamic approach by analyzing no thermodynamic forces that generate the fluxes under consideration. Instead, all the regularities are derived on the basis of conservation and linearity reasons. The terminology "Kinematics of Fluxes" is proposed for such an analysis on the basis of the analogy with Mechanics where Kinematics deals with regularities of motion by considering no mechanic forces. The only thermodynamic steps of the analysis relate to the discussion on the partial molar volumes of electrolyte and ions that are the equilibrium thermodynamic parameters of the adjacent solutions. These parameters are important for interrelating between the transmembrane fluxes of the solution components and the transmembrane volume flow. The paper contains short literature reviews concerned with the partial molar volumes of electrolyte and ions: the methods of measurement, the obtained results and their theoretical interpretations. It is concluded from the reviews that the classical theories should be corrected to make them applicable for sufficiently concentrated solutions, 1M or higher. The proposed correction is taken into account in the kinematic analysis.

We present a comprehensive study of cross-flow ultrafiltration (UF) of charge-stabilized suspensions, under low-salinity conditions of electrostatically strongly repelling colloidal particles. The axially varying permeate flux, near-membrane concentration-polarization (CP) layer and osmotic pressure profiles are calculated using a macroscopic diffusion-advection boundary layer method, and are compared with filtration experiments on aqueous suspensions of charge-stabilized silica particles. The theoretical description based on the one-component macroion fluid model (OCM) accounts for the strong influence of surface-released counterions on the renormalized colloid charge and suspension osmotic compressibility, and for the influence of the colloidal hydrodynamic interactions and electric double layer repulsion on the concentration-dependent suspension viscosity $\eta$, and collective diffusion coefficient $D_c$. A strong electro-hydrodynamic enhancement of $D_c$ and $\eta$, and likewise of the osmotic pressure is predicted theoretically, as compared with their values for a hard-sphere suspension. We also point to the failure of generalized Stokes-Einstein relations describing reciprocal relations between $D_c$ and $\eta$. According to our filtration model, $D_c$ is of dominant influence, giving rise to an only weakly developed CP layer having practically no effect on the permeate flux. This prediction is quantitatively confirmed by our UF measurements of the permeate flux using an aqueous suspension of charged silica spheres as the feed system. The experimentally detected fouling for the largest considered transmembrane pressure values is shown not to be due to filter cake formation by crystallization or vitrification., Comment: 17 pages, 9 figures

Membrane ultrafiltration (UF) is a pressure driven process allowing for the separation and enrichment of protein solutions and dispersions of nanosized microgel particles. The permeate flux and the near-membrane concentration-polarization (CP) layer in this process is determined by advective-diffusive dispersion transport and the interplay of applied and osmotic transmembrane pressure contributions. The UF performance is thus strongly dependent on the membrane properties, the hydrodynamic structure of the Brownian particles, their direct and hydrodynamic interactions, and the boundary conditions. We present a macroscopic description of cross-flow UF of non-ionic microgels modeled as solvent-permeable spheres. Our filtration model involves recently derived semi-analytic expressions for the concentration-dependent collective diffusion coefficient and viscosity of permeable particle dispersions [Riest et al., Soft Matter, 2015, 11, 2821]. These expressions have been well tested against computer simulation and experimental results. We analyze the CP layer properties and the permeate flux at different operating conditions and discuss various filtration process efficiency and cost indicators. Our results show that the proper specification of the concentration-dependent transport coefficients is important for reliable filtration process predictions. We also show that the solvent permeability of microgels is an essential ingredient to the UF modeling. The particle permeability lowers the particle concentration at the membrane surface, thus increasing the permeate flux., Comment: 19 pages, 11 figures (Electronic Supplementary Information included: 2 pages, 1 figure)