Extended Nernst-Planck equation

T. Tsuru, S. Nakao and S. Kimura, Calculation of ion rejection by extended Nernst-Planck equation with charged reverse osmosis membranes for single and mixed electrolytes. J. Chem. Eng. Japan 24 (1991) 511-517. 

UF and RO models may all apply to some extent to NF. Charge, however, appears to play a more important role than for other pressure driven membrane processes. The Extended-Nemst Planck Equation (equation (3.28)) is a means of describing NF behaviour. The extended Nernst Planck equation, proposed by Deen et al. (1980), includes the Donnan expression, which describes the partitioning of solutes between solution and membrane. The model can be used to calculate an effective pore size (which does not necessarily mean that pores exist), and to determine thickness and effective charge of the membrane. This information can then be used to predict the separation of mixtures (Bowen and Mukhtar (1996)). No assumptions regarding membrane morphology ate required (Peeters (1997)). The terms represent transport due to diffusion, electric field gradient and convection respectively. Jsi is the flux of an ion i, Di,i> is the ion diffusivity in the membane, R the gas constant, F the Faraday constant, y the electrical potential and Ki,c the convective hindrance factor in the membrane. 

Assuming that the active layer of the nanofiltration membrane is porous in nature, the extended Nernst-Planck equation is apphcable to describe the transport of multieomponent systems in nanofiltration membranes. It represents transport due to diffusion, eleetrieal potential gradient and convection. The equations can be written as ... 

The Nernst-Planck equation (3.1.106) is valid in the absence of any convection. In the presence of convection, one can use the extended Nernst-Planck equation ... 

Integrate the extended Nernst-Planck equation for any counterion i in the membrane in the z-direction perpendicular to the membrane surface when the membrane excludes co-ions perfectly. Determine the constant of integration in terms of counterion concentration and resin-phase potentitd just inside the membrane, CiR (0-t-) and r (0-I-). Use now the condition of thermodynamic equilibrium for the counterion at the interface (between C,t)(0-I-) and C, , via relation (3.3.118b)) and the assumption that , = 0 to relate Caj(z) to Ciw Obtain r z] by using the electroneutrality condition (3.3.30b). 

Last but not least, the fundamental understanding of the permeation mechanisms within perovsldte and more extensively MIEC membranes is still in its infancy. The most extended models are based on the Nernst-Planck equations (e.g., the Wagner equation) providing a macroscopic view of the permeation process itself. These models usually cannot afford the description of heterogeneous materials including impurities and occluded bubbles, as is the case for most real perovskite layers. To this aim, the development of meso- or microscale models with a proper description of diffusion effects and vacancy generation would be desirable. 

The Nemst-Planck theory (under the Nemst-Einstein Eq. 4) can be derived from the extended Stefan-Maxwell equation by taking O to be a quasi-electrostatic potential referred to one ion m and taking the limit of extreme dilution. Thus it can be seen formally that Nernst-Planck theory neglects solute-solute interactions, and applies strictly only in the limit of infinite dilution. In an n-component electrolytic phase, transport can be quantified using n(n — 1) independent species mobilities, which quantify the binary interactions between each pair of species. 

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