Nernst film model

The concentration polarization occurring in electrodialysis, that is, the concentration profiles at the membrane surface can be calculated by a mass balance taking into account all fluxes in the boundary layer and the hydrodynamic conditions in the flow channel between the membranes. To a first approximation the salt concentration at the membrane surface can be calculated and related to the current density by applying the so-called Nernst film model, which assumes that the bulk solution between the laminar boundary layers has a uniform concentration, whereas the concentration in the boundary layers changes over the thickness of the boundary layer. However, the concentration at the membrane surface and the boundary layer thickness are constant along the flow channel from the cell entrance to the exit. In a practical electrodialysis stack there will be entrance and exit effects and concentration... 

A brief comment should be made concerning the use of the Nernst-Planck equations for ion transport across the liquid film (e.g., Copeland and Marchello [1969], Kataoka et al. [1987]). This is a nonlinear, three-ion problem because of the presence of at least one coion at comparable concentration. The Nernst film model relies on the assumption of a linear concentration gradient in the liquid film. The film has no physical reality, and the calculation of nonlinear concentration profiles in it overburdens the model and offers little improvement over the much simpler linear driving force approximation. For higher accuracy, more refined and complex hydrodynamic models would have to be used (Van Brocklin and David, 1975). 

The Nernst film model is used to quantify diffusional transport through the static boundary layer. This model approximates the low velocity boundary layer as a thin, static film between the surface and the free fiowing solution. Pick s first law gives the diffusional flux (/, mol/m sec) through this film (Figure 7.6). 

Early investigators assumed that this so-called diffusion layer was stagnant (Nernst-Whitman model), and that the concentration profile of the reacting ion was linear, with the film thickness <5N chosen to give the actual concentration gradient at the electrode. In reality, however, the thin diffusion layer is not stagnant, and the fictitious t5N is always smaller than the real mass-transfer boundary-layer thickness (Fig. 2). However, since the actual concentration profile tapers off gradually to the bulk value of the concentration, the well-defined Nernst diffusion layer thickness has retained a certain convenience in practical calculations. 

The film model referred to in Chapters 2 and 5 provides, in fact, an oversimplified picture of what happens in the vicinity of interface. On the basis of the film model proposed by Nernst in 1904, Whitman [2] proposed in 1923 the two-film theory of gas absorption. Although this is a very useful concept, it is impossible to predict the individual (film) coefficient of mass transfer, unless the thickness of the laminar sublayer is known. According to this theory, the mass transfer rate should be proportional to the diffusivity, and inversely proportional to the thickness of the laminar film. However, as we usually do not know the thickness of the laminar film, a convenient concept of the effective film thickness has been assumed (as... 

A more general theoretical approach for dissolution modeling called the Film-Model Theory was postulated by Nernst (1904) and expanded upon by Brunner (1904) in an effort to deconvo-lute the components of the dissolution constinIBoth Nernst and Brunner made the following assumptions ... 

Nonlinear concentration gradients in Nernst-Planck film model. 

The mechanism of dissolution was proposed by Nernst (1904) using a film-model theory. Under the influence of non-reactive chemical forces, a solid particle immersed in a liquid experiences two consecutive processes. The first of these is solvation of the solid at the solid-liquid interface, which causes the formation of a thin stagnant layer of saturated solution around the particle. The second step in the dissolution process consists of diffusion of dissolved molecules from this boundary layer into the bulk fluid. In principle, one may control the dissolution through manipulation of the saturated solution at the surface. For example, one might generate a thin layer of saturated solution at the solid surface by a surface reaction with a high energy barrier (Mooney et al., 1981), but this application is not commonly employed in pharmaceutical applications. 

Commonly, mass transfer in the liquid is modeled as diffusion in a fic-litious liquid film adhering to the particle Nernst film (Nernst, 1889 Planck, 1890). Intraparticle diffusion and film diffusion are the two possible rate-controlling steps (Helfferich, 1962a). 

In most common separation processes, the main mass transfer is across an interface between a gas and a liquid or between two liquid phases. At fluid-fluid interfaces, turbulence may persist to the interface. A simple theoretical model for turbulent mass transfer to or from a fluid-phase boundary was suggested in 1904 by Nernst, who postulated that the entire resistance to mass transfer in a given turbulent phase lies in a thin, stagnant region of that phase at the interface, called a him, hence the name film theory.2 4,5 Other, more detailed, theories for describing the mass transfer through a fluid-fluid interface exist, such as the penetration theory.1,4... 

An important example of the system with an ideally permeable external interface is the diffusion of an electroactive species across the boundary layer in solution near the solid electrode surface, described within the framework of the Nernst diffusion layer model. Mathematically, an equivalent problem appears for the diffusion of a solute electroactive species to the electrode surface across a passive membrane layer. The non-stationary distribution of this species inside the layer corresponds to a finite - diffusion problem. Its solution for the film with an ideally permeable external boundary and with the concentration modulation at the electrode film contact in the course of the passage of an alternating current results in one of two expressions for finite-Warburg impedance for the contribution of the layer Ziayer = H(0) tanh(icard)1/2/(iwrd)1/2 containing the characteristic - diffusion time, Td = L2/D (L, layer thickness, D, - diffusion coefficient), and the low-frequency resistance of the layer, R(0) = dE/dl, this derivative corresponding to -> direct current conditions. 

Concentration Polarisation is the accumulation of solute due to solvent convection through the membrane and was first documented by Sherwood (1965). It appears in every pressure dri en membrane process, but depending on the rejected species, to a very different extent. It reduces permeate flux, either via an increased osmotic pressure on the feed side, or the formation of a cake or gel layer on the membrane surface. Concentration polarisation creates a high solute concentration at the membrane surface compared to the bulk solution. This creates a back diffusion of solute from the membrane which is assumed to be in equilibrium with the convective transport. At the membrane, a laminar boundary layer exists (Nernst type layer), with mass conservation through this layer described by the Film Theory Model in equation (3.7) (Staude (1992)). cf is the feed concentration, Ds the solute diffusivity, cbj, the solute concentration in the boundary layer and x die distance from the membrane. 

The contribution of electric field to lithium transport has been considered by a few authors. Pyun et argued on the basis of the Armand s model for the intercalation electrode that lithium deintercalation from the LiCo02 composite electrode was retarded by the electric field due to the formation of an electron-depleted space charge layer beneath the electrode/electrolyte interface. Nichina et al. estimated the chemical diffusivity of lithium in the LiCo02 film electrode from the current-time relation derived from the Nernst-Planck equation for combined lithium migration and diffusion within the electrode. 

Two types of model have been used to describe the charge transport through electroactive films continuous models, considering ionic transport in a compact film based on the Nernst-Planck equations, and porous models, whose transport is described by transmission line equivalent circuits. 

The simplest theory for interfacial mass transfer, shown schematically in Fig. 9.1-1, assumes that a stagnant film exists near every interface. This film, also called an unstirred layer, is almost always hypothetical, for fluid motions commonly occur right up to even a solid interface. Nonetheless, such a hypothetical film, suggested first by Nernst in 1904, gives the simplest model of the interfacial region. 

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