Nemst-Planck equation diffusive term

Unfortunately, concentration gradients develop under most experimental conditions, and the motion of charge carriers takes place by both diffusion and migration as described by the Nemst-Planck equation, Eq. (7). In this case, Eqs. (14 and 15) provide the local (i.e. position-dependent) values of the conductivity and the transport number of species i, respectively. By writing the Nemst-Planck equation in terms of ti as [1,14]... 

These three terms represent contributions to the flux from migration, diffusion, and convection, respectively. The bulk fluid velocity is determined from the equations of motion. Equation 25, with the convection term neglected, is frequently referred to as the Nemst-Planck equation. In systems containing charged species, ions experience a force from the electric field. This effect is called migration. The charge number of the ion is Eis Faraday s constant, is the ionic mobiUty, and O is the electric potential. The ionic mobiUty and the diffusion coefficient are related ... 

Here, n x) is the carrier concentration at a distance x from the surface, R is the coefficient of reflection, lv is the intensity of illumination at a frequency v and a, is the absorption coefficient for photons, while De is the diffusion coefficient of electrons. This equation can be understood if one recalls Fick s second law, which gives the rate of change in concentration during diffusion in the absence of an electric field (Vol. 1, Section 4.2). The other terms on the left represent the application of the Nemst-Planck equation (Section 4.4). The term on the right represents the rate of absorption of light, taking into account a reflection coefficient, Rv at a frequency V. 

Moreover the electrodiffusion potential gradient is likely to cause electroosmotic transfer of the solution, whose local content is not in equilibrium with that of the counterions [5]. In this case, as it is pointed out in Ref. 5, the ion mobility and concentration depend on the prior history of the process which can bring about non-Fickian diffusion. The application of Nemst-Planck equations to the real system may require inclusion of additional terms that account for the effect of activity coefficient gradients which may be important in IE with zeolites [4,5]. 

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. 

Equation 1 is the Poisson equation. This equation should be solved in order to obtain the electric potential distribution in the computational domain. On the right hand of this equation, the term F Z] iZ,c, shows the gradient influence of the co-ions and counterions on the electric potential inside the domain. The electric field is the gradient of the electric potential (Eq. 2). Equation 3 is the Nemst-Planck equation, where the definition of ionic flux is given by Eq. 4. On the right-hand side of this equation, (m c,), (D, Vc,), and (z,/t,c,V( ) represent flow field (the electroosmosis), diffusion, and electric field (the electrophoresis), respectively, which contribute to the ionic mass transfer. The ionic concentrations of each species can be found by solving these two equations. Equations 5 and 6 are the Navier-Stokes and the continuity equations, respectively, which describe the velocity field and the pressure gradient in the computational domain. 

It was shown in Section 2.1.2.4 that the general flux equations (e.g. the Nemst-Planck equation) contain, in addition to the diffusion terms, a contribution from migration, that is the movement of charged particles under the influence of an electric field. Under certain circumstances it is quite possible to carry out experiments in which the field is negligibly small compared to the concentration or activity driving force. 

The Nemst-Planck equation is often employed by practitioners because of its similarity to Pick s law and its convenient separation of diffusion and migratiOTi terms. It should be borne in mind, however, that the theory is inconsistent with the basic requirements of irreversible thermodynamics [6, 7]. Nemst-Planck theory uses n + k properties to characterize transport in an isothermal, isobaric n-species system containing... 

This is not the standard form of the Nemst-Planck equation for a diffusion/migration process. This point was first realized by Saveant. The first term in Eqn. 64 is simply due to diffusion. The second term carries the effect of the electric field and also reflects the bimolecular nature of the intersite electron exchange. Note that because of the coupled reaction between A and B, the term in ab/cj passes through a maximum when E = E A/B). At this potential a = b = cyi. There is maximum redox conductivity in this region. The term ablc is very small for either a fully oxidized or a fully reduced layer. This characteristic feature of redox conduction is illustrated in Fig. 1.12. 

Nemst-Planck equations that describe the flux of chemical species through diffusion, migration, and convection, see the equations below. The material balance includes the accumulation or depletion term, the difference between the flux in and out in an infinitesimally small volume element (second term), and the production or consumption terms for species in reactions (third term) ... 

The first term represents the flux by diffusion, the second term the flux by migration of ions in the electric field, while the third term represents convective flux. Combining the material balance and the Nemst-Planck equation yields the model equation for each species material balance ... 

One-dimensional mass transfer is operative once the geometries of WE and of the cell are suitable, as described below for pure diffusion. In the case that all three mass transfer processes are operative, it is accounted for by the Nemst—Planck equation. The first term accounts for diffusion, the second one for migration, and the third one for convection ... 

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