## 3 Irreversible Thermodynamics Approach
### The Nernst-Planck Equation
In this case, the electrolyte diffusion coefficient in solution is used instead of \(P\). Equation (18) is equivalent to the Nernst-Planck equation, if the LEN assumption is accepted [80]. While Equation (18) does not describe the pressure-driven transport, the impact of current-induced (electroosmosis) or diffusion-induced (osmosis) water transport can be implicitly taken into account [88]. Water transport gives rise to the ion convection transport. Therefore, electromigration and/or diffusion of ions occur in a mobile medium, which affects the effective value of the diffusion coefficient. This effect in the thermodynamics of irreversible processes is called the frame of reference effect [92]. It is easy to see from Equation (18), which is valid in the framework of the \"solution-diffusion\" approach, that the electrolyte permeability coefficient, \(P\), is related to the electrolyte diffusion coefficient, \(\overline{D}\) and the electrolyte partition coefficient, \(K_{s}\), as follows [85]:  
\[\overline{D}=P\frac{c_{-}}{\overline{c}_{-}}=P/K_{s} \tag{19}\]  
The definition of \(K_{s}\) (i.e., \(K_{s}=\overline{c}_{-}/c_{-}\)) is clear from Equation (19). It is taken into account that the electrolyte concentration in the membrane is equal to the co-ion concentration \(\overline{c}_{-}\) co-ions. In strongly charged IEMs, \(K_{s}\) is essentially less than 1 due to Donnan exclusion of co-ions. With increasing external concentration, the Donnan exclusion gets less strong and \(K_{s}\) increases [46; 48; 93; 94]. The electrolyte diffusion coefficient and the ion transport number in the membrane are linked with the individual ion diffusion coefficients via the following relations:  
\[\overline{D}=\frac{\overline{D}_{+}\overline{D}_{-}\left(z_{+}^{2}\overline{c} _{+}+z_{-}^{2}\overline{c}_{-}\right)}{\overline{D}_{+}z_{+}^{2}\overline{c} _{+}+\overline{D}_{-}z_{-}^{2}\overline{c}_{-}},\overline{t}_{i}=\frac{ \overline{D}_{i}z_{i}^{2}\overline{c}_{i}}{\overline{D}_{+}z_{+}^{2}\overline{c }_{+}+\overline{D}_{-}z_{-}^{2}\overline{c}_{-}} \tag{20}\]Equations (20) can be simplified, if the LEN assumption is used:  
\[z_{+}\vec{e}_{+}+z_{-}\vec{e}_{-}=Q \tag{21}\]  
where \(Q\) is the concentration of functional fixed groups (in meq. cm\({}^{-3}\)), that is, the ion-exchange capacity (IEC). There are several generalizations of the Nernst-Planck equation including described above the Kedem-Katchalsky and Onsager equations. The following equation takes into account the concentration variation of the activity coefficients (through the introduction of the activity factor, \(g\), [defined just after Equation (11)] and the convective flow (by adding the term with the centre of mass velocity, \(V\)) [80]:  
\[J_{i}=-D_{i}\bigg{(}g\frac{dc_{i}}{dx}+z_{i}c_{i}\frac{F}{RT}\frac{d\varphi}{ dx}\bigg{)}+c_{i}V \tag{22}\]  
Equation (22) is called the extended Nernst-Planck equation. Convective transfer is the main transport mechanism in pressure-driven processes [95; 96], in dialysis across biological or biomimetic membranes [97]. In ED, water transport has to be taken into account in the applications aimed at the production of concentrated solutions (brines), since water moving into the concentration compartment dilutes the brine. Water transport across IEMs affects also the characteristics of energy generation from salinity differences via reverse ED [98]. When applying the approach of the virtual solution, \(g\) can be found from the literature data [78] on the activity coefficient as a function of concentration [46; 75; 76]. Another approach is developed by [PERSON] et al. [48; 88], who calculated \(g\) for a membrane considered homogeneous by applying [PERSON]'s counter-ion condensation theory [51]. The theory provides expressions for individual ion activity coefficients as functions of ion concentration in the membrane.