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𝜃 = 𝐶⁄𝐶 (17) 𝜃 = 𝐶⁄𝐶 (18) where 𝐶 is the electrolyte concentration at the solution (dilute or concentrate) bulk. In a cell pair, there are four polarization coefficients, one for each IEM-solution interface. Values of 𝜃 can be derived from eq. (14), though a more rigorous approach relates them to eq. (11). The membrane potential can then be split into two terms, one accounting for the potential of a membrane ideally in contact with the bulk solutions (∆𝜑), and the other accounting for the concentration overpotential due to polarization in the DBLs (𝜂 ) [168, 179, 190, 202]: ∆𝜑 = ∆𝜑 + 𝜂 = 𝛼 𝑙𝑛 𝐹 𝐶 𝛾 where 𝜉 = 𝛾⁄𝛾 and 𝜉 = 𝛾⁄𝛾 (often assumed equal to 1). 𝑅𝑇 𝐶 𝛾 − 𝑙𝑛𝜃𝜃𝜉𝜉 (19) Interestingly, focusing on hydrodynamics-related features, the DBLs contribute to the total potential drop by means of both Ohmic and non-Ohmic phenomena, which can easily be calculated assuming a linear concentration profile. 𝜂 can be computed by Ohm’s law integrating the resistivity over δ and multiplying by the current density (1-dimensional approach, where i = cost). 𝜂 , also called concentration potential or junction potential, is strictly non-Ohmic and originates from the difference in the ionic diffusion coefficients and the need to maintain the local electroneutrality [62]. Note that 𝜂 has a dissipative nature, i.e. it is caused by the irreversible process involving Joule’s effect [62]; conversely, non-Ohmic phenomena are (at least ideally) reversible and, ultimately, convert the electrical energy into the chemical energy of a salinity gradient. However, under the hypothesis of a linear variation of concentration within the DBL and of an equivalent conductivity (𝜆) independent of concentration, 𝜂 and 𝜂 can be expressed as [169, 170, 199]: 𝜂 =∓(𝑡 −𝑡 )𝑅𝑇𝑙𝑛𝜃 𝐹 𝜂 = − 𝐹𝐷 𝑙𝑛𝜃 𝑡 − 𝑡 𝜆 (20) (21) where 𝑡 and 𝑡 are the transport numbers of anions and cations, respectively. In eq. (20) the sign – is valid when the concentration gradient is negative, as in Figure 9 (thus 𝜂 is positive, i.e. there is a potential drop), while the sign + is valid when the concentration gradient is positive (thus 𝜂 is negative, i.e. there is a potential increase). In very recent years, researchers have shown a wide interest on transport phenomena and DBL effects in IEMs-based systems. The resistance associated with the boundary layer has been measured by 33 PDF Image | Electrodialysis for water desalination
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