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Figure 9. Segmentation of the system composed by a membrane and two bathing solutions, according to the TMS theory, showing concentration of counter-ions and electric potential changes in each layer. The example in the picture shows a CEM immersed between a dilute solution and a concentrate solution. Some aspects related to the total voltage drop were treated in the first experimental and theoretical studies by Cooke [168] and Sonin and Probstein [236]. The topic was then treated more extensively by some authors in the early 1970s [169, 170] and many further studies have since been performed [174, 199, 202, 205, 247–254]. However, by pursuing the aim of a general discussion based on these works, the total voltage drop including all the possible contributions can be expressed as ∆𝑉 = 𝜂 + 𝜂 + 𝜂 + ∆𝜑 + 𝜂 + 𝜂 + 𝜂 + 𝜂 (15) where the different terms represent: Ohmic voltage drop within the left solution bulk, Ohmic voltage drop within the left DBL, (non-Ohmic) diffusion potential within the left DBL, (non-Ohmic) membrane potential, Ohmic voltage drop within the membrane, (non-Ohmic) diffusion potential within the right DBL, Ohmic voltage drop within the right DBL, Ohmic voltage drop within the right solution bulk. In the case of a binary monovalent electrolyte, the membrane potential (eqs. (3)-(5)) acting as a back electromotive force [199, 255–258] can be expressed as: ∆ 𝜑 𝑅 𝑇 𝐶 𝛾 ( 1 6 ) = 𝛼 𝐹 𝑙 𝑛 𝐶 𝛾 where 𝐶 and 𝛾 represent the electrolyte concentration and the electrolyte activity coefficient at the IEM-solution interface (solution side) and the subscripts C and D refer to the concentrate and the dilute channel, respectively. It is useful to resort to polarization coefficients [6, 193], whose values range from 0 (maximum polarization) to 1 (no polarization), defined as: 32PDF Image | Electrodialysis for water desalination
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