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Both cells have identical geometries and flow condition; The stack operates in a co-current flow; The unit works below the limiting current density; The membranes potential (back electro-motive force) is neglected; Concentration polarisation is neglected; Diffusion of ions and water transport are neglected; The current utilisation factor is fixed. On this basis, the “degree of desalination” (i.e. the difference between the inlet and the outlet concentration of the diluate stream) can be expressed as: 𝑑𝐶∆ = 𝑖𝜁𝑁 𝑑𝐴 𝑧𝐹𝑄 (47) where 𝐶∆ is the degree of desalination, 𝑖 is the current density, 𝜁 is the current utilisation factor (already defined in Section 5.2.4), 𝑁 is the number of cell pairs, 𝑧 is the valence, 𝑄 the volume flow rate and A is the effective membrane area. The current density is given by: 𝑖 = 𝜅 ∆𝑉 (48) 2h𝑁 where ∆𝑉 is the applied voltage, h is the channel thickness and 𝜅 is the average electrical conductivity, which can be expressed in terms of solution conductivities and membrane resistances: By substituting eqs. (48) and (49) in eq. (47), rearranging and integrating this equation (with the boundary conditions 𝐶∆ = 0 at 𝐴 = 0 and 𝐶∆ = 𝐶 − 𝐶 at the generic A) in order to express the effective area as a function of the other parameters, one has: (50) where 𝜆 is the equivalent conductivity, 𝐶 and 𝐶 are the outlet concentrations and 𝐶 and 𝐶 are the feed concentrations. 𝜅 = 2h (49) h + h + 𝑟 + 𝑟 𝜅 𝜅 where 𝜅 is the concentrate conductivity, 𝜅 is the diluate conductivity, 𝑟 and 𝑟 are the membrane’s resistances. h 𝐶𝐶 𝑧𝐹𝑄 𝐴 = 𝑙 𝑛 + ( 𝑟 + 𝑟 ) 𝐶 − 𝐶 𝜆 𝐶𝐶 𝜁∆𝑉 64PDF Image | Electrodialysis for water desalination
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