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Due to the fact that membranes are not perfectly permselective there is also a water flux through them. Two phenomena contribute to this, namely osmosis and electroosmosis, and can be quantified as [6, 141, 199, 254]: 𝐽 ( 𝑥 ) = 𝐿 , [ 𝜋 ( 𝑥 ) − 𝜋 ( 𝑥 ) ] + 𝐿 , [ 𝜋 ( 𝑥 ) − 𝜋 ( 𝑥 ) ] 𝐽(𝑥) = 𝑛 𝐽(𝑥) 𝐽(𝑥) = 𝐽 (𝑥) + 𝐽(𝑥) ( 3 5 ) (36) (37) where 𝐿 and 𝐿 are the water (osmotic) permeability coefficients of IEMs, 𝜋 denotes the , , osmotic pressure of the solutions at the membrane-solution interface, 𝑛 is the number of water molecules in the solvation shell of salts [6, 141] and 𝐽 is the total water molar flux. Some models totally neglect osmotic phenomena [297], others simplify eq. (35) by using only one proportionality factor (Lp) for both IEMs [141, 199, 254] and/or the bulk osmotic pressures instead of the ones at the membrane-solution interface [141, 199]. In ED units, osmosis and electroosmosis are both from the diluate channel to the concentrate one. Moreover, the flux of water across membranes can be of primary importance for the process performance, especially in stacks with a large membrane area and long residence time. One-dimensional mass balance equations (global and for the solute) along the flow direction x can be expressed by the following differential equations: 𝑑𝑄(𝑥)𝐶(𝑥) =−𝑑𝑄(𝑥)𝐶(𝑥)= −𝑏𝐽(𝑥) (38) 𝑑𝑥 𝑑𝑥 𝑑𝑄(𝑥) =−𝑑𝑄(𝑥)=−𝑏𝐽 (𝑥) (39) 𝑑𝑥 𝑑𝑥 where 𝑄 represents the volumetric flow rate, 𝑏 the membrane width and 𝐽 the volumetric flux of the solvent through IEMs. In order to take the effect of solution density variation into account in eq. (39), more rigorous mass balances should be written [6], though such effect is typically negligible. Differential mass balance equations are typically solved by numerical methods in several different platforms, thus allowing an easy and reliable characterisation of the unit behaviour along the flow direction in co-current and counter-current stacks. A particular type of semi-empirical models is represented by the time dependent models. These are used to simulate batch or semi-batch operation [199, 351, 352], usually taking place in laboratory test rigs or in food industry applications. In most cases, batch ED is simulated using a quasi-steady state approach, where a stationary lumped model is coupled with time dependent mass balances applied to recirculation tanks. Although the lumped-parameters approach is less accurate than distributed- 59PDF Image | Electrodialysis for water desalination
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