Electrodialysis Stack

What is modelled

Applied voltage drives a current through N cell pairs. In each pair cations cross the cation-exchange membrane towards the cathode and anions cross the anion-exchange membrane towards the anode, so the dilute channel loses salt and the concentrate channel gains it.

Equations

• Salt transfer: dn/dt = η N I / F (F = 96485 C/mol, η current efficiency).
• Ohmic resistance: R = N (d/(κdA) + d/(κcA) + 2 Rm/A) with conductivity κ = Λ c.
• Concentration (Nernst) back-EMF: Eb = 2 N η (RT/F) ln(cc/cd); current I = (V - Vel - Eb)/R.
• Limiting current: Ilim = F km cd A / (tm - ts).

Checks run in the page

• Total salt in both tanks is constant.
• Constant 1 A removes exactly η N I t / F moles.
• Work against the back-EMF equals the ideal Gibbs energy of un-mixing.
• Initial current equals a separate hand calculation of Ohm's law; a starved channel is clipped to I_lim.

Honest simplifications

Plug-flow channels are lumped into well-mixed tanks, activity coefficients are ideal, water transport through the membranes and co-ion leakage are ignored, and the limiting current uses one fixed boundary-layer coefficient. Real stacks use staged flow and reversal.

• Press "Starved cell": the dilute tank drains until the current is clipped to the limiting value.
• Raise N: the same voltage gives a smaller current but each ampere moves more salt.
• Watch the back-EMF grow as the concentrate gets saltier.