where 𝑖 is the applied current, 𝐹 is the Faraday's constant, and 𝐴 is the surface area of the electrodes. The initial pH for this simulation was set to pH 7 and the proton diffusion coefficient used in this model was 9.31 × 10 -5 𝑐𝑚 2 𝑠 -1 (Lee and Rasaiah, 2011). Diffusion of protons near the protonator electrodes were conducted based on Fick's second law.