A mathematical model to represent threedimensional bipolar electrodes is proposed taken into account the leakage current. The model is analytically solved when the electrochemical reaction has a masstransfer control at low overpotentials, which represents a limiting case of the general mathematical treatment. For this simplified situation, expressions are deduced to evaluate the current and potential distribution and to calculate the leakage current. The effect on the leakage current of kinetic, electrochemical and geometric variables, which are lumped into one dimensionless number, is discussed. The influence of the leakage current on the optimal bed depth under limiting current conditions is also analyzed. Likewise, experimental data, using the deposition-dissolution of copper as test reaction, are compared with the theoretical prediction of the general treatment achieving a good agreement between both.
List of symbols VariablesA s Surface area per unit electrode volume (m -1 ) C Concentration (mol m -3 , mol dm -3 or ppm) d Wire diameter (m) D Diffusion coefficient (m 2 s -1 ) E 0 Reversible electrode potential (V) E 0 0 Reversible electrode potential under standard conditions (V) F Faraday constant (C mol -1 ) i Current density (A m -2 ) I Total current (A) I * Leakage current (A) j Reaction rate (A m -2 ) j d j-factor for mass transfer = k m /(u/e) Sc 2/3 j 0 Exchange current density (A m -2 ) j L Limiting current density (A m -2 ) k m Mass-transfer coefficient (m s -1 ) L Thickness of the bipolar electrode (m) L c Thickness of the cathodic part at the bipolar electrode (m) M Atomic weight (kg mol -1 ) RT/F Constant (0.0261 V at 30°C) (V) Re Reynolds number = ud/(em) R p Polarization resistance (X) R s Electrolyte resistance (X) S Membrane area (m 2 ) Sc Schmidt number = m/D t Time of the experiment (s or min) u Superficial electrolyte flow velocity (m s -1 ) Wa Wagner number x Axial coordinate (m) y Axial coordinate (m) z Axial coordinate (m) Greek characters b Charge-transfer coefficient c Activity coefficient d Density (kg m -3 ) Dr Change in the wire radius (m)