Journal of Applied Electrochemistry

, Volume 44, Issue 3, pp 361–381 | Cite as

Electrochemical modelling using electroneutrality equation as a constraint

  • Frédérick GagnonEmail author
  • Donald Ziegler
  • Mario Fafard
Research Article


This paper deals with ionic species migration based on classical mass balance equations in conjunction with a charge balance equation and the electroneutrality condition. Two new methods are proposed to apply the electroneutrality condition, avoiding the elimination of one of the conservation equations of chemical species and permitting to explicitly apply boundary conditions on all species. The first method is based on variational principles and the second on a Lagrange multiplier. It is shown that the new methods are analytically equivalent to the more standard method in which one conservation equation is eliminated without any other consideration. The variational method is more complex to implement than the Lagrange multiplier method. The new methods were applied to a multi-ionic problem together with the charge conservation equation without capacitive effects. Different kinds of boundary conditions were applied: Neumann, Dirichlet and a nonlinear case based on Butler–Volmer kinetics. All methods gave the same results for non-complex problems. In the case of complex problems including chemical equilibrium between each ionic species, it was found that more investigations are necessary even for more conventional methods.


Electroneutrality Finite element method Lagrange multiplier Electrochemical engineering Variational method 

List of symbols

\( \vec{x} \)

Position (m)


Time (s)


Domain size (m)


Volume (m3)


Area (m2)


Boundary area (m2)

\( c_{i} (\vec{x},t) \)

Concentration of species i (mol m−3)


Total number of ionic species


Rate of reaction (mol m−3 s−1)


Stoichiometric coefficient for species i


Diffusion coefficient of species i (m2 s−1)

\( \vec{J}_{i} \)

Molar flux of species i (mol m−2 s−1)

\( \bar{J}_{i} \)

Boundary molar flux of species i (mol m−2 s−1)


Mobility of ionic species i (J m2 s−1 mol−1)


Charge number of species i


Faraday’s constant (C mol−1)


Solution potential (V)


Electrode surface potential (V)


Equilibrium potential (V)

\( \bar{R} \)

Gas constant (J mol−1 K−1)


Temperature (K)


Exchange current density (A m−2)


Surface overvoltage (V)


Apparent anodic transfer coefficient


Apparent cathodic transfer coefficient

\( \vec{i} \)

Current density (A m−2)

\( \bar{i} \)

Boundary current density (A m−2)


Total weighted residual


Species i conservation equation weighted residual


Charge conservation equation weighted residual


Lagrange multiplier


Ionic equilibrium constant



The authors gratefully acknowledge the financial support provided by Alcoa Inc. and the Natural Sciences and Engineering Research Council of Canada.


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Frédérick Gagnon
    • 1
    Email author
  • Donald Ziegler
    • 2
  • Mario Fafard
    • 1
  1. 1.Département de génie civil et de génie des eaux, NSERC/Alcoa Industrial Research Chair MACE3 and Aluminium Research Centre, REGALUniversité LavalQuebecCanada
  2. 2.Alcoa Primary MetalsAlcoa Technical CenterAlcoa CenterUSA

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