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A numerical method of calculating secondary current distributions in electrochemical cells

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A numerical method is proposed based on the analogy between the potential distribution in an electrolytic solution and the temperature distribution in a heat-conducting medium. Thus the equation of non-steady-state heat conduction which contains a hypothetical temperaturev(x, y, t) is solved numerically with appropriate boundary conditions. In the steady state the distribution ofv(x, y, t) corresponds to the distribution of potentialφ s (x,y) which satisfies Laplace's equation. The method is useful not only for conventional electrochemical cells but also for complicated systems such as a bipolar electrode for which boundary conditions provide neither the potential nor the current density at the electrode surface.

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a :

length of unit cell (see Fig. 1)

b, c, d :

geometric parameters of rectangular cell (see Fig. 2a)

C m :

heat capacity of metal

E :

average electric field in solution or average temperature gradient in medium

I :

total current in unit cell

I F :

faradaic current in unit cell

I S :

by-pass current in solution in unit cell

i e :

cathodic limiting current density

j :

current density

n :

normal distance from the electrode surface

r, θ:

polar coordinates

r 0 :

radius of cylindrical electrode

S :

surface area of electrode

t :


V :

cell voltage

V 0 :

threshold voltage or theoretical decomposition voltage

v(x, y, t) :

hypothetical temperature which, fort=∞, corresponds toφ s (x,y)

v m(t):

hypothetical temperature which, fort=∞, correspouds toφ m

w :

complex number defined in Fig. 2c

x, y :

Cartesian coordinates

z :

complex number defined in Fig. 2a


thermal diffusivity


complex number defined in Fig. 2b

η c :

cathodic overpotential


electric conductivity of solution or thermal conductivity of medium

φ m :

potential of metal

φ s (x,y):

potential of solution

i, j, k:

ordinal numbers of division ofx, y, t


  1. [1]

    C. Wagner,J. Electrochem. Soc. 98 (1951) 116.

  2. [2]

    C. W. Tobias and R. Wijsman,J. Electrochem Soc. 100 (1953) 459.

  3. [3]

    F. Hine, S. Yoshizawa and S. Okada,J. Electrochem. Soc. 103 (1956) 186.

  4. [4]

    J. A. Klingert, S. Lynn and C. W. Tobias,Electrochim. Acta 9 (1964) 297.

  5. [5]

    J. Newman, ‘Electrochemical Systems,’ Prentice Hall, Engelwood Cliffs, NJ (1973).

  6. [6]

    S. Yoshizawa (editor), ‘Denki Kagaku,’ Vol. 3, Kyoritsushuppan, Tokyo (1974).

  7. [7]

    R. Alkire, T. Bergh and R. L. Sani,J. Electrochem. Soc. 125 (1978) 1981.

  8. [8]

    M. Takahashi and N. Masuko, ‘Kogyodenkai no Kagaku’, Agune, Tokyo (1979).

  9. [9]

    N. Ibl. ‘Comprehensive Treatise of Electrochemistry,’ Vol. 6, Plenum Press, New York (1983) pp. 239–315.

  10. [10]

    Y. Nishiki, K. Aoki, K. Tokuda and H. Matsuda,J. Appl. Electrochem. 14 (1984) 653.

  11. [11]

    F. Hine, ‘Electrode Processes and Electrochemical Engineering’, Plenum Press, New York (1985).

  12. [12]

    H. Kawamoto,Denki Kagaku 53 (1985) 98.

  13. [13]

    H. Shih and H. W. Pickering,J. Electrochem. Soc. 134 (1987) 551.

  14. [14]

    E. C. Dimpault-Darcy and R. E. White,J. Electrochem. Soc. 135 (1988) 656.

  15. [15]

    S. Yoshizawa, Y. Mayazaki and A. Katagiri,Nippon Kagaku Kaishi (1977) 19.

  16. [16]

    S. Yoshimura, A. Katagiri and S. Yoshizawa.Nippon Kagaku Kaishi (1978) 1144.

  17. [17]

    Y. Miyazaki, A. Katagiri, S. Yoshizawa and Z. Takehara,Mem. Fac. Eng. Kyoto Univ. 49 (1987) 162.

  18. [18]

    Y. Miyazaki, A. Katagiri and S. Yoshizawa,J. Appl. Electrochem. 17, (1987) 113.

  19. [19]

    Y. Miyazaki, A. Katagiri and S. Yoshizawa,J. Appl. Electrochem. 17 (1987) 877.

  20. [20]

    A. Katagiri and Y. Miyazaki, Abstract No. G324, presented at the 54th Meeting of the Electrochemical Society of Japan, 5–7 April 1987, Osaka.

  21. [21]

    A. Katagiri, Abstract No. D105 presented at the 55th Meeting of the Electrochemical Society of Japan, 5–7 April 1988, Tokyo.

  22. [22]

    H. S. Carslaw and J. C. Jaeger ‘Conduction, of Heat in Solids’, Oxford University Press London (1959).

  23. [23]

    H. Fletcher Moulton,Proc. London Math. Soc., Ser. 2 3 (1905) 104.

  24. [24]

    M. Abramowitz and I. A. Stegun, ‘Handbook of Mathematical Functions’, Dover New York (1970) p. 591.

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Katagiri, A., Miyazaki, Y. A numerical method of calculating secondary current distributions in electrochemical cells. J Appl Electrochem 19, 281–286 (1989). https://doi.org/10.1007/BF01062313

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  • Boundary Condition
  • Physical Chemistry
  • Steady State
  • Temperature Distribution
  • Heat Conduction