Skip to main content
SpringerLink
Log in

An analysis of random perturbations in electrolyzers

  • Papers
  • Published:
Journal of Applied Electrochemistry Aims and scope Submit manuscript

Abstract

A method of analysis of the stochastic behaviour of electrolyzers due to stationary random fluctuations in current and inlet electrolyte concentration is presented. The analysis is illustrated by means of a numerical example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

c :

electrolyte concentration;c i inlet electrolyte concentration;c E exit electrolyte concentration (mol dm−3)

F :

Faraday's constant (96487 C mol−1)

I :

current (A)

j :

imaginary unit = √(−1)

K :

equivalent gain

N(A) :

nonlinear function of amplitudeA of an independent variable

n :

number of electrons participating in the electrode process

P(z):

probability density function of random variablez

Q :

electrolyte volumetric flow rate (dm3 min−1)

R z(λ):

autocorrelation function

S z(ω):

power spectrum of random variablez

T 0,T 1 :

Frequency characteristic parameters (obtained from process reaction curve)

u :

shorthand for the product αz 1 (see Equation 30)

V :

active volume of electrolyzer (dm3)

v :

shorthand for the product βz 2 (see Equation 30)

χ:

dimensionless exit electrolyte concentration

z 1 :

dimensionless electrolyte inlet concentration

z 2 :

dimensionless current

α:

lumped parameter defined as-c *i /c *E

β:

lumped parameter defined asI */nFQc *E

γ:

lumped parameter defined as aI */nFVc *E

λ:

dummy' integration variable

σ 2x :

variance (or mean power) of random variablez

τ:

mean residence time in electrolyzer, equal toV/Q (min)

ϕ(jω):

frequency spectrum

ω:

angular frequency

*:

steady state (superscript)

erf:

error function, defined as\(erf (z) = \frac{2}{{\surd \pi }}\int_0^z {\exp ( - \lambda ^2 ) d\lambda }\)

o:

step magnitude (superscript)

H :

Heaviside's shifting function, defined asH(t−τ)=0 fort<τ;H(t−τ)=1 fort≥τ

Γ:

gamma function, defined as\(\Gamma (z) = \int_0^\infty { \lambda ^{z - 1} } \exp ( - \lambda ) d\lambda\)

CSTER:

acronym for continuous flow stirred tank electrolytic reactor

PFER:

acronym for plug-flow electrolytic reactor

References

  1. V. V. Solodovnikov, ‘Introduction to the Statistical Dynamics of Automatic Control Systems’, Dover, New York (1960).

    Google Scholar 

  2. V. S. Pugachev, ‘Teoriya Sluchainikh Funkciy’ (Random Function Theory) Fizmatgiz, Moscow (1960).

    Google Scholar 

  3. A. A. Pervozvanskii, ‘Random Processes in Nonlinear control Systems’, Academic Press, New York (1965).

    Google Scholar 

  4. J. H. Laning Jr. and R. H. Battin, ‘Random Processes In Automatic Control’, McGraw Hill, New York (1956).

    Google Scholar 

  5. H. W. Smith, ‘Approximate Analysis of Randomly Excited Nonlinear Controls’, MIT. Press, Cambridge, Mass. (1966).

    Google Scholar 

  6. D. Graham and D. McRuer, ‘Analysis of Nonlinear Control Systems’, Dover, New York (1961).

    Google Scholar 

  7. J. C. West, ‘Analytical Techniques for Non-Linear Control Systems’, Van Nostrand, Princeton, NJ (1960).

    Google Scholar 

  8. T. Z. Fahidy,J. Appl. Electrochem. 17 (1987) 57.

    Google Scholar 

  9. 17 (1987) 841.

    Google Scholar 

  10. ,Symp. Series, Inst. Chem. Engrs. (UK)112 (1989) 119.

    Google Scholar 

  11. Idem, J. Appl. Electrochem. The Effect of Random Perturbations On The Performance of Tank Electrolyzers, in press.

  12. J.-C. Gille, M. J. Pélegrin and P. Decaulne, ‘Feedback Control Systems’, McGraw Hill, NY (1959).

    Google Scholar 

  13. K. H. Lee and T. Z. Fahidy,Instrum. Control Syst. 40 (1967) 141.

    Google Scholar 

  14. Y. Sawaragi, N. Sugai and Y. Sunahara, ‘Statistical Studies of Nonlinear Control Systems’, Nippon Printing and Publishing Co. Osaka, Japan (1962).

    Google Scholar 

  15. J. G. Ziegler and N. B. Nichols,Trans. ASME 64 (1942) 759.

    Google Scholar 

  16. C. L. Smith, ‘Digital Computer Process Control’, Intext Scranton PA (1972).

  17. C. A. Smith and A. B. Corripio, ‘Principles and Practice of Automatic Process Control’, John Wiley and Sons, NY (1985).

    Google Scholar 

  18. G. H. Cohen and G. A. Coon,Trans. ASME 75 (1953) 827.

    Google Scholar 

  19. T. W. Weber, ‘An Introduction to Process Dynamics and Control’, John Wiley and Sons, NY (1973).

    Google Scholar 

  20. T. Z. Fahidy, ‘Principles of Electrochemical Reactor Analysis’, Elsevier, Amsterdam (1985).

    Google Scholar 

  21. R. W. Yeo and T. Z. Fahidy,Electrochim. Acta 31 (1986) 1397.

    Google Scholar 

  22. M. Klerer and F. Grossman, ‘A New Table of Indefinite Integrals’, Dover, New York (1971).

    Google Scholar 

  23. I. S. Gradshtein and I. M. Ryzhik, ‘Table of Integrals, Series and Products’, 2nd edition, Academic Press, New York (1980).

    Google Scholar 

  24. R. W. Yeo and T. Z. Fahidy,Electrochim. Acta 32 (1987) 277.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical Engineering, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    T. Z. Fahidy

Authors
  1. T. Z. Fahidy
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fahidy, T.Z. An analysis of random perturbations in electrolyzers. J Appl Electrochem 20, 901–906 (1990). https://doi.org/10.1007/BF01019563

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01019563

Keywords

Search

Navigation