Journal of Applied Electrochemistry

, Volume 24, Issue 8, pp 737–744 | Cite as

Turbulent mass transfer in small radius ratio annuli

  • J. Legrand
  • S. A. Martemyanov


Mass transfer in annuli for both fully developed laminar and turbulent flow conditions has been studied with respect to available experimental data. It is shown that prediction of the Sherwood number for the inner annular wall based on the hypothesis of coincidence of the zero shear stress position for laminar and turbulent flows leads to serious error in the case of small radius ratio. Also it is shown that in contrast with plain tubes the curvature in small radius ratio annuli should be taken into account for the case of small Reynolds numbers. In consequence, the well-known Leveque equation can be used for the calculation of the mass transfer coefficient in annuli only under certain conditions. Possibilities of electrodiffusion diagnostics for the precise determination of the zero shear stress position in annuli are discussed.


Experimental Data Shear Stress Mass Transfer Reynolds Number Transfer Coefficient 

List of symbols


cross-section flow area (m2)

a =r1/r2

annular radius ratio (−)

\(\bar c,c',c_0 \)

mean fluctuation and bulk concentration (mol m−3)


molecular diffusivity (m2s−1)


hydraulic diameter (m)


overall, inner and outer wall friction factors (−)

fτ = τ1

near wall velocity gradient (s−1)

\(\bar \iota \)

pressure drop per unit of length (Pam−1)


average mass transfer coefficient (ms−1 )

k =r0/r0,L

ratio of zero shear stress position in turbulent and laminar flows (−)


mass transfer surface length (m)


diffusion ‘leading edge’ length (m)


diffusion entrance length (m)


wetted perimeter (m)

Re =Uavdh

Reynolds number (−)


radial distance from conduit axis (m)


radial distance of zero shear stress position in turbulent and laminar flows (m)


radius of inner and outer annular cylinders (m)

Sc = ν/D

molecular Schmidt number (−)

Sh =KLdh/D

Sherwood number (−)


average liquid velocity (ms−1)


mean and fluctuation axial velocity (ms−1)

υ, υ′

mean and fluctuation radial velocity (ms−1)

y = r − r1

distance from the inner wall (m)

yτ = ν(ρ/τ1)1/2

dynamic length (m)


distance in direction of the flow (m)

Greek symbols


diffusion layer thickness (m)


dynamic viscosity (Pa s)


kinematic viscosity (m2s−1)


density (kgm−3)


shear stress (Pa)


wall shear stress for tube and plate channel (Pa)

τ1, τ2

wall shear stress for inner and outer annular cylinders (Pa)


Geometrical factor with respect to k-function (−)

ϕR, ϕK

geometrical factor with respect to Rothfus or Kays-Leung equations (−)


ratio of radial distance of zero shear stress position to outer radius in laminar flow (−)


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    T. K. Ross and A. A. Wragg,Electrochim. Acta 10 (1965) 1093.Google Scholar
  2. [2]
    A. A. Wragg and T. K. Ross,ibid. 12 (1967) 1421.Google Scholar
  3. [3]
    and T. K. Ross,ibid. 13 (1968) 2192.Google Scholar
  4. [4]
    V. T. Turitto,Chem. Eng. Sci. 30 (1975) 503.Google Scholar
  5. [5]
    D. J. Pickett, ‘Electrochemical Reactor Design’, Elsevier, Amsterdam (1977).Google Scholar
  6. [6]
    M. R. Remorino, P. D. Tonin and U. Bohm, AIChE. J.25 (1979) 368.Google Scholar
  7. [7]
    U. K. Ghosh and S. N. Upadhydy,ibid. 31 (1985) 1721.Google Scholar
  8. [8]
    J. A. Brighton and J. B. Jones,J. Basic Engng. 86 (1964) 835.Google Scholar
  9. [9]
    A. Quarmby,Int. J. Mech. Sci. 9 (1967) 205.Google Scholar
  10. [10]
    K. Rehme,J. Fluid Mech. 64 (1974) 263.Google Scholar
  11. [11]
    C. J. Lawn, C. J. Elliott,J. Mech. Eng. Sci. 14 (1972) 195.Google Scholar
  12. [12]
    B. Kjellstrom, S. Hedberg, AB Atomenergi, Stockholm Report Ae-243 (1966).Google Scholar
  13. [13]
    H. Barthlels, K. F. K. Julich Report Jul-506-RB (1967).Google Scholar
  14. [14]
    S. L. Smith, C. J. Lawn and M. J. Hamlin, CEGB Report RD/B/N 1231 (1968).Google Scholar
  15. [15]
    N. W. Wilson and J. O. Medwell, Paper No. 67-WA/HT-4, Winter Ann. Meeting, ASME, Heat Transfer Division, Pittsburgh, 12–17 Nov. (1967).Google Scholar
  16. [16]
    W. M. Kays and E. Y. Leung,Int. J. Heat Mass Transf. 6 (1963) 537.Google Scholar
  17. [17]
    A. Quarmby,Appl. Sci. Res. 19 (1968), 250.Google Scholar
  18. [18]
    S. Levy,J. Heat Transfer 89 (1967) 25.Google Scholar
  19. [19]
    W. Eifler,Warme & Stoffubertragung 2 (1969) 36.Google Scholar
  20. [20]
    P. S. V. K. Rao and P. K. Sarma, Indian J. Technol.14 (1976) 59.Google Scholar
  21. [21]
    R. R. Rothfus, C. C. Monrad and K. E. Senecal,Ind. Engng. Chem. 42 (1950) 2511.Google Scholar
  22. [22]
    R. R. Rothfus, C. C. Monrad, K. G. Sikchi and W. J. Heideger,Ind. Engng. Chem. 47 (1955) 913.Google Scholar
  23. [23]
    J. Newman,in ‘Electroanalytical Chemistry’, (edited by A. J. Bard), Marcel Dekker, N.Y., 6 (1973) p. 187.Google Scholar
  24. [24]
    S. G. Springer and T. J. Pedley,Proc. Roy. Soc. Lond. A333 (1973) 347.Google Scholar
  25. [25]
    S. G. Springer,ibid. A337 (1974) 395.Google Scholar
  26. [26]
    S. C. Ling,Trans. ASME C: J. Heat Transf. 85 (1963) 230.Google Scholar
  27. [27]
    S. A. Martemyanov, M. A. Vorotyntsev and B. M. Grafov,Sov. Electrochem. 16 (1980) 612.Google Scholar
  28. [28]
    T. J. Hanratty and J. A. Campbell, ‘Fluid Mechanics Measurements’ (edited by R. J. Goldstein), Hemisphere, Washington (1983) p. 559.Google Scholar
  29. [29]
    J. Leveque,Ann. Mines 13 (1928).Google Scholar
  30. [30]
    H. Lamb, ‘Hydrodynamics’, 5th edn, Cambridge University Press, London (1924).Google Scholar
  31. [31]
    E. S. Davis,Trans. ASME. 65 (1943) 755.Google Scholar
  32. [32]
    Y. Lee and H. Barrow,Proc. Inst. Mech. Engrs. 178 (1964) 1.Google Scholar
  33. [33]
    V. K. Jonsson and E. M. Sparrow,J. Mech. Eng. Sci. 25 (1972) 65.Google Scholar
  34. [34]
    H. Schlichting, ‘Boundary Layer Theory’, 7th edn, McGrawHill, New York (1979) p. 596.Google Scholar
  35. [35]
    J. G. Knudsen,Am. Inst. Chem. Engnrs J. 8 (1962) 565.Google Scholar
  36. [36]
    F. R. Lorenz, Mitt. Inst. Stromungsmaschinen, TH Karlsruhe No. 2 (1932).Google Scholar
  37. [37]
    A. A. Nicol and J. O. Medwell,J. Mech. Engng. Sci. 6 (1964) 110.Google Scholar
  38. [38]
    R. B. Crookston, R. R. Rothfus and R. I. Kermode,Int. J. Heat Mass Transf. 11 (1968) 415.Google Scholar

Copyright information

© Chapman & Hall 1994

Authors and Affiliations

  • J. Legrand
    • 1
  • S. A. Martemyanov
    • 2
  1. 1.Laboratoire de Génie des ProcédésSaint-Nazaire CedexFrance
  2. 2.A.N. Frumkin Institute of Electrochemistry, Russian Academy of SciencesMoscowRussia

Personalised recommendations