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On time-dependent settling of a dilute suspension in a rotating conical channel

Published online by Cambridge University Press:  21 April 2006

Gustav Amberg ,
Anders A. Dahlkild ,
Fritz H. Bark  and
Dan S. Henningson
Gustav Amberg
Affiliation:
Department of Mechanics, The Royal Institute of Technology, Stockholm, S-100 44 Sweden Present address: Department of Hydromechanics, The Royal Institute of Technology, Stockholm, S-10044 Sweden.
Anders A. Dahlkild
Affiliation:
Department of Mechanics, The Royal Institute of Technology, Stockholm, S-100 44 Sweden Present address: Department of Hydromechanics, The Royal Institute of Technology, Stockholm, S-10044 Sweden.
Fritz H. Bark
Affiliation:
Department of Mechanics, The Royal Institute of Technology, Stockholm, S-100 44 Sweden Present address: Department of Hydromechanics, The Royal Institute of Technology, Stockholm, S-10044 Sweden.
Dan S. Henningson
Affiliation:
Department of Mechanics, The Royal Institute of Technology, Stockholm, S-100 44 Sweden
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Abstract

The time-dependent settling of a dilute monodisperse suspension in a centrifugal force field is considered. The settling takes place between two axisymmetric narrowly spaced conical disks that are rotating rapidly. Clear fluid and suspension are assumed to behave as Newtonian fluids of different densities. The viscosities are, for simplicity, assumed to be the same. All effects of the sediment are neglected. The fluid motion is assumed to be almost parallel with the disks and is computed by using lubrication theory. This leads to a nonlinear hyperbolic equation of first order for the location of the interface between clear fluid and suspension. Local multivaluedness of the solution is removed by inserting shocks. Such a shock is a model for a small region where the slope of the interface, as scaled in the lubrication approximation, is large. Two problems are solved: batch settling and a case where the suspension is pumped into a conical channel that is initially filled with clear fluid. In the batch-settling case, the solution is quite similar to that computed by Herbolzheimer & Acrivos for settling due to a constant gravity field in a narrow tilted channel. For large values of the Taylor number, it is found that the blocking of radial flow outside the Ekman layers leads to a somewhat slower separation process than expected. In the filling problem, the character of the solution is distinctly different for Taylor numbers of order unity and large values of this parameter.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 166 , May 1986 , pp. 473 - 502
Copyright
© 1986 Cambridge University Press

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References

Acrivos, A. & Herbolzheimer, H. 1979 J. Fluid Mech. 92, 435.
Anestis, G. & Schneider, W. 1983 Ingenieur-Archiv 83, 399.
Bark, F. H., Johansson, A. V. & Carlsson, C. G. 1984 J. Méc. Théorique et Appliquée 3, 861.
Barnea, E. & Mizrahi, J. 1973 Chem. Engng J. 5, 171.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Boycott, A. E. 1920 Nature 104, 532.
Chan, D. & Powell, R. L. 1984 J. Non-Newtonian Fluid Mech. 15, 165.
Cramer, M. S. & Kluwick, A. 1984 J. Fluid Mech. 142, 9.
Davis, R. H., Herbolzheimer, E. & Acrivos, A. 1982 Intl J. Multiphase Flow 8, 571.
Diprima, R. C. 1969 J. Lubrication Tech. 91, 45.
Drew, D. A. 1983 Ann. Rev. Fluid Mech. 15, 261.
Engquist, B. & Osher, S. 1981 Math. Comp. 36, 125.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Greenspan, H. P. 1983 J. Fluid Mech. 127, 91.
Greenspan, H. P. 1985a J. Fluid Mech. 157, 359.
Greenspan, H. P. 1985b Intl J. Multiphase Flow 11, 825.
Herbolzheimer, E. & Acrivos, A. 1981 J. Fluid Mech. 108, 485.
Hsu, H.-W. 1981 Separation by Centrifugal Phenomena. Wiley.
Ishii, M. 1975 Thermo-fluid Dynamic Theory of Two-phase Flow. Eyrolles.
Jeffrey, A. 1976 Quasilinear Hyperbolic Systems and Waves. Pitman.
Kluwick, A. 1977 Acta Mech. 26, 15.
Kynch, G. J. 1952 Trans. Faraday Soc. 48, 166.
Lax, P. D. 1973 SIAM Regional Conf. Series, in Appl. Math. No. 11 (1973).
Leung, W.-F. & Probstein, R. F. 1983 I & EC Process Design & Development 22, 58.
Lighthill, M. J. & Whitham, G. B. 1955 Proc. R. Soc. Lond. A 299, 281.
Nakamura, H. & Kuroda, K. 1937 Keijo J. Med. 8, 256.
Ponder, E. 1925 Q. J. Exp. Physiol. 15, 235.
Probstein, R. F., Yung, D. & Hicks, R. E. 1977 Paper presented at Theory, Practice and Process Principles for Physical Separation, Eng. Foundation Conf., Asilomar, Calif.
Schaflinger, U. 1984 Intl J. Multiphase Flow 11, 189.
Schneider, W. 1982 J. Fluid Mech. 120, 323.
Sokolov, W. J. 1971 Moderne Industrizentrifugen. VEB Verlag Technik.
Svarovsky, L. 1981 Solid—Liquid Separation, 2nd edn. Butterworths.
Ungarish, M. & Greenspan, H. P. 1986 J. Fluid Mech. 162, 117.
Wendroff, B. 1972a J. Math. Anal. Appl. 38, 454.
Wendroff, B. 1972b J. Math. Anal. Appl. 38, 640.
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.