Skip to main content
SpringerLink
Log in

Sedimentation and suspension flows: Historical perspective and some recent developments

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

Sedimentation and suspension flows play an important role in modern technology. This special issue joins nine recent contributions to the mathematics of these processes. The Guest Editors provide a concise account of the contributions to research in sedimentation and thickening that were made during the 20th century with a focus on the different steps of progress that were made in understanding batch sedimentation and continuous thickening processes in mineral processing. A major breakthrough was Kynch's kinematic sedimentation theory published in 1952. Mathematically, this theory gives rise to a nonlinear first-order scalar conservation law for the local solids concentration. Extensions of this theory to continuous sedimentation, flocculent and polydisperse suspensions, vessels with varying cross-section, centrifuges and several space dimensions, as well as its current applications are reviewed.

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

References

  1. G. J. Kynch, A theory of sedimentation. Trans. Faraday Soc. 48 (1952) 166–176.

    Google Scholar 

  2. E. Ardaillon, Les Mines du Laurion dans l'Antiquité. Paris: Thorin (1897) 216 pp.

    Google Scholar 

  3. A. J. Wilson, The Living Rock. Cambridge: Woodland Publishing Ltd (1994) 310 pp.

    Google Scholar 

  4. G. Agricola, De Re Metallica. Translated from Latin by H.C. Hoover. New York: Dover Publ. Inc. (1950) 670 pp.

    Google Scholar 

  5. F. Concha and R. Bürger, A century of research in sedimentation and thickening. KONA Powder and Particle, to appear.

  6. A. Hazen, On sedimentation. Trans. ASCE 53 (1904) 45–71.

    Google Scholar 

  7. J. V. N. Dorr, The use of hydrometallurgical apparatus in chemical engineering. J. Ind. Eng. Chem. 7 (1915) 119–130.

    Google Scholar 

  8. R. T. Mishler, Settling slimes at the Tigre Mill. Eng. Min. J. 94 (1912) 643–646.

    Google Scholar 

  9. R. T. Mishler, Methods for determining the capacity of slime-settling tanks. Trans. AIME 58 (1918) 102–125.

    Google Scholar 

  10. M. C. Bustos, F. Concha, R. Bürger and E. M. Tory, Sedimentation and Thickening. Dordrecht: Kluwer Academic Publishers (1999) 304 pp.

    Google Scholar 

  11. A. Clark, A note on the settling of slimes. Eng. Min. J. 99 (1915) 412.

    Google Scholar 

  12. H. S. Coe and G. H. Clevenger, Methods for determining the capacities of slime-settling tanks. Trans. AIME 55 (1916) 356–385.

    Google Scholar 

  13. C. B. Egolf and W. L. McCabe, Rate of sedimentation of flocculated particles. Trans. Amer. Inst. Chem. Engrs. 33 (1937) 620–640.

    Google Scholar 

  14. C. S. Robinson, Some factors influencing sedimentation. Ind. Eng. Chem. 18 (1926) 869–871.

    Google Scholar 

  15. H. T. Ward and K. Kammermeyer, Sedimentation in the laboratory: Design data from laboratory experimentation. Ind. Eng. Chem. 32 (1940) 622–626.

    Google Scholar 

  16. L. T. Work and A. S. Kohler, Sedimentation of suspensions. Ind. Eng. Chem. 32 (1940) 1329–1334.

    Google Scholar 

  17. E. W. Comings, Thickening calcium carbonate slurries. Ind. Eng. Chem. 32 (1940) 663–668.

    Google Scholar 

  18. E.W. Comings, C. E. Pruiss and C. De Bord, Continuous settling and thickening. Ind. Eng. Chem. 46 (1954) 1164–1172.

    Google Scholar 

  19. H. H. Steinour, Rate of sedimentation. Nonflocculated suspensions of uniform spheres. Ind. Eng. Chem. 36 (1944) 618–624.

    Google Scholar 

  20. H. H. Steinour, Rate of sedimentation. Suspensions of uniform-size angular particles. Ind. Eng. Chem. 36 (1944) 840–847.

    Google Scholar 

  21. H. H. Steinour, Rate of sedimentation. Concentrated flocculated suspensions of powders. Ind. Eng. Chem. 36 (1944) 901–907.

    Google Scholar 

  22. E. J. Roberts, Thickening—art or science? Mining Eng. 1 (1949) 61–64.

    Google Scholar 

  23. J. F. Richardson and W. N. Zaki, Sedimentation and fluidization: Part I. Trans. Instn. Chem. Engrs. (London) 32 (1954) 35–53.

    Google Scholar 

  24. A. S. Michaels and J. C. Bolger, Settling rates and sediment volumes of flocculated Kaolin suspensions. Ind. Eng. Chem. Fund. 1 (1962) 24–33.

    Google Scholar 

  25. P. T. Shannon, E. P. Stroupe and E. M. Tory, Batch and continuous thickening. Ind. Eng. Chem. Fund. 2 (1963) 203–211.

    Google Scholar 

  26. E. M. Tory, Batch and Continuous Thickening of Slurries. West Lafayette: PhD Thesis, Purdue University (1961).

    Google Scholar 

  27. K. E. Davis, W. B. Russel and W. J. Glantschnig, Settling suspensions of colloidal silica: observations and X-ray measurements. J. Chem. Soc. Faraday Trans. 87 (1991) 411–424.

    Google Scholar 

  28. D. Chang, T. Lee, Y. Jang, M. Kim and S. Lee, Non-colloidal sedimentation compared with Kynch theory. Powder Technol. 92 (1997) 81–87.

    Google Scholar 

  29. R. J. LeVeque, Numerical Methods for Conservation Laws. Basel: Birkhäuser Verlag, Second Ed. (1992) 223 pp.

    Google Scholar 

  30. R. Bürger and E. M. Tory, On upper rarefaction waves in batch settling. Powder Technol. 108 (2000) 74–87.

    Google Scholar 

  31. G. B. Wallis, A simplified one-dimensional representation of two-component vertical flow and its application to batch sedimentation. In: Proc. of the Symposium on the Interaction between Fluids and Particles, London, June 20–22, 1962. London: Instn. Chem. Engrs. (1962) pp. 9–16.

    Google Scholar 

  32. P. Grassmann and R. Straumann, Entstehen und Wandern von Unstetigkeiten der Feststoffkonzentration in Suspensionen. Chem.-Ing.-Techn. 35 (1963) 477–482.

    Google Scholar 

  33. S. N. Kružkov, First order quasilinear equations in several independent variables. Math. USSR Sbornik 10 (1970) 217–243.

    Google Scholar 

  34. A. Bressan, Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem. Oxford: Oxford University Press (2000) 262 pp.

    Google Scholar 

  35. C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Berlin: Springer-Verlag (2000) 459 pp.

    Google Scholar 

  36. D. Serre, Systems of Conservation Laws 1. Cambridge: Cambridge University Press (1999) 275 pp.

    Google Scholar 

  37. M. C. Bustos, On the Existence and Determination of Discontinuous Solutions to Hyperbolic Conservation Laws in the Theory of Sedimentation. Darmstadt: Doctoral thesis, TU Darmstadt (1984) 168 pp.

    Google Scholar 

  38. D. P. Ballou, Solutions to non-linear hyperbolic Cauchy problems without convexity conditions. Trans. Amer. Math. Soc. 152 (1970) 441–460.

    Google Scholar 

  39. K. S. Cheng, Asymptotic behaviour of solutions on a conservation law without convexity conditions. J. Diff. Eqns. 40 (1981) 343–376.

    Google Scholar 

  40. K. S. Cheng, Constructing solutions of a single conservation law. J. Diff. Eqns. 49 (1983) 344–358.

    Google Scholar 

  41. T. P. Liu, Invariants and asymptotic behavior of solutions of a conservation law. Proc. Amer. Math. Soc. 71 (1978) 227–231.

    Google Scholar 

  42. C. A. Petty, Continuous sedimentation of a suspension with a nonconvex flux law. Chem. Eng. Sci. 30 (1975) 1451–1458.

    Google Scholar 

  43. M. C. Bustos, F. Concha and W. L. Wendland, Global weak solutions to the problem of continuous sedimentation of an ideal suspension. Math. Meth. Appl. Sci. 13 (1990) 1–22.

    Google Scholar 

  44. M. C. Bustos, F. Paiva and W. L. Wendland, Entropy boundary conditions in the theory of sedimentation of ideal suspensions. Math. Meth. Appl. Sci. 19 (1996) 679–697.

    Google Scholar 

  45. S. Diehl, On boundary conditions and solutions for ideal clarifier-thickener units. Chem. Eng. J. 80 (2000), 119–133.

    Google Scholar 

  46. R. Bürger, K. H. Karlsen, C. Klingenberg and N. H. Risebro, A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units. Los Angeles: UCLA Computational and Applied Mathematics Report (2001) 21 pp.

    Google Scholar 

  47. H. Holden and N. H. Risebro, Front Tracking for Conservation Laws. Berlin: Springer-Verlag, to appear.

  48. K. J. Scott, Experimental study of continuous thickening of a flocculated silica slurry. Ind. Eng. Chem. Fund. 7 (1968) 582–595.

    Google Scholar 

  49. R. Bürger, W. L. Wendland and F. Concha, Model equations for gravitational sedimentation-consolidation processes. Z. Angew. Math. Mech. 80 (2000) 79–92.

    Google Scholar 

  50. R. Bürger, K. H. Karlsen, E. M. Tory and W. L. Wendland, Model equations and instability regions for the sedimentation of polydisperse suspensions of spheres. Los Angeles: UCLA Computational and Applied Mathematics Report (2001) 29 pp.

    Google Scholar 

  51. G. Anestis, Eine eindimensionale Theorie der Sedimentation in Absetzbehältern veränderlichen Querschnitts und in Zentrifugen. Vienna: Doctoral Thesis, Technical University of Vienna (1981) 145 pp.

    Google Scholar 

  52. G. Anestis and W. Schneider, Application of the theory of kinematic waves to centrifugation. Ingenieur-Archiv 53 (1983) 399–407.

    Google Scholar 

  53. R. Bürger, C. Liu and W. L. Wendland, Existence and stability for mathematical models of sedimentationconsolidation processes in several space dimensions. J. Math. Anal. Appl., to appear.

  54. W. P. Talmage and E. B. Fitch, Determining thickener unit areas. Ind. Eng. Chem. 47 (1955) 38–41.

    Google Scholar 

  55. F. Concha and A. Barrientos, A critical review of thickener design methods. KONA Powder and Particle 11 (1993) 79–104.

    Google Scholar 

  56. Outukumpu Mintec: Supaflo Thickeners & Clarifiers. Brochure distributed at the XX International Mineral Processing Congress, Aachen, Germany, September 1997.

  57. L. Lapidus and J. C. Elgin, Mechanics of vertical-moving fluidized systems. AIChE J. 3 (1957) 63–68.

    Google Scholar 

  58. E. W. Lewis and E. W. Bowerman, Fluidization of solid particles in liquids. Chem. Eng. Progr. 48 (1952) 603–609.

    Google Scholar 

  59. T. S. Mertes and H. B. Rhodes, Liquid-particle behavior. Chem. Eng. Progr. 51 (1955) 429–432 and 517–522.

    Google Scholar 

  60. T. V. Thelen and W. F. Ramirez, Modeling of solid-liquid fluidization in the Stokes flow regime using twophase flow theory. AIChE J. 45 (1999) 708–723.

    Google Scholar 

  61. R. Jackson, The Dynamics of Fluidized Particles. Cambridge: Cambridge University Press (2000) 351 pp.

    Google Scholar 

  62. F. M. Tiller, Revision of Kynch sedimentation theory. AIChE J. 27 (1981) 823–829.

    Google Scholar 

  63. R. Bürger, F. Concha and F. M. Tiller, Applications of the phenomenological theory to several published experimental cases of sedimentation processes. Chem. Eng. J. 80 (2000) 105–117.

    Google Scholar 

  64. R. Font and A. Hernández, Filtration with sedimentation: application of Kynch's theorems. Sep. Sci. Technol. 35 (2000) 183–210.

    Google Scholar 

  65. R. Bürger, F. Concha and K. H. Karlsen, Phenomenological model of filtration processes: 1. Cake formation and expression. Chem. Eng. Sci. 56 (2001) 4537–4553.

    Google Scholar 

  66. C. H. Lee, Modeling of Batch Hindered Settling. University Park: PhD Thesis, College of Earth and Mineral Sciences, Pennsylvania State University (1992) 168 pp.

    Google Scholar 

  67. L. G. Austin, C. H. Lee, F. Concha and P. T. Luckie, Hindered settling and classification partition curves, Minerals & Metallurgical Process. 9 (1992) 161–168.

    Google Scholar 

  68. J. P. Chancelier, M. Cohen de Lara, C. Joannis and F. Pacard, New insights in dynamic modeling of a secondary settler—I. Flux theory and steady-state analysis. Wat. Res. 31 (1997) 1847–1856.

    Google Scholar 

  69. S. H. Cho, F. Colin, M. Sardin and C. Prost, Settling velocity model of activated sludge. Wat. Res. 27 (1993) 1237–1242.

    Google Scholar 

  70. K. Grijspert, P. Vanrolleghem and W. Verstraete, Selection of one-dimensional sedimentation: models for on-line use. Wat. Sci. Tech. 31(2) (1995) 193–204.

    Google Scholar 

  71. J. R. Karl and S. A. Wells, Numerical model of sedimentation/thickening with inertial effects. J. Environ. Eng. 125 (1999) 792–806.

    Google Scholar 

  72. P. Krebs, Success and shortcomings of clarifier modeling. Wat. Sci. Tech. 31(2) (1995) 181–191.

    Google Scholar 

  73. R. W. Watts, S. A. Svoronos and B. Koopman, One-dimensional modeling of secondary clarifiers using a concentration and feed velocity-dependent dispersion coefficient. Wat. Res. 30 (1996) 2112–2124.

    Google Scholar 

  74. T. S. Tan, K. Y. Yong, E. C. Leong and S. L. Lee, Sedimentation of clayey slurry. J. Geotech. Eng. 116 (1990) 885–898.

    Google Scholar 

  75. E. A. Toorman, Sedimentation and self-weight consolidation: constitutive equations and numerical modelling. Géotechnique 49 (1999) 709–726.

    Google Scholar 

  76. T. H. Druitt, Settling behaviour of concentrated dispersions and some volcanological applications. J. Volcanol. Geotherm. Res. 65 (1995) 27–39.

    Google Scholar 

  77. P. M. Biesheuvel and H. Verweij, Calculation of the composition profile of a functionally graded material produced by continuous casting. J. Amer. Ceram. Soc. 83 (2000) 743–749.

    Google Scholar 

  78. P. M. Biesheuvel, V. Breedveld, A. P. Higler and H. Verweij, Graded membrane supports produced by centrifugal casting of a slightly polydisperse suspension. Chem. Eng. Sci. 56 (2001) 3517–3525.

    Google Scholar 

  79. C. Puccini, D. M. Stasiw and L. C. Cerny, The erythrocyte sedimentation curve: a semi-empirical approach. Biorheology 14 (1977) 43–49.

    Google Scholar 

  80. W. K. Sartory, Prediction of concentration profiles during erythrocyte sedimentation by a hindered settling model. Biorheology 11 (1974) 253–264.

    Google Scholar 

  81. W. K. Sartory, Three-component analysis of blood sedimentation by the method of characteristics. Math. Biosci. 33 (1977) 145–165.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics A, University of Stuttgart, Pfaffenwaldring 57, 70569 , Stuttgart, Germany

    Raimund Bürger & Wolfgang L. Wendland

Authors
  1. Raimund Bürger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wolfgang L. Wendland
    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

Bürger, R., Wendland, W.L. Sedimentation and suspension flows: Historical perspective and some recent developments. Journal of Engineering Mathematics 41, 101–116 (2001). https://doi.org/10.1023/A:1011934726111

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1011934726111

Search

Navigation