Skip to main content

Advertisement

Springer Nature Link
Log in

On the settling of a bidisperse suspension with particles having different sizes and densities

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

The settling of a bidisperse suspension with small particles having different sizes and densities can be described by an initial value problem for a system of two non-linear, first-order conservation laws. Solutions to this problem are in general discontinuous and exhibit kinematic shocks that separate areas of different composition. The solution requires the construction of so-called elementary curves in phase space, which are determined from eigenvector fields of the Jacobian of the flux function. Differences in solution behavior to the previously analyzed equal-density case are due to an umbilic point, which appears for different densities only. The initial value problem is eventually solved by the front tracking method, which generates a series of Riemann problems. It turns out that the solution of the problem predicts a fairly complex process of sediment formation, and that the stationary solution can consist of non-constant smooth transitions. This observation is of interest for manufacturing of functionally graded materials.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Bargieł M., Tory E.M.: An extension of the particle-based approach to simulating the sedimentation of polydisperse suspensions. Int. J. Mineral Process. 79, 235–252 (2006)

    Article  Google Scholar 

  2. Batchelor G.K., Janse van Rensburg R.W.: Structure formation in bidisperse sedimentation. J. Fluid Mech. 166, 379–407 (1986)

    Article  Google Scholar 

  3. Berres S., Bürger R.: On Riemann problems and front tracking for a model of sedimentation of polydisperse suspensions. ZAMM Z. Angew. Math. Mech. 87, 665–691 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berres S., Bürger R., Karlsen K.H.: Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions. J. Comput. Appl. Math. 164/165, 53–80 (2004)

    Article  Google Scholar 

  5. Berres S., Bürger R., Karlsen K.H., Tory E.M.: Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math. 64, 41–80 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Berres S., Bürger R., Tory E.M.: Mathematical model and numerical simulation of the liquid fluidization of polydisperse solid particle mixtures. Comput. Vis. Sci. 6, 67–74 (2004)

    MATH  MathSciNet  Google Scholar 

  7. Biesheuvel P.M.: Particle segregation during pressure filtration for cast formation. Chem. Eng. Sci. 55, 2595–2606 (2000)

    Article  Google Scholar 

  8. Biesheuvel P.M., Verweij H.: Calculation of the composition profile of a functionally graded material produced by centrifugal casting. J. Am. Ceram. Soc. 83, 743–749 (2000)

    Article  Google Scholar 

  9. Bürger R., Fjelde K.-K., Höfler K., Karlsen K.H.: Central difference solutions of the kinematic model of settling of polydisperse suspensions and three-dimensional particle-scale simulations. J. Eng. Math. 41, 167–187 (2001)

    Article  MATH  Google Scholar 

  10. Bürger R., Karlsen K.H., Tory E.M., Wendland W.L.: Model equations and instability regions for the sedimentation of polydisperse suspensions of spheres. ZAMM Z. Angew. Math. Mech. 82, 699–722 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bürger R., Kozakevicius A.: Adaptive multiresolution WENO schemes for multi-species kinematic flow models. J. Comput. Phys. 224, 1190–1222 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dafermos C.M.: Polygonal approximations of solutions of the initial value problem for a conservation law. J. Math. Anal. Appl. 38, 33–41 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dafermos C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2005)

    MATH  Google Scholar 

  14. El-Genk M.S., Kim S.-H., Erickson D.: Sedimentation of binary mixtures of particles of unequal densities and of different sizes. Chem. Eng. Commun. 36, 99–119 (1985)

    Article  Google Scholar 

  15. Fried E., Roy B.C.: Gravity-induced segregation of cohesionless granular mixtures. In: Hutter, K., Kirchner, N.(eds) Dynamic Response of Granular and Porous Materials Under Large and Catastrophic Deformations, pp. 393–421. Springer, Berlin (2003)

    Google Scholar 

  16. Greenspan H.P., Ungarish M.: On hindered settling of particles of different sizes. Int. J. Multiphase Flow 8, 587–602 (1982)

    Article  MATH  Google Scholar 

  17. Holden H., Holden L., Høegh-Krohn R.A.: A numerical method for first order nonlinear scalar conservation laws in one dimension. Comput. Math. Appl. 15, 595–602 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  18. Holden H., Risebro N.H.: Front tracking for hyperbolic conservation laws. Springer, New York (2002)

    MATH  Google Scholar 

  19. Isaacson E., Marchesin D., Plohr B., Temple B.: The Riemann problem near a hyperbolic singularity: the classification of solutions of quadratic Riemann problems I. SIAM J. Appl. Math. 48, 1009–1032 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  20. Isaacson, E., Temple, B.: The Riemann problem near a hyperbolic singularity. II, III. SIAM J. Appl. Math. 48, 1287–1301, 1302–1318 (1988)

    Google Scholar 

  21. Lockett M.J., Bassoon K.S.: Sedimentation of binary particle mixtures. Powder Technol. 24, 1–7 (1979)

    Article  Google Scholar 

  22. Masliyah J.H.: Hindered settling in a multiple-species particle system. Chem. Eng. Sci. 34, 1166–1168 (1979)

    Article  Google Scholar 

  23. Mityushev V., Jaworska L., Rozmus M., Królicka B.: Compaction of the diamond—Ti3SiC2 graded material by the high-speed centrifugal compaction process. Arch. Mat. Sci. Eng. 28, 677–682 (2007)

    Google Scholar 

  24. Moritomi H., Iwase T., Chiba T.: A comprehensive interpretation of solid layer inversion in liquid fluidised beds. Chem. Eng. Sci. 37, 1751–1757 (1982)

    Article  Google Scholar 

  25. Moritomi H., Yamagishi T., Chiba T.: Prediction of complete mixing of liquid-fluidized binary solid particles. Chem. Eng. Sci. 41, 297–305 (1986)

    Article  Google Scholar 

  26. Phillips C.R., Smith T.N.: Modes of settling and relative settling velocities in two-species suspensions. Ind. Eng. Chem. Fund. 10, 581–587 (1971)

    Article  Google Scholar 

  27. Richardson J.F., Zaki W.N.: Sedimentation and fluidization: Part I. Trans. Inst. Chem. Eng. (Lond.) 32, 35–53 (1954)

    Google Scholar 

  28. Schneider W., Anestis G., Schaflinger U.: Sediment composition due to settling of particles of different sizes. Int. J. Multiphase Flow 11, 419–423 (1985)

    Article  Google Scholar 

  29. Simura R., Ozawa K.: Mechanism of crystal redistribution in a sheet-like magma body: constraints from the Nosappumisaki and other shoshonite intrusions in the Nemuro peninsula, Northern Japan. J. Petrol. 47, 1809–1851 (2006)

    Article  Google Scholar 

  30. Stevenson P., Mantle M.D., Hicks J.M.: NMRI studies of the free drainage of egg white and meringue mixture froths. Food Hydrocoll. 21, 221–229 (2007)

    Article  Google Scholar 

  31. Weiland R.H., Fessas Y.P., Ramarao B.V.: On instabilities arising during sedimentation of two-component mixtures of solids. J. Fluid Mech. 142, 383–389 (1984)

    Article  Google Scholar 

  32. Xue B., Sun Y.: Modeling of sedimentation of polydisperse spherical beads with a broad size distribution. Chem. Eng. Sci. 58, 1531–1543 (2003)

    Article  Google Scholar 

  33. Xue B., Tong X., Sun Y.: Polydisperse model for the hydrodynamics of expanded-bed adsorption systems. AIChE J. 49, 2510–2518 (2003)

    Article  Google Scholar 

  34. Zeidan A., Rohani S., Bassi A.: Dynamic and steady-state sedimentation of polydisperse suspension and rpediction of outlets particle-size distribution. Chem. Eng. Sci. 59, 2619–2631 (2004)

    Article  Google Scholar 

Download references

Author information

Author notes

  1. Stefan Berres

    Present address: Departamento de Ciencias Matemáticas y Fí sicas, Universidad Católica de Temuco, Temuco, Chile

Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Stefan Berres & Raimund Bürger

Authors
  1. Stefan Berres
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Raimund Bürger
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Raimund Bürger.

Additional information

Dedicated to Professor Wilhelm Schneider on the occasion of his 70th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berres, S., Bürger, R. On the settling of a bidisperse suspension with particles having different sizes and densities. Acta Mech 201, 47–62 (2008). https://doi.org/10.1007/s00707-008-0072-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-008-0072-0

Keywords