Advertisement

Granular Matter

, 21:31 | Cite as

Flow dynamics of spherical grains through conical cardboard hoppers

  • Antonio ParrettaEmail author
  • Pietro Grillo
Original Paper
  • 13 Downloads

Abstract

The gravity-driven flow of monodisperse spherical grains of different nature and diameter d, through conical cardboard hoppers, has been studied as function of the orifice diameter D for different values of the aperture angle α (~ 3° ÷ 15°) at large grains conditions (D ≤ 10d). The mass flow rate trend function has displayed, at the lowest angles, a series of linear tracts, with increasing slope, delimited by approximately odd integers of the grains diameter. The linear tracts have been associated to different flow rate regimes, governed by the formation, at the bottom of the granular column, of short-lived arches of “quantized” size (~ 5d, ~ 7d, ~ 9d, …), acting as brakes to flow, by their detachment and ejection from the hopper. This mechanism of events should give rise to a modulation of the flow whose frequency was effectively measured, for the arches of ~ 5d size, by analyzing the signal produced by the falling grains on a microphone. The data of mass flow rate W, as function of the orifice diameter D, have shown, on average, a growth following the 5/2 power-law function, as foreseen by the well-known Beverloo law. Here we analyze the simplified expression of the mass flow rate with the dimension of the square root of the acceleration of gravity, which shows only a slight dependence on the aperture angle of the hopper. The jamming of grains at the outlet opening has been also investigated throughout the transition region at D ~ 3d  ÷ 4d, which characterizes the passage from the blocked to the continuous flow for few tens thousand grains, by an optical method and by measuring the frequency of the clogging events.

Keywords

Granular matter Flow rate Kinetics Spherical grains Conical hopper Cardboard wall Jamming Clogging 

Notes

Acknowledgements

We sincerely thank Nicola Ricci for the characterization of the samples at the optical microscope, and Achille Monegato, of Favini S.r.l., for the measurements on the cardboard.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Beverloo, W.A., Leniger, H.A., van de Velde, J.: The flow of granular solids through orifices. Chem. Eng. Sci. 15, 260–269 (1961)CrossRefGoogle Scholar
  2. 2.
    Nedderman, R.M., Tüzün, U.: The flow of granular materials I. Discharge rates from hoppers. Chem. Eng. Sci. 37, 1597–1609 (1982)CrossRefGoogle Scholar
  3. 3.
    Tüzün, U., Houlsby, G.T., Nedderman, R.M., Savage, S.B.: The flow of granular materials II. Velocity distribution in slow flow. Chem. Eng. Sci. 37, 1691–1709 (1982)CrossRefGoogle Scholar
  4. 4.
    Savage, S.B., Nedderman, R.M., Tüzün, U., Houlsby, G.T.: The flow of granular materials III. Rapid Shear Flows. Chem. Eng. Sci. 38, 189–195 (1983)CrossRefGoogle Scholar
  5. 5.
    Jaeger, H.M., Nagel, S.R.: Physics of the granular state. Science 255, 1523–1531 (1992)ADSCrossRefGoogle Scholar
  6. 6.
    Jaeger, H.M., Nagel, S.R., Behringer, R.P.: Granular solids, liquids, and gases. Rev. Mod. Phys. 68, 1259–1273 (1996)ADSCrossRefGoogle Scholar
  7. 7.
    Herrmann, H.J., Hovi, J.-P., Luding, S.: Physics of Dry Granular Media. Springer, Dordrecht (1998)CrossRefGoogle Scholar
  8. 8.
    Kadanoff, L.P.: Built upon sand: theoretical ideas inspired by granular flows. Rev. Mod. Phys. 71, 435–444 (1999)ADSCrossRefGoogle Scholar
  9. 9.
    De Gennes, P.G.: Granular matter: a tentative view. Rev. Mod. Phys. 71, S374–S382 (1999)CrossRefGoogle Scholar
  10. 10.
    Yersel, M.: The flow of sand. Phys. Teach. 38, 290–291 (2000)ADSCrossRefGoogle Scholar
  11. 11.
    Duran, J.: Sand, Powders and Grains. An Introduction to the Physics of Granular Materials. Springer, New York (2000)CrossRefGoogle Scholar
  12. 12.
    Herrmann, H.J.: Granular matter. Phys. A 313, 188–210 (2002)CrossRefGoogle Scholar
  13. 13.
    Kozicki, J., Tejchman, J.: Application of a cellular automaton to simulations of granular flow in silos. Granul. Matter 7, 45–54 (2005)CrossRefGoogle Scholar
  14. 14.
    Mankoc, C., Janda, A., Arévalo, R., Pastor, J.M., Zuriguel, I., Garcimartín, Á., Maza, D.: The flow rate of granular materials through an orifice. Granul. Matter 9, 407–414 (2007)CrossRefGoogle Scholar
  15. 15.
    Janda, A., Zuriguel, I., Garcimartín, Á., Pugnaloni, L.A., Maza, D.: Jamming and critical outlet size in the discharge of a two-dimensional silo. Eur. Phys. Lett. 84, 44002 (2008)ADSCrossRefGoogle Scholar
  16. 16.
    Franklin, S.V., Shattuck, M.D.: Handbook of Granular Materials. CRC Press, Taylor & Francis Group, Boca Raton (2016)Google Scholar
  17. 17.
    To, K., Lai, P.-Y.: Jamming pattern in a two-dimensional hopper. Phys. Rev. E 66, 011308 (2002)ADSCrossRefGoogle Scholar
  18. 18.
    Zuriguel, I., Garcimartín, Á., Maza, D., Pugnaloni, L.A., Pastor, J.M.: Jamming during the discharge of granular matter from a silo. Phys. Rev. E 71, 051303 (2005)ADSCrossRefGoogle Scholar
  19. 19.
    Rubio-Largo, S.M., Janda, A., Maza, D., Zuriguel, I., Hidalgo, R.C.: Disentangling the free-fall arch paradox in silo discharge. Phys. Rev. Lett. 114, 238002 (2015)ADSCrossRefGoogle Scholar
  20. 20.
    To, K.: Jamming transition in two-dimensional hoppers and silos. Phys. Rev. E 71, 060301(R) (2005)ADSCrossRefGoogle Scholar
  21. 21.
    Clement, E., Reydellet, G., Rioual, F., Parise, B., Fanguet, V., Lanuza, J., Kolb, E.: Jamming patterns and blockade statistics in model granular flows. In: Helbing, D., Herrmann, H., Schreckenberg, M., Wolf, D. (eds.) Traffic and Granular Flow’99, pp. 457–468. Springer, Berlin (2000)CrossRefGoogle Scholar
  22. 22.
    Nedderman, R.M.: Statics and Kinematics of Granular Materials. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  23. 23.
    Helbing, D., Johansson, A.: Analytical approach to continuous and intermittent bottleneck flows. Phys. Rev. Lett. 97, 168001 (2006)ADSCrossRefGoogle Scholar
  24. 24.
    Janda, A., Zuriguel, I., Maza, D.: Flow rate of particles through apertures obtained from self-similar density and velocity profiles. Phys. Rev. Lett. 108, 248001 (2012)ADSCrossRefGoogle Scholar
  25. 25.
    Gella, D., Maza, D., Zuriguel, I.: Role of particle size in the kinematic properties of silo flow. Phys. Rev. E 95, 052904 (2017)ADSCrossRefGoogle Scholar
  26. 26.
    Koivisto, J., Durian, D.J.: Effect of interstitial fluid on the fraction of flow microstates that precede clogging in granular hoppers. Phys. Rev. E 95, 032904 (2017)ADSCrossRefGoogle Scholar
  27. 27.
    Wójcik, M., Sondej, M., Rejowski, K., Tejchman, J.: Full-scale experiments on wheat flow in steel silo composed of corrugated walls and columns. Powder Technol. 311, 537–555 (2017)CrossRefGoogle Scholar
  28. 28.
    Thomas, C.C., Durian, D.J.: Intermittency and Velocity Fluctuations in Hopper Flows Prone to Clogging. arXiv:1604.08081v1 [cond-mat.soft] 27 Apr 2016
  29. 29.
    Cambou, B.: Behaviour of Granular Materials. Springer, Wien (1998)CrossRefGoogle Scholar
  30. 30.
    Brown, R.L., Richards, J.C.: Principles of Powder Mechanics. International Series of Monographs in Chemical Engineering, vol. 10. Pergamon press, Oxford (1970)Google Scholar
  31. 31.
    Sheldon, H.G., Durian, D.J.: Granular discharge and clogging for tilted hoppers. Granul. Matter 12, 579–585 (2010)CrossRefGoogle Scholar
  32. 32.
    Thomas, C.C., Durian, D.J.: Geometry dependence of the clogging transition in tilted hoppers. Phys. Rev. E 87, 052201 (2013)ADSCrossRefGoogle Scholar
  33. 33.
    Lozano, C., Lumay, G., Zuriguel, I., Hidalgo, R.C., Garcimartín, Á.: Breaking arches with vibrations: the role of defects. Phys. Rev. Lett. 109, 068001 (2012)ADSCrossRefGoogle Scholar
  34. 34.
    Lozano, C., Zuriguel, I., Garcimartín, Á.: Stability of clogging arches in a silo submitted to vertical vibrations. Phys. Rev. E 91, 062203 (2015)ADSCrossRefGoogle Scholar
  35. 35.
    Garcimartín, Á., Lozano, C., Lumay, G., Zuriguel, I.: Avoiding clogs: the shape of arches and their stability against vibrations. In: Powders and Grains 2013 AIP Conference Proceedings, vol. 1542, pp. 686–689 (2013)Google Scholar
  36. 36.
    Guerrero, B., Lozano, C., Zuriguel, I., Garcimartín, Á.: Dynamics of breaking arches under a constant vibration. In: EPJ Web of Conferences, vol. 140, p. 03016 (2017)Google Scholar
  37. 37.
    Zuriguel, I., Janda, A. Arévalo, R., Maza, D., Garcimartín, Á.: Clogging and unclogging of many-particle systems passing through a bottleneck. In: Powders and Grains 2017 EPJ Web of Conferences, vol. 140, p. 01002 (2017)Google Scholar
  38. 38.
    Mankoc, C., Garcimartín, Á., Zuriguel, I., Maza, D.: Role of vibrations in the jamming and unjamming of grains discharging from a silo. Phys. Rev. E 80, 011309 (2009)ADSCrossRefGoogle Scholar
  39. 39.
    Janda, A., Maza, D., Garcimartín, Á., Kolb, E., Lanuza, J., Clément, E.: Unjamming a granular hopper by vibration. Eur. Phys. Lett. 87, 24002 (2009)ADSCrossRefGoogle Scholar
  40. 40.
    Lindemann, K., Dimon, P.: Two-dimensional granular flow in a vibrated small-angle funnel. Phys. Rev. E 62, 5420 (2000)ADSCrossRefGoogle Scholar
  41. 41.
    Evesque, P., Meftah, W.: Mean flow of a vertically vibrated hour-glass. Int. J. Mod. Phys. B 7, 1799 (1993)ADSCrossRefGoogle Scholar
  42. 42.
    Wassgren, C.R.: Effects of vertical vibration on hopper flows of granular material. Phys. Fluids 14, 3439 (2002)ADSCrossRefGoogle Scholar
  43. 43.
    Chen, K., Stone, M.B., Barry, R., Lohr, M., McConville, W., Klein, K., Sheu, B.L., Morss, A.J., Scheidemantel, T., Schiffer, P.: Flux through a hole from a shaken granular medium. Phys. Rev. E 74, 011306 (2006)ADSCrossRefGoogle Scholar
  44. 44.
    Pacheco-Martinez, H., van Gerner, H.J., Ruiz-Suárez, J.C.: Storage and discharge of a granular fluid. Phys. Rev. E 77, 021303 (2008)ADSCrossRefGoogle Scholar
  45. 45.
    Nicolas, A., Garcimartín, Á.: Zuriguel, I: trap model for clogging and unclogging in granular hopper flows. Phys. Rev. Lett. 120, 198002 (2018)ADSCrossRefGoogle Scholar
  46. 46.
    Kruelle, C.A.: Physics of granular matter: pattern formation and applications. Rev. Adv. Mater. Sci. 20, 113–124 (2009)Google Scholar
  47. 47.
    Hunt, M.L., Weathers, R.C., Lee, A.T., Brennen, C.E.: Effects of horizontal vibration on hopper flows of granular materials. Phys. Fluids 11, 68–75 (1999)ADSCrossRefGoogle Scholar
  48. 48.
    Zuriguel, I., Janda, A., Garcimartín, Á., Lozano, C., Arévalo, R., Maza, D.: Silo clogging reduction by the presence of an obstacle. Phys. Rev. Lett. 107, 278001 (2011)ADSCrossRefGoogle Scholar
  49. 49.
    Lozano, C., Janda, A., Garcimartín, Á., Maza, D., Zuriguel, I.: Flow and clogging in a silo with an obstacle above the orifice. Phys. Rev. E 86, 031306 (2012)ADSCrossRefGoogle Scholar
  50. 50.
    Saraf, S., Franklin, S.V.: Power-law flow statistics in anisometric (wedge) hoppers. Phys. Rev. E 83, 030301(R) (2011)ADSCrossRefGoogle Scholar
  51. 51.
    Parisi, D.R., Hidalgo, R.C., Zuriguel, I.: Active particles with desired orientation flowing through a bottleneck. Sci. Rep. 8, 9133 (2018)ADSCrossRefGoogle Scholar
  52. 52.
    Börzsönyi, T., Somfai, E., Szabó, B., Wegner, S., Mier, P., Rose, G., Stannarius, R.: Packing, alignment and flow of shape-anisotropic grains in a 3D silo experiment. New J. Phys. 18, 093017 (2016)ADSCrossRefGoogle Scholar
  53. 53.
    Saraf, S., Franklin, S.: Jamming of rod-like granular materials in hoppers. Bull. Am. Phys. Soc. 45, J14.005 (2009)Google Scholar
  54. 54.
    Desmond, K., Franklin, S.V.: Jamming of three-dimensional prolate granular materials. Bull. Am. Phys. Rev. E 73, 031306 (2006)ADSCrossRefGoogle Scholar
  55. 55.
    Tighe, B.P., Sperl, M.: Pressure and motion of dry sand: translation of Hagen’s paper from 1852. Granul. Matter 9, 141–144 (2007)CrossRefGoogle Scholar
  56. 56.
    Janssen, H.A.: Tests on grain pressure silos. Zeits. d. Vereins Deutsch Ing. 39(35), 1045–1049 (1985). (in German) Google Scholar
  57. 57.
    Sperl, M.: Experiments on corn pressure in silo cells—translation and comment of Janssen’s paper from 1895. Granul. Matter 8, 59–65 (2006)CrossRefGoogle Scholar
  58. 58.
    Madrid, M.A., Darias, J.R., Pugnaloni, L.A.: Forced flow of granular media: breakdown of Beverloo scaling. Eur. Phys. Lett. 123, 14004 (2018)ADSCrossRefGoogle Scholar
  59. 59.
    Wu, X-l, Måløy, K.J., Hansen, A., Ammi, M., Bideau, D.: Why hour glasses tick. Phys. Rev. Lett. 71, 1363–1366 (1993)ADSCrossRefGoogle Scholar
  60. 60.
    Veje, C.T., Dimon, P.: The dynamics of granular flow in an hourglass. Granul. Matter 3, 151–164 (2001)CrossRefGoogle Scholar
  61. 61.
    Hirshfeld, D., Radzyner, Y., Rapaport, D.C.: Molecular dynamic studies of granular flow through an aperture. Phys. Rev. E 56, 4404–4415 (1997)ADSCrossRefGoogle Scholar
  62. 62.
    Hirshfeld, D., Rapaport, D.C.: Granular flow from a silo: discrete-particle simulations in three dimensions. Eur. Phys. J. E 4, 193–199 (2001)CrossRefGoogle Scholar
  63. 63.
    Mankoc, C., Janda, A., Arévalo, R., Pastor, J.M., Zuriguel, I., Garcimartín, Á., Maza, D.: Erratum: the flow rate of granular materials through an orifice. Granul. Matter 10, 469 (2008)CrossRefGoogle Scholar
  64. 64.
    Zuriguel, I., Pugnaloni, L.A., Garcimartín, Á., Maza, D.: Jamming during the discharge of grains from a silo described as a percolating transition. Phys. Rev. E 68, 030301(R) (2003)ADSCrossRefGoogle Scholar
  65. 65.
    Garcimartín, Á., Zuriguel, I., Maza, D., Pastor, J.M., Pugnaloni, L.A.: Jamming in granular matter. In: AIP Conference Proceedings, vol. 742, no. 279, pp. 279–288 (2004)Google Scholar
  66. 66.
    Janda, A., Harich, R., Zuriguel, I., Maza, D., Cixous, P., Garcimartín, Á.: Flow-rate fluctuations in the outpouring of grains from a two-dimensional silo. Phys. Rev. E 79, 031302 (2009)ADSCrossRefGoogle Scholar
  67. 67.
    To, K., Lai, P.-Y., Pak, H.K.: Jamming of granular flow in a two-dimensional hopper. Phys. Rev. Lett. 86, 71–74 (2001)ADSCrossRefGoogle Scholar
  68. 68.
    Garcimartín, Á., Zuriguel, I., Pugnaloni, L.A., Janda, A.: Shape of jamming arches in two-dimensional deposits of granular materials. Phys. Rev. E 82, 031306 (2010)ADSCrossRefGoogle Scholar
  69. 69.
    Tang, J., Behringer, R.P.: How granular materials jam in a hopper. Chaos 21, 041107 (2011)ADSCrossRefGoogle Scholar
  70. 70.
    Sakaguchi, H., Ozaki, E., Igarashi, T.: Plugging of the flow of granular materials during the discharge from a silo. Int. J. Mod. Phys. B 7, 1949–1963 (1993)ADSCrossRefGoogle Scholar
  71. 71.
    Janda, A., Zuriguel, I., Garcimartín, Á., Maza, D.: Clogging of granular materials in narrow vertical pipes discharged at constant velocity. Granul. Matter 17, 545–551 (2015)CrossRefGoogle Scholar
  72. 72.
    Tang, J., Sagdiphour, S., Behringer, R.P.: Jamming and flow in 2D hoppers. In: AIP Conference Proceedings, vol. 1145, pp. 515–518 (2009)Google Scholar
  73. 73.
    Thomas, C.C., Durian, D.J.: Fraction of clogging configurations sampled by granular hopper flow. Phys. Rev. Lett. 114, 178001 (2015)ADSCrossRefGoogle Scholar
  74. 74.
    Török, J., Lévay, S., Balázs, S., Somfai, E., Wegner, S., Stannarius, R., Börzsönyi, T.: Arching in three-dimensional clogging. In: EPJ Web of Conferences vol. 140, p. 03076 (2017)Google Scholar
  75. 75.
    Manna, S.S., Herrmann, H.J.: Intermittent granular flow and clogging with internal avalanches. Eur. Phys. J. E 1, 341–344 (2000)CrossRefGoogle Scholar
  76. 76.
    Bouchaud, J.-P., Claudin, P., Clément, E., Otto, M., Reydellet, G.: The stress response function in granular materials. C. R. Phys. 3, 141–151 (2002)ADSCrossRefGoogle Scholar
  77. 77.
    Bouchaud, J.-P., Claudin, P., Clément, E., Otto, M., Reydellet, G.: The stress response function in granular materials. Eur. Phys. J. E 1, 341–344 (2000)CrossRefGoogle Scholar
  78. 78.
    Bouchaud, J.-P., Claudin, P., Levine, D., Otto, M.: Force chain splitting in granular materials: a mechanism for large-scale pseudo-elastic behavior. Eur. Phys. J. E 4, 451–457 (2001)CrossRefGoogle Scholar
  79. 79.
    www.favini.com. Accessed 5 Mar 2019
  80. 80.
    Scott, G.D.: Packing of spheres. Nature 188, 908–909 (1960)ADSCrossRefGoogle Scholar
  81. 81.
    Scott, G.D., Kilgour, D.M.: The density of random close packing of spheres. Br. J. Appl. Phys. (J. Phys. D) 2, 863–866 (1969)ADSCrossRefGoogle Scholar
  82. 82.
    Aste, T., Weaire, D.: The pursuit of perfect packing. Taylor & Francis Group, New York (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Physics and Earth ScienceUniversity of FerraraFerraraItaly
  2. 2.ENEA Research Center ArcadesPorticiItaly

Personalised recommendations