Skip to main content
Log in

The role of the hopper angle in silos: experimental and CFD analysis

  • Original Paper
  • Published:
Granular Matter Aims and scope Submit manuscript

Abstract

In this work, we reported experimental and numerical results of granular flows in silos and hoppers. We used a very flexible experimental setup, allowing us to explore the entire domain of the hopper angles. In addition, the granular flow was also studied numerically using Computational Fluid Dynamics. First, the numerical protocol was validated, comparing the output with experimental data of mass flow rate. In general, we obtained a good quantitative agreement between numerical and experimental results using a single set of the model parameters. Remarkably, the numerical results reproduced very well the weak non-monotonic behavior of the mass flow rate dependence on the hopper angle obtained experimentally. Stepping forward, we examined the scaling properties of the simulated velocity v(r) and density \(\phi (r)\) profiles at the outlet region. Finally, we also analyzed the velocity and volume fraction field inside the silo. The outcomes suggested that fast dynamics at orifice perturbs the system distinctly, depending on the hopper angle. Interestingly, small and large angles showed a larger zone of influence in comparison with intermediate angles.

Graphic abstract

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Nedderman, R.M.: Statics and Kinematics of Granular Materials. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  2. Forterre, Y., Pouliquen, O.: Flows of dense granular media. Ann. Rev. Fluid Mech. 40(1), 1–24 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  3. Andreotti, B., Forterre, Y., Pouliquen, O.: Granular Media: Between Fluid and Solid. Cambridge University Press, Cambridge (2013)

    Book  Google Scholar 

  4. Bagnold, R.A.: Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. London Ser. A Math. Phys. Sci. 225(1160), 49–63 (1954)

    ADS  Google Scholar 

  5. Chapman, S., Cowling, T.G.: The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge, Eng.: Cambridge University Press, 3rd ed ed., (1970)

  6. Savage, S.B., Sayed, M.: Gravity flow of coarse cohesionless granular materials in conical hoppers. Zeitschrift für angewandte Mathematik und Physik ZAMP 32(2), 125–143 (1981)

    Article  ADS  Google Scholar 

  7. Jenkins, S.S.J., Jenkins, J.T., Savage, S.B.: A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202 (1983)

    Article  ADS  Google Scholar 

  8. Lun, C.K.K., Savage, S.B., Jeffrey, D.J., Chepurniy, N.: Kinetic theories for granular flow: inelastic particles in couette flow and slightly inelastic particles in a general flowfield. J. Fluid Mech. 140, 223–256 (1984)

    Article  ADS  Google Scholar 

  9. Gidaspow, D., Bezburuah, R., Ding, J.:“Hydrodynamics of circulating fluidized beds: kinetic theory approach,” tech. rep., Illinois Inst. of Tech., Chicago, IL (United States). Dept. of Chemical, (1991)

  10. Johnson, P.C., Jackson, R.: Frictional-collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 176, 67–93 (1987)

    Article  ADS  Google Scholar 

  11. Syamlal, M., Rogers, W., O’Brien, T.J.: “Mfix documentation: Volume 1, theory guide,” National Technical Information Service, Springfield, VA, (1993)

  12. Gidaspow, D.: Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. Academic press, Cambridge (1994)

    MATH  Google Scholar 

  13. Benyahia, S.: Validation study of two continuum granular frictional flow theories. Ind. Eng. Chem. Res. 47(22), 8926–8932 (2008)

    Article  Google Scholar 

  14. Staron, L., Lagrée, P.-Y., Popinet, S.: The granular silo as a continuum plastic flow: the hour-glass vs the clepsydra. Phys. Fluids 24(10), 103301 (2012)

    Article  ADS  Google Scholar 

  15. Dunatunga, S., Kamrin, K.: Continuum modelling and simulation of granular flows through their many phases. J. Fluid Mech. 779, 483–513 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  16. Zheng, Q., Xia, B., Pan, R., Yu, A.: Prediction of mass discharge rate in conical hoppers using elastoplastic model. Powder Technol. 307, 63–72 (2017)

    Article  Google Scholar 

  17. Zhou, Y., Lagrée, P.-Y., Popinet, S., Ruyer, P., Aussillous, P.: Experiments on, and discrete and continuum simulations of, the discharge of granular media from silos with a lateral orifice. J. Fluid Mech. 829, 459–485 (2017)

    Article  ADS  Google Scholar 

  18. Luo, Q., Zheng, Q., Yu, A.: Quantitative comparison of hydrodynamic and elastoplastic approaches for modeling granular flow in silo. AIChE J. 65(5), e16533 (2019)

    Article  Google Scholar 

  19. Fullard, L., Holland, D.J., Galvosas, P., Davies, C., Lagrée, P.-Y., Popinet, S.: Quantifying silo flow using MRI velocimetry for testing granular flow models. Phys. Rev. Fluids 4(7), 074302 (2019)

    Article  ADS  Google Scholar 

  20. Zhou, Y., Lagrée, P.-Y., Popinet, S., Ruyer, P., Aussillous, P.: Gas-assisted discharge flow of granular media from silos. Phys. Rev. Fluids 4(12), 124305 (2019)

    Article  ADS  Google Scholar 

  21. Beverloo, W.A., Leniger, H.A., Van de Velde, J.: The flow of granular solids through orifices. Chem. Eng. Sci. 15(3–4), 260–269 (1961)

    Article  Google Scholar 

  22. Mankoc, C., Janda, A., Arevalo, R., Pastor, J.M., Zuriguel, I., Garcimartín, A., Maza, D.: The flow rate of granular materials through an orifice. Granul. Matter 9(6), 407–414 (2007)

    Article  Google Scholar 

  23. Aguirre, M.A., Grande, J.G., Calvo, A., Pugnaloni, L.A., Géminard, J.-C.: Granular flow through an aperture: pressure and flow rate are independent. Phys. Rev. E 83(6), 061305 (2011)

    Article  ADS  Google Scholar 

  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(24), 248001 (2012)

    Article  ADS  Google Scholar 

  25. 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(23), 238002 (2015)

    Article  ADS  Google Scholar 

  26. Koivisto, J., Durian, D.J.: The sands of time run faster near the end. Nat. commun. 8(1), 1–6 (2017)

    Article  Google Scholar 

  27. Darias, J., Madrid, M.A., Pugnaloni, L.A.: Differential equation for the flow rate of discharging silos based on energy balance. Phys. Rev. E 101(5), 052905 (2020)

    Article  ADS  Google Scholar 

  28. Huang, X., Zheng, Q., Yu, A., Yan, W.: Shape optimization of conical hoppers to increase mass discharging rate. Powder Technol. 361, 179–189 (2020)

    Article  Google Scholar 

  29. Danczyk, M., Meaclem, T., Mehdizad, M., Clarke, D., Galvosas, P., Fullard, L., Holland, D.: Influence of contact parameters on discrete element method (dem) simulations of flow from a hopper: Comparison with magnetic resonance imaging (mri) measurements. Powder Technol. 372, 671–684 (2020)

    Article  Google Scholar 

  30. Huang, X., Zheng, Q., Yu, A., Yan, W.: Optimised curved hoppers with maximum mass discharge rate—an experimental study. Powder Technol. 377, 350–360 (2021)

    Article  Google Scholar 

  31. Brown, R.: Minimum energy theorem for flow of dry granules through apertures. Nature 191(4787), 458 (1961)

    Article  ADS  Google Scholar 

  32. Darias, J., Gella, D., Fernández, M., Zuriguel, I., Maza, D.: The hopper angle role on the velocity and solid-fraction profiles at the outlet of silos. Powder Technol. 366, 488–496 (2020)

    Article  Google Scholar 

  33. ANSYS, Inc, ANSYS Fluent Theory Guide, (2018)

  34. Ng, B.H., Ding, Y., Ghadiri, M.: “Assessment of the kinetic–frictional model for dense granular flow,” Particuology, vol. 6, no. 1, pp. 50 – 58, (2008). Selected papers from 1st UK-China Particle Technology Forum

  35. Busch, A., Johansen, S.T.: On the validity of the two-fluid-ktgf approach for dense gravity-driven granular flows as implemented in ansys fluent r17.2. Powder Technol. 364, 429–456 (2020)

    Article  Google Scholar 

  36. Schaeffer, D.G.: Instability in the evolution equations describing incompressible granular flow. J. Differ. Equ. 66(1), 19–50 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  37. Chialvo, S., Sundaresan, S.: A modified kinetic theory for frictional granular flows in dense and dilute regimes. Phys. Fluids 25(7), 070603 (2013)

    Article  ADS  Google Scholar 

  38. Johnson, P.C., Nott, P., Jackson, R.: Frictional—collisional equations of motion for participate flows and their application to chutes. J. Fluid Mech. 210, 501–535 (1990)

    Article  ADS  Google Scholar 

  39. Boemer, A., Qi, H., Renz, U.: Eulerian simulation of bubble formation at a jet in a two-dimensional fluidized bed. Int. J. Multiphase Flow 23(5), 927–944 (1997)

    Article  Google Scholar 

  40. Syamlal, M.: “A review of granular stress constitutive relations,” tech. rep., EG and G Washington Analytical Services Center, Inc., Morgantown, WV (USA), 1 (1987)

  41. Wachem, B. G. M., van, Schouten, J. C., Krishna, R., Bleek, C. M., van den: “Comparative analysis of CFD models for dense gas-solid systems. In: Proc. of the AIChE 1999 Annual Meeting, Fluidization and Fluid-Particle Systems (L. Glicksman, ed.), p. 79, (1999)

  42. D. Fletcher, Mcclure, D., Kavanagh, J., Barton, G.: “Cfd Simulation of Industrial Bubble Columns : Numerical and Modelling Challenges and Successes. In: 11th International Conference on CFD in the Minerals and Process Industries, vol. 3, no. December, pp. 1–6, (2015)

  43. Rubio-Largo, S., Maza, D., Hidalgo, R.C.: Large-scale numerical simulations of polydisperse particle flow in a silo. Comp. Part. Mech. 4, 419–427 (2017)

    Article  Google Scholar 

  44. Brown, R.L., Richards, J.C.: Principles of Powder Mechanics: Essays on the Packing and Flow of Powders and Bulk Solids, vol. 10. Elsevier, Amsterdam (2016)

    Google Scholar 

  45. Nedderman, R., Tüzün, U.: A kinematic model for the flow of granular materials. Powder Technol. 22(2), 243–253 (1979)

    Article  Google Scholar 

  46. Zuriguel, I., Maza, D., Janda, A., Hidalgo, R.C., Garcimartín, A.: Velocity fluctuations inside two and three dimensional silos. Granul. Matter 21, 47 (2019)

    Article  Google Scholar 

  47. Choi, J., Kudrolli, A., Bazant, M.Z.: Velocity profile of granular flows inside silos and hoppers. J. Phys. Condens. Matter 17, S2533–S2548 (2005)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was funded by Ministerio de Economía y Competitividad (Spanish Government) through Projects No. FIS2017-84631-P, MINECO/AEI/FEDER, UE.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to David Méndez.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest, and the work is original and have not been published elsewhere in any form or language.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Méndez, D., Hidalgo, R.C. & Maza, D. The role of the hopper angle in silos: experimental and CFD analysis. Granular Matter 23, 34 (2021). https://doi.org/10.1007/s10035-021-01094-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10035-021-01094-6

Keywords

Navigation