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.
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Nedderman, R.M.: Statics and Kinematics of Granular Materials. Cambridge University Press, Cambridge (1992)
Forterre, Y., Pouliquen, O.: Flows of dense granular media. Ann. Rev. Fluid Mech. 40(1), 1–24 (2008)
Andreotti, B., Forterre, Y., Pouliquen, O.: Granular Media: Between Fluid and Solid. Cambridge University Press, Cambridge (2013)
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)
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)
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)
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)
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)
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)
Johnson, P.C., Jackson, R.: Frictional-collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 176, 67–93 (1987)
Syamlal, M., Rogers, W., O’Brien, T.J.: “Mfix documentation: Volume 1, theory guide,” National Technical Information Service, Springfield, VA, (1993)
Gidaspow, D.: Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. Academic press, Cambridge (1994)
Benyahia, S.: Validation study of two continuum granular frictional flow theories. Ind. Eng. Chem. Res. 47(22), 8926–8932 (2008)
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)
Dunatunga, S., Kamrin, K.: Continuum modelling and simulation of granular flows through their many phases. J. Fluid Mech. 779, 483–513 (2015)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
Koivisto, J., Durian, D.J.: The sands of time run faster near the end. Nat. commun. 8(1), 1–6 (2017)
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)
Huang, X., Zheng, Q., Yu, A., Yan, W.: Shape optimization of conical hoppers to increase mass discharging rate. Powder Technol. 361, 179–189 (2020)
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)
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)
Brown, R.: Minimum energy theorem for flow of dry granules through apertures. Nature 191(4787), 458 (1961)
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)
ANSYS, Inc, ANSYS Fluent Theory Guide, (2018)
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
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)
Schaeffer, D.G.: Instability in the evolution equations describing incompressible granular flow. J. Differ. Equ. 66(1), 19–50 (1987)
Chialvo, S., Sundaresan, S.: A modified kinetic theory for frictional granular flows in dense and dilute regimes. Phys. Fluids 25(7), 070603 (2013)
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)
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)
Syamlal, M.: “A review of granular stress constitutive relations,” tech. rep., EG and G Washington Analytical Services Center, Inc., Morgantown, WV (USA), 1 (1987)
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)
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)
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)
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)
Nedderman, R., Tüzün, U.: A kinematic model for the flow of granular materials. Powder Technol. 22(2), 243–253 (1979)
Zuriguel, I., Maza, D., Janda, A., Hidalgo, R.C., Garcimartín, A.: Velocity fluctuations inside two and three dimensional silos. Granul. Matter 21, 47 (2019)
Choi, J., Kudrolli, A., Bazant, M.Z.: Velocity profile of granular flows inside silos and hoppers. J. Phys. Condens. Matter 17, S2533–S2548 (2005)
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This work was funded by Ministerio de Economía y Competitividad (Spanish Government) through Projects No. FIS2017-84631-P, MINECO/AEI/FEDER, UE.
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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
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DOI: https://doi.org/10.1007/s10035-021-01094-6