Skip to main content Accessibility help

Steady compressible flow of cohesionless granular materials through a wedge-shaped bunker

  • J. Ravi Prakash (a1) (a2) and K. Kesava Rao (a1)


A continuum model based on the critical-state theory of soil mechanics is used to generate stress, density, and velocity profiles, and to compute discharge rates for the flow of granular material in a mass flow bunker. The bin–hopper transition region is idealized as a shock across which all the variables change discontinuously. Comparison with the work of Michalowski (1987) shows that his experimentally determined rupture layer lies between his prediction and that of the present theory. However, it resembles the former more closely. The conventional condition involving a traction-free surface at the hopper exit is abandoned in favour of an exit shock below which the material falls vertically with zero frictional stress. The basic equations, which are not classifiable under any of the standard types, require excessive computational time. This problem is alleviated by the introduction of the Mohr–Coulomb approximation (MCA). The stress, density, and velocity profiles obtained by integration of the MCA converge to asymptotic fields on moving down the hopper. Expressions for these fields are derived by a perturbation method. Computational difficulties are encountered for bunkers with wall angles θw [ges ] 15° these are overcome by altering the initial conditions. Predicted discharge rates lie significantly below the measured values of Nguyen et al. (1980), ranging from 38% at θw = 15° to 59% at θw = 32°. The poor prediction appears to be largely due to the exit condition used here. Paradoxically, incompressible discharge rates lie closer to the measured values. An approximate semi-analytical expression for the discharge rate is obtained, which predicts values within 9% of the exact (numerical) ones in the compressible case, and 11% in the incompressible case. The approximate analysis also suggests that inclusion of density variation decreases the discharge rate. This is borne out by the exact (numerical) results – for the parameter values investigated, the compressible discharge rate is about 10% lower than the incompressible value. A preliminary comparison of the predicted density profiles with the measurements of Fickie et al. (1989) shows that the material within the hopper dilates more strongly than predicted. Surprisingly, just below the exit slot, there is good agreement between theory and experiment.



Hide All
Ackermann, N. L. & Shen, H., 1982 Stresses in rapidly sheared fluid-solid mixtures. J. Engng Mech. Div. ASCE 108, 95113.
Airey, D. W., Budhu, M. & Wood, D. M., 1985 Some aspects of the behaviour of soils in simple shear. In Developments in Soil Mechanics and Foundation Engineering-2 (ed. P. K. Bannerjee & R. Butterfield), pp. 185213. Elsevier.
Atkinson, J. H. & Bransby, P. L., 1978 The Mechanics of Soils. McGraw-Hill.
Bagnold, R. A.: 1954 Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. Lond. A 225, 4963.
Blair-Fish, P. M. & Bransby, P. L. 1973 Flow patterns and wall stresses in a mass flow bunker. Trans. ASME B: J. Engng Ind. 95, 1726.
Blight, G. E.: 1986 Pressures exerted by materials in silos: Part I, coarse materials. Geotech. 36, 3346.
Bolton, M. D.: 1986 The strength and dilatancy of sands. Geotech. 36, 6578.
Bosley, J., Schofield, C. & Shook, C. A., 1969 An experimental study of granule discharge from model hoppers. Trans. Inst. Chem. Engrs 47, 147153.
Bransby, P. L. & Blair-Fish, P. M. 1974 Wall stresses in mass flow bunkers. Chem. Engng Sci. 29, 1061 1074.
Bransby, P. L. & Blair-Fish, P. M. 1975 Deformations near rupture surfaces in flowing sand. Geotech. 25, 384389.
Brennen, C. & Pearce, J. C., 1978 Granular material flow in two-dimensional hoppers. Trans. ASME E: J. Appl. Mech. 45, 4350.
Bridgwater, J. & Scott, A. M., 1983 Flow of solids in bunkers. In Handbook of Fluids in Motion (ed. N. P. Chemeserinoff & R. Gupta), pp. 807846. Butterworths.
Brown, R. L. & Richards, J. C., 1970 Principles of Powder Mechanics. Pergamon.
Campbell, C. S. & Brennen, C. E., 1985 Chute flows of granular materials: some computer simulations. Trans. ASME E: J. Appl. Mech. 52, 172178.
Cole, E. R. L.: 1967 The behaviour of soils in the simple shear apparatus. Ph.D. thesis, University of Cambridge;
Courant, R. & Hilbert, D., 1962 Methods of Mathematical Physics, Vol. 2. Interscience.
Davidson, J. F. & Nedderman, R. M., 1973 The hour-glass theory of hopper flow. Trans. Inst. Chem. Engrs 51, 2935.
Desai, C. S. & Siriwardane, H. J., 1984 Constitutive Laws for Engineering Materials. Prentice-Hall.
Drescheb, A., Cousens, T. W. & Beansby, P. L., 1978 Kinematics of the mass flow of granular material through a plane hopper. Geotech. 28, 2712.
Drucker, D. C. & Prager, W., 1952 Soil mechanics and plastic analysis or limit design. Q. Appl. Maths 10, 157165.
Enstad, G.: 1975 On the theory of arching in mass flow hoppers. Chem. Engng Sci. 30, 12731283.
Fickie, K. E., Mehrabi, R. & Jackson, R., 1989 Density variations in a granular material flowing from a wedge-shaped hopper. AIChE J. 35, 853855.
Gebogiannopoulos, N. G. & Bbown, E. T., 1978 The critical state concept applied to rock. Intl J. Rock Mech. Min. Sci. (Geomech. Abstr.) 15, 110.
Handley, M. F. & Perry, M. G., 1967 Stresses in granular materials flowing in converging hopper sections. Powder Technol. 1, 245251.
Hill, R.: 1950 The Mathematical Theory of Plasticity. Clarendon.
Horne, R. M. & Neddeeman, R. M., 1978 An analysis of switch stresses in two-dimensional bunkers. Powder Technol. 19, 235241.
Hunter, S. C.: 1983 Mechanics of Continuous Media, 2nd edn. Ellis Horwood.
Jackson, R.: 1983 Some mathematical and physical aspects of continuum models for the motion of granular materials. In Theory of Dispersed Multiphase Flow (ed. R. E. Meyer), pp. 291337. Academic.
Jackson, R.: 1986 Some features of the flow of granular materials and aerated granular materials. J. Rheol. 30, 907930.
Jenike, A. W.: 1964a Storage and Flow of Solids. Bulletin 123, Univ. Utah Engng Expt. Station.
Jenike, A. W.: 1964b Steady gravity flow of frictional cohesive solids in converging channels. Trans. ASME E: J. Appl. Mech. 31, 511.
Jenike, A. W.: 1965 Gravity flow of frictional cohesive solids — convergence to radial stress fields. Trans. ASME E: J. Appl. Mech. 32, 205207.
Jenike, A. W.: 1987 A theory of flow of particulate solids in converging and diverging channels based on a conical yield function. Powder Technol. 50, 229236.
Jenike, A. W. & Johanson, J. R., 1968 Bin loads. J. Struct. Div. A8CE 94, 10111041.
Johanson, J. R.: 1964 Stress and velocity fields in the gravity flow of bulk solids. Trans. ASME E: J. Appl. Mech. 31, 499506.
Johnson, P. C. & Jackson, R., 1987 Frictional—collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 176, 6793.
Kaza, K. R.: 1982 The mechanics of flowing granular materials. Ph.D. thesis, University of Houston.
Kaza, K. R. & Jackson, R., 1982a The rate of discharge of coarse granular material from a wedge-shaped mass flow hopper. Powder Technol. 33, 223237.
Kaza, K. R. & Jackson, R., 1982b A problem in the flow of granular materials. Proc. 9th US Natl Congress on Applied Mechanics.
Kaza, K. R. & Jackson, R., 1984 Boundary conditions for a granular material flowing out of a hopper or bin. Chem. Engng Sci. 39, 915916.
Lade, P. V. & Duncan, J. M., 1975 Elastoplastic stress-strain theory for cohesionless soil. J. Geotech. Engng Div. ASCE 101, 10371053.
Lee, J., Cowin, S. C. & Templeton, J. S., 1974 An experimental study of the kinematics of flow through hoppers. Trans. Soc. Rheol. 18, 247269.
Manjunath, K. S.: 1988 Silo/feeder interfacing. Rep. CMI No. 88/03150–1, Chr. Michelsen Institute, Bergen.
Marchello, J. M.: 1976 In Gas—Solids Handling in the Process Industries (ed. J. M. Marchello & A. Gomezplata), pp. 112. Marcel Dekker.
Meric, R. A. & Tabarrok, B., 1982 On the gravity flow of granular materials. Intl J. Mech. Sci. 24, 469478.
Michalowski, R. L.: 1984 Flow of granular material through a plane hopper. Powder Technol. 39, 2940.
Michalowski, R. L.: 1987 Flow of granular media through a plane parallel/converging bunker. Chem. Engng Sci. 42, 25872596.
Mroz, Z. & Szymanski, C., 1978 Non-associated flow rules in description of plastic flow of granular materials. In Limit Analysis and Rheological Approach in Soil Mechanics (ed. W. Olszak & L. Suklje), pp. 5194. Springer.
Naylor, D. J.: 1978 Stress-strain laws for soil. In Developments in Soil Mechanics (ed. C. R. Scott), vol. 1, pp. 3968. Applied Science Publishers.
Nedderman, R. M.: 1988 The measurement of the velocity profile in a granular material discharging from a conical hopper. Chem. Engng Sci. 43, 15071516.
Nedderman, R. M. & Laohakul, C., 1980 The thickness of the shear zone of flowing granular materials. Powder Technol. 25, 91100.
Nguyen, T. V., Brennen, C. E. & Sabersky, R. H., 1980 Funnel flow in hoppers. Trans. ASME E: J. Appl. Mech. 47, 729735.
Perry, M. G. & Handley, M. F., 1967 The dynamic arch in free flowing granular material discharging from a model hopper. Trans. Inst. Chem. Engrs 45, 367371.
Pitman, E. B.: 1986 Stress and velocity fields in two– and three-dimensional hoppers. Powder Technol. 47, 219231.
Phakash, J. R.: 1989 Steady compressible plane flow of cohesionless granular materials in hoppers and bunkers. Ph.D. thesis, Indian Institute of Science.
Prakash, J. R. & Rao, K. K., 1988 Steady compressible flow of granular materials through a wedge-shaped hopper; the smooth wall, radial gravity problem. Chem. Engng Sci. 43, 479494.
Prasad, P. & Ravindran, R., 1985 Partial Differential Equations. Wiley Eastern.
Rao, L. & Venkateswarlu, D., 1974 Static and dynamic wall pressures in experimental mass flow hoppers. Powder Technol. 10, 143152.
Roscoe, K. H.: 1970 The influence of strains in soil mechanics. Geotech. 20, 129170.
Roscoe, K. H. & Burland, J. B., 1968 On the generalized stress—strain behaviour of wet clay. In Engineering Plasticity (ed. J. Heyman & R. A. Leckie), pp. 535609. Cambridge University Press.
Roscoe, K. H., Schofield, A. N. & Wroth, C. P., 1958 On the yielding of soils. Geotech. 8, 2253.
Savage, S. B.: 1965 The mass flow of granular material derived from coupled velocity—stress fields. Brit. J. Appl. Phys. 16, 18851888.
Savage, S. B.: 1983 Granular flows at high shear rates. In Theory of Dispersed Multiphase Flow (ed. R. E. Meyer), pp. 339358. Academic.
Savage, S. B., Nedderman, R. M., Tüzüun, U. & Houlsby, G. T. 1983 The flow of granular materials — III. Chem. Engng Sci. 38, 189195.
Savage, S. B. & Sayed, M., 1979 Gravity flow of cohesionless granular materials in wedge shaped hoppers. In Mechanics Applied to the Transport of Bulk Materials (ed. S. C. Cowin), pp. 124.
Asme, Savage, S. B. & Yong, R. N. 1970 Stresses developed by cohesionless granular materials in bins. Intl J. Mech. Sci. 12, 675693.
Sayed, M. & Savage, S. B., 1983 Rapid gravity flow of cohesionless granular materials down inclined chutes. Z. Angew Math. Phys. 34, 8499.
Schaeffer, D. G. & Pitman, E. B., 1988 Ill-posedness in three-dimensional plastic flow. Commun. Pure Appl. Maths 41, 879890.
Schofield, A. N. & Wroth, C. P., 1968 Critical State Soil Mechanics. McGraw-Hill.
Shamlou, P. A.: 1988 Handling of Bulk Solids. Butterworths.
Shield, R. T.: 1953 Mixed boundary value problems in soil mechanics. Q. Appl. Maths 11, 6175.
Slattery, J. C.: 1981 Momentum, Energy and Mass Transfer in Continua, 2nd edn. Huntington: Robert E. Kreiger.
Sokolovskii, V. V.: 1965 Statics of Granular Media. Pergamon.
Spencer, A. J. M.: 1982 Deformation of an ideal granular material. In Mechanics of Solids (ed. H. G. Hopkins & M. J. Sewell), pp. 607652. Pergamon.
Spink, C. D. & Nedderman, R. M., 1978 Gravity discharge rate of fine particles from hoppers. Powder Technol. 21, 245261.
Stroud, M. A.: 1971 The behaviour of sand at low stress levels in the simple shear apparatus. Ph.D. thesis, University of Cambridge.
Sundaram, V. & Cowin, S. C., 1979 A reassessment of static bin pressure experiments. Powder Technol. 22, 2332.
Tüzün, U., Adams, M. J. & Briscoe, B. J., 1988 An interface dilation model for the prediction of wall friction in a particulate bed. Chem. Engng Sci. 43, 10831098.
Tüzün, U., Houlsby, G. T., Nedderman, R. M. & Savage, S. B., 1982 The flow of granular materials — II. Chem. Engng Sci. 37, 16911709.
Tüzün, U. & Nedderman, R. M. 1979 Experimental evidence supporting kinematic modelling of the flow of granular media in the absence of air drag. Powder Technol. 24, 257266.
Tüzün, U. & Nedderman, R. M. 1982 An investigation of the flow boundary during steady-state discharge from a funnel flow bunker. Powder Technol. 31, 2713.
Tüzün, U. & Nedderman, R. M. 1985 Gravity flow of granular materials round obstacles — II. Chem. Engng Sci. 40, 337351.
Van Zuilichem, D. J., Van Egmond, N. D. & de Swart, J. G. 1974 Density behaviour of flowing granular material. Powder Technol. 10, 161169.
Villadsen, J. & Michelsen, M. L., 1978 Solution of Differential Equation Models by Polynomial Approximation. Prentice-Hall.
Walker, D. M.: 1966 An approximate theory for pressures and arching in hoppers. Chem. Engng Sci. 21, 975997.
Walters, J. K.: 1973 A theoretical analysis of stresses in axially-symmetric hoppers and bunkers. Chem. Engng Sci. 28, 779789.
Williams, J. C.: 1977 The rate of discharge of coarse granular materials from conical mass flow hoppers. Chem. Engng Sci. 32, 247255.
Wilms, H. & Schwedes, J., 1985 Analysis of the active stress field in hoppers. Powder Technol. 42, 1525.
Zienkiewicz, O. C., Humpheson, C. & Lewis, R. W., 1977 A unified approach to soil mechanics problems (including plasticity and visco-plasticity). In Finite Elements in Geomechanics (ed. G. Gudehus), pp. 151177. John Wiley.
Zucrow, M. J. & Hoffman, J. D., 1976 Gas Dynamics, Vol. 2. John Wiley.
MathJax is a JavaScript display engine for mathematics. For more information see


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed