Flows of Granular Materials

In this chapter the derivation of the governing macroscopic equations of granular flow are examined. 

In a granular material there are primarily three mechanisms by which the bulk stresses are generated. These are (i) dry friction, (ii) transport of momentum by particle translation like in dilute gases, and (hi) momentum 

Jakobsen, Chemical Reactor Modeling, doi 10.1007/978-3-540-68622-4.4, Springer-Verlag Berlin Heidelberg 2008 

In many ways the thermodynamic and granular temperatures play similar roles in governing the behavior of their respective systems. In particular, both temperatures generate pressures and govern the internal transport rates of mass, momentum, and energy. 

It is emphasized that in the pioneering papers, mentioned above, simplified kinetic theories based on mechanically derived or intuitive relationships 

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Example 5.3 Consider the two-dimensional gravity flow of granular materials down an inclined plane, as shown in Fig. E5.1 [Savage, 1983]. Assume that the flow is fully developed and collision-dominated with a free surface. 

Whitaker, S. (1969). Advances in Theory of Fluid Motion in Porous Media. I EC., 61,4. Zeininger, G. and Brennen, C. E. (1985). Interstitial Fluid Effects in Hopper Flows of Granular Materials. ASME Cavitation and Multiphase Flow Forum, Albuquerque, N. Mex. 

Nedderman, R. M. and Ttiztin, U. (1979). A Kinematic Model for the Flow of Granular Materials. 

Nguyen, T. V., Brennen, C. and Sabersky, R. H. (1979). Gravity Flow of Granular Materials in Conical Hoppers. Trans. ASME, J. Appl. Mech., 46,529. 

Savage SB (1989) Flow of Granular Materials. In Germain P, Piau M, Caillerie D (eds). Theoretical and Apphed Mechanics. Elsevier Science Publishers B. V. (North-Holland)... 

Shahinpoor M (1983) Advances in the Mechanics and the flow of Granular Materials. Trans Tech Publications, Volume II, First edition, Clausthal-ZeUerfeld, Germany... 

T. Tamura and H. Haze, Determination of the flow of granular materials in silos, Bulk Solids Handling, 5(3) (June 1985), 633-40. 

For the rheology of granular materials, see Shahinpoor, M., (ed.) Advances in the Mechanics and the Flow of Granular Materials, vol. I and II. Trans. Tech. Publications, Clausthal-Zellerfeld, Germany (1983). 

In the analysis of flow of granular material, two types of flow can be distinguished. The first is slow frictional flow where the particles remain in continuous contact with each other the internal forces result from Coulomb friction between contacting particles. The second type of flow is much more rapid the particles are not in constant contact with their neighbors. The energy associated with the velocity fluctuations is comparable to that of the mean motion. In this type of flow, the internal forces arise because of the transfer of momentum during collisions between particles. The constitutive relations for this rapid flow are rate-dependent. This type of flow, therefore, is referred to as viscous flow (sometimes just rapid flow). Steady,... 

Wassgren, C.R. Hunt, M.L. Freese, P.J. Palamara, J. Brennen, C.E. (2002) Effects of vertical vibration on hopper flows of granular material. Physics of Fluids 14, 3439-3448. 

These fundamental aspects of the flow of granular material or bulk solids, similar to the flow in rotary kilns, have been reviewed by Savage (1979). Many theories have been considered in the effort to establish appropriate constitutive relations for such flows. As mentioned earlier, these theories extend from traditional soil mechanics to all types of viscous flows. The most important conclusions that can be drawn from these works to describe granular flow include the following ... 

Equations of motion and the pertinent constitutive equations for the flow of granular materials have been developed by Lun et al. (1984) using the hard sphere kinetic theory of dense gas approach. In this derivation a fixed control volume was considered in which a discrete number of smooth but inelastic particles are undergoing deformation. The resulting system of equations was given as follows ... 

H. Ahn, C. E. Brennen, and R. H. Sabersky. "Measurements of velocity, velocity fluctuation, density, and stresses in chute flows of granular materials," J. Applied Mech., 58, 792-803,1991. 

S. C. Cowin. "A theory for the flow of granular materials," Powder TechnoL, 9, 62-69, 1974. 

G. Mandl and R. F. Luque. "Fully developed plastic shear flow of granular materials," /. Geotechnique, 20(3), 277-307, 1970. 

Ackerman NL, Shen H Flow of granular material as a two-component system. In Cowin SC, Satake M, editors Proc. U. S.-Japan Symp. on continuum mechanics and statistical approaches in the mechanics of granular materials, Tokyo, 1978, Gakujutsu Bunken Fukyu-kai,... 

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