Granular flow, kinetic theory

Rahaman M, NaserJ, Witt P An unequal granular temperature kinetic theory description of granular flow with multiple particle classes. Powder Technol 138 82—92, 2003. 

Bokkers, G. A., Van Sint Annaland, M., and Kuipers, J. A. M., Comparison of continuum models using kinetic theory of granular flow with discrete particle models and experiments extent of particle mixing induced by bubbles. Proceedings of Fluidization XI, May 9-14, 2004, 187-194, Naples, Italy (2004). 

Ding J, Gidaspow D. A bubbling fluidization model using the kinetic theory of granular flow. AIChE J 1990 36 523-538. 

Lun, C. K. K., Savage, S. B. and Jeffery, D. J. (1984). Kinetic Theories for Granular Flow Inelastic Particles in Couette Flow and Slightly Inelastic Particles in a General Flow Field. J. Fluid Mech., 140, 223. 

Ding, J. and Gidaspaw, D., Bubbling fluidization model using kinetic theory of granular flows, AIChE J, 36, 523, 1990. 

Lun, C.K.K. Savage, S.B. A simple kinetic theory for granular flow of rough, inelastic, spherical particles. J. Appl. Mech. 1987, 54, 47-61. 

The first term on the right-hand side represents momentum exchange between solid phases I and s and Kis is the solid-solid exchange coefficient. The last term represent additional shear stresses, which appear in granular flows (due to particle translation and collisions). Expressions for solids pressure, solids viscosity (shear and bulk) and solid-solid exchange coefficients are derived from the kinetic theory of granular flows. 

Several different expressions have been derived for solids pressure, solids shear viscosity and solids bulk viscosity, employing different approximations and assumptions while applying the kinetic theory of granular flows. Some of the commonly used equations are described below (see Gidaspow, 1994 and a review given by Peirano, 1998) Solids pressure ... 

In the limit of maximum packing, granular flow becomes incompressible. Under such conditions, the kinetic contribution to viscosity is replaced by a friction contribution. Theories of soil mechanics may be used to estimate such friction contributions (Schaeffer, 1987). 

It must be noted here that most industrial fluidized bed reactors operate in a turbulent flow regime. Trajectory simulations of individual particles in a turbulent field may become quite complicated and time consuming. Details of models used to account for the influence of turbulence on particle trajectories are discussed in Chapter 4. These complications and constraints on available computational resources may restrict the number of particles considered in DPM simulations. Eulerian-Eulerian approaches based on the kinetic theory of granular flows may be more suitable to model such cases. Application of this approach to simulations of fluidized beds is discussed below. 

Sinclair and Jackson (1989) used the kinetic theory of granular flows to simulate gas-solid flows in risers. Their model was found to exhibit extreme sensitivity with respect to the value of restitution coefficient, e, . Nieuwland et al. (1996) also observed such an extreme sensitivity. Bolio et al. (1995) reported that such extreme sensitivity could be overcome by including a gas phase turbulence model. Despite these studies, there are no systematic guidelines available to make appropriate selection of models and model parameters (such as laminar versus turbulent, values of... 

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