Granular kinetic theory

Generalized source term in granular kinetic theory Generalized flux term in granular kinetic theory Dense gas collisional transfer flux Dense gas kinetic transfer flux... 

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 ... 

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... 

The science of mechanics constitutes a vast number of sub-disciplines commonly considered beyond the scope of the standard chemical engineering education. However, when dealing with kinetic theory-, granular flow- and population balance modeling in chemical reactor engineering, basic knowledge of the principles of mechanics is required. Hence, a very brief but essential overview of the disciplines of mechanics and the necessary prescience on the historical development of kinetic theory are given before the more detailed and mathematical principles of kinetic theory are presented. 

Besides, to understand the basic principles of kinetic theory, granular flows and population balances we need to widen our knowledge of classical mechanics. Newton s mathematical formulation of the laws of motion is perhaps the most intuitive point of view considering familiar quantities like mass, force, acceleration, velocity and positions and as such preferred by chemical engineers. However, this framework is inconvenient for mathematical generalizations as required describing the motion of large populations of particles for which it is necessary to take into account the constraints that limit the motion of the... 

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