Kinetic theory for granular flow

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. 

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 severe implications of these facts have been partially uncovered in reference [5] as a result of formulating a kinetic theory for granular flow without interaction with the ambient medium. These implications, as well as additional difficulties due to the necessity to calculate the energy supply to the particle fluctuations, make somewhat problematic, at the present state of the art, the formulation of a reliable and sufficiently simple hydrodynamic model even for coarse dispersions. We have succeeded in this respect only at the expense of making certain supplementary assumptions. These assumptions cne ... 

Lun CKK, Savage SB, Jeffrey DJ, Chepumiy 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. 

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. 

To model the particle velocity fluctuation covariances caused by particle-particle collisions and particle interactions with the interstitial gas phase, the concept of kinetic theory of granular flows is adapted (see chap 4). This theory is based on an analogy between the particles and the molecules of dense gases. The particulate phase is thus represented as a population of identical, smooth and inelastic spheres. In order to predict the form of the transport equations for a granular material the classical framework from the kinetic theory of... 

Chapter 10 contains a literature survey of the basic fluidized bed reactor designs, principles of operation and modeling. The classical two- and three phase fluidized bed models for bubbling beds are defined based on heat and species mass balances. The fluid dynamic models are based on kinetic theory of granular flow. A reactive flow simulation of a particular sorption enhanced steam reforming process is assessed. 

The TFM has been widely used for simulation of gas—solid fluidization. Based on the local equihbrium assumption, it treats the collective behavior of sohd particles as a pseudo-fluid, whose strain—stress relation can be closed with the kinetic theory of granular flow (KTGF). The homogenous drag is further used, which is actually a requirement of the local equihbrium assumption. 

In the past two decades, the kinetic theory of granular flow has developed rapidly (see, e.g., Jenkins Savage [1], Lun et al. [2]), and numerical results for flows with nearly elastic particles, where the particle concentrations range from small to moderate, have shown quite good agreement with experimental data. But the effect of the interstitial gas has generally been regarded as too complex to consider. 

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