Schematic representation of different regimes of granular flow regimes.

solid phases. The solids stress can be written as:

asTs = -PSI + 2as^sS + as - 2 /j,s^ V ■ USI (4.30)

where Ps is solids pressure, is solids (shear) viscosity and is solids bulk viscosity. SI is given by:

In addition to these solids stresses, the solids momentum equation may have the following terms:

The first term on the right-hand side represents momentum exchange between solid phases l and s and Kls 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.

The solids pressure, Ps consists of a collisional and a kinetic part:

where PC,sn is the pressure caused by collisions between the solid phases s and n and 0s is the granular temperature. Granular temperature is a measure of the kinetic energy contained in the fluctuating velocity of particles, and is defined:

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 Gidaspaw, 1994 and a review given by Peirano, 1998): Solids pressure:

For dense suspensions (Gidaspaw, 1994):

and for dilute suspensions, Bolio et al. (1995) used

es is the value of the restitution coefficient of solid particles (for elastic particles, the restitution coefficient is unity) and g0s is a radial distribution function. The restitution coefficient is defined as the ratio of normal relative velocity after the collision and before the collision. The value of restitution coefficient may depend on particle size and relative velocity, however, generally it is assumed that it has a constant value for a given material. Measured restitution coefficients for different types of materials are given by Foerster et al. (1994). In the framework of the kinetic theory, the radial distribution function accounts for the increase in probability of collisions when the gas becomes denser. Analogously, when particles are closely packed (when motion is almost impossible), the radial distribution function tends to infinity. Chapman and Cowling (1970) derived an expression for the radial distribution function. However, this expression is not consistent with the asymptotic behavior of dense gases at extreme high concentrations. Several empirical correlations have been proposed. Lun and Savage (1986) proposed:

Ogawa et al. (1980) proposed:

Ma and Ahmadi (1986) proposed a more complicated expression, which shows good agreement with simulations. However, in most engineering simulations, the above two empirical expressions are used and may be considered adequate.

Solids shear viscosity also comprises a kinetic contribution and collisional contributions. Commonly used expressions for viscosity are:


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