14.5.16. Granular Temperature

The granular temperature for the solids phase is proportional to the kinetic energy of the particles’ random motion. The formal expression is:

(14–373)

In Equation 14–373 represents the component of the fluctuating solids velocity in the Cartesian coordinate system. This is defined as an ensemble average of the particles’ random velocity within a finite volume and time period. The averaging basis is particle number per unit volume following [99] and [142].

The transport equation derived from kinetic theory takes the form [142]:

(14–374)

where

= the generation of energy by the solid stress tensor

= the diffusion of energy ( is the diffusion coefficient)

= the collisional dissipation of energy

= the energy exchange between the fluid or solid phase and the solid phase

Equation 14–374 contains the term describing the diffusive flux of granular energy. When the default Syamlal et al. model [640] is used, the diffusion coefficient for granular energy, is given by

(14–375)

where

Ansys Fluent uses the following expression if the optional model of Gidaspow et al. [205] is enabled:

(14–376)

The collisional dissipation of energy, , represents the rate of energy dissipation within the solids phase due to collisions between particles. This term is represented by the expression derived by Lun et al. [397]

(14–377)

The transfer of the kinetic energy of random fluctuations in particle velocity from the solids phase to the fluid or solid phase is represented by [205] :

(14–378)

Ansys Fluent allows you to solve for the granular temperature with the following options:

  • algebraic formulation (the default)

    It is obtained by neglecting convection and diffusion in the transport equation, Equation 14–374 [640].

  • partial differential equation

    This is given by Equation 14–374 and it is allowed to choose different options for it properties.

  • dpm-averaged granular temperature

    An alternative formulation available only with the Dense Discrete Phase Model (DDPM).

  • constant granular temperature

    This is useful in very dense situations where the random fluctuations are small.

  • UDF for granular temperature

For a granular phase , we may write the shear force at the wall in the following form:

(14–379)

Here is the particle slip velocity parallel to the wall, is the specularity coefficient between the particle and the wall, is the volume fraction for the particles at maximum packing, and is the radial distribution function that is model dependent.

The general boundary condition for granular temperature at the wall takes the form: [282]

(14–380)