ANSYS FLUENT 12.0 Theory Guide - 16.5.8 Granular Temperature

16.5.8 Granular Temperature

The granular temperature for the $s^{\rm th}$ solids phase is proportional to the kinetic energy of the random motion of the particles. The transport equation derived from kinetic theory takes the form [ 71]

$\frac{3}{2} \left[ \frac{\partial}{\partial t} (\rho_{s} \a... ...k_{\Theta_s} \nabla \Theta_s ) - \gamma_{\Theta_s} + \phi_{ls}$

(16.5-94)

where
	$(-p_s \overline{\overline{I}}+\overline{\overline{\tau}}_s):\nabla \vec v_s$	=	the generation of energy by the solid stress tensor
	$k_{\Theta_s} \nabla \Theta_s$	=	the diffusion of energy ( $k_{\Theta s}$ is the diffusion coefficient)
	$\gamma_{\Theta_s}$	=	the collisional dissipation of energy
	$\phi_{ls}$	=	the energy exchange between the $l^{\rm th}$
			fluid or solid phase and the $s^{\rm th}$ solid phase

Equation 16.5-94 contains the term $k_{\Theta_s} \nabla \Theta_s$ describing the diffusive flux of granular energy. When the default Syamlal et al. model [ 343] is used, the diffusion coefficient for granular energy, $k_{\Theta_s}$ is given by

$k_{\Theta_s} = \frac{15 d_s \rho_s \alpha_s \sqrt{\Theta_s \... ... + \frac{16}{15\pi}(41-33\eta) \eta \alpha_s g_{0,ss}) \right]$

(16.5-95)

where

$\eta = \frac{1}{2}(1+e_{ss})$

ANSYS FLUENT uses the following expression if the optional model of Gidaspow et al. [ 110] is enabled:

$k_{\Theta_s} = \frac{150\rho_s d_s \sqrt{(\Theta \pi)}} {384... ...alpha_s}^2 d_s (1+e_{ss}) g_{0,ss} \sqrt{\frac{\Theta_s}{\pi}}$

(16.5-96)

The collisional dissipation of energy, $\gamma_{\Theta_s}$ , represents the rate of energy dissipation within the $s^{\rm th}$ solids phase due to collisions between particles. This term is represented by the expression derived by Lun et al. [ 208]

$\gamma_{\Theta m} = \frac{12(1-e_{ss}^2) g_{0,ss}}{d_{s} \sqrt{\pi}} \rho_{s} \alpha_{s}^2 \Theta_{s}^{3/2}$

(16.5-97)

The transfer of the kinetic energy of random fluctuations in particle velocity from the $s^{\rm th}$ solids phase to the $l^{\rm th}$ fluid or solid phase is represented by $\phi_{ls}$ [ 110]:

$\phi_{ls} = - 3 K_{ls} \Theta_s$

(16.5-98)

ANSYS FLUENT allows the user 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 16.5-94 [ 343].
partial differential equation
This is given by Equation 16.5-94 and it is allowed to choose different options for it properties.
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:

$\vec {\tau_{s}} = - \frac{\pi}{6}\sqrt{3} \phi \frac{\alpha_... ...a_{s,max}} \rho_s g_0 \sqrt{\Theta_s} \vec{ U_{s,\vert\vert} }$

(16.5-99)

Here $\vec {U_{s,\vert\vert}}$ is the particle slip velocity parallel to the wall, $\phi$ is the specularity coefficient between the particle and the wall, $\alpha_{s,max}$ 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 [ 151]

$q_s = \frac{\pi}{6}\sqrt{3} \phi \frac{\alpha_s}{\alpha_{s,m... ...\alpha_{s,max}}(1-e_{sw}^2) \rho_s g_0 \Theta^{\frac{3}{2}}_s$

(16.5-100)

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Up: 16.5 Eulerian Model Theory
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