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23.5.8 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 [ 81]


 \frac{3}{2} \left[ \frac{\partial}{\partial t} ( \rho_{s} \a... ...k_{\Theta_s} \nabla \Theta_s ) - \gamma_{\Theta_s} + \phi_{ls} (23.5-75)


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  23.5-75 contains the term $ k_{\Theta_s} \nabla \Theta_s$ describing the diffusive flux of granular energy. When the default Syamlal et al. model [ 364] 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] (23.5-76)

where


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

FLUENT uses the following expression if the optional model of Gidaspow et al. [ 121] 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}} (23.5-77)

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. [ 222]


 \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} (23.5-78)

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 $\phi_{ls}$ [ 121]:


 \phi_{ls} = - 3 K_{ls} \Theta_s (23.5-79)

FLUENT allows the user to solve for the granular temperature with the following options:

For a granular phase $s$, 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} } (23.5-80)

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 $g_0$ is the radial distribution function that is model dependent.

The general boundary condition for granular temperature at the wall takes the form  [ 166]


 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 (23.5-81)


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Up: 23.5 Eulerian Model Theory
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