Granular Temperature

The granular temperature, ${\theta }_p$, can be estimated by solving the conservation equation for granular energy.

The conservation equation for the fluctuating granular kinetic energy is given as:

$$\frac{3}{2}[\frac{\partial }{\partial t}\underset{V}{\int }{\alpha }_p{\rho }_p{\theta }_{p}dV+\underset{A}{\oint }{\alpha }_p{\rho }_p{\theta }_p{\mathbf{\text{v}}}_p\cdot da]=\underset{V}{\int }{\tau }_p\text{:}\nabla {\mathbf{\text{v}}}_{p}dV+\underset{A}{\oint }{\kappa }_{eff}\nabla {\theta }_p\cdot da-\underset{V}{\int }\gamma dV-\underset{V}{\int }{J}_{p}dV+\underset{V}{\int }{\alpha }_p{\rho }_p{ϵ}_{p}dV$$

(2366)

The first term in Eqn. (2366) (above) is the granular energy production. Wall treatment is used to compute a modified value of granular energy production in the first cell adjacent to the wall.

The effective granular diffusion coefficient, ${\kappa }_{eff}$, is:

$${\kappa }_{eff}=\kappa +\frac{3{\mu }_{pt}}{2{\sigma }_{pt}}$$

(2367)

where:

• $\kappa$ is the granular diffusion coefficient
• ${\mu }_{pt}$ is the turbulent viscosity of the granular phase
• ${\sigma }_{pt}$ is the turbulent granular diffusion Prandtl number. The default value is 1.

The granular diffusion coefficient, $\kappa$, comes in two forms:

• The Gidaspow form:

$$\kappa =\frac{150{\rho }_p{d}_p\sqrt{\pi {\theta }_p}}{384(1+e)g_0}[\frac{{\alpha }_p}{{\alpha }_{p,sum}}+\frac{12}{5}{\alpha }_p(1+e)g_0+\frac{36}{25}{\alpha }_p{\alpha }_{p,sum}{(1+e)}^2{g_0}^2]+2{\alpha }_p{\alpha }_{p,sum}{\rho }_p{d}_p(1+e)g_0\sqrt{\frac{{\theta }_p}{\pi }}$$

(2368)

• The Syamlal form:

$$\kappa =\frac{15{\alpha }_p{\rho }_p{d}_p\sqrt{\pi {\theta }_p}}{4\left(41-33\eta \right)}[1+\frac{12}{5}{\eta }^2\left(4\eta -3\right){\alpha }_{p,sum}{g}_0+\frac{16}{15\pi }\left(41-33\eta \right)\eta {\alpha }_{p,sum}{g}_0]$$

(2369)

where:

• ${\rho }_p$ is the particle density.
• $d_p$ is the particle diameter. The interaction length scale is set as equal to the particle diameter.
• $\eta =\frac{1}{2}(1+e)$

The dissipation of granular energy, $\gamma$, is:

$$\gamma =\frac{12\left(1-e^2\right)g_0}{d_p\sqrt{\pi }}{\alpha }_p{\alpha }_{p,\text{sum}}{\rho }_p{{\theta }_p}^{3/2}$$

(2370)

The dissipation of granular energy due to inter-phase drag, $J_p$, is:

$$J_p=A_D[3{\theta }_p-\frac{A_D{d}_p{(v_g-v_p)}^2}{4{\alpha }_p{\rho }_p{g}_0\sqrt{\pi {\theta }_p}}]$$

(2371)

where $A_D$ is the interphase momentum transfer coefficient.

The first term is attributed to the work of Gidaspow [466] and is always included. The second term is attributed to the work of Louge and others [510] and is included only if the Cross-Correlation Term property of the Granular Energy Transfer model is set to Louge.

The last term in Eqn. (2366), ${\alpha }_p{\rho }_p{ϵ}_p$, is a source term that is due to the dissipation of granular phase turbulent kinetic energy. This term is present only if the granular phase is modeled using the Turbulent viscous regime.

Algebraic Model for Granular Temperature

By assuming local equilibrium, where energy dissipation and production balance each other, the granular energy equation Eqn. (2366) can be reduced to a quadratic equation which is solved directly for ${\theta }_p$ :

$${\alpha }_p{K}_{4p}{\theta }_p+{\alpha }_p{K}_{1p}tr\left(S\right)\sqrt{{\theta }_p}-\{K_{2p}{\left[tr\right(S\left)\right]}^2+2K_{3p}(S:S)\}=0$$

(2372)

where:

$$K_{1p}=2g_0{\rho }_p(1+e)+3(e^2-1){\rho }_p{g}_0$$

(2373)

$$K_{2p}=\frac{4}{3\sqrt{\pi }}{\alpha }_p{\rho }_p{d}_p{g}_0(1+e)-\frac{2}{3}{K}_{3p}$$

(2374)

$$K_{3p}=\frac{4}{5\sqrt{\pi }}{\alpha }_p{\rho }_p{d}_p{g}_0(1+e)+\frac{{\mu }_p^K}{\sqrt{{\theta }_p}}$$

(2375)

$$K_{4p}=\frac{12\left(1-e^2\right)}{d_p\sqrt{\pi }}{\rho }_p{g}_0$$

(2376)

where $S$ is the strain rate tensor, $tr\left(S\right)$ is the trace of the strain rate tensor, ${\rho }_p$ is the particle density, and $d_p$ is the particle diameter. The interaction length scale is equal to the particle diameter. The kinetic contribution to effective granular viscosity, ${\mu }_p^K$ , is defined using one of the forms in Eqn. (2384) or Eqn. (2385).

The last term in Eqn. (2375) is neglected when using the Algebraic Model for Granular Temperature:

$$$$ $$K_{3p}=\frac{4}{5\sqrt{\pi }}{\alpha }_p{\rho }_p{d}_p{g}_0(1+e)$$

(2377)

The minimum granular temperature in the domain is set to $1\times {10}^{-10}{m}^2/s^2$ by default but you can change the value if necessary. To estimate the granular temperature at boundaries, either extrapolate the granular temperature at the face cell or use a user-defined value. The specified granular temperature value is used to calculate the solid pressure at the boundaries, which has a significant influence on results.

Expressions for the solid pressure, bulk viscosity, and effective granular viscosity are obtained as functions of the particle granular temperature,   ${\theta }_p$ . The solid pressure is composed of collisional and kinetic contributions, [564]:

$$P_p=P_p^C+P_p^K$$

(2378)

$$P_p^C=2g_0{\rho }_p{\alpha }_p^2{\theta }_p(1+e)$$

(2379)

$$P_p^K={\rho }_p{\alpha }_p{\theta }_p$$

(2380)

The bulk viscosity is [542]:

$${\xi }_p=\frac{4}{3}{{\alpha }_p^2{\rho }_p{d}_{p}g}_0(1+e)\sqrt{\frac{{\theta }_p}{\pi }}$$

(2381)

The effective granular viscosity ${\mu }_p$ is composed of collisional, kinetic, and frictional contributions:

$${\mu }_p={\mu }_p^C+{\mu }_p^K+{\mu }_p^f$$

(2382)

A user-defined Maximum Solid Viscosity limit is applied to ${\mu }_p$ .

$${\mu }_p^C=\frac{4}{5}{{\alpha }_p^2{\rho }_p{d}_{p}g}_0(1+e)\sqrt{\frac{{\theta }_p}{\pi }}$$

(2383)

You can select between two forms of the particle kinetic viscosity:

• The form from Gidaspow [466] is:

  $${\mu }_p^K=\frac{10{\rho }_p{d}_p\sqrt{{\pi \theta }_p}}{96(1+e)g_0}{[1+\frac{4}{5}{g}_0{\alpha }_p(1+e)]}^2$$

  (2384)

• The form from Syamlal [433] is:

  $${\mu }_p^K=\frac{{{\alpha }_p\rho }_p{d}_p\sqrt{{\pi \theta }_p}}{6\left(3-e\right)}[0.5(3e+1)+\frac{2}{5}(1+e\left)\right(3e-1\left)g_0{\alpha }_p\right]$$

  (2385)

The Syamlal version predicts lower viscosity at particle volume fractions below 0.3 [564]. You are advised to use the corresponding linearized drag law (Gidaspow or Syamlal) in accordance with the version of particle kinetic viscosity selected.