Turbulence Transfer between Phases

The interphase turbulence transfer model accounts for turbulence interactions between the continuous and the dispersed phase due to drag. Note that the model is applicable only for a continuous gas phase with a solid particle dispersed phase. The model adds source terms to the turbulent kinetic energy and the dissipation equation for both the continuous and the dispersed phase. The unclosed or modeled term (within the source terms) is the cross-correlation between fluid and particle fluctuating velocities evaluated along the particle trajectory.

These source terms are a result of the work of the drag force on the fluctuating components of velocity and are formulated according to Fox [459].

This model is applicable for gas-particle flows carrying fine particles (that is, Geldart A type particles, having a size between 20 and 100 $\mu m$ , and a density typically less than 1.4g/cm³). A typical application for this model is a fluidized catalytic cracking (FCC) system.

The interphase turbulence transfer model is available only when the dispersed phase is a solid particle phase. Simcenter STAR-CCM+ restricts particle phases to use only the standard $k-ϵ$ two-layer turbulence model, so this formulation applies only to the scenario where both the continuous phase and the dispersed phase use the $k-ϵ$ model. $S_{kc}$ and $S_{kd}$ are the source terms for the continuous phase and dispersed phase turbulent kinetic energy equations respectively:

$S_{ϵc}$ and $S_{ϵd}$ are the source terms for the continuous and dispersed phase turbulence dissipation rate equations respectively:

where:

• $C_0$ is a constant that lets you calibrate the model. (This constant is not part of the original model formulation.)
• $C_{ϵ3}$ defaults to 1.54.
• $A_{cd}$ is the linearized drag coefficient.
• ${\beta }_{cd}$ is the velocity cross-correlation coefficient and defaults to 0.7.

  In Simcenter STAR-CCM+, ${\beta }_{cd}$ can be set as a constant, specified with a field function, or specified with the Zaichik method:

• $k_k=\frac{1}{2}{〈{{\mathbf{\text{v}}}_k}^{"}\cdot {{\mathbf{\text{v}}}_k}^{"}〉}_k$

  where ${\mathbf{\text{v}}}_k$ is the instantaneous velocity of phase $k$ .

• ${\epsilon }_k=\frac{{\mu }_k}{{\rho }_k}{〈2{S_k}^{"}\cdot {S_k}^{"}〉}_k$

  where ${S_k}^{"}=\frac{1}{2}\{\nabla {{\mathbf{\text{v}}}_k}^{"}+{(\nabla {{\mathbf{\text{v}}}_k}^{"})}^T\}$

Zaichik Method

Zaichik [577] provides a model for the unclosed velocity cross-correlation coefficient term ${\beta }_{cd}$ that is valid for arbitrary values of particle-to-fluid density and the ratio between particle size and fluid turbulence length scale:

$${\beta }_{cd}=\frac{(1-A)f_{u\alpha }+A}{\sqrt{(1-A^2)f_{u\alpha }+A^2}}$$

(2517)

with:

$$A=\frac{(1+C_{VM}){\rho }_c}{{\rho }_d+C_{VM}{\rho }_c}$$

(2518)

and:

$$f_{u\alpha }=\frac{2{\mathrm{\Omega }}_{\alpha }+Z^2}{2{\mathrm{\Omega }}_{\alpha }+2{\mathrm{\Omega }}_{\alpha }^2+Z^2}$$

(2519)

where:

• $C_{VM}$ is the virtual mass coefficient
• $f_{u\alpha }$ is the response coefficient
• ${\mathrm{\Omega }}_{\alpha }$ is the particle inertia parameter
• $Z$ is the ratio between the Taylor differential and the Lagrangian integral timescales of the continuous phase

The particle inertia parameter is defined as:

$${\mathrm{\Omega }}_{\alpha }=\frac{{\tau }_p}{T_{Lp}}$$

(2520)

with:

$${\tau }_p=\frac{{\alpha }_d{\rho }_d}{A_{cd}}(1+C_{VM}\frac{{\rho }_c}{{\rho }_d})$$

(2521)

where ${\tau }_p$ is the particle response time. $T_{Lp}$ is the eddy particle interaction timescale and is given by:

$$T_{Lp}=\frac{T_{Lp}^l+2T_{Lp}^n}{3}$$

(2522)

with:

$$T_{Lp}^l=\frac{T_L}{\sqrt{1+C_{\beta }{\zeta }^2}}$$

$$T_{Lp}^n=\frac{T_L}{\sqrt{1+{4C}_{\beta }{\zeta }^2}}$$

$$\begin{array}{ccc}{T}_L=\frac{2{\tau }_k({\mathrm{Re}}_{\lambda }+C_1)}{\sqrt{15}{C}_{0\infty }},& C_{0\infty }=7,& C_1=32\end{array}$$

where:

• $T_{Lp}^l$ and $T_{Lp}^n$ are the Lagrangian integral timescales of fluid velocities viewed by particles, respectively, in the longitudinal and normal directions to the relative velocity vector
• $T_L$ is the Lagrangian integral timescale
• $\zeta$ is the particle slip velocity, scaled by turbulent fluctuation velocity
• $C_{\beta }$ is a calibration coefficient for the crossing-trajectories effect. Following Deutsch and Simonin [448], it defaults to 0.45.

The scaled particle slip velocity is given by:

$$\zeta =\frac{\left|v_r\right|}{\sqrt{\frac{2}{3}{k}_c}}$$

where $v_r$ is the relative velocity between phases.

The ratio between the Taylor differential and the Lagrangian integral timescales of the continuous phase is:

$$Z=\frac{{\tau }_T}{T_L}$$

with:

$${\tau }_T={{\tau }_k\left(\frac{2{\mathrm{Re}}_{\lambda }}{\sqrt{15}{a}_0}\right)}^{1/2}$$

where:

• ${\tau }_T$ is the Taylor differential timescale
• ${\tau }_k$ is the Kolmogorov timescale
• ${\mathrm{Re}}_{\lambda }$ is the Taylor-scale Reynolds number

The Kolmogorov timescale is defined as:

$$\begin{array}{c}{\tau }_k=\sqrt{\frac{{\nu }_c}{{ϵ}_c}}\end{array}$$

The Taylor-scale Reynolds number is given by:

$\begin{array}{c}{\mathrm{Re}}_{\lambda }=\sqrt{\frac{20k_c^2}{3{\nu }_c{ϵ}_c}}\end{array}$

where ${\nu }_c$ is

$\begin{array}{cccc}{a}_0=\frac{a_{01}+a_{0\infty }{\mathrm{Re}}_{\lambda }}{a_{02}+{\mathrm{Re}}_{\lambda }},& a_{01}=11,& a_{02}=205,& a_{0\infty }=7\end{array}$