Thermal Diffusion

The heat transfer by thermal diffusion between phase i and phase j is modeled as:

Figure 1. EQUATION_DISPLAY

Q_{i j} = - Q_{j i} = h^{(i j)} a_{i j} (T_{j} - T_{i})

(2041)

where:

$Q_{i j}$ is the interphase heat transfer rate to phase $i$ from phase $j$
$h^{(i j)}$ is the mean surface average heat transfer coefficient
$a_{i j}$ is the interfacial area per unit volume

If c and d represent the continuous and dispersed phases in the phase interaction pair (ij), the mean heat transfer coefficient is computed as a function of the continuous phase thermal conductivity, the Nusselt number, and the interaction length scale:

Figure 2. EQUATION_DISPLAY

h^{(i j)} = \frac{k_{c} N u}{l_{c d}}

(2042)

Ranz-Marshall Correlation: According to Ranz-Marshall [532], the Nusselt number is computed as:

In Simcenter STAR-CCM+ the Nusselt number is:

Figure 3. EQUATION_DISPLAY

N u = 2 + 0.6 R e_{d}^{0.5} P r_{c}^{0.3}

(2043)

where:

$P r_{c}$ is the continuous phase Prandtl number
$R e_{d}$ is the dispersed phase Reynolds number

For a multiple flow regime phase interaction, the calculation of the interphase heat transfer is analogous to drag.

The total source term for the energy equation is calculated as:

Figure 4. EQUATION_DISPLAY

\begin{matrix} S_{T} = \sum_{t = f r, i r, s r} W_{t} h_{T, t} T_{r} & T_{r} = T_{s} - T_{p} \end{matrix}

(2044)

The interphase heat transfer coefficient (h_T) for the first dispersed regime is calculated in the same way as for the continuous-dispersed phase interaction. The primary phase is considered as the continuous phase and the secondary phase is considered as the dispersed phase. The interphase heat transfer coefficient for the second dispersed regime is calculated considering the secondary phase as the continuous phase and the primary phase as the dispersed phase.

The Nusselt number, Nu, is also calculated in the same manner:

Figure 5. EQUATION_DISPLAY

h_{T, f r} = \frac{k_{p} {N u}_{f r} a_{i, f r}}{l_{i, f r}}

(2045)

Figure 6. EQUATION_DISPLAY

h_{T, s r} = \frac{k_{s} {N u}_{s r} a_{i, s r}}{l_{i, s r}}

(2046)

For the intermediate regime, the following approach is used:

Figure 7. EQUATION_DISPLAY

\begin{matrix} h_{T, i r} \end{matrix} = \frac{k_{m} {N u}_{i r} a_{i, i r}}{l_{i, i r}}

(2047)

$\begin{matrix} k_{m} = α_{p} k_{p} + α_{s} k_{s} \end{matrix}$

$a_{i, i r} = 0.5 | \nabla α_{p} | | \nabla α_{s} | / (| \nabla α_{p} | + | \nabla α_{s} |)$

$l_{i, i r} = \sqrt[3]{0.5 {v o l}_{c}}$ for a 3D simulation, or $l_{i, i r} = \sqrt[2]{0.5 {v o l}_{c}}$ for a 2D simulation. ${v o l}_{c}$ is the volume of the cell.

This definition is consistent with the form of the interphase heat transfer coefficient, while providing the freedom to control the slip in temperature by setting Nu_ir.