## 4 Radiation Heat Transfer in Crystal Growth Chamber  
### Surface Visibility Checking Algorithm for S2S Radiation Model  
The classical surface to surface (S2S) radiation model is  
\[\left\{ \begin{array}{l}q_{r}=\left(A^{-1}.B\right)\cdot\left(\sigma T^{4} \right)\\ A_{jk}=\frac{\delta_{jk}}{\varepsilon_{k}}-F_{jk}\frac{1-\varepsilon_{k}}{ \varepsilon_{k}},\\ B_{jk}=\delta_{jk}-F_{jk}\\ F_{jk}=\frac{1}{\Lambda_{j}}\int_{\Lambda_{j}}\int_{\Lambda_{k}}\frac{\cos \theta_{j}\cos\theta_{k}}{\pi d^{2}}\eta_{jk}dA_{j}dA_{k}\end{array} \right. \tag{7}\]  
where \(A\) and \(B\) are coefficient matrixes of view factors (\(F\)); when \(j=k\), \(\delta_{jk}=1\), otherwise \(\delta_{jk}=0\); \(T\) is wall temperature; \(\eta_{jk}=1\) in \(F_{ij}\) means face \(j\) can see face \(k\), otherwise \(\eta_{jk}=0\). So \(\eta_{jk}\) should be determined before calculating \(F_{ij}\). The radiation source term, \(Q_{rad}\), is added into energy equation by \(q_{r}\) as the following forms [13]  
\[\left\{ \begin{array}{l}solid:S_{C,ad,rad}=-\frac{A}{\Delta V_{r}}\frac{q_{r} \delta_{ff}/\lambda_{f}}{\delta_{rf}\lambda_{s}+\delta_{rf}/\lambda_{f}}\\ fluid:S_{C,ad,rad}=\frac{A}{\Delta V_{r}}\frac{q_{r}\delta_{rf}/\lambda_{s}}{ \delta_{rf}\lambda_{s}+\delta_{rf}/\lambda_{s}}\\ T_{sf}=\frac{T_{A_{k}}/\delta_{sf}+T_{rf}\lambda_{f}/\delta_{rf}}{\lambda_{s} /\delta_{rf}+\lambda_{rf}/\delta_{rf}}\end{array} \right. \tag{8}\]  
where \(\delta_{sf}\) and \(\delta_{ff}\) are distances from the center of solid mesh cell and fluid mesh cell to the fluid-solid interface center. Least distance method (LDM) is proposed to determine the value of \(\eta_{jk}\) between two wall faces. Figure 2 exhibits a schematic diagram and a flowchart of LDM. An example (\(i\)th cell face) is given to illustrate the principle of LDM: (1) give the mathematical forms (\(F\)(R)) of block faces, such as growth surface and cylindrical surface of silicon carbide crystal; (2) search all other \(j\)th cell faces and save their face information as shown in Figure 2a, such as normal vector \(\mathbf{n}\), distance vector \(\mathbf{d}\), position coordinates of face center, line equation \(\mathrm{L}(i,j)\) from \(i\)th face center to \(j\)th face center, angle \(\theta_{i}\) and \(\theta_{j}\) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Density (\(\rho\))** & **Heat Capacity (\(\epsilon_{p}\))** & **Conductivity (\(\mathbf{a}\))** \\
**Name** & **kg/m\({}^{3}\)** & **J(Rg-K)** & **W/(m-K)** \\ \hline SiC crystal & 3140.0 & \(\frac{1}{37.4^{10.7}\times 18.3^{10.7}\times 18.3^{10.7}\times 17.4^{10.7}}\) & \(\frac{61100}{7-115}\) \\ \hline Graphite & 1750.0 & \(37.7e^{19.66\times 10^{-4}T}\) & \(\frac{1}{441.127^{-31.7}\times 19.75\times 10^{-4}\times 14^{-4}\times 10^{-4}}\) \\ \hline Felt & 170.0 & 2100.0 & \(\left\{\begin{array}{ll}8.17\times 10^{-2}+2.49\times 10^{-4}T,&T\leq 1473K\\ 4.19\times 10^{-4}T^{0.87},&T>1473K\end{array}\right.\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical properties of materials of SiC growth furnace. Figure 1: Silicon carbide crystal growth furnace and its three-dimensional mesh. (\(\mathbf{a}\)) Physical model for electromagnetic field; (\(\mathbf{b}\)) physical model for temperature field; (\(\mathbf{c}\)) mesh for the furnace. between n and d; (3) calculate and save all intersection points (P(L, F)) between line L(\(i,j\)) and face \(F\)(R); (4) calculate all distances from \(i\)th face center to P(L, F) and record the point owning a minimum distance; (5) check whether the recorded point coincides with \(j\)th face center or not, if not then \(\eta_{ij}=1\), otherwise \(\eta_{ij}=0\). Two wall faces (located on SiC powder surface 1# mesh face and graphite crucible inner surface (2# mesh face)) are selected to verify LDM as shown in Figure 3. The red regions as shown in Figure 3a,c are ideal visible region for 1# mesh face and 2# mesh face, respectively.