## 2 Modeling of Induction Heating and Heat Transfer  
Radio frequency (RF) induction heating technology is used to heat the silicon carbide crystal growth furnace. The reasonable temperature field is premised on an accurate electromagnetic field. Generally, mathematical models for the electromagnetic field are developed based on the Maxwell Equation (1)  
\[\left\{ \begin{array}{l}\
abla\cdot E=\frac{\rho}{\mu_{0}}\\ \
abla\cdot B=0\\ \
abla\times E=-\frac{\partial B}{\partial t}\\ \
abla\times B=\mu_{0}e_{0}\frac{\partial E}{\partial t}+\mu_{0}(J_{c}+J_{i}) \end{array} \right. \tag{1}\]  
where \(E\) and \(B\) are electric field intensity and magnetic induction intensity, respectively; \(\mu_{0}\) (\(4\pi\times 10^{-7}\) H/m) is permeability of free space; \(e_{0}\) (\(8.855\times 10^{-12}\) F/m in vacuum) is permittivity of free space; \(J_{c}\) is current flux in coil; and \(J_{i}\) (A/m\({}^{2}\)) is induction current flux in graphite crucible. To simplify Maxwell equations, magnetic potential, \(A\) (\(B=\
abla\times A\)), is introduced and the simplified equation is proposed in 3D cylindrical coordinates  
\[\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}r\frac{\partial A}{\partial r} \bigg{)}+\frac{1}{r^{2}}\frac{\partial^{2}A_{0r}}{\partial\theta^{2}}+\frac{ \partial^{2}A_{0r}}{\partial z^{2}}-\frac{A_{0r}}{r^{2}}+\mu_{0}\sigma\omega A _{0i}+\mu_{0}J_{c}=0 \tag{2}\]  
\(A\) can be expressed by (\(0\), \(A_{0}\), \(0\)), and \(A_{0}=A_{0r}+iA_{0i}\), so Equation (2) can be rewrote as [7; 17]  
\[\left\{ \begin{array}{l}\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}r\frac {\partial A_{0r}}{\partial r}\bigg{)}+\frac{1}{r^{2}}\frac{\partial^{2}A_{0r} }{\partial\theta^{2}}+\frac{\partial^{2}A_{0r}}{\partial z^{2}}-\frac{A_{0r}}{ r^{2}}+\mu_{0}\sigma\omega A_{0i}+\mu_{0}J_{c}=0\\ \frac{1}{r}\frac{\partial}{\partial r}\bigg{(}r\frac{\partial A_{0i}}{\partial r }\bigg{)}+\frac{1}{r^{2}}\frac{\partial^{2}A_{0i}}{\partial\theta^{2}}+\frac{ \partial^{2}A_{0i}}{\partial z^{2}}-\frac{A_{0i}}{r^{2}}-\mu_{0}\sigma\omega A _{0r}=0\end{array} \right. \tag{3}\]  
where \(\omega\) is angular velocity (rad/s, \(\omega=2\pi\)F) and \(F\) is current frequency (Hz); \(J_{c}\) is current flux in coil (A/m\({}^{2}\), \(J_{c}=I/A_{coil}\)) and \(I\) is current (A), \(A_{coil}\) is area of coil cross section; \(\sigma\) (\(1.1\times 10^{5}\)\(\Omega^{-1}\cdot\)m\({}^{-1}\)) is coil electric conductivity. For the different subsystems--such as crystals, silicon carbide powder, graphite crucible, insulation layer, and air of outside furnace--the heat transfer process can be described as the following energy Equation (4)  
\[\lambda\left[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{ \partial r}\right)+\frac{1}{r^{2}}\frac{\partial^{2}T}{\partial\theta^{2}}+ \frac{\partial^{2}T}{\partial z^{2}}\right]+Q_{eddy}+Q_{IF}=0 \tag{4}\]  
where \(Q_{eddy}\) is induction heating source term in the graphite crucible; \(Q_{IF}\) is heating source term acting on cell interfaces, such as radiation and convection heat exchange (\(Q_{insu}\)) on felt walls, radiation heat exchange (\(Q_{rad}\)) on growth chamber walls. Equations (1) and (2) are discretized by the finite volume method, and solved by the SIMPLE algorithm. \(Q_{eddy}\) as the induction heating source is calculated by [2; 18]  
\[Q_{eddy}=\frac{1}{2}\sigma\omega^{2}\left(A_{0r}^{2}+A_{0i}^{2}\right) \tag{5}\]  
The third boundary condition is used to describe heat release on the felt walls, and the heating source can be added into Equation (2) with the additional source terms method  
\[\left\{ \begin{array}{l}-\lambda\frac{\partial T}{\partial n}\Big{|}_{wall}=h \Big{(}T-T_{f}\Big{)}+\sigma\varepsilon\Big{(}T^{4}-T_{f}^{4}\Big{)}\\ =\underbrace{\Big{[}h+\sigma\varepsilon\Big{(}T+T_{f}\Big{)}\Big{(}T^{2}+T_{ f}^{2}\Big{)}\Big{]}}_{h_{eff}}\Big{(}T-T_{f}\Big{)}\\ S_{C,ad,insu}=\frac{A}{\Delta V}\frac{T_{f}}{\Gamma/h_{eff}+\delta/\Lambda},\\ S_{P,ad,insu}=-\frac{A}{\Delta V}\frac{1}{\Gamma/h_{eff}+\delta/\lambda}\end{array} \right.