## 2 Numerical Methods
### Governing Equations
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Component**} & \multirow{2}{*}{**Density**} & \multirow{2}{*}{**Heat Capacity**} & \multicolumn{2}{c}{**Thermal**} & \multirow{2}{*}{**Heating Power Per Volume**} \\  & & & & & \\ \cline{2-2} \cline{5-6}  & **(kg/m3)** & & & & \\ \hline \multirow{3}{*}{Coils} & \multirow{3}{*}{2554} & \multirow{3}{*}{832.5} & \multirow{3}{*}{3.6} & Primary magnet units and & \multirow{3}{*}{154} \\  & & & & core-elevated magnet units & \\ \cline{1-1} \cline{4-6}  & & & & End magnet units & \\ \cline{1-1} \cline{4-6} Iron core & \multirow{3}{*}{7450} & \multirow{3}{*}{430} & \multirow{3}{*}{17} & \multirow{3}{*}{0} \\ \cline{1-1} \cline{4-6} Epoxy & \multirow{3}{*}{1150} & \multirow{3}{*}{2410} & \multirow{3}{*}{0.4} & \multirow{3}{*}{0} \\ \cline{1-1} \cline{4-6} Flange & \multirow{3}{*}{2800} & \multirow{3}{*}{880} & \multirow{3}{*}{160} & \multirow{3}{*}{0} \\ \cline{1-1} \cline{4-6} Glass fiber & \multirow{3}{*}{2600} & \multirow{3}{*}{670} & \multirow{3}{*}{0.4} & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of solid components. where \(\mathbf{r}\) is the position vector and \(\overset{\wedge}{\mathbf{s}}\) is the unit vector into a given direction. \(I\) and \(I_{b}\) are the radiative intensity and black body intensity, respectively. \(I_{b}=n^{2}\sigma T_{s}{}^{4}/\pi\), \(n\) is the absolute refractive index. For atmospheric and subatmospheric conditions, the refractive index of air is approximately 1. \(\sigma\) is the Stefan-Boltzmann constant and \(k_{a}\) and \(k_{s}\) are the absorption coefficient and scattering coefficient of the transport medium, respectively. Since air is the transport medium in this study, the value of \(k_{a}\) and \(k_{s}\) are assumed to be zero. \(\beta_{r}\) is the extinction coefficient defined as \(k_{a}+k_{s}\). \(\overset{\wedge}{\mathbf{s}}^{\prime}\) denotes an incoming radiation direction. \(\Omega^{\prime}\) is the solid angle of \(\mathbf{s}^{{}^{\prime}}\). The boundary condition for radiative heat transfer is defined in the format of Equation (7),  
\[I\left(\mathbf{r}_{w},\overset{\wedge}{\mathbf{s}}\right)=\varepsilon_{w}( \mathbf{r}_{w})I_{b}(\mathbf{r}_{w})+\frac{\rho_{w}(\mathbf{r}_{w})}{\pi}\int_ {(\mathbf{n}\ast<0)}I\left(\mathbf{r}_{w},\overset{\wedge^{\prime}}{\mathbf{ s}}\right)\mathbf{n}\cdot\overset{\wedge^{\prime}}{\mathbf{s}}\left|d\Omega^{\prime}\right. \tag{7}\]  
where \(\mathbf{r}_{w}\) is the wall position vector and \(\varepsilon_{w}\) is the surface emissivity. In this study, the surface emissivity of all solids is approximated to be 0.8, which is the weighted average of surface emissivity among all components. \(\rho_{w}\) is the reflectivity and \(\varepsilon_{w}+\rho_{w}=1\). \(\mathbf{n}\) is the surface normal. In the numerical model, the heat transfer within both fluid and solid domain obeys [PERSON]'s law, which is formulated as \(\mathbf{q}_{cd}=-\mathbf{k}\
abla T\), where \(\mathbf{q}_{cd}\) is the conduction heat flux, \(\mathbf{k}\) is the material's thermal conductivity tensor, and \(\
abla T\) is the spatial temperature gradient. The conjugate heat transfer method is applied to the fluid-solid interfaces [35]. The energy equation is then solved throughout the solid and fluid domain with implicit thermal coupling. Based on the conjugate heat transfer method, the convection heat transfer coefficient between fluid and solid is not required as the numerical model input. However, the convection heat transfer coefficient is very useful in evaluating the cooling performance of a heated object and can be directly applied to the numerical model when the computational capacity is limited. Therefore, in this study, the convection heat transfer coefficient is derived based on the numerical results according to Equation (8):  
\[h=\frac{\mathbf{q}_{cv}\cdot\mathbf{a}}{|\mathbf{a}|\left(T_{s}-T_{ref}\right)} \tag{8}\]  
where \(h\) is the convection heat transfer coefficient, \(\mathbf{q}_{cv}\) is the convection heat flux.