## 6 Thermal System Design
### Defining the Thermal Load
To ensure steady-state operation, a basic energy balance can be applied to the system factoring solar radiation, Albedo, Earth infra-red, and radiation from the body to space:  
\[ \begin{split}\varepsilon_{IR}^{*}*\sigma*5*A_{P}*\left(T^{4}-T_{ space}^{4}\right)+\varepsilon_{IR}*\sigma*A_{P}*\left(T^{4}-T_{E}^{4}\right)\\ =\alpha_{s_{s}}*I_{s}*A_{P}+0.34*\alpha_{s_{B}}*I_{S}*A_{P}+\alpha _{IR}*I_{E}*A_{P}\end{split}  \tag{70}\]  
It is assumed \(\left(T^{4}-T_{e}^{4}\right)\cong 0\) therefore \(\varepsilon_{IR}*\sigma*A_{P}*\left(T^{4}-T_{E}^{4}\right)\) can be assumed to also be equal to zero. The equation shown above can be rearranged to solve the equilibrium temperature for the hottest worst-case scenario:  
\[T=\sqrt[4]{\frac{\alpha_{S_{2}}}{\varepsilon_{IR}^{*}}*\frac{I_{S}}{\sigma}* \frac{A_{p}}{5*A_{p}}+\frac{\alpha_{s1}}{\varepsilon_{IR}^{*}}*\frac{0.34*I_{s }}{\sigma}*\frac{A_{p}}{5*A_{p}}+\frac{\alpha_{IR}}{\varepsilon_{IR}^{*}}* \frac{A_{P}}{5*A_{p}}*\frac{I_{E}}{\sigma}+T_{Space}^{4}} \tag{71}\]  
The worst case hot temperature for the CubeSat's components giving during the heating phase is given by:  
\[T_{max}=\left(T_{space}^{4}+\frac{I_{s}}{\sigma}*\frac{A_{p}}{5*A_{p}}*\frac{ \alpha_{ss}}{\varepsilon_{IR}^{*}}+\frac{I_{E}}{\sigma}*\frac{A_{p}}{5*A_{p}}* \frac{\varepsilon_{IR}}{\varepsilon_{IR}^{*}}+\frac{0.34*I_{s}}{\sigma}*\frac{ A_{p}}{5*A_{p}}*\frac{\alpha_{SB}}{\varepsilon_{IR}^{*}}\right)^{0.25} \tag{72}\]  
Applying the materials shown in the face identification table, the worst case hot temperature is found to be 315.018 K (42.018 \({}^{\circ}\)C). This matches extremely close to temperatures published in literature. For example, Compass-1 encountered a theoretical hot-case temperature of 320.8 K (47.8 \({}^{\circ}\)C) at an altitude of 600 km. A temperature difference of 5.782 \({}^{\circ}\)C represents an extremely accurate estimation, the difference in these temperatures is accounted for by differences in material selection, for example compass-1's bottom face was assumed to be 70% solar cells and 30% aluminum whereas this CubeSat's bottom face was purely 6061-T6 aluminum. The difference in absorptivity of these materials will account for this difference, as more flux enters in the higher absorptivity Solar and Black paint configuration causing a relative temperature increase compared to aluminum. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cube Surface Number** & **Emissivity (\(e\))** & **Absorptivity (\(\alpha\))** &
\begin{tabular}{c} **Material** \\ **Composition** \\ \end{tabular} \\ \hline \(1\) & & & \\ (100 mm \(\times\) 300 mm) & 0.62 & 0.76 & 70\% Solar Cell, 30\% Aluminum \\ Sun-facing & & & 30\% Aluminum \\ \hline \(2\) & & & \\ (100 mm \(\times\) 300 mm) & 0.654 & 0.7927 & 70\% Black Paint, 30\% Aluminum \\ Exposed to Background Space, 3 K radiation & & & \\ \hline \(3\) & & & \\ (100 mm \(\times\) 300 mm) & 0.08 & 0.379 & 100\% Aluminum \\ Earth-facing, Exposed to IR and Albedo & & & \\ \hline \(4\) & & & \\ (100 mm \(\times\) 300 mm) & 0.654 & 0.7927 & 70\% Black Paint, 30\% Aluminum \\ Exposed to Background Space, 3 K radiation & & & 30\% Aluminum \\ \hline \(5\) & & & \\ (100 mm \(\times\) 100 mm) & 0.654 & 0.7927 & 70\% Black Paint, 30\% Aluminum \\ Antenna Face & & & 30\% Aluminum \\ \hline \(6\) & & & \\ (100 mm \(\times\) 100 mm) & 0.08 & 0.379 & 100\% Aluminum \\ Thruster Exhaust & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 29: Material Selection for the CubeSat Faces. Alternative material choices can be made for the structure, for example changing the background space faces to aluminum to observe the impact upon the hot-case temperature. For example, converting the background space-facing surfaces with aluminum instead of 30% aluminum and 70% black paint yields a maximum hot-case steady-state temperature of 500.682 K (227.682 \({}^{\circ}\)C) which would lead to critical failure of the CubeSat components. This highlights the critical importance of material selection for the external CubeSat faces.