Appendix C: Thermal Loads for Lilium-1 CubeSat (Tze Yee)
Fig C-1. Isometric View of Lilium-1 CAD Model
For the approximation of deep space as a 4K heat sink, it is set on all exposed surfaces on the CAD model. However, different emissivity values are applied.
External surfaces facing outwards are given an emissivity value of 0.88, representing black paint such that these surfaces are able to absorb heat quickly during the Sunlit phase. The internal surfaces are given an emissivity value of 0.03, as their coating ensures that there is minimal heat transfer in and out of the material, keeping internal components within a relatively constant temperature.
Thermal loads in the + and - X faces only receive Solar Constant from the Sun during the first or second half of the Sunlit phase respectively. For example, the +X face’s incident area facing the sun decreases from maximum to 0 within the first quarter of the orbital period, before emerging in the last quarter at the end of the Eclipse phase.
Fig C-2. Graph of Solar Constant received by the +X face as a fraction of incident Solar Constant (1361 W/m2)
Fig C-3. Graph of Solar Constant received by the -X face as a fraction of incident Solar Constant (1361 W/m2)
For the +/- Y faces, there is no heat load applied to it as they are facing neither the Earth nor the Sun, which are assumed to be point sources. Hence, the only thermal load applied is the radiative heat sink to deep space with a temperature of 4K.
On the +Z face, heat load is a constant (within Sunlit and Eclipse phase respectively) as there is no change in the incident angle of the CubeSat face with respect to Earth. However, the -Z face’s incident angle towards the sun changes according to orbital position.
Fig C-4. Graph of Earth Albedo + IR Heat Flux on +Z face, showing a constant load of 628 W/m2 during Sunlit phase and 208 W/m2 during Eclipse
Unique to Lilium-1, given its extended solar panels angled at 45 degrees to the axes, its incident area must be taken into account when applying heat loads. For example, the solar panels facing Earth in the +Z direction would have the same heat load but multiplied with a factor of sin(45°) applied on the same area.
Fig C-5. Heat Load on the angled Solar Panels in +Z
On the contrary, the -Z has exposure to the sun only during the first half of the orbit, where the extent of exposure can be approximated to a Sin Curve that changes according to the orbital position of the CubeSat.
Fig C-6. Graph of incident Solar Constant on the -Z Face as a fraction of incident Solar Constant (1361 W/m2). A factor of sin(45°) is added to the heat load applied on the solar panels in the -Z face.
One limitation of the plotting of the sine or cosine curve is that I was unable to directly input a sin curve into the thermal simulation. Hence, it was approximated using 10 points for each quarter of a period (1440s).
| Time in s | Sin (x) | Cos (x) |
|---|---|---|
| 0 | 0 | 1 |
| 144 | 0.156434465 | 0.9876883406 |
| 288 | 0.3090169944 | 0.9510565163 |
| 432 | 0.4539904997 | 0.8910065242 |
| 576 | 0.5877852523 | 0.8090169944 |
| 720 | 0.7071067812 | 0.7071067812 |
| 864 | 0.8090169944 | 0.5877852523 |
| 1008 | 0.8910065242 | 0.4539904997 |
| 1152 | 0.9510565163 | 0.3090169944 |
| 1296 | 0.9876883406 | 0.156434465 |
| 1440 | 1 | 0 |
| 1584 | 0.9876883406 | 0.156434465 |
| 1728 | 0.9510565163 | 0.3090169944 |
| 1872 | 0.8910065242 | 0.4539904997 |
| 2016 | 0.8090169944 | 0.5877852523 |
| 2160 | 0.7071067812 | 0.7071067812 |
| 2304 | 0.5877852523 | 0.8090169944 |
| 2448 | 0.4539904997 | 0.8910065242 |
| 2592 | 0.3090169944 | 0.9510565163 |
| 2736 | 0.156434465 | 0.9876883406 |
| 2880 | 0 | 1 |
| 3024 | 0.156434465 | 0.9876883406 |
| 3168 | 0.3090169944 | 0.9510565163 |
| 3312 | 0.4539904997 | 0.8910065242 |
| 3456 | 0.5877852523 | 0.8090169944 |
| 3600 | 0.7071067812 | 0.7071067812 |
| 3744 | 0.8090169944 | 0.5877852523 |
| 3888 | 0.8910065242 | 0.4539904997 |
| 4032 | 0.9510565163 | 0.3090169944 |
| 4176 | 0.9876883406 | 0.156434465 |
| 4320 | 1 | 0 |
| 4464 | 0.9876883406 | 0.156434465 |
| 4608 | 0.9510565163 | 0.3090169944 |
| 4752 | 0.8910065242 | 0.4539904997 |
| 4896 | 0.8090169944 | 0.5877852523 |
| 5040 | 0.7071067812 | 0.7071067812 |
| 5184 | 0.5877852523 | 0.8090169944 |
| 5281 | 0.4990550155 | 0.8665703038 |
| 5328 | 0.4539904997 | 0.8910065242 |
| 5472 | 0.3090169944 | 0.9510565163 |
| 5616 | 0.156434465 | 0.9876883406 |
| 5760 | 0 | 1 |
Table C-1. Sin and Cos values used at each stage of the orbit in terms of time in seconds