## 6 Thermal System Design
### Defining the Thermal Load
\[\frac{t_{E}}{t_{satellite}}=\frac{2*\alpha}{360^{\circ}}\]  
where \(\alpha\) equals:  
Figure 47:0  
Therefore \(\alpha\):  
Figure 47:1Figure 47:2 Figure 47:3 Figure 47:4  
During the CubeSat's 96 min 30 s orbit, approximately 35 min and 38 s will be spent in the shade cooling due to net heat dissipation as the CubeSat will only be exposed to a very small amount of radiation as no direct sunlight will contact the body and the body would attempt to reach thermal equilibrium with the 3 K background space temperature. However, for the remaining 60 min and 52 s the CubeSat will be exposed to intense solar radiation which will raise component temperatures potentially causing mission failure. It is vital to select materials and develop temperature regulation system to prevent this from occurring. Heating of the CubeSat will occur through 3 thermodynamic transfer mechanisms, conduction, convection, and radiation. Convection can be classed as negligible for this system, as the atmosphere is extremely thin meaning very little particle mass is available to drive this process. Per the Zeroth Law, energy will always flow from a region of high temperature to a region of low temperature to reach equilibrium. This heat transfer is expressed mathematically in Fourier's Law as shown below:  
Figure 47:5  
To make Fourier's Law more suited to its application, the geometrical and material properties of the CubeSat can be factored into the equation by replacing \(\dot{Q}_{cond}\) with the resistance factor \(R_{th,\ cond}\). This gives the following equation:  
Figure 47:6  
Using this equation, Fourier's Law can be written as:  
Figure 47:7  
Heat transfer due to Radiation is extremely significant in space craft design. No medium (i.e., atmosphere) is required for this process to occur as energy is transferred in the form of electromagnetic radiation. A body which has a non-zero temperature will always emit radiation, the magnitude will depend upon the bodies geometry, material composition, relative position to other thermally heated bodies and the bodies surface temperature. The following equation numerically  
Figure 47:8 Shadow “eclipse zone” for Sun-synchronous Orbit [10]. analyses the significance of these stated factors to allow the rate of Radiation emission from a body to be calculated:  
Figure 47:9  
This incoming radiation can be absorbed, reflected, and transmitted. All energy must be conserved in any interaction therefore:  
\[t_{E}=\frac{1}{180^{\circ}}*\cos^{-1}(\frac{\left(H^{2}+2*R*H\right)^{0.5}}{ \left(R+H\right)*\cos(\beta)}) \tag{35}\]0  
Most CubeSat bodies used on CubeSats, for example the Compass-1 craft, are opaque to thermal radiation, therefore transmission can be neglected. \[t_{E}=\frac{1}{180^{\circ}}*\cos^{-1}(\frac{\left(H^{2}+2*R*H\right)^{0.5}}{ \left(R+H\right)*\cos(\beta)}) \tag{35}\]1  
To achieve a thermal balance for the CubeSat to prevent net heating and failure of components, the amount of incoming heat must equal the amount of heat exiting the CubeSat system. Radiation is the most significant source of heat incoming to the body, and radiation is the main driving force causing heat exhaustion from the system.