# Thermal Optimization Design for a Small Flat-Panel Synthetic Aperture Radar Satellite
## 3 Thermal Design Strategy
### Methods to Reduce Heaters  
The dimensions of the support plate are 0.78 m \(\times\) 2.77 m, and it needs to be maintained at 20 \({}^{\circ}\)C. A dense arrangement of heaters and thermostats are capable of achieving a temperature control of 20 \({}^{\circ}\)C; however, it is not an optimized design and will inevitably result in a waste of satellite resources. The improved method is presented in Figure 3. The connection points between the support plate and the SAR antenna, as well as other panels, are located at its circumferential positions, resulting in significant heat leakage at these locations. Twelve groups of heaters are arranged in the circumferential direction of the support plate, with a controlled temperature of 20 \({}^{\circ}\)C, leaving the center area of the support plate blank (without heaters). Assuming that both the antenna and the platform are in a low-temperature state, we take the longitudinal section of the support plate and simplify it into an infinite flat plate model; that is, assuming there is no temperature gradient in the thickness and width directions within the analysis region, it can be regarded as a one-dimensional model with a single-direction temperature gradient. The temperature at a distance of \(x\) from the edge of the heater is defined as \(T_{x}\), so the heat conduction differential equation is expressed as follows [28].  
\[\frac{d^{2}T_{x}}{dx^{2}}=\frac{[K_{equ}(T-T_{outu})+K_{eyd}(T-T_{outd})]}{ \delta_{p}\lambda_{p}} \tag{1}\]  
\[K_{eq}=\varepsilon_{p}\sigma(T_{0}^{2}+T_{out}^{2})(T_{0}+T_{out}) \tag{2}\]  
where \(K_{eq}\) represents the equivalent heat transfer coefficient of the plate surface, \(T_{out}\) is the ambient temperature outside the support plate, \(\delta\) signifies the thickness, \(\lambda\) stands for thermal conductivity, \(\varepsilon\) indicates infrared emissivity, \(\sigma\) represents the Stefan-Boltzmann constant, the subscript \(u\) designates the upper surface of the flat plate, while the subscript \(d\) designates the lower surface, the subscript \(p\) designates the support plate, and \(T_{0}\) represents the temperature of heaters. The boundary conditions of this model can be described as follows.  
\[x=0,T_{x}=T_{0};x=0.5d_{\text{max}},\frac{dT}{dx}=0 \tag{3}\]  
Figure 2: Heat transfer pathway. In which dotted arrows stands for the direction of heat transfer.  
In the formula, \(d_{\max}\) represents the maximum distance left blank. The excess temperature \(\theta_{x}\) is defined as follows.  
\[\theta=T_{x}-\frac{K_{equ}T_{outu}+K_{eqd}T_{outd}}{K_{equ}+K_{eqd}} \tag{4}\]  
Formula (1) can be rewritten as follows:  
\[\frac{d^{2}\theta}{dx^{2}}=n^{2}\theta_{x} \tag{5}\]  
\[n=\sqrt{\frac{K_{equ}+K_{eqd}}{\delta_{p}\lambda_{p}}} \tag{6}\]  
\(n\) is a constant. The boundary conditions of the model are described as follows.  
\[x=0,\theta_{x}=\theta_{0}=T_{0}-\frac{K_{equ}T_{outu}+K_{eqd}T_{outd}}{K_{ equ}+K_{eqd}};x=0.5d_{\max},\frac{dT}{dx}=0 \tag{7}\]  
Solving differential equations yields the following results.  
\[\theta_{x}=\frac{\theta_{0}}{ch(nx)} \tag{8}\]  
By utilizing the aforementioned analytical method, we can determine the maximum distance \(d_{\max}\) for leaving blank space when the lowest temperature at the center of the support plate exceeds 15 \({}^{\circ}\)C. This approach reduces the heaters layout area by 30%, indicating that more heaters and thermostats can be allocated towards ensuring the temperature of other components, such as batteries.