# Implementation of a Cryogenic Facility for Space Debris Analysis
## 3 Prototype Design
### Thermal Modeling
Finite elements and nodes define the basic geometry of the physical structure being modeled, splitting the geometry into relatively small and simple geometric entities called finite elements. Each element in the model represents a discrete portion of the physical structure, which is represented by many interconnected elements. Shared nodes connect these elements to one another. Several mesh densities were evaluated, and the analysis results were checked to converge to a unique solution. The solution obtained from the numerical model is generally a good approximation of the solution of the physical problem being simulated. An important parameter in the assessment of mesh quality is aspect ratio checks [15]. Simulation accuracy is best achieved by a mesh with uniform-size elements whose edges are equal in length. Therefore, by definition, the aspect ratio of a perfect tetrahedral is 1.0. Table 6 summarizes the mesh used in the simulation. Since 89% of the elements have an aspect ratio below 3.0, the mesh is considered of good quality. In the simulations, only thermal conductivity was considered, which is a good approximation since there is no convection in vacuum and all thermal radiation from the wall of the chamber, at room temperature, is assumed to be shielded by the MLI blankets. \begin{table}
\begin{tabular}{c c} \hline \hline
**Mesh Information** & **Values** \\ \hline Mesh type & Standard mesh \\ Total nodes & 36,158 \\ Total elements & 18,379 \\ Percentage of elements with aspect ratio \(<\) 3 & 89 \\ Percentage of elements with aspect ratio \(>\) 10 & 0.0326 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Mesh used in the simulation. Figure 8: Representation of the mesh used in the simulation. Figure 7: 3D model used to simulate the applied thermal loads. On the baseplate, the thermal loads applied for this simulation are \(-150\)\({}^{\circ}\)C and \(150\)\({}^{\circ}\)C.  
The simulations are driven by the heat balance equation given by  
\[Q-\frac{\partial q_{x}}{\partial x}-\frac{\partial q_{y}}{\partial y}-\frac{ \partial q_{z}}{\partial z}=cp\frac{\partial T}{\partial t}, \tag{9}\]  
where \(q_{x}\), \(q_{y}\), and \(q_{z}\) are the heat flux in the \(x\), \(y\), and \(z\) directions, \(Q\) is the internal heat generated/absorbed by the body (volume), \(c\) is the specific heat, \(\rho\) is the mass density, and \(T\) is the temperature. [PERSON]'s law of heat conduction relates heat flux and temperature and is proportionality constant to \(k\), the thermal conductivity. For an orthotropic body (volume element), with the axes of orthotropy coinciding with the coordinate axes, Fourier heat conduction is given by  
\[\stackrel{{\rightarrow}}{{q}}\;=\;-k\overset{\rightarrow}{ \
abla}T \tag{10}\]  
Considering Equations (9) and (10), is given by  
\[Q=\;c\rho\frac{\partial T}{\partial t}-k\overset{\rightarrow}{\
abla}^{2}T \tag{11}\]  
Analysis of steady-state heat conduction problems involves setting the term \(\partial T/\partial t\) equal to zero and assuming \(Q\) to be independent of time. Taking this into account, in the first simulation, the baseplate is considered at 150 \({}^{\circ}\)C. As shown in Figure 8, the temperature applied on the left side is the flange temperature (room temperature, 25 \({}^{\circ}\)C), and on the right side, 150 \({}^{\circ}\)C is applied. Figure 9 shows the results of this simulation. The top plot shows the temperature gradient on the epoxy, and on the bottom the heat flux. The important result here is the total power transferred through the epoxy, which should be the minimum. The simulation shows 0.02 W being transferred. Since the model is the same, the same mesh was used to simulate it with a different temperature in the extremity; the result is as plotted below. Again, the top of Figure 10 shows the temperature gradient on the epoxy, and on the bottom the heat flux. The power transferred was 0.015 W.  
Figure 9: Thermal analysis of the outer epoxy support when the large clutch is at \(-150\)\({}^{\circ}\)C. (**a**) A temperature map and (**b**) the heat flux.