# Implementation of a Cryogenic Facility for Space Debris Analysis
## 3 Prototype Design
### Thermal Modeling
There are two solutions for \(a\), given by  
\[\text{a}=\ \frac{k_{0}\big{[}C_{p}R_{t}+C_{sh}(R_{c}+R_{t})\big{]}+C_{sh}+C_{p}}{2C_ {p}C_{sh}R_{c}(k_{0}R_{t}+1)}\pm\frac{\sqrt{\big{[}C_{p}R_{t}+C_{sh}(R_{c}+R_{ t})\big{]}+C_{sh}+C_{p})^{2}-4k_{0}C_{p}C_{sl}R_{c}(k_{0}R_{t}+1)}}{2C_{p}C_{sh}R_{c}(k_{0 }R_{t}+1)} \tag{8}\]  
In Equation (8), the constants \(R_{t}\) and \(R_{c}\) were calculated as the inverse of thermal conductivity and \(R_{t}\) was used as the length of the section of copper straps. \(R_{c}\) (surface-to-surface contact coefficient) was assumed to be limited by the thermal conductivity of the copper part of the pistons. \(C_{sh}\) was determined by the heat capacity of the sample holder, made entirely of aluminum. \(C_{p}\) was calculated by adding the heat capacity of the stainless steel and copper of the piston. Table 5 shows the computed constants. Replacing the constants of Table 5 in Equation (8) provides two solutions. The solutions are as follows: \(a\)\(\rightarrow\)\(\{6.979\ \times\ 10^{-1}\), \(1.396\ \times\ 10^{-4}\}\), meaning the solutions are a linear combination of two exponential functions over time. This result was expected since there are two mechanisms involved in the cooling process, one from the thermal mass of the pistons, a very quick process, and the other from the conduction on the straps. The lumped model provides a first approach to system dimensioning that should be complemented with a more accurate model (i.e., the finite element method). * Finite element method  
The finite element method (FEM) is the dominant discretization technique in structural mechanics. The basic concept in the physical interpretation of the FEM is the subdivision of the mathematical model into disjoint (non-overlapping) components of simple geometry, called finite elements, or elements in short. The response of each element is expressed in terms of a finite number of degrees of freedom characterized as the value of an unknown function, or functions, at a set of nodal points. The response of the mathematical model is then considered to be approximated by that of the discrete model obtained by connecting or assembling the collection of all elements. All analyses were performed using the Simulation solution implemented using the SolidWorks framework. For the TVC system, the main concern was thermal conductivity, either from the hot baseplate to the flange, dissipating the needed power to raise the baseplate temperatures, or from the flange to the cold baseplate. These simulations are referred to as static analyses. A dynamic simulation was also accomplished in order to assess the time needed for a full cooldown of the system from 150 \({}^{\circ}\)C to \(-\)150 \({}^{\circ}\)C.  
* FEM, static analyses  
A static simulation was performed on fiberglass epoxies; these parts are the support structure of the cold pistons and sample holder. Therefore, these parts interface with extreme cold/hot parts (\(\pm\) 150 \({}^{\circ}\)C) at one extreme and the vacuum chamber at the other extreme (ambient temperature). To assess the thermal isolation performance of these parts, static simulations in the extreme temperature range were considered as a worst-case scenario. * Outer epoxy \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline \(C_{p}\) & 832.500 \\ \(C_{sh}\) & 291.350 \\ \(R_{t}\) & 4.867 \\ \(R_{c}\) & \(6.64^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters used in the lumped model. The outer epoxy experiences both high and low temperatures at one end, and the other is attached to the flange, which gives a good thermal connection, so it is always at room temperature. A computer-aided design (CAD) model was 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, as shown in Figure 7.  
A mesh is a discretization of the CAD model.