# Implementation of a Cryogenic Facility for Space Debris Analysis
## 3 Prototype Design
### Thermal Modeling  
In this subsection, the thermal analysis of the chosen architecture, using both an analytical lumped circuit and the numerical finite element method, is presented. * Approximated lumped network analysis  
A simplified scheme of the thermal system is shown in Figure 5. As a rough approximation, the thermal contact resistance between the sample holder and the pistons is not considered; however, the heat flux is defined by the area and length of the copper part of the pistons. The thermal straps also add resistance to the heat flux. As a convention, heat flows from the sample holder (at a higher temperature) to the cryocooler (at a lower temperature). Here, \(V_{c}\) is the temperature in the cryocooler at the generator terminals \(G\) (Figure 6), \(R_{t}\) is the sum of the internal resistance of the cryocooler and the thermal conduction of the copper straps, \(C_{p}\) is the heat capacity of the pistons, \(R_{c}\) is the contact resistance between the pistons and the sample holder, \(C_{sh}\) is the heat capacity of the sample holder, \(i_{sh}\) is the heat flow between the sample holder and the pistons, and \(i_{c}\) is the heat flow between the pistons and the cryocooler. Figure 4: Variation in the thermal conductivity of the ISOVAL epoxy [12]. \begin{table}
\begin{tabular}{c c} \hline
**Physical Property** & **Value** \\ \hline Density & 2000 kg/m\({}^{3}\) \\ Tensile strength & 374 MPa \\ Elastic modulus & 18.9 GPa \\ Thermal expansion & \(30\times 10^{-6}\)/K \\ Thermal conductivity & 0.36 W/m K \\ \hline \end{tabular}
\end{table} 
Table 4: Properties of the ISOVAL FR4-HF epoxy at 25 \({}^{\circ}\)C.  
An effective way to better understand the thermal problem is to use the analogy between thermal and electrical networks. The voltage (representing the temperature) at various nodes can be calculated, together with the current (representing the heat) flow between nodes. For transient analysis, electrical capacitances (associated with thermal capacitances) are added to account for the change in the internal energy of the body with time. Thermal resistances result from geometric dimensions, thermal properties of the materials, and heat transfer coefficients [13; 14]. The equivalent circuit diagram that relates the heat path from the sampler holder to the cryocooler is shown in Figure 6.  
Considering Figure 6, the modeling heat conduction is given by  
\[i_{\rm sh}=-C_{\rm sh}\frac{dV_{\rm sh}}{dt} \tag{1}\]  
\[V_{\rm sh}-V_{\rm p}=R_{\rm c}\;i_{\rm sh} \tag{2}\]  
\[i_{\rm sh}-i_{\rm c}=C_{\rm p}\;\frac{dV_{\rm p}}{dt} \tag{3}\]  
\[V_{\rm p}-V_{\rm c}=R_{\rm t}\;i_{\rm c} \tag{4}\]  
\[i_{\rm c}=k_{0}\;V_{\rm c}+k_{1} \tag{5}\]  
The particular solution is given by  
\[\stackrel{{\rightarrow}}{{X}}=\;\stackrel{{ \rightarrow}}{{C}}\stackrel{{\rightarrow}}{{a}}\stackrel{{ \rightarrow}}{{e}}-\stackrel{{\rightarrow}}{{a}} \tag{6}\]  
\[\left| \begin{array}{c}0\\ 0\\ 0\\ 0\\ k_{1}\end{array} \right|=\left| \begin{array}{cccc}0&0&C_{\rm sh}a&1&0\\ 0&-1&1&-R_{\rm c}&0\\ 0&-C_{\rm p}a&0&1&-1\\ -1&1&0&0&-R_{\rm t}\\ -k_{0}&0&0&0&1\end{array} \right|\left| \begin{array}{c}V_{\rm c}^{0}\\ V_{\rm p}^{0}\\ V_{\rm sh}^{0}\\ i_{\rm sh}^{0}\\ i_{\rm c}^{0}\end{array} \right| \tag{7}\]  
Figure 5: Simplified scheme of the thermal system for lumped network analysis. Figure 6: Electrical transmission line equivalent circuit diagram for modeling heat conduction properties; the physical variables are specified in their thermal equivalents. To determine the exponential argument, \(e^{-\mu t}\), the determinant must be set to zero.