## 3 Results  
In Figure 13, the resulting Hampson-Linde cryogenic cooler model diagram is presented. The schematic from this figure is linked to the Simulink/Simscape model from Figure 14, where all sections were highlighted.  
Figure 12: Nitrogen density as a function of pressure and temperature.  
Figure 13: Hampson-Linde process block diagram.  
The following sections can be distinguished from Figure 14:  
1. Pressurized gas heat transfer;
2. External heat exchange;
3. External heat exchange;
4. Heat exchanger heat transfer;
5. Infinite resistance was used to set the bias (initial point of simulation) of temperature to the ambient temperature;
6. Decompressed gas heat transfer;
7. Decompression at the Joule-Thomson valve.  
The previous part of the model was used to predict the temperature change, while the part from Figure 15 modeled the behavior of decompression and other prototypes previously implemented during experimental stages.  
As for the optimization criteria, the heuristic optimization was run to superimpose the model's output characteristic with a reference signal, an asymptotic signal which decreased gradually to a target value. The section used in determining the objective function value through the absolute value of an integral is presented below in Figure 16, highlighted in green.  
The output temperature characteristic is presented below in Figure 17. The chart shows an asymptotic decrease in temperature at the Joule-Thomson valve, which corresponded to a heat transfer model based on differential equations. Due to the absence of data from Figure 12, the linear lookup table extrapolation caused the last illustrated characteristic to vary non-linearly.  
Figure 14: Simulink/Simscape model of the Joule-Thomson-based cryogenic cooler process. Three sections of discrete heat exchanger thermal transfer can be distinguished, each containing a thermal capacitance. All corresponding sections are explained below.  
Figure 16: Optimization criteria.  
Figure 15: Density mapping and decompression behavior. All outputs were filtered using first order filters for further smoothing of characteristics, having the first filtered output, “Temp_K”, corresponding to the “Temp_Out” tag.  
The superconductive reference compound was Y-Ba-Cu-O, having a critical temperature of 92 K [19]. The previously presented result showed that the superconductive state for this category II superconductor could be obtained since its temperature was below the previously mentioned value.  
All values resulting from the particle swarm optimization algorithm are listed below in Table 2.  
Comparing the obtained characteristics and results with studies mentioned in [20], even with the limitation, which will be further discussed within the next section, results were comparable to cryogenic coolers using all or any Brayton cycle, Claude and Kapitza cycle. Figures 17 and 18 show a characteristic that opposed external heat transfer (non-adiabatic gas expansion) and an adiabatic gas expansion temperature characteristic.  
Following the description of the prototype model and results after PSO, consistent results were obtained from a heat transfer point of view, for example, high thermal resistor values for external heat exchanger and cooling dynamics corresponding to existing studies. However, several limitations occurred using these methods, covered in the next section.  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Nitrogen thermal resistance & 8.0540612 K/W \\ Heat exchanger internal thermal capacitance & 7.1426119 J/K (428.556714 J/K) 1  \\ Heat Exchanger internal thermal resistance & 53.550687 K/W \\ External heat exchange thermal resistance & 9135.2208 K/W \\ Compressor volumetric flow & 3682.3641 L/h \\ Square of compensation density & 33.2700 (kg/m\({}^{3}\))2  \\ \hline \end{tabular}
\end{table}
Table 2: Cryogenic cooler model parameters.  
Figure 17: Cooling characteristic of the model prototype at expander outlet.