# Experimental Identification of Characteristic Curves of
## 4 Results  
### Experimental Results  
As explained in Section 3.1, two types of experiments were conducted to collect the data. The first was used to obtain the characteristic curve \(m_{s}(p)\), while the second was used to obtain the polytropic curve \(n(p)\). Multiple setups were used to gradually improve the quality of the data, as explained in Section 3.2. These data were used to determine the fitting polynomial coefficients of \(m_{s}(p)\) and \(n(p)\), both of which depend on the secondary pressure.  
Figure 6 shows the refinement of the data used to capture the characteristic curve of \(m_{s}\) with setups 1, 2 and 3. Setup 2 uses the data obtained in setup 1 (in blue) and adds the newly captured data (in red). The same procedure for setup 3, which used the data captured by setup 1 and 2 (in blue) and added the new data (in red). Setup 3, described in Section 2, shows the evidence of the bP and was used in the piece-wise fitting equation.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Setup** & **a** & **b** & **c** & **d** & \(\mathbf{bP_{p}}\) & \(\mathbf{bP_{m}}\) \\ \hline
5 & \(-2.35\times 10^{-3}\) & \(1.31\times 10^{-7}\) & \(-1.74\times 10^{-3}\) & \(1.05\times 10^{-7}\) & \(2.38\times 10^{4}\) & \(7.51\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Coefficients used for setup 5 [5].  
Figure 5: Flowmeter _LRN_: (**a**) Used in setup 1 and 4; (**b**) Used in setup 2 and 3; (**c**) Drawing of (**b**).  
Figure 7 shows the polytropic curve \(n(p)\) and the corresponding fitting equations for each setup. The curve transitions from approximately 1.4 (adiabatic behavior) at the start of evacuation (secondary pressure around 1 bar) toward approximately 1 (isothermal behavior), as outlined in [6]. The data captured by setups 1, 2 and 3 does not show a second-order polynomial trend.  
Figure 8 shows on the left the characteristic curve \(\dot{m}_{s}(p)\) captured with setup 4. In the same graph, setup 4 and setup 5 are compared, which are taken from [5]. The data are presented along with its fitting curves, its equations, and the pressure breakpoint. On the right, there is the polytropic curve \(n(p)\) captured with setup 4 and its fitting curve. Capturing this curve using setup 4 shows a perfect second-order polynomial trend.  
Table 3 shows the progression of the coefficients for the characteristic curve \(\dot{m}_{s}(p)\): \(a\), \(b\), and, if used, \(c\) and \(d\) as a piece-wise curve. The coefficients for the polytropic curve \(n(p)\) use the parameters \(e\), \(f\), and, if used, \(g\). The \(n\_\text{exp1}\) shows the number of data used to obtain the coefficients [6].  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Setup** & **a** & **b** & **\(\mathbf{\epsilon}\)** & **d** & **e** & **f** & **g** & **bP\({}_{P}\)** & **bP\({}_{m}\)** & **\(n\_\text{exp1}\)** \\ \hline
1 & \(-1.83\times 10^{-3}\) & \(1.05\times 10^{-7}\) & – & – & 0.854 & \(5.15\times 10^{-6}\) & – & – & – & 12 \\
2 & \(-1.59\times 10^{-3}\) & \(1.02\times 10^{-7}\) & – & – & 0.954 & \(8.69\times 10^{-6}\) & – & – & – & 21 \\
3 & \(-3.88\times 10^{-3}\) & \(2.00\times 10^{-7}\) & \(-1.58\times 10^{-3}\) & \(1.02\times 10^{-7}\) & \(0.690\) & \(7.21\times 10^{-6}\) & – & \(2.34\times 10^{4}\) & \(8.05\times 10^{-4}\) & 30 \\
4 & \(-2.79\times 10^{-3}\) & \(1.47\times 10^{-7}\) & \(-1.76\times 10^{-3}\) & \(1.04\times 10^{-7}\) & \(1.080\) & \(-5.21\times 10^{-6}\) & \(8.17\times 10^{-11}\) & \(2.41\times 10^{4}\) & \(7.50\times 10^{-4}\) & 95 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summarized coefficients per setup.  
Figure 6: Characteristic curve, \(\dot{m}_{s}(p)\), setups 1, 2 and 3. bP was introduced in setup 3.  
Figure 7: Polytropic coefficient evolution, \(n(p)\), for setups 1, 2 and 3.