# Experimental Identification of Characteristic Curves of
## 3 Mathematical Method and Empirical Data Adquisition
### Experimental Data Acquisition Setups  
Two types of experiments are required to capture the characteristic and polytropic curves. Both use the same setup with different configurations. Figure 2 shows the experimental schematic, and Figure 3 shows the laboratory setup used to obtain data for fitting to the curves explained in Section 3.1.  
Closing valve \(VI\) starts both experiments. The flow meter \(DP\), a differential manometer, measures the primary flow rate and has the same error along the setups of \(-1.10\times 10^{-4}\) kg s\({}^{-1}\) (1 %). \(LRN\), a long radius nozzle, measures the secondary flow rate of the secondary inlet. They were both constructed as instructed in ISO 5167-2:2022 [33]. Pressure sensor \(P1\) measures the primary inlet pressure and has the same error along the setups of 0.0175 bar (0.25%). A vacuum pressure sensor \(P2\) measures the tank pressure, which ranges from 1 bar(a) to 0.2 bar(a). The first experiment captures the characteristic curve. The two-way flow restrictor valve \(V2\) is closed incrementally and flow rates are measured with the flow meter, \(LRN\). The second experiment captures the evolution of secondary pressure over time with the valve \(V2\) closed, which prevents atmospheric air from entering.  
In this study, two different sensor approaches are used for the experiments, and up to four setups are used to capture the required data, as explained in Section 3.1. The first approach is used in setups 1, 2 and 3. It uses hydrostatic liquid column manometer sensors. The second approach was only used in setup 4. It uses electronic WIKA transducers (A-10) connected to a Keysight DAQ970A/DAQ973A, a Data Acquisition System (DAS), which is shown in Figure 4.  
Figure 2: Schematic of the setups used for experiments. \(V1\) is a closing valve, \(V2\) is a two-way flow restrictor valve, \(DP\) is the differential flow meter, \(P1\) and \(P2\) are pressure sensors, \(LRN\) is the long radius nozzle flowmeter, and \(V\) is the volume used in the research.  
For the first approach, the data were taken manually, which had an uncalculated impact on the exact error, and the chronometer, not shown in Figure 2, is used to capture time. The chronometer is already incorporated into the DAS system, the setup 4. _P2_ is a heavier-density fluid manometer. _LRN_ uses a manometer to capture the flow rate and its design was different along the setups as detailed in the list.  
* Setup 1: Used a long-radius nozzle of \(\diameter\) 10.5 mm, shown in Figure 5. Water is used.
* Setup 2: Used a long-radius nozzle of \(\diameter\) 5.15 mm, shown in Figure 5. Water is used.
* Setup 3: The same as in setup 2. Alcohol is used, with \(\rho_{r}=0.7918\).
* Setup 4: WIKA A-10 transducer sensors are used for _P1_, _LRN_, and _P2_. All are connected to the DAS, as shown in Figure 4. _LRN_ used a long-radius nozzle of \(\diameter\)10.5 mm.  
Table 1 outlines the errors introduced per setup and sensor devices. The implementation of the multiplexer enabled simultaneous data capture across multiple instances, resulting in greater accuracy. However, quantifying the exact improvement remains a challenge. Although the electronic device of the WIKA transducer provided slightly less precision, it facilitated the capture of data points simultaneously, enhancing overall data acquisition efficiency.  
Figure 4: Electronic circuit used to connect the WIKA sensors to the multiplexer.  
Figure 3: Laboratory view of the experimental setup. Acronyms are explained in the caption of Figure 2.