# Experimental Identification of Characteristic Curves of
## 5 Discussion  
The aim of this research was to analytically predict the TET by creating mathematical models and using the characteristic \(\mu_{s}(p)\) and polytropic \(n(p)\) curves. The fitting of  
Figure 11: Time deviation in seconds between experimental and predicted TET for setup 5 for a tank with a volume of 0.5.  
Figure 9: Percentage deviation or error between experimental and predicted evacuation time for setups 1, 2, 3 and 4, for a tank with a volume of 0.5.  
experimental data to polynomial curves is used as a method to empirically predict TET, as explained in Section 3.1. The R-squared values, which reflect the accuracy of the fit, are directly related to the refinement of the polynomial coefficients. Figure 6 presents the R-squared values for setups 1, 2 and 3. Setup 1, despite having less data, shows a strong fit with an R-squared value rounded to 0.99805. In contrast, setup 2, which includes more data but has an ambiguous bP, displays greater variation and is the least consistent, with a rounded R-squared of 0.99794. Setup 3, where the bP is evident, demonstrates a better fit, yielding a rounded R-squared of 0.99876. Setup 4 achieves the highest data precision, as shown in Figure 8, where the R-squared value is rounded to 1.0, indicating an almost perfect fit. Table 3 presents the refinement of the polynomial coefficients and the appearance of the bP in setups 3 and 4. Table 4 highlights the significance of the polynomial coefficients and R-squared values, showing deviations in the PTET ranging from 7.6% to 1.4%. Incorporation of bP into \(m_{s}(p)\) and in the mathematical model has a significant impact on the PTET by almost halving it: in setup 2, the PTET against experimental TET deviation is 5.3% whereas in setup 3, it is 2.9%.  
Polytropic \(n(p)\) curve data from setups 1, 2 and 3 were fitted to a first-order polynomial due to the absence of a second-order trend (see Figure 7). In contrast, the setup 4 exhibited a second-order polynomial trend in its polytropic data (see Figure 8), which contributed to a better fit. The second-order polynomial trend has a clear impact on PTET, as shown in Figure 9. In setup 3, the deviation between PTET and experimental TET is 2.9%, whereas in setup 4 halves this deviation to 1.4%. The PTET deviation of 1.4% (2.6 s in absolute terms) demonstrates that the model successfully matches the TET, confirming its validity.  
After model validation, a new optimized ejector, setup 5, was tested using data from an author's previous work [5]. Figure 10 shows a setup 5 being the best with an overall performance improvement by reducing the minimum secondary pressure achieved and lowering the PTET by 4% (8 s). The \(n(p)\) curve coefficients obtained in setup 4 are used to calculate the PTET for setup 5 due to the similarity of \(m_{s}(p)\). Figure 11 indicates that while PTET is lower, evacuation time is higher at mid-range working pressures, peaking at 2 s at 0.5 bar(a). Nonetheless, this is not a significant issue as the main objective of this research is to improve PTET relative to experimental TET.  
Finally, the PTET of the experimental setup 4 shows a deviation error of 1.4%, and the tested setup 5 showed an increase of performance by about 4%. This difference exceeds the experimental error of1.4%, indicating its significance. These results highlight the importance of the mathematical models developed, which can be used in the future to compare ejectors and reduce the need for experimental data.