# Experimental Identification of Characteristic Curves of
## 3 Mathematical Method and Empirical Data Adquisition  
### Model for TET Prediction  
As stated in the Section 1, TET is a key metric for comparing the performance of vacuum ejectors. For predicting the TET, the PTET, it is necessary to establish a relation between the decrease in pressure with time (\(dp/dt\)) and both curves: the characteristic or secondary flow \(\dot{m}_{s}(p)\) and the polytropic coefficient evolution \(n(p)\). Both curves depend on the working pressure of the ejector [6]. The relation can be found by taking the derivative of the definition of density and reorganizing terms:  
\[\dot{m}_{s}(p)=V\cdot\frac{d}{dt}(\rho) \tag{1}\]  
where the volume (\(V\)), considered constant at 0.5 m\({}^{3}\) in this work. Rearranging the polytropic equation, we obtain the following:  
\[\frac{p}{\rho^{n}}=k\rightarrow\rho=\frac{p^{1/n}}{k^{1/n}} \tag{2}\]where the \(k\) is a generic constant and the \(n\) is the polytropic coefficient. Combining Equation (2) with the perfect gas equation yields the following:  
\[\frac{1}{rT}=\frac{p^{\frac{1-n}{n}}}{k^{1/n}} \tag{3}\]  
where the \(r\) is the specific gas constant for dry air, which equals to \(287\,\mathrm{J}\,\mathrm{kg}^{-1}\,\mathrm{K}^{-1}\) and the temperature \(T\) is approximated as constant as 298 K. Combining Equation (1) and Equation (2) results in the following:  
\[\dot{m}_{\mathrm{s}}(p)=V\frac{d}{dt}\left(\frac{p^{1/n}}{k^{1/n}}\right)= \frac{V}{n}\frac{p^{\frac{1-n}{n}}}{k^{1/n}}\frac{dp}{dt} \tag{4}\]  
Introducing Equation (3) in Equation (4) can be used to derive the following differential equation:  
\[\frac{-rT}{V}dt=\frac{dp}{\dot{m}_{\mathrm{s}}(p)\cdot n(p)} \tag{5}\]  
The characteristic curve, \(\dot{m}_{\mathrm{s}}(p)\), is the relation between the secondary flow and the working pressure of an ejector. This relation is linear, according to [1,6], and it is parameterized by the first-order polynomial equation:  
\[\dot{m}_{\mathrm{s}}(p)=a+b\cdot p \tag{6}\]  
where \(a\) is the offset and \(b\) is the multiplier. The improved experimental setups refined the data and exposed the transition or breakpoint (bP), as divulged by [6,15,16], where \(\mathrm{bP}_{p}\) is the value of the bP's pressure and \(\mathrm{bP}_{m}\) is the bP's flow rate. Beyond bP, the behavior of the secondary flow changes, and the parameters of the linear equation adjust, which are captured piece-wise. This research used a machine learning for piece-wise linear regression modeling with automatic breakpoint detection [6,31]. \[ \begin{cases}\dot{m}_{\mathrm{s}}(p)=a+b\cdot p&\text{if }\mathrm{bP}_{p}\leq p \\ \dot{m}_{\mathrm{s}}(p)=c+d\cdot p&\text{if }\mathrm{bP}_{p}>p\end{cases}  \tag{7}\]  
The polytropic curve, \(n(p)\), relates the evolution of the polytropic coefficient to the working pressure of the ejector or the secondary pressure. Authors past work [6] showed that near atmospheric pressure, the polytropic coefficient is adiabatic (\(n=1.4\)), and near the minimum working pressure of the ejector it is isothermal (\(n=1\)). Depending on the data trend, the fitting curve is a first-order polynomial  
\[n(p)=e+f\cdot p+g\cdot p^{2} \tag{8}\]  
or a second-order polynomial; its terms appear in blue. The evacuation time and PTET are predicted using Equation (5), with the empirical curve formula, Equation (6) or Equation (7) for \(\dot{m}_{\mathrm{s}}(p)\), and Equation (8) for \(n(p)\), through a piecewise integration framework. The PTET represents the time required to reach the ejector's minimum working pressure, which is \(p_{\mathrm{min}}=20\mathrm{kPa}\). \[\mathrm{PTET}=\int_{0}^{PTET}dt=\frac{-V}{rT}[I_{1}+I_{2}] \tag{9}\]  
For each setup, \(I_{1}\) and \(I_{2}\) are explicitly written as follows.