## 2 Materials and Methods
### Experimental Studies
\[\frac{dp_{1}}{dt}=-\frac{\kappa RT}{V_{1}}\bigg{(}(\mu A)_{ev}\sqrt{\kappa RT }\frac{p_{1}}{RT}\varphi_{max}a\frac{p_{1}-p_{2}}{ap_{1}-p_{2}}\bigg{)} \tag{9}\]  
\[\frac{dp_{2}}{dt}=\frac{\kappa RT}{V_{2}}\bigg{(}(\mu A)_{ev}\sqrt{\kappa RT} \frac{p_{1}}{RT}\varphi_{max}a\frac{p_{1}-p_{2}}{ap_{1}-p_{2}}-(\mu A)_{nozzle }\sqrt{\kappa RT}\frac{p_{2}}{RT}\varphi_{max}a\frac{p_{2}-p_{3}}{ap_{2}-p_{3 }}\bigg{)} \tag{10}\]  
\[\frac{dp_{3}}{dt}=\frac{\kappa RT}{V_{3}}\bigg{(}(\mu A)_{nozzle}\sqrt{\kappa RT }\frac{p_{2}}{RT}\varphi_{max}a\frac{p_{2}-p_{3}}{ap_{2}-p_{3}}\bigg{)} \tag{11}\]  
where \(\kappa\)--the adiabatic exponent; \((\mu A)_{ev}\)--the throughput of the electrovalve; \((\mu A)_{nozzle}\)--the throughput of the nozzle; \(R\)--the gas constant; \(\varphi_{\max}\)--the maximum value of the Saint-Venant-Wantzel theoretical function; and \(a=1.13\)--the Myetluk-Avtushko coefficient. According to Figure 7, a solenoid valve with a throughput of \((\mu A)_{ev}\) was used in the measurement system to start and stop the measurement. In the course of identifying the parameter \((\mu A)_{nozzle}\), only the recorded \(p_{2}\) pressure waveforms were used (the pressure transducer mounted directly downstream of the control solenoid valve) and \(p_{3}\) (11 on Figure 7). The determination of the parameter, \((\mu A)_{nozzle}\), was restricted to the numerical solution of Equation (11) with the implicit trapezoidal method with reverse differentiation. The determination of the throughput of the tested nozzle, \((\mu A)_{nozzle}\), was carried out from the method of least squares, Equation (12). The minimization of the squared deviations was carried out using the Nelder-Mead downhill simplex method [63; 64], by comparing the experimental pressure waveform, \(p_{3 cap}\), with the model waveform, \(p_{3 model}\), in concentrated volume, \(V_{3}\), calculated from the Equation (11). \[\sum_{i=1}^{n}\left(p_{3 exp}-p_{3 model}\right)^{2}\Rightarrow min \tag{12}\]  
Figure 8: Mathematical description of the stand—schematic. A qualitative assessment of the fit between the model run and the experimental run was carried out using the _MAPE_ (Mean Absolute Percentage Error) index, according to Equation (13). In addition, for each waveform, the minimum and maximum values of the fitting error were determined, \(\mathit{FPE}_{\text{min}}\) and \(\mathit{FPE}_{\text{max}}\), as well as the coefficient of determination, \(R^{2}\), corrected for the degrees of freedom. \[MAPE=\frac{1}{n}\sum_{i=1}^{n}\left|\frac{p_{3 exp}-p_{3 model}}{p_{3 exp}}\right| \tag{13}\]  
In the course of identifying the nozzle throughput, \((\mu A)_{\text{nozzle}}\), the area of stabilized flow through the local resistance, indicated by the vertical dashed lines in Figure 9a, was isolated, and only this time interval was considered, excluding transient processes from the identification (Figure 9b). In the area considered, a normalization of points was carried out, limiting the number of points to 100, and the nozzle throughput was sought on this basis. Analyzing the recorded experimental runs shown in Figure 9b, it is noticeable that the throughput, \((\mu A)_{\text{ee}}\), has little effect on the pressure waveforms, \(p_{1}\) and \(p_{2}\). Using the presented methodology, the throughput values of the nozzles were determined using the waveforms recorded during the five repetitions of the experimental tests. By substituting the values obtained for Equation (7), the experimental MFR values were calculated, and the results are presented and analyzed in Section 3.