## 2 Materials and Methods
### Experimental Research
\[Q=\frac{dm}{dt}=(\mu A)_{i}v_{max}\varphi\varphi(\sigma)_{max}f(\sigma) \tag{1}\]  
where \((\mu A)_{i}\)--throughput of pneumatic resistance; \(v_{max}\)--maximum air speed; \(\rho\)--gas density (air); \(\varphi(\sigma)_{max}\)--Saint-Venant-Wantzel function maximum value;\(f(\sigma)\)--dimensionless Myetluk-Avtushko flow function. By adopting idealized flow conditions and a simplification excluding heat exchange with the environment during the flow (adiabatic process), it is possible to simplify Equation (1) to Equation (2) by describing the value of the adiabatic mass flow rate through the nozzle. \[Q=\frac{dm}{dt}=\frac{V}{\kappa RT}\frac{dp}{dt} \tag{2}\]  
where: \(V\)--volume of the node. Figure 7 shows the assumed instantaneous values of pressure and temperature in the lumped volumes; in accordance with the lumped method, the mass balance of the component streams in each of the three lumped volumes was described using differential equations by taking into account the flow direction (in accordance with the direction of the arrow in Figure 7), as shown in Equations (3)-(5). \[\frac{dp_{1}}{dt}=-\frac{\kappa RT}{V_{1}}\bigg{(}(\mu A)_{evalc}\sqrt{\kappa RT }\frac{p_{1}}{RT}\varphi_{max}a\frac{p_{1}-p_{2}}{ap_{1}-p_{2}}\bigg{)} \tag{3}\]  
\[\frac{dp_{2}}{dt}=\frac{\kappa RT}{V_{2}}\bigg{(}(\mu A)_{evalc}\sqrt{\kappa RT }\frac{p_{1}}{RT}\varphi_{max}a\frac{p_{1}-p_{2}}{ap_{1}-p_{2}}-(\mu A)_{truct }\sqrt{\kappa RT}\frac{p_{2}}{RT}\varphi_{max}a\frac{p_{2}-p_{3}}{ap_{2}-p_{3} }\bigg{)} \tag{4}\]  
\[\frac{dp_{3}}{dt}=\frac{\kappa RT}{V_{3}}\bigg{(}(\mu A)_{truct}\sqrt{\kappa RT }\frac{p_{2}}{RT}\varphi_{max}a\frac{p_{2}-p_{3}}{ap_{2}-p_{3}}\bigg{)} \tag{5}\]  
where \(\kappa\)--adiabatic exponent; \((\mu A)_{evalc}\)--electrovalcy throughput; \((\mu A)_{truct}\)--throughput of brake valve tract; \(R\)--gas constant (air); \(\varphi_{\max}\)--Saint-Venant-Wantzel theoretical function maximum value; \(a\) = 1.13--Myetluk-Avtushko coefficient value. The use of the solenoid valve controlling the start and end of the measurement (7 in Figure 5) with the capacity \((\mu A)_{evalc}\) (Figure 7), which is not the subject of the research,  
Figure 7: Mathematical diagram of the test rig using the lumped method: \(p_{1},p_{2},p_{3}\)—instantaneous pressure value, \(V_{1}\), \(V_{2}\), \(V_{3}\)—concentrated volume, \(T=T_{1}=T_{2}=T_{3}\)—temperature (constant). resulted in us only taking into account the pressure courses recorded using the transducer placed behind the start solenoid valve \(p_{2}\) and the measuring tank \(p_{3}\) to identify the capacity of the valve \((\mu A)_{tract}\). In connection with the above, the search for the capacity of the tested valve tract (depending on the supply and outlet connections, and consequently the flow direction) was limited to determining the capacity \((\mu A)_{tract}\) based on Equation (5), using the implicit trapezoidal integration method used in the numerical solution of ordinary differential equations. The implicit trapezoidal method [31] can be considered in the context of the Runge-Kutta method and the multistage linear method. The tract capacity \((\mu A)_{tract}\) was determined using the least squares regression method in Equation (6), while the minimization of squared deviations was performed using the downhill simplex method [32; 33] to find the minimum value in the multidimensional space. Figure 4:0  
The methodology of searching for the capacity value \((\mu A)_{tract}\) according to Equation (6) consisted in comparing the experimental pressure curve in the measuring tank \(p_{3 xperimental}\) with the model curve \(p_{3 model}\) obtained from the solution of Equation (5) in the lumped volume \(V_{3}\). The assessment of the fit of both curves was made using the MAPE index. The identification of the valve capacity required the recorded pressure time course to be limited to the stabilized flow area, where the influence of transient processes was excluded.