## 2 Materials and Methods
### Experimental Studies  
Based on the available literature [58], a concept test bench was developed for the determination of the static flow characteristics [59; 60]. The operating principle of the test bench was based on the indirect measurement of the mass flow rate. Based on the assumptions contained in the concept [61], a test bench was constructed with a few design and functional modifications, a schematic diagram of which is shown in Figure 7.  
A Raspberry Pi microcomputer with a measurement program developed in Python using the PyQT graphics library was used to operate the test stand and record the pressure waveforms in both tanks. The Raspberry Pi controls the operation of the station in a loop with a defined number of repetitions of measurement cycles, successively filling tank 6 from the compressed air preparation station after opening solenoid valve 3. After filling the supply tank, solenoid valve 3 is closed, and the pressure recording begins, with a sampling time of 100 ms. After the time required to stabilize the pressure in reservoir 6, solenoid valve 7 opens, and compressed pressure flows from supply reservoir 6 to measurement reservoir 9, caused by the pressure difference in the two reservoirs through nozzle 8. Once the pressures in both reservoirs have equalized, data acquisition stops, and the measurement file is saved in CSV format. Solenoid valves 5 and 10 are then left open, and the pressure in the tanks and the entire system is equalized to atmospheric pressure. After this time, the solenoid valves are closed, and the measurement sequence is restarted, saving the results in a separate file with a CSV extension. The output file is the input data for the identification of flow parameters calculated in the MATLAB environment. The pressure recording of the measurement cycle for each nozzle was repeated five times. The recorded time courses of pressure variations in the compressed air tank chambers, based on the lumped method, allowed the effective flow field to be determined \((\mu A)_{nozzle}\). The identification of the nozzle flow parameters was carried out in the MATLAB environment, where a mathematical description of the process built from the structural diagram of the bench (Figure 7), based on the lumped method [62], was implemented. All this allowed for the search for the throughput from the runs recorded on the test bench. Figure 7: Structural diagram of the test stand: 1—compressor; 2—pressure regulator; 3, 5, 7, 10—solenoid valve; 4—choke valve; 6, 9—40.23 \(\times\) 10\({}^{-3}\) m\({}^{3}\) tank; 8—tested nozzle; 11–13—pressure transducer; 14—measuring section; 15—Raspberry Pi; 16—MATLAB software, R2024b. The description of the mass flux of the fluid flowing through the local resistance, based on the lumped method (Figure 8), can be represented as Equation (7). \[Q=\frac{dm}{dt}=\left(\mu A\right)_{element}v_{max}\rho\varphi(\sigma)_{max}f(\sigma) \tag{7}\]  
where \((\mu A)_{element}\)--the throughput of element; \(v_{max}\)--the maximum air speed; \(\rho\)--the air density; \(\varphi(\sigma_{kr})\)--the maximum value of the Saint-Venant-Wantzel function; and \(f(\sigma)\)--the dimensionless flow function (Myetluk-Avtushko). Assuming a simplification of the adiabatic flow through the nozzle, the description of the mass flow rate (Equation (7)) takes the form of Equation (8). \[Q=\frac{dm}{dt}=\frac{V}{\kappa RT}\frac{dp}{dt} \tag{8}\]  
where \(V\)--tank volume. In order to create a system of equations describing the pressure changes in all the concentrated volumes, it is necessary to assume the values of instantaneous pressure equal to \(p_{1}\), \(p_{2}\), and \(p_{3}\) in concentrated volumes \(V_{1}\), \(V_{2}\), and \(V_{3}\) at a constant temperature throughout the system \(T=T_{1}=T_{2}=T_{3}\). Assuming the above, a system of differential equations can be written in all the concentrated volumes, according to the mass balance of the component fluxes, Equations (9)-(11).