## 5 Conclusions  
The use of experimental methods based on the lumped method and numerical CFD in the assessment of the throughput and mass flow rate provides similar results, which was the basis for undertaking this research, with the adoption of certain simplifications enabling the analytical determination of throughput from recorded pressure curves and differential equations based on the lumped method, which has been used repeatedly in the literature to determine pneumatic throughput. No attempts to determine the static characteristics of a pneumatic brake valve using numerical and experimental approaches or to compare both methods were found in the literature. * It was confirmed that simplifying the flow characteristics of the tracts to a rectangular shape does not reflect the properties of the valves. The applicability of the numerical approach to assessing the flow of brake valve tracts was verified by comparing the values obtained using computer simulation with those obtained using the analytical approach, which has been repeatedly verified in the literature. In earlier studies by the authors, where the nozzle throughput was subjected to experimental and numerical assessment, the differences between the two methods were smaller, mainly due to the geometry of the system, the presence of seals, and the need to introduce a control system to simulate control operations. * The movement of the control piston, caused by a sudden increase in pressure at the valve supply port and the piston surface (depending on the flow direction), was observed during the tests. This was the result of the clearance of the threaded connection. Nevertheless, this was the least invasive solution, enabling manual control. It could therefore have had a small impact on the value of the results obtained by the experimental method of this study. * Preparing the valve's supply section for experimental tests required the use of additional components to ensure regulation in line with the assumed test conditions. For this purpose, a spring supporting the control piston was used inside the valve body, which ensured the precise adjustment of the piston position and, consequently, the degree of opening of the main or additional supply path. * Numerical tests using a polyhedral mesh with 500 declared iterations and dimensionless residual indicators at \(10^{-6}\), representing imbalance in the solver's differential equations, produced a stable solution, which formed the basis for comparing the results obtained in the numerical and experimental approaches. In numerical modelling, adiabatic flow was assumed, which in real conditions, occurring during experimental tests and the normal operation of the brake valve during service, is a certain simplification. A broader discussion of the impact of temperature on valve operation will form the basis of a dynamic simulation based on multiphysics. * The discrepancies between both methods assessed using the mean absolute percentage error MAPE were within the range of 0-24%, while the mean value in the entire measurement range was 9.43%, which was mainly influenced by the high MAPE value (17.25%) of the main supply tract during the flow direction resulting from the release of the brake. In the case of a larger sample size, statistical analysis would be necessary, but the aim of the numerical and experimental studies was to assess the possibility of using numerical simulation to determine the pneumatic output of the brake valve, which was achieved by comparing five repetitions of each experimental measurement. * For the other flow directions of both tracts, the values of the average MAPE index were equal to 5.88%, 9.40% and 5.20%, which proves that, after taking all simplifications into account, the use of simulation methods to evaluate the operation of pneumatic brake valves in a static approach provides satisfactory results.