## 3 Determining the Pressure Wave Speed in the Tested System
### Verification of the Numerical Method Using the Quasi-Steady Friction Loss Model  
The calculations performed to carry out the comparative analysis were made for run no. 4 (the initial conditions are presented in Table 1) and together with the measured pressure heads are shown in Figure 8.  
Since the analysis of the pressure wave propagation speed determined by the method #1 and method #2 showed a high degree of their compatibility, the verification of calculations made using friction loss models was carried out for the results obtained with the method easier to use, i.e., the proposed method #2 (based on Equation (9)--Table 3): \(a_{1}=1223\) m/s and \(a_{2}=1254\) m/s for a pipe with a larger and smaller diameter, respectively. The numerical grid used in the calculations (a regular, rectangular grid) had 446 nodes along the pipes \(x\) axis (206 for pipe No. 1 and 240 for pipe No. 2) and 76,642 nodes along the time axis (simulation time \(t_{max}=15\) s). Details regarding the method of discretization of the solution region are included in Appendix B. According to Figure A1, the grid within each pipeline is identical, but differs in the dimensions of the grid cells. The dimensions of the cells along the \(x\)-axis (pipelines axis), defined according to the relationship \(\Delta x_{i}=\Delta t_{i}\cdot a_{i\_{\text{grid}}}\), where \(a_{i\_{\text{grid}}}\) is defined in Equation (A25), were as follows:  
* For pipe #1: \(\Delta x_{1}=0.2393\) m
* For pipe #2: \(\Delta x_{2}=0.2454\) m The grid dimensions along the \(t\)-axis for both pipelines (\(\Delta t_{1}=\Delta t_{2}=\Delta t\)) were equal \(\Delta t=1.957\cdot 10^{-4}\)s. Thanks to such selection of grid size, the relative error of mapping the wave speed in individual pipes was \(|\varepsilon_{a_{1}}|=0.02\%\), \(|\varepsilon_{a_{2}}|=0.01\%\) for the respective pipes (Equation (A26) in Appendix B). It should be emphasized here that the relative errors of mapping the wave speed in individual pipes presented above do not generate numerical damping, but only affect the discrepancies between the set (\(a_{i}\)) and the calculated (\(a_{i\_{\text{grid}}}\)) values of pressure wave propagation speed. The errors associated with this are not less than ten times smaller than the uncertainty of methods #1 and #2 of determining the speed a_i from the measurement data, so they do not have a significant impact on the calculation results and conclusions formulated at a later stage of the analysis. In order to check the calculation algorithm, calculations were made using a quasi-steady model of friction losses (calculations carried out using the Hagen-Poiseuille law and the Colebrook-White formula were performed with the value of density \(\rho=998.97\) kg/m\({}^{3}\) and dynamic viscosity \(\mu=0.938\cdot 10^{-3}\) Pa\(\cdot\)s for water at 23 \({}^{\circ}\)C and assuming the absolute roughness of the inner surface of the copper pipe \(k_{r}=2\cdot 10^{-6}\) m--see Equations (A11)-(A13) in Appendix B). The comparison of the measurement and calculation results made with the use of the quasi-steady model of friction losses, presented in Figure 8, showed a lack of agreement  
Figure 8: Comparison of the pressure oscillations measured at the valve (test No. 4) with the calculated pressure wave obtained with the use of the quasi-steady model of friction losses for the wave speed in pipes connected in series determined on the basis of Equation (9). between them in terms of changes in pressure amplitudes over time and the shape of the pressure wave. Numerous wave reflections persisting throughout the simulated time are characteristic for the calculated pressure wave, while the measured wave shows a gentle, sinusoidal-like course in the developed phase of the pressure oscillation. These observations confirm the commonly known features of the quasi-steady model of friction losses, which, apart from the first few pressure amplitudes, does not allow for an accurate simulation of the water hammer phenomenon.