**5. Model Validation and Analysis of Experimental Data**

Numerical calculations were performed to simulate all the conducted experiments. Calibration of the numerical model was performed by adjusting the viscoelastic parameters in order to obtain the result of calculations that match the observed pressure changes. The retardation time is a quantity that characterises one of the viscoelastic properties of the polymers. It is the duration of the retardation phenomenon that describes the moment when the stress becomes zero. Based on the constitutive linear viscoelasticity equations, it is possible to separate the creep compliance into an elastic part independent of time *J*<sup>0</sup> and a time-dependent creep function *J*(*t*). Due to the fact that the numerical model takes into account a single value of pressure wave velocity for the entire pipeline system, it also includes the value of the elastic part of compliance. As mentioned earlier, only steady friction was taken into account in the momentum Equation (2). Thus, matching of the observed and calculated pressure changes was obtained by calibrating the parameters of the creep function, i.e., *J* and *τ* and the number of elements *N* in the Kelvin–Voigt viscoelastic model. The values of the viscoelastic parameters were chosen by trial and error to minimise the mean squared error between the calculated and observed pressure samples. In order to simplify the model calibration, if the pipeline system consisted of three pipes (experiments no. 5 and 6), the same values of viscoelastic parameters for each pipe were assumed. Moreover, if more than one Kelvin–Voight element was used in the calculations, identical viscoelastic parameters for each element were used. This simple approach, as presented later, made it possible to obtain satisfactory compliance between the calculation results and experimental data. More advanced methods of calibrating the viscoelastic parameters are presented in [25–27].

In all the conducted numerical simulations, the time step and spatial step were selected to ensure a Courant number as close as possible to 1. The parameters of the pipeline system and initial conditions listed in Table 1 were used as an input. The values of the viscoelastic parameters determined for all cases are presented in Table 2.


**Table 2.** Viscoelastic and transient flow parameters used as an input for the numerical model.

It should be noted that the parameters related to Kelvin–Voigt elements, such as the retardation time and creep compliance, are purely mathematical. Dashpots and springs are conceptual elements with no strict attitude to the physical side of the water hammer phenomenon [17].

Furthermore, the creep and retardation of an HDPE pipe depend not only on the pipe stress history (specifically the frequency and amplitude of the load), but also on the axial and radial limitations of the pipeline system. Thus, the values of the viscoelastic parameters (Table 2) are influenced not only by a single water hammer experimental test, but also by the entire cycle of performed measurements. The calculated and observed pressure oscillations for all experiments are presented in Figures 4–9.

**Figure 4.** Calculated and observed pressure oscillations during experiment no. 1.

**Figure 5.** Calculated and observed pressure oscillations during experiment no. 2.

**Figure 6.** Calculated and observed pressure oscillations during experiment no. 3.

**Figure 7.** Calculated and observed pressure oscillations during experiment no. 4.

**Figure 8.** Calculated and observed pressure oscillations during experiment no. 5.

**Figure 9.** Calculated and observed pressure oscillations during experiment no. 6.

Figures 4–9 show that the pressure wave is strongly damped and time-dispersed. A particularly smooth *h*(*t*) function was recorded during experiments no. 1 and no. 3 (Figures 4 and 6, respectively). For these tests, measurements were conducted using a pipe with a wall thickness of 2.4 mm (with an inner diameter of 35.2 mm) and a pipe with a wall thickness of 3.0 mm (with an inner diameter of 44 mm). For both of these

tests (experiments no. 1 and no. 3), the ratio of the inside diameter to the wall thickness of the pipes was equal to *D*/*s* = 15. For other experiments (Table 1), the wall thickness to internal diameter ratio was constant and equalled *D*/*s* = 9. It is apparent from Figures 4 and 6 that during the water hammer experiments in the pipeline system with thinner pipe walls, a stronger damping of the pressure wave can be observed. Additionally, it can be noticed that the values of the pressure wave velocity for pipes with thinner walls are lower compared to those for the rest of the experiments (Table 2). These lower values of the pressure wave velocity confirm a faster attenuation of disturbances in viscoelastic pipes with thinner walls.

In the hydraulic transient data (Figures 4–9), additional disturbances typical of seriesconnected pipes are visible for the first pressure increase. The use of the MacCormack time-marching scheme along with the presented connection node equations satisfactorily reflects the pressure disturbances recorded during the measurements (zoom window in Figures 4 and 5). It can be noted that the connection sequence of the pipes with various inner diameters has an influence on the pressure disturbance during the first pressure increase. In the case of the combination of tank–smaller-diameter-pipe–larger-diameterpipe–outlet-valve (Figures 4 and 5), the first pressure disturbance is not the maximum increase. This effect can also be observed for the water hammer test with three pipes in series (experiment no. 5). In the reverse configuration, the first recorded disturbance is also the maximum pressure increase (Figures 6, 7 and 9).
