Comparative assessment of the inline and branching design strategies based on the compound technique

The inline or branching water hammer control strategies, which are based on the insertion of compound plastic short-penstock or inline section at the transient-induced region of main pipes, illustrated a promising ability to upgrade steel pipe-based hydraulic systems concerning the extension of admissible pressure level. In this respect, prior results suggested that the specific layout utilizing an (HDPE–LDPE) compound short-penstock (where the (HDPE) sub-short-penstock is attached to the main steel pipe and the (LDPE) sub-short-penstock corresponds to the shortpenstock dead-end side) provided significant attenuation of pressure magnitude. Concurrently, recent studies concluded that the (HDPE–LDPE) compound short-section-based inline strategy provided substantial attenuation of pressure magnitude. However, these strategies illustrated a drawback relying on the expansion of the period of pressure wave oscillations. Accordingly, this study assessed and compared the capacities of the compound technique concerning the trade-off between the magnitude-attenuation and the period-expansion of pressure wave oscillations. The findings of these analyses showed that the (HDPE–LDPE) compound short-penstock particular setup of the branching strategy allowed the best trade-off between the attenuation of magnitude and the period expansion of pressure wave oscillations. Furthermore, results showed the competitiveness of the latter upgrading strategy as compared to the (HDPE) or (LDPE) main pipe-based renewed hydraulic systems.


GRAPHICAL ABSTRACT INTRODUCTION
Water supply systems operate over a broad range of operating regimes. Occasionally, the improper setting of hydraulic parts or the breakdown of hydraulic machinery leads to large magnitudes of pressure wave fluctuations and may even cause the onset of a cavitating flow regime. Depending on the magnitude of these pressure surges, commonly referred to as water hammer surge-waves, the hydraulic system may experience undesirable effects (e.g. perturbation in serviceability, structural vibrations, and excessive noise) or extensive costly damages (e.g. pipe collapse or bursting, rupture of the piping system); and the operators' safety may even be risked (Bergant &  ). It is, hence, essential to anticipate and mitigate excessive water hammer surges in the design stage of water supply systems and to define safe operation guidelines of these systems in advance. Generally, the prediction of ultimate transient pressure wave magnitudes is used to verify whether the selected pipe material, thicknesses, and pressure class are appropriate to withstand predicted pressure loads to avoid pipe rupture or system damage. Besides, the period value of pressure wave oscillation is used to set out the operational procedures of hydraulic parts. Alternatively to the classic design measures cited above, and benefiting from the mechanical behavior of plastic materials, certain concepts of water hammer control strategies have been addressed in the literature. Incidentally, these studies aimed at upgrading the capacities of existing pressurized steel-piping systems in terms of admissible pressure level. Principally, these strategies included the inline concept (Figure 1(a)), which is based on the substitution of a short section of the main steel-piping system by another one made of plastic material; and the branching concept (Figure 1(b)), which is based on adding a plastic   In the next section, the methodology used for approximating the flow parameters is briefly outlined. and Triki (, a, b)):

METHODOLOGY
The retarded radial strain may be written according to Aklonis et al. (): In the above equation, the creep-compliance function where j ¼ pipe number (1 j np); i ¼ section index (1 i n j s ), n j s ¼ number of sections of the jth pipe, n p ¼ number of pipes, and Δt ¼ time-step increment chosen referring to the CFL rule.
Besides, the STS-MOC-based solver is combined with the discrete gas cavity model (DGCM) to describe the cavi- For instance, the cavity volume obtained from the discretization of the continuity equation, associated with the cavity zone, is: where Q and Q u ¼ average discharges at the up-and downstream side of the cavity zone during the Δt period, respectively; and ψ ¼ weighting factor (0:5 ψ 1).
In addition, the discretized form of the perfect gas law for the isothermic evolution of the cavity reads: It is worth noting that the cavity collapses inasmuch as Incidentally, the hydraulic parameters at an inline or Accordingly, the discharge and pressure head parameters at the connection of the plastic short-penstock with the main steel pipe are linked as follows: Similarly, the flow parameters at the inline connection of (sub-)short-penstocks or -sections, are evaluated as: and Hj Ultimately, the stability condition of the STS-MOCbased solver outlined above is ensured based on the Courant-Friedrichs-Lewy criterion: One notes that the next numerical computations were carried out using the set of input parameters for the STS-

Case 1
The original hydraulic system considered in this subsection is sketched in Figure 3    complete renovation of the hydraulic system. Jointly, the data in Table 3 specify completely the features of the first cycle of the wave curves plotted in Figure 4.
At first glance, the pressure wave patterns corresponding to the upgraded and renewed system cases illustrate amortized trends of first peaks and crests accompanied by   Figure 5. Table 3

Inspection of Figures 4 and 5 and
In this case, the implementations of the compound technique-based inline and branching concepts are schematized in Figure 6(b) and 6(c), respectively. It is interesting to delineate that the (HDPE) sub-short-inline section or  (12), the length and diameter values of the short-inline section or short-penstock used in the conventional technique framework are equal to: l conventional shortÀsection ¼ 10 m and d conventional shortÀsection ¼ 53:2 mm.   Table 4.
At first sight, Figure 7 shows that the cavitating flow regime is established in the original system case. For instance, the pressure head signal first drops to the saturated pressure head value of the liquid (i.e. H min steel ¼ À10:2 m); and, subsequently, rises to H max steel ¼ 63:7 m, due to the superposition of the surge wave involved by the valve closure and the wave generated by the collapse of the vapor cavity. In this regard, the up-and down-pressure surge magnitudes are ΔH steel ¼ 41:7 m and ΔH steel ¼ 32:2 m, respectively.
Alternatively, Figure 7 suggests that the cavitation is removed from all upgraded system cases. Furthermore, the pressure wave signals illustrated attenuated and expanded profiles. As for the first case study, to classify the different upgraded system layouts, the magnitude-period nexus is shown in Figure 8

CONCLUSIONS
Overall, the present research verified the key advantage of the compound technique-based inline and branching concepts over the conventional technique-based ones, which lies in the trade-off between the attenuation of the first pressure head peak and crest and the period of pressure wave oscillations. In this regard, the upgraded system layout devised upon an (HDPE-LDPE) compound shortpenstock-based branching strategy illustrated the best trade-off between the two last parameters. Although this study investigated the case of a single pipe system, extended simulation of pipe networks may be addressed as a perspective to the present study.

DISCLOSURE STATEMENT
The author declares that there are no conflicts of interest.

DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.