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

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 & Simpson 1999; Thorley 2004; Besharat et al. 2015; Zhang et al. 2018; Du et al. 2020). 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.

From a design standpoint, water hammer surges should be kept within a prescribed limit using a combination of different classical design measures ranging from the adjustment of operational procedures and the installation of surge control devices to the redesign of the original pipeline layout. It is interesting to highlight herein that the cumulative use of different control devices may adversely affect the water hammer courses, due to inconsistency between the nature of included devices (Boulos et al. 2005; Jung et al. 2007, 2008, 2009; Seog-Jung & Karney 2009; Chen et al. 2015). Besides, the prediction of flow parameters is complicated furthermore by the nonlinear behavior of embedded surge control devices. In this regard, Jung et al. (2007, 2008, 2009) concluded that even modern computer systems cannot cope with the computational challenges brought by a strict and systematic search for the best system protection in most practical field applications.

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 short-penstock at the transient-induced region of the main pipe (Massouh & Comolet 1984; Pezzinga & Scandura 1995; Triki 2016, 2017, 2018a, 2018b; Gong et al. 2018; Fersi & Triki 2019; Triki & Chaker 2019; Trabelsi & Triki 2019, 2020; Chaker & Triki 2020a, 2020b; Kubrak & Kodura 2020). Results have shown that these two concepts allow promising improvement of the admissible pressure of existing hydraulic installations without making significant modifications. Nevertheless, previous investigations unveiled that the use of plastic material induces an expansion effect (i.e. an increase) of the period of pressure wave oscillations. Incidentally, the expansion effect of the period of pressure wave oscillations may negatively affect the operational procedures of hydraulic services, such as the increase in critical time for closing the valve. Consequently, the use of plastic material types leads to a conflict between two design factors including the attenuation of magnitude and the expansion of the period of pressure wave oscillation. Physically, the attenuation of pressure head magnitude is mainly attributed to the reduced modulus of the used material for the short-section, and the expansion effect of the period of pressure wave oscillations is due to the retarded response of viscoelastic materials (Ramos et al. 2004; Brinson & Brinson 2008; Duan et al. 2010a; Mitosek & Szymkiewicz 2012; Evangelista et al. 2015; Ferrante & Capponi 2017; Pan et al. 2020, 2021). Precisely, previous researches showed that the LDPE plastic material type allows more attenuation of the pressure surge magnitude than the HDPE one; and inversely, the latter concept offers less expansion of the period of pressure wave oscillations than the first one.

Subsequently, in order to address the foregoing drawback of the conventional technique of implementation of the inline and branching concepts, Triki & Chaker (2019) and Chaker & Triki (2020a) proposed the compound technique, which is based on splitting the single short-section or short-penstock, used in the conventional technique-based inline or branching strategy, into two sub-short-sections made up of two distinct plastic material types. This technique was intended to combine the merits from the large pressure attenuation allowed by the LDPE material type and the low expansion of pressure wave oscillation period capacities provided by the HDPE material type. Triki & Chaker (2019) demonstrated that the upgraded system layout based on an (HDPE–LDPE) compound inline-short-section (where the HDPE sub-short-section is attached to the hydraulic parts, while the LDPE one is attached to the main steel pipe) leads to the best trade-off between the magnitude-attenuation and period-expansion factors, within the compound technique-based branching strategy framework. Concurrently, Chaker & Triki (2020a) concluded that the particular setup of the compound technique-based branching strategy employing an (HDPE–LDPE) compound short-penstock allows the best trade-off (i.e. magnitude-attenuation and period-expansion), within the compound technique-based branching strategy framework. Accordingly, this research aims at gaining further insight into the water hammer control topic by comprehensively assessing and comparing the capacities of the compound technique-based inline and branching strategies in terms of the trade-off (between the magnitude-attenuation and period-expansion effects of transient pressure wave oscillations) provided by an (HDPE–LDPE) compound inline-short-section and that allowed by an (HDPE–LDPE) compound short-penstock. Additionally, this investigation looks into the benefits of the former upgrading strategies in comparison with the total renewal of the main-piping systems using HDPE or LDPE plastic materials.

In the next section, the methodology used for approximating the flow parameters is briefly outlined.

From a computational point of view, different approaches can be used for hydraulic transient analysis, including simplified approaches used for extreme pressure surge magnitude estimation (such as the Joukowsky law for rapid maneuvers, and the Michaud formula for slow maneuvers), classical transient solvers which are based on a set of simplifications, and complete transient solvers which account for different dynamic effects (e.g. unsteady friction loss, non-elastic pipe-wall behavior, fluid-structure interaction, or cavitation). In this line, the 1-D Extended Water Hammer Model (1-D-EWHM) embedding the Kelvin–Voigt (Aklonis et al. 1972) and Vitkovsky et al. (2000) formulations, is typically implemented to analyze fast transient events in plastic pipes (Covas et al. 2004; Ghidaoui et al. 2005; Wood et al. 2005; Duan et al. 2010b; Duan et al. 2012; 2018; 2020; Carriço et al. 2016; Bertaglia et al. 2018; Walters & Leishear 2018; Cao et al. 2020; Lashkarbolok & Tijsseling 2020; Warda et al. 2020). Accordingly, this formulation is selected in the present study.

The 1-D-EWHM (Equation (1)) may be solved using a Method of Characteristics-based algorithm, established upon a specified time-step rectangular grid (STS-MOC). It is worth noting that this algorithm was validated earlier by the author against experimental data presented by Covas et al. (2004) (e.g. Triki 2017; Triki & Chaker 2019; Chaker & Triki 2020a, 2020b)).

It is worth noting that the cavity collapses inasmuch as ⁠.

Data from laboratory experiments conducted by Covas et al. (2004) are used to validate the numerical model developed above. The experimental apparatus investigated by the authors pertains to a reservoir-HDPE pipe-valve system. The pipe characteristics are {⁠; length: ⁠; pipe-wall thickness: ⁠; and diameter: }; and the coefficients of the Kelvin–Voigt formulation of HDPE material are listed in Table 1 (Keramat & Haghighi 2014). The initial steady-state discharge and pressure head at the upstream reservoir are q₀ = 1.008 L/s and ⁠, respectively. Results are carried out for a transient flow initiated by the abrupt closure of the downstream valve. One notes that the next numerical computations are performed by the STS-MOC procedure developed earlier, using a specified time step and a Courant number ⁠.

Figure 1 compares the measured pressure head signals and computed one involved by the water hammer solver embedding the Vitkovsky and Kelvin–Voigt formulations. This figure shows that the pressure head signal predicted by the 1-D-EWH equations based on the Vitkovsky and Kelvin–Voigt formulations complies well with the measured data, in terms of magnitude and the phase shift of the first cycle of wave oscillation.

Incidentally, it is worth pointing out that the pressure wave magnitudes calculated based on the Joukowsky law are greater than the observed and computed magnitudes using the 1-D-EWH equations. In this respect, the Joukowsky pressure magnitudes corresponding to the HDPE main-piping system is ⁠, while the observed magnitude is equal to and the predicted one using the 1-D-EWH equations is equal to ⁠. Likewise, the theoretical period values computed for the HDPE or LDPE main-piping systems (i.e. ⁠) are significantly lower than those collected from the pressure wave curves issued from the observed signal and the predicted one using the 1-D-EWH equations (i.e. and ⁠, respectively). Physically, these dispersions are substantially due to the retarded response of plastic materials. Indeed, the Joukowsky and theoretical period laws are based on the elastic creep-compliance value and hence could not account for the retarded response involved by the viscoelastic behavior of plastic material.

From the numerical side, exploration of the sensitivity of the values of pressure head magnitudes to the time-step size, listed in Table 1, suggests that the decrease in the time step does not significantly affect the value of the first pressure peak.

In the following, the performances of the compound technique are assessed within the inline and branching strategy concepts. The HDPE or LDPE plastic material types are employed for the (sub-)short-sections or -penstocks. The mechanical characteristics of employed pipe-wall materials are listed in Table 2 (Keramat & Haghighi 2014).

One notes that the next numerical computations were carried out using the set of input parameters for the STS-MOC-based procedure: {⁠; ⁠; and }.

The next results interpreting the up- or down-surge pressure wave characteristics are denoted by the symbols (+) or (–), respectively; and, the symbol prime (′) is assigned to the branching concept. Furthermore, the following definitions are used in the next result interpretations: (i) the magnitude of up- or down-surge pressure wave is evaluated as: ⁠; (ii) the attenuation of the up- or down-surge pressure wave involved the controlled system cases as compared to those associated with original system case are computed as: ⁠; and (iii) the phase shift between the first cycle of pressure wave oscillations involved in the controlled system cases and their counterparts associated with the original system case is computed as ⁠.

In such a situation, the compound technique-based inline strategy consists of substituting a downstream short-section of the original steel-piping system by an (HDPE–LDPE) compound short-inline section, where the (HDPE) and (LDPE) sub-short-sections are attached to the main steel pipe and valve, respectively (Figure 3(b)). However, the compound technique-based branching strategy consists of installing an (HDPE–LDPE) compound short-penstock at the downstream extremity of the piping system, where the (HDPE) sub-short-penstock is attached to the main steel pipe and the (LDPE) sub-short-penstock corresponds to the dead-end side of the short-penstock (Figure 3(c)).

The diameter and length values of the sub-short-sections or penstocks are selected equal to: and ⁠. Thereupon, referring to Equation (12), the results associated with the conventional technique-based inline or branching concepts are addressed for the short-section or penstock diameters and lengths equal to: and ⁠.

Figure 3 illustrates the estimates of downstream pressure wave signals predicted into the original system case and their counterparts involved by the upgraded system cases based on an (HDPE–LDPE) compound short-inline section or short-penstock, and (HDPE) and (LDPE) conventional short-sections or -penstocks. Besides, the renewed hydraulic system cases corresponding to main pipe systems made of HDPE, or LDPE plastic pipe, are also reported in this figure to check the usefulness of the inline upgrading strategy for the 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 expansions of the period values pressure wave oscillations, as compared with the original system case. Thereby, to classify the different system layouts, the magnitudes of up- and down-pressure surges versus the period for the first cycle of pressure wave oscillations are reported in Figure 5.

Inspection of Figures 4 and 5 and Table 3 reveals that the larger up- and down-pressure surge magnitudes are associated with the original system case: and ⁠, respectively; however, less important magnitudes are involved by the upgraded system cases utilizing an (HDPE–LDPE) compound short-inline section or short-penstock. In this regard, the attenuations of up- and down-pressure surge magnitudes involved by the upgraded system cases utilizing an (HDPE–LDPE) compound short-inline section or short-penstock are { and } or { and }, respectively.

On the other hand, Figures 4 and 5 and Table 3 reveal that the periods of the first cycle of pressure wave oscillations estimated in the upgraded system cases built upon an (HDPE–LDPE) compound short-inline section or short-penstock are or ⁠, respectively; while the corresponding period, involved by the original system case, is equal to: ⁠. This implies that the plastic material types of the compound short-inline section or short-penstock induce the expansion of the pressure wave oscillation period. Specifically, the phase shifts between an (HDPE–LDPE) compound short-inline section or short-penstock-based layouts of the upgraded system and their counterpart predicted into the original system case are or ⁠, respectively. These results imply the ratios between the attenuations of up- and down-surge magnitudes and the phase shift involved by the (HDPE–LDPE) compound short-inline section or short-penstock-based setups of upgraded systems: { or } and { or }, respectively.

On this point, it may be concluded that the compound technique-based branching concept utilizing an (HDPE–LDPE) compound short-penstock leads to a better trade-off between the attenuation of the magnitude and the expansion of the period of pressure wave oscillations than the compound technique-based inline concept.

Incidentally, the former compound technique setup provides more important amortization of magnitudes of up- and down-pressure surges and more important expansion of the period of pressure wave oscillations, than the (HDPE) or (LDPE) short-penstock-based setups of conventional technique. Indeed, the attenuations of up- and down-pressure surge magnitudes involved by upgraded system cases build upon (HDPE) or (LDPE) short-penstocks are { and } or { and }, respectively. Additionally, the phase shifts calculated between an (HDPE) or (LDPE) short-penstock-based layout of upgraded systems, and their counterpart predicted into the original system case are or ⁠, respectively. In return, the ratios between the attenuations of up- and down-surge magnitudes and the phase shift involved by the (HDPE) or (LDPE) short-penstock-based upgraded systems are { or } and {}, respectively.

Contrarily, the (HDPE–LDPE) compound short-penstock-based branching concept provides less important amortization of magnitudes of up- and down-pressure surges and more important expansion of the period of pressure wave oscillations, than the (HDPE) and (LDPE) setups of the conventional technique-based inline concept. Indeed, the attenuations of up- and down-pressure surge magnitudes involved by the upgraded system cases built upon an (HDPE) or (LDPE) short-inline section are { and } or { and }, respectively.

Additionally, the phase shifts calculated between the (HDPE) or (LDPE) short-inline section-based setups of the upgraded system and their counterpart predicted into the original system case are or ⁠, respectively. In return, the ratios between the attenuations of up- and down-surge magnitudes and the phase shift involved by the (HDPE) or (LDPE) short-penstocks-based upgraded systems are { or } and {}, respectively.

Incidentally, the renewed system cases utilizing HDPE or LDPE main pipe-based renewed system cases lead to a more substantial reduction of pressure wave magnitude than that involved by the original system case. In this respect, the attenuations of up- and down-pressure surge magnitudes involved by an HDPE or LDPE main pipe are { and } or { and }, respectively. However, the foregoing renewed system layouts induce significant expansion of the period of the first cycle of pressure wave oscillations, as compared to upgraded system setup based on an (HDPE–LDPE) compound short-penstock. Specifically, the phase shifts between an (HDPE) or (LDPE) main pipe-based layout of the renewed systems, and their counterparts predicted into the original system case are or ⁠, respectively.

In summary, the above results attest that the (HDPE–LDPE) compound short-penstock-based setup of the upgraded system allows more attenuation of the first pressure head peak and crest as compared to the original system case and the upgraded systems cases involving an (HDPE) or (LDPE) setup of the conventional technique-based branching or inline concepts. Conversely, the former compound technique setup leads to more important expansion of the period of pressure wave oscillations than the original system case and the conventional technique setups-based branching or inline concepts.

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 short-penstock are attached to the main steel pipe; however, the (LDPE) sub-short-inline section is attached to the main steel pipe and the (LDPE) sub-short-penstock corresponds to the dead-end side of the short-penstock.

The investigation addresses the length and diameter values of the sub-short-inline sections or short-penstocks equals to: and ⁠. Hence, as per Equation (12), the length and diameter values of the short-inline section or short-penstock used in the conventional technique framework are equal to: and ⁠.

Figure 7 shows the upstream pressure wave signals, computed into the original system case along with their counterparts estimated into the upgraded system cases based on an (HDPE–LDPE) compound short-inline section or short-penstock, and (HDPE) and (LDPE) short-sections or short-penstocks. Additionally, the renewed hydraulic system cases corresponding to main pipe systems made of HDPE, or LDPE plastic pipe, are also reported in this figure to check the usefulness of the inline upgrading strategy for the complete renovation of the hydraulic system. Jointly, the main features of the first cycle of the wave patterns plotted in Figure 7 are enumerated in 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. ⁠); and, subsequently, rises to ⁠, 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 and ⁠, 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, for the first cycle of pressure wave oscillations.

Concerning the compound technique-based inline or branching concepts, the amortization of up- and down-pressure surge magnitudes involved by an (HDPE–LDPE) compound short-inline section or short-penstock are { and } or { and }, respectively.

On the other hand, Figures 7 and 8 and Table 4 suggest that the upgraded system-based branching strategy setup using an (HDPE–LDPE) compound short-penstock induces less expansion of the period of pressure wave oscillations than the corresponding setup utilizing an (HDPE–LDPE) compound short-inline section For example, the period of the first cycle of pressure wave oscillations predicted into the upgraded systems based on an (HDPE–LDPE) compound short-inline section or short-penstock is or ⁠, respectively; however, the period value corresponding to the original system case is equal to ⁠. This in turn implies that the phase shifts between an (HDPE–LDPE) compound short-inline section or short-penstock-based layout of the upgraded system and their counterpart predicted into the original system are or ⁠, respectively. On this point, it may be delineated that the phase shift obtained in this test case is more important than the one deduced, previously, in the first test case. These results suggest that the (HDPE–LDPE) compound short-penstock-based upgraded system provides more important ratios (between the attenuation of up- and down-pressure surge and the phase shift) than the upgraded system utilizing an (HDPE–LDPE) compound short-inline section: { and } or { and }, respectively.

Further interpretations concerning the (HDPE) short-inline section or short-penstock-based setup of the conventional technique-based inline or branching concepts suggest that the attenuation values of the first pressure head peak or crest are {} or { and }, respectively. Besides, this conventional technique-based setup of the inline or branching concepts leads to phase shift values: or ⁠, respectively. In other words, the ratios between the attenuation of up- and down-pressure surge and the phase shift obtained using an (HDPE) short-inline section or short-penstock are { and and }, respectively.

Similarly, much lower ratio values are obtained in the upgraded system cases using an (LDPE) short-inline section or short-penstock. In particular, the attenuation values of the first pressure head peak or crest provided by these setups: { and } or { and }, respectively. Furthermore, these setups of the conventional technique lead to phase shift values equal to or ⁠, respectively. Consequently, the ratios between the attenuation of up- and down-pressure surge and the phase shift obtained using an (LDPE) inline-short-section or short-penstock are { and } or {⁠, respectively.

Incidentally, the renewed system cases utilizing an HDPE or LDPE main pipe-based renewed system cases lead to a more important reduction of pressure wave magnitude than that involved by the original system case. In this respect, the attenuations of up- and down-pressure surge magnitudes involved by an HDPE or LDPE main pipe are { and } or { and }, respectively. However, the foregoing renewed system layouts induce significant expansion of the period of the first cycle of pressure wave oscillations, as compared to upgraded system setup based on an (HDPE–LDPE) compound short-penstock. Specifically, the phase shifts between an (HDPE) or (LDPE) main pipe-based layouts of the renewed systems and their counterpart predicted into the original system case are or ⁠, respectively.

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 short-penstock-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.

The author declares that there are no conflicts of interest.

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