Recent studies have proved that the utilization of polyethylene (PE) short-section or penstock is a promising water hammer control tool. However, the interplay between the magnitude attenuation and the phase offset of pressure-wave oscillations remains challenging. This study aimed at inspecting the capacity of a dual PE penstock/short-section-based control technique, with regard to the aforementioned interplay. In this technique, a PE penstock was lumped to the transient initiating zone of the main pipe and a short-section of the counter extremity of the pipe was replaced with PE. The transient pressure-wave behavior in a gravitational viscoelastic pipe involving cavitation was described by the extended 1D water hammer equations embedding the Vitkovsky and Kelvin–Voigt add-ons. The numerical solution was performed by the fixed grid method of characteristics. The high- (HDPE) and low-density (LDPE) were demonstrated in this study. Analysis revealed that upgrading techniques based on LDPE enabled a desirable tradeoff between the magnitude attenuation and the phase offset of pressure-wave oscillations. Particularly, the dual penstock/short-section specific upgrading technique allowed a more important attenuation magnitude of pressure peak (or crest), and led to a similar expansion of the wave oscillation period. Furthermore, results evidenced that the proposed technique outperformed the renewal of the original piping system.

  • A dual polyethylene (PE) penstock upstream and a PE inline short-section downstream are addressed to palliate cavitation.

  • The rheological mechanical behavior of PE is modeled referring to the generalized linear Kelvin–Voigt approach.

  • High- and low-density PE are utilized for the short-section or penstock.

  • The tradeoff between the wave magnitude attenuation and the wave period relaxation is inspected.

Hydraulic systems are inevitably subjected to a wide range of operational conditions. In principle, the inappropriate switching process of a valve or premature pump shutdown can have devastating consequences, including backflow and system overpressure or pressure drop. In particular, the undesirable occurrence of a hydraulic cavitation regime may have a significantly adverse impact on the hydraulic system (e.g. severe structural and hydraulic vibrations, erosion, collapse, and even the rupture of the pipeline) and can involve risks for operators (Wood et al. 2005; Bergant et al. 2006; Martins et al. 2015; Essaidi & Triki 2020; Triki & Essaidi 2022). For exemplification purposes, Figure 1 shows a pipe failure pictorial during a transient pressure event involving a cavitation regime. Physically, this regime is established when, at specific locations, such as closed ends or high points, the pressure drops to the vapor pressure of the liquid (i.e. changes in the pipe slope). Besides, the cavitation phenomenon corresponds to the separation of the liquid column by vapor cavities that grow and collapse during the flow dynamic processes. This phenomenon is characterized by high-magnitude pressure pulses, which leads to a critical load for supporting structures, individual piping systems, and hydraulic machinery. Consequently, cavitation control is crucial to protect hydraulic systems and ensure the operator's safety. Normally, hydraulic systems are designed and equipped with several water hammer control devices to withstand the first transient pressure peak/crest which is unaffected by dissipation. In particular, the piping system component of a hydraulic system is sized based on the fixed design demand considering a peak (or crest) factor and design period. In general, common design tools include system modification, operational considerations, surge protection devices, emergency control procedures, and inter-system protection. Incidentally, the designers consider several hydraulically feasible alternatives and determine the most cost-effective one. A holistic review of various water hammer control design measures is presented in Boulos et al. (2005), Ghidaoui et al. (2005), Chen et al. (2015), Besharat et al. (2016a, 2016b, 2017), and Mery et al. (2021).
Figure 1

Elucidation of experienced pipe collapse during a cavitation process.

Figure 1

Elucidation of experienced pipe collapse during a cavitation process.

Close modal

Currently, polyethylene (PE) materials are increasingly used for hydraulic piping systems thanks to their low cost, easy processing, lightweight, and high corrosion resistance. In addition, these materials have the potential capacity of dampening high- (and low-) pressure loads. The physical explanation for this property relies on the viscoelastic behavior of these materials which has dissipative and dispersive effects on the pressure wave (Ferry 1970; Güney 1983; Brinson & Brinson 2008; Tjuatja et al. 2023). These effects, which are similar to unsteady friction losses, result in pressure-wave oscillation pattern characterized by a small magnitude and a high oscillation period. Thereupon, based on the viscoelastic property of PE, several design measures were addressed in recent literature as alternatives to classical ones.

For example, Ghilardi & Paoletti (1986), Triki (2016), Gong et al. (2018), Triki & Chaker (2019), Fersi & Triki (2019a, 2019b, 2020, 2023), Trabelsi & Triki (2021), and Kraiem & Triki (2023) investigated the inline technique which is conceptualized upon the substitution of a short-section of the transient source zone of an existing steel-made main pipe with its rubber-made counterpart (Figure 2(a)). Concurrently, Pezzinga & Scandura (1995), Pezzinga (2002), Triki (2017), Triki & Fersi (2018), and Chaker & Triki (2020a, 2020b, 2020c, 2021) introduced the branching technique based on lumping a PE-made penstock at the transient source zone of an existing steel-made pipe (Figure 2(b)). In summary, these studies evidenced that the use of PE short-sections or penstocks could significantly reduce the magnitudes of pressure-surge-rise or -drop. Nonetheless, despite the practical relevance of such techniques, the PE material induced a relaxation effect of the pressure-wave oscillation period, which could unfavorably lengthen the critical time for valve closure and, hence, retrogress the operational procedure of hydraulic systems (Trabelsi & Triki 2020a). In order to address the aforementioned limitation of these conventional inline or branching techniques, Triki (2018a, 2018b), Trabelsi & Triki (2019, 2020b, 2020c), BenIffa & Triki (2019), BenAmira & Triki (2021), Triki & Trabelsi (2021), and Triki (2021) examined, subsequently, the benefit of splitting the short-section or penstock used in the last techniques into two sub-short-sections or -penstocks placed at each hydraulically connected region of the original steel main pipe. The authors showed that such a technique, called henceforth the ‘dual technique’, had more advantages than the conventional design ones. Indeed, the implementation of the dual technique resulted in a better tradeoff between the attenuation magnitude and the period relaxation of the transient pressure-wave response (Triki & Trabelsi 2021). Accordingly, thanks to reliable preliminary results already attained, this study planned to investigate a design tool based on the use of a PE penstock upstream and a short-section downstream of a main piping system. The key point intended from this strategy is to combine the merits of the high magnitude attenuation and the low relaxation of the transient pressure-wave oscillation period provided by the inline and branching strategies, respectively. In addition, this study inspects the usefulness of such a strategy compared to the renewal of the main piping system using PE materials.
Figure 2

Schematic overview of (a) the inline and (b) the branching techniques.

Figure 2

Schematic overview of (a) the inline and (b) the branching techniques.

Close modal

Within this framework, this paper initially gives a brief overview of the fixed grid-based method of characteristics (FG-MOC) used for the numerical discretization of the extended one-dimensional water hammer equations (E-1D-WH Eqs), then the proposed control design tool is studied and analyzed for a test case involving cavitation and, finally, a summary and conclusions are provided.

The E-1D-WH Eqs coupled with the Vitkovsky et al. and Kelvin–Voigt add-ons are (Pezzinga 2014, 2023; Triki 2018a, 2018b):
formula
(1)
where h is the pressure head; q is the discharge; A is the sectional area of the pipe; g is the gravitational constant; x is the axial distance; and t is the time.
In the above equations, the elastic pressure wave speed is expressed as:
formula
(2)
where is the mass density of the fluid (, for water); K is the bulk elasticity modulus of the fluid (, for water); is the dimensionless parameter accounting for the pipe constraint condition (, for thin-wall elastic pipes (Wylie & Streeter 1993)); e is the pipe wall thickness; D is the pipe diameter; is the elastic creep compliance of the pipe wall material, and is Young's modulus of the pipe wall material.
The quasi-steady head loss per unit length is determined based on the Hagen–Poiseuille or Colebrook–White formulas for laminar or turbulent flow, respectively:
formula
(3)
where is the kinematic viscosity of the fluid and R is the Darcy–Weisbach friction factor.
The unsteady head loss per unit length is computed as per Vitkovsky et al. (2000):
formula
(4)
where is the Vitkovsky decay constant, and or for or , respectively.
The retarded radial-strain is computed according to ‘Boltzmann superposition principle’ sketched in Figure 3:
formula
(5)
where is the creep compliance function associated with the rheological behavior of the pipe wall, and is the total circumferential stress, which is defined by:
formula
(6)
where p is the pressure and R is the Darcy–Weisbach friction factor.
Figure 3

Illustration of (a) stress ‘’ and (b) radial-strain ‘’ evolutions due to an instantaneous and constant pressure load in a viscoelastic material-based piping system.

Figure 3

Illustration of (a) stress ‘’ and (b) radial-strain ‘’ evolutions due to an instantaneous and constant pressure load in a viscoelastic material-based piping system.

Close modal
The typical procedure for solving the E-1D-WH Eqs within a multiple pipe framework is based on the method of characteristics established upon a fixed grid (MOC-FG). Briefly, the discretization of the equations set (1) using the MOC-FG leads to the following compatibility equations, along the characteristic lines (Wylie & Streeter 1993; Kaveh et al. 2010; Pothof & Karney 2012; Triki 2017, 2018a, 2018b):
formula
(7)
where j is the pipe number (); i is the section index of the pipe (); is the number of sections of the pipe; is the number of pipes; is the Courant number; is the time step; and is the space increment.
The discretization of the radial-strain is established from the linear viscoelastic Kelvin–Voigt model (Figure 4) as follows (Gally et al. 1979; Franke & Seyler 1983; Ramos et al. 2004; Brinson & Brinson 2008; Ferrante & Capponi 2017):
formula
(8)
where is the elastic creep compliance; and () are the creep compliance and retardation time coefficients associated with Kelvin–Voigt element; and is the number of Kelvin–Voigt elements.
Figure 4

Sketch of the generalized linear viscoelastic Kelvin–Voight model.

Figure 4

Sketch of the generalized linear viscoelastic Kelvin–Voight model.

Close modal
Hence, the positive and negative characteristic equations are expressed as (e.g. Wylie & Streeter 1993; Covas et al. 2004; Triki 2018a, 2018b; Trabelsi & Triki 2019, 2020a, 2020b, 2020c):
formula
(9)
where ; ; ; the quasi-steady friction coefficients: , , and , the unsteady friction coefficients: , , and ; (, a relaxation coefficient), and the coefficients associated with the viscoelastic pipe wall behavior: , , and (Triki 2018b).
So far, the MOC-FG procedure has been established for a one-phase flow regime. For a cavitating flow regime, the discrete gas cavity model (DGCM) may be adopted for a cavitating flow (Wylie & Streeter 1993; Urbanowicz et al. 2021). In such a situation, the volume of each gas pocket is:
formula
(10)
where is the void fraction at the initial pressure head ; is the elevation of the ith section of the jth pipe, and the subscripts ‘0’ and ‘v’ denote the reference and vapor conditions, respectively.
The discretization of Equation (10) using the FG-MOC leads to:
formula
(11)
where is the weighting factor.

It is worth noting that the one-phase water hammer solution (9) is re-established whenever the cavity collapses (i.e. ).

Finally, the hydraulic parameters at the inline or branched connection may be expressed for no flow storage and common hydraulic grade-line elevation modeling assumptions (Wylie & Streeter 1993):
formula
(12)
formula
(13)

Finally, the stability of the numerical solution (Equation (9)) is ensured according to the Courant–Frederic–Lewy rule: (Wylie & Streeter 1993).

It should be pointed out, herein, that the suitability and accuracy of the MOC-FG model were previously validated by the authors against the experiment of Covas et al. (2004) (e.g. Triki 2018a, 2018b).

The next section is dedicated to the examination of the key advantage of the proposed dual branching/inline technique-based upgrading design strategy for the attenuation of pressure-surge-wave magnitude, and the expansion of the pressure-wave oscillation period.

The downward-sloping main pipe (length , diameter , and thickness ), linking two pressure-controlled tanks and equipped with a control valve upstream (Figure 5(a)), is considered in this study. The steady-state regime corresponds to a constant value of flow velocity: and a pressure head value . The transient event relates to the full and instantaneous closure of the valve upstream, while maintaining a constant pressure head value at the downstream reservoir. The boundary equations corresponding to such a scenario can be expressed as:
formula
(14)
Figure 5

Basic layouts of (a) the original system and (b) the implementation of the dual PE upstream penstock and downstream inline short-section-based controlled system.

Figure 5

Basic layouts of (a) the original system and (b) the implementation of the dual PE upstream penstock and downstream inline short-section-based controlled system.

Close modal

For such a scenario, the dual penstock and short-section technique-based upgraded system setup consists in lumping a PE penstock upstream of the main piping system and substituting a short-section downstream of the main pipe with PE material (Figure 5(b)). For comparison purposes, the results related to the conventional and dual techniques built upon the inline or branching strategies are also presented (Triki 2017, 2018a; Trabelsi & Triki 2020c). The (dual sub-)short-section or (dual sub-)penstock lengths are: or ), respectively, and the diameters of the dual short-sections or dual penstocks are: .

The PE penstock and short-section types utilized in this study are the low-density PE (LDPE) and high-density PE (HDPE) whose creep compliance and retardation time coefficients embedded in the Kelvin–Voigt formula are: and , respectively (Keramat & Haghighi 2014). Besides, the elastic creep compliance of steel material is , which involves a wave speed value: .

The transient pressure-surge-wave responses are predicted using a MOC-FG-based algorithm for a specified time step input: .

Hereafter, the interpretation of the results uses the following rules:

The magnitude of the pressure-surge-wave is:
formula
(15)
The attenuation capacity of 1st pressure-wave peak or crest of the upgraded or renewed hydraulic system cases as compared with the original system based on the steel main piping system benchmark is:
formula
(16)
Besides, the phase shift between the upgraded or renewed hydraulic system cases and that involved by the steel main pipe-based original system benchmark is evaluated for the 1st cycle of pressure-wave oscillations as follows:
formula
(17)
Finally, the ratio of the 1st pressure head crest and phase shift associated with upgraded or renewed hydraulic system cases is:
formula
(18)

It is interesting to highlight, herein, that the last rule (Equation (18)) is used to evaluate the tradeoff between the attenuation effect of the magnitude of up- or down-surge and the expansion effect of the period of pressure-wave oscillations.

Figure 6 depicts the upstream pressure head traces corresponding to (i) the original hydraulic system cases utilizing steel, HDPE, or LDPE material types for the main piping system, and the upgraded system cases built upon: (ii) the conventional (or dual) PE inline (sub-)short-sections; (iii) the conventional (or dual) PE (sub-)penstocks, and (iv) the dual PE upstream penstock and downstream short-section. Besides, Table 1 appends the main key characteristics of the pressure-wave curves in Figure 6.
Table 1

Summary of 1st cycle water hammer wave characteristics in Figure 7 

System layoutsMaterialsT1 (s)Δhup-surge (m)Δhdown-surge (m)
Original main piping system Steel 0.472 41.8 32.6 
HDPE 2.321 5.8 8.5 
LDPE 4.351 2.6 5.2 
Inline technique HDPE short-section 0.798 27.9 32.6 
LDPE short-section 1.31 16.5 19.6 
Dual HDPE sub-short-sections 0.817 27.5 32.6 
Dual LDPE sub-short-sections 1.235 17.7 20.7 
Branching technique HDPE penstock 0.986 27.9 22.6 
LDPE penstock 1.875 16.5 13.2 
Dual HDPE sub-penstocks 0.817 27.5 32.6 
Dual LDPE sub-penstocks 1.26 17.7 20.7 
Dual branching and inline techniques Dual HDPE penstock/short-section 0.836 27.2 32.5 
Dual LDPE penstock/short-section 1.273 17.6 20.7 
System layoutsMaterialsT1 (s)Δhup-surge (m)Δhdown-surge (m)
Original main piping system Steel 0.472 41.8 32.6 
HDPE 2.321 5.8 8.5 
LDPE 4.351 2.6 5.2 
Inline technique HDPE short-section 0.798 27.9 32.6 
LDPE short-section 1.31 16.5 19.6 
Dual HDPE sub-short-sections 0.817 27.5 32.6 
Dual LDPE sub-short-sections 1.235 17.7 20.7 
Branching technique HDPE penstock 0.986 27.9 22.6 
LDPE penstock 1.875 16.5 13.2 
Dual HDPE sub-penstocks 0.817 27.5 32.6 
Dual LDPE sub-penstocks 1.26 17.7 20.7 
Dual branching and inline techniques Dual HDPE penstock/short-section 0.836 27.2 32.5 
Dual LDPE penstock/short-section 1.273 17.6 20.7 
Figure 6

Comparison of upstream pressure head signals, for the original hydraulic system and upgraded system cases.

Figure 6

Comparison of upstream pressure head signals, for the original hydraulic system and upgraded system cases.

Close modal

As can be seen from Figure 6, two issues arise during the mechanism of surge formation and propagation across the original system case utilizing the steel material for the main piping system: an excessive pressure head drop appears and the cavitation phenomenon develops as a result of the valve operation, which leads to severe pulsating water level oscillation. In this regard, the pressure head firstly drops to the saturated pressure head value of the liquid (i.e. ), and then rises sharply to the 1st pressure head peak , associated with the compression of the air pockets. This leads to down- and up-surge magnitudes of: and , respectively.

Figure 6 further illustrates that the pressure-wave curves associated with the upgraded systems employing HDPE unique or dual short-sections, or HDPE upstream short-penstock and downstream short-section are almost undistinguishable during the 1st cycle of pressure-wave oscillations. In addition, a perusal of Figure 6 reveals a slight attenuation of pressure-wave crest allowed by these upgraded system layouts. Consequently, these layouts are ruled out from further interpretation.

Incidentally, the remaining system setups perform a noticeable attenuation of 1st up- and down-surge magnitudes and, conjointly, lead to the expansion of the 1st period of the pressure-wave oscillations. In this regard, the rates of the last two effects depend upon the piping system layout and the employed PE type used for the short-section or penstock. Thereupon, to comprehensively explore the interplay between the aforementioned two effects, the magnitude-period nexus is reported in Figure 7.
Figure 7

Behavior of up- and down-surge magnitudes versus the phase shift, for the 1st cycle of pressure-wave oscillation (: phase shift; : ratios of the attenuations of pressure-surge-wave magnitudes and the relaxation of the period of pressure-wave oscillations).

Figure 7

Behavior of up- and down-surge magnitudes versus the phase shift, for the 1st cycle of pressure-wave oscillation (: phase shift; : ratios of the attenuations of pressure-surge-wave magnitudes and the relaxation of the period of pressure-wave oscillations).

Close modal

Unlike the steel material-based main piping system, a close examination of Figures 6 and 7 confirms that the cavitation phenomenon is mitigated in the renewed system cases utilizing HDPE or LDPE material for the main piping system. Specifically, the preceding system setups lead to a substantial attenuation of 1st pressure-wave magnitude. In this respect, the attenuation capacities of 1st pressure-surge crest and peak involved by an HDPE or LDPE main piping system are: { and } or { and }, respectively. Nonetheless, the foregoing system cases lead to a considerable relaxation of the 1st period of pressure-wave oscillations. Specifically, the phase shifts between the HDPE or LDPE type-based main pipe system layouts and their counterpart involved by the steel material-based pipe are: or , respectively. This leads to these ratios of 1st pressure-surge crest (or peak) and phase shift: { or }, and { or }, respectively.

Likewise, as per Figures 6 and 7, the cavitation is also palliated in the controlled system cases involving an HDPE or LDPE penstock, namely, as per Table 1, the attenuations of 1st pressure-surge crest (and peak) corresponding to upgraded systems made of an HDPE or LDPE penstock are: { and } or { and }, respectively. Conjointly, these upgraded system layouts lead to a significant relaxation of the 1st period of pressure-wave oscillations. Specifically, the phase shifts between HDPE or (LDPE) penstock-based controlled layouts and their respective case corresponding to the steel main pipe are: or , respectively. This, in turn, involves the ratios of the 1st pressure-surge crest (or peak) and phase shift: { or }, or { or }, respectively (Table 1).

On the other hand, Figures 6 and 7 illustrate practically undistinguishable pressure-wave curves for the upgraded system layouts based upon LDPE dual sub-short-sections, LDPE sub-penstocks, and LDPE upstream penstock and downstream short-section. Specifically, as per Table 1, these setups allow similar values of up- and down-surge magnitudes: and , respectively. However, these setups involve different period values of the 1st cycle of pressure-wave oscillations: , , and (Table 1). In other words, these upgraded system setups allow the attenuation capacity values: { and }, and phase shift values , , and , respectively. Consequently, they allow the magnitude ratios of the up- or down-surge to the 1st phase shift: { or }, { or }, and { or }, respectively.

A priori, this wave group allows an acceptable tradeoff between the 1st pressure-magnitude attenuation and the 1st cycle of pressure-wave oscillation period. Perhaps, the specific setup designed upon an LDPE upstream penstock and downstream short-section slightly outperforms other setups concerning the compromise between the two design criteria (i.e. magnitude attenuation and relaxation of oscillation period).

Based on the simulation results, the findings proved that the dual technique utilizing a PE penstock upstream and an inline short-section downstream could improve the steel material-based main piping system capacity, while providing a satisfactory tradeoff between the attenuation magnitude of pressure head, on the one hand, and the relaxation of the 1st period of pressure-wave oscillations, on the other hand. Precisely, the comparison between the different upgrading techniques or the renewal of the piping system and the original piping system suggests that the specific layout of the dual technique utilizing an LDPE penstock upstream and an inline short-section downstream is superior to other upgraded or renewed systems.

Additionally, the key advantage of the proposed technique relies on the easy and practical implementation merit as compared with other conventional water hammer control devices. For example, the replacement of burst pipe sections of an existing steel piping system with their respective ones composed of HDPE or LDPE types constitutes a straightforward application of this control measure.

The above-mentioned conclusions can provide some insights into developing practical design measures within the hydraulic network framework.

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

The authors declare there is no conflict.

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