On the transient flow behavior in pressurized plastic pipe-based water supply systems

Symbols

elastic wave-speed (m/s)

⁠,

constants for pump ⁠) curve (–)

⁠,

constants for pump (⁠⁠) curve (–)

cross-sectional area of the pipe (m²)

internal diameter of the pipe (m)

Young's modulus of the pipe-wall material (Pa)

pipe-wall thickness (m)

gravity acceleration (m/s²)

pressure-head (m)

dimensionless pressure-head (–)

pressure-head loss per unit length (–)

moment of inertia of the pump (kg.m²)

radius of gyration of the rotating mass (m)

_,

Ramos's unsteady decay coefficients (–)

pipe length (m)

rotational speed of the pump (rpm)

average flow-rate (m³/s)

resistance coefficient of the pipe (–)

instantaneous torque of the pump (Nm)

time (s)

weight of rotating parts plus entrained liquid (kg)

coordinate along the pipe centerline (m)

z

pipe centerline elevation with respect to pump intake level (m)

Greek symbols

dimensionless speed ratio (–)

dimensionless torque ratio (–)

dimensionless velocity (–)

time or space step increment (–)

valve closure function

Subscripts

0

steady-state/rated value

section index

number of sections

R

rated values

(+)/(–)

up-/down-surge

Superscripts

mesh index of the piping system

Acronyms

MOC

method of characteristics

HDPE/LDPE

high-/low-density polyethylene

1-D-EWHM

one-dimensional extended waterhammer model

Water supply systems are frequently subjected to waterhammer surge-waves due to temporal changes in flow conditions at any region in these systems. These changes may be intentional (e.g., startup, stoppage, or control of pump operation, and the setting of their respective valves), or accidental (e.g., failure of power to pumping stations). Waterhammer surge-waves can lead to serious mechanism failure, and can even jeopardize the operator's safety (Besharat et al. 2015; Coronado-Hernández et al. 2019). Therefore, they should be correctly anticipated in the design stage of water supply systems. Furthermore, the particular transient scenario associated with the failure of power to pumps, also referred to as ‘pump failure’, needs special consideration, since the hydraulic machine may go through abnormal operation modes during the transient scenario (Donsky 1961; Thorley & Faithfull 1992; Thanapandi & Prasad 1995; Almeida & Ramos 2010; Wan & Zhang 2011; Adamkowski et al. 2016; Wan et al. 2019). In this regard, the proper analysis of the transient behavior of the hydraulic machine may result in the abandonment of additional surge control devices (e.g., surge tanks) and, hence, may be economically advantageous.

On the other hand, plastic pipes are nowadays preferred to steel and cast-iron ones thanks to their reduced acquisition cost and easy installation, besides their advantageous chemical resistance proprieties. Compared with polyvinyl chloride (PVC), the high- or low-density polyethylene (HDPE or LDPE) is the most widely used. These materials provide a more significant attenuation of transient pressure fluctuations during high- and low-pressure surge loading than elastic ones (Aklonis et al. 1972). Physically, this effect is expected due to the viscous properties of plastic materials characterized by a retarded deformation component in conjunction with the immediate one, and observed in the case of elastic materials (such as steel and cast-iron materials). Indeed, during high- and low-pressure-surge loads, the viscoelasticity effects are exhibited by a delayed response of the radial expansion or contraction of the pipe-wall, which results in the magnitude attenuation and the period expansion of the pressure-wave oscillations. Incidentally, several numerical and experimental studies delineated that the visco-elastic properties of plastic pipe-wall materials could be beneficially used for suppressing excessive transient pressure-wave oscillations from water supply systems (Güney 1983; Ghilardi & Paoletti 1986; Triki 2016, 2017, 2018; Fersi & Triki 2019; Triki & Chaker 2019; Trabelsi & Triki 2019, 2020; Chaker & Triki 2020).

From a computational point of view, the analysis of fast transient waves in plastic pipes requires the implementation of a complete transient solver accounting for unsteady friction loss and non-elastic pipe-wall behavior (Ghidaoui et al. 2005; Duan et al. 2010; El-Gazzar 2017; Bertaglia et al. 2018).

Several approaches have been presented in the literature to enhance both friction losses and pipe-wall material effects which are identified as being the main sources of surge damping. For example, Triki (2017, 2018) used the extended waterhammer solver embedding the Kelvin–Voigt (Aklonis et al. 1972) and Vitkovský et al. (2000) formulations to account for the pipe-wall viscoelastic behavior and unsteady friction loss, respectively. Incidentally, Triki (2017, 2018) delineated that the combination of the two effects (i.e., pipe-wall viscoelasticity and unsteady friction) leads to a satisfactory agreement between measured data and numerical results. However, the consideration of a single effect could not reproduce experimental results. Subsequently, Ramos et al. (2004) introduced an alternative formulation, built upon the Vitkovský et al. (2000) approach, coupling the damping effect due to the pipe-wall behavior with the unsteady friction-loss effect. Incidentally, the Ramos et al. approach was based on splitting the unique coefficient of the Vitkovsky formulation (⁠⁠) into two distinct decay coefficients, applied to the convective and local acceleration terms in the momentum equation of the classical waterhammer solver. The authors (Ramos et al. 2004) verified that these coefficients were practically constant for different Reynold number values. Furthermore, they proved that their numerical solver reasonably reproduced experimental observations with regards to the damping and phase-shift pressure-wave. Compared with the Vitkovsky et al. formulation, that of Ramos could characterize unsteady frictions and pipe-wall viscoelastic behavior in simple and rapid implementation. Accordingly, the latter approach will be adopted in this work.

As previously stated, the detailed analysis of waterhammer transients caused by the power failure to pumping stations is crucial in the design stage of water supply systems, which are nowadays increasingly relying on plastic pipe-wall materials. Accordingly, exploration of the transient pressure-wave behavior due to pump failure in a plastic pipe-based water supply system was planned in this paper.

The following section outlines the discretization of the Ramos et al. formulation-based one-dimensional extended waterhammer model (1-D-EWHM) using the method of characteristics (MOC).

The numerical solution of the 1-D-EWHM (1) is commonly derived using the MOC.

In order to proceed further, it is necessary to stipulate boundary conditions. The conditions used for this study include the following.

(i) Power loss to a pump upstream:

The major problem in this case consists in the handling of pump characteristics and the resulting boundary conditions at the suction and discharge junctions of the pumping station. Principally, four parameters are required for the mathematical characterization of the pump behavior and including the discharge Q, the pressure-head H, the rotational speed N of the pump, and the net shaft-torque T. These parameters are commonly utilized in a dimensionless form relative to the rated values benchmark (i.e., ⁠, ⁠, ⁠, and ⁠, respectively). They are characterized basing on the Suter's head and torque curves which are generally collected for different specific speeds (i.e., and vs ⁠).

The boundary equations describing the pump failure condition may be established according to Figure 2.

Finally, Equations (11) and (12) are solved for and using a Newton–Raphson method-based numerical procedure.

Incidentally, the pressure-head value at the upstream extremity may be determined from the backward characteristic equation ⁠.

(ii) Constant level reservoir downstream:

Besides, the discharge value is determined using the forward characteristic equation ⁠.

To ascertain the validity of the developed MOC-based algorithm, the numerical solution is then compared with experimental data available from the literature.

First, data from laboratory experiments, conducted by Ramos et al. (2004), are used to validate the developed numerical model, within a reservoir-high-performance-polyethylene-pipe-valve system whose characteristics are: length ⁠, diameter ⁠, and wave-speed ⁠. The initial steady-state regime corresponds to a discharge value ⁠. Experimental data relate to a transient flow regime initiated by the rapid closure of the downstream valve.

Figure 3 provides a comparison of measured and numerical results computed from the 1-D-EWHM based on the Ramos et al. (2004) or the Vitkovský et al. (2000) formulations.

At first glance, it illustrates clearly that the wave signal computed based on the Ramos et al. formulation agrees well with the observed one, with regard to the magnitude and phase-shift, for the first and second cycles of pressure-head-wave oscillations. Contrarily, it suggests that the magnitude of the pressure-wave signal computed using the Vitkovsky et al. formulation exceeds the observed one. This, in turn, suggests that the use of two separate weighting coefficients (i.e., and ⁠), instead of a single one (i.e., ⁠), improved, significantly, the prediction accuracy of the unsteady friction and damping effects on the plastic pipe.

Second, the numerical solution is compared with the Hollander experimental data measured at the California Institute of Technology (Streeter & Wylie 1967). The hydraulic system consists of two identical centrifugal pumps in parallel (⁠⁠, ⁠, ⁠, ⁠, and dimensionless-homologous head and torque curves for the pump-specific speed in Figure 4, as taken from Streeter & Wylie's data) discharging in a single common buried concrete pipe (⁠⁠, ⁠). The experimental test conducted by Hollander corresponds to the simultaneous failure of both pumps.

Figure 5 compares the dimensionless pressure-head and speed signals recorded by Hollander and their counterparts provided by the numerical solution. It shows acceptable agreement between experimental and computed signals within the normal operation zone of the hydraulic machine (i.e., and ⁠). However, discrepancies appear between measured and computed results in the turbine zone (i.e., and ⁠). As per Streeter & Wylie (1967) and Wylie et al. (1993), these discrepancies are associated with the assumed pump characteristic data which are available only for the normal operating zone of the hydraulic machine.

In the following section, the developed solver is implemented to describe the waterhammer wave behavior initiated by the power failure to a pumping system.

The test case, considered in this study, relates to the pumping system shown in Figure 6. The elevation of the upstream reservoir above the downstream one is equal to: ⁠. The pipe and pump characteristics are summarized in Table 1, and the Suter's head and torque curves, corresponding to the pump-specific speed ⁠, are plotted in Figure 7. Initially, the pump operates at rated conditions, and the transient regime is caused by the failure of power to the upstream pump. Meanwhile, the check-valve begins to close after ⁠.

To highlight the effect of the pipe-wall material on the transient pressure-surge behavior, the steel, HDPE and LDPE materials are addressed in this study (⁠⁠, or ⁠, respectively (Keramat & Haghighi 2014)).

In the following, the characteristics of positive and negative pressure-surges are denoted by the symbols (+) and (−), respectively. Besides, indicators linked to the first cycle of pressure-wave oscillations are used to interpret different attributes of the pressure-head signals, including: (i) the magnitude of positive or negative pressure-surges: ⁠; (ii) the attenuations of positive and negative pressure-surge magnitudes involved by the HDPE or LDPE material-based piping system relatively to those corresponding to the steel material-based system: ⁠; (iii) the phase-shift between the pressure-wave oscillation signals performed by an HDPE or LDPE material-based pipe and the signal involved by a steel material pipe: ⁠; and (iv) the ratios between the relative attenuation of positive or negative pressure-surge and the phase-shift: ⁠.

Figure 8(a)–8(c) illustrate, respectively, the dimensionless pressure-head signals and valve position, the dimensionless velocity and speed signals, predicted at the upstream extremity of the piping system, and the zones of pump operation during the transient process involved by steel, HDPE or LDPE materials-based piping systems. Jointly, the main features of the pressure-head signals displayed in Figure 8(a) are listed in Table 2.

At first glance, the general wave pattern of Figure 8 indicates that the flow rapidly diminishes to zero and then reverses (Figure 8(b)). As the pump rapidly loses its forward rotation and reverses shortly after the flow reversal, a negative pressure-wave is propagated downstream from the pump (Figure 8(a)) and a positive pressure-wave is propagated upstream through the suction pipe. As the pump increases in speed in the reverse direction, it causes greater resistance to flow, which produces a high pressure in the discharge line (Figure 8(a)).

On the other hand, Figure 8(a) clearly illustrates expanded values of the wave oscillation periods involved in plastic pipe-based hydraulic systems as compared with their counterpart observed in a steel pipe-based system. Overall, the magnitudes of positive and negative pressure-surges and the values of wave oscillation periods depend on the pipe-wall material used for the piping system.

Referring to Figure 8(a)–8(c) and the data listed in Table 2, the following interpretations may be carried out for the first cycle of pressure-wave oscillations.

First, Figure 8(a) and Table 2 suggest that the larger positive and negative pressure-head magnitudes are involved by the case utilizing a steel pipe: and ⁠, respectively. Inversely, this case involves the shorter wave oscillation period: ⁠.

On the other hand, concerning the operating zones of the hydraulic machine during the transient process, Figure 8(c) shows that the machine operates in the normal pump zone (i.e., zone I and ⁠) up to ⁠, and then operates in the dissipation zone (zone IV) up to due to flow reversal (⁠⁠), while the inertia of the rotating parts maintains a positive rotation (⁠⁠). Subsequently, the hydraulic machine experiences reversal in both flow and rotational speed (i.e., and ⁠) beyond the time ⁠. In this case, the machine operates in the turbine zone (zone III).

Second, Figure 8(a) and Table 2 demonstrate that the HDPE material-based piping system case leads to less important magnitudes of positive and negative pressure-head surges and a more important wave oscillation period compared with the case utilizing a steel pipe. Specifically, the relative attenuations of positive and negative pressure-surge magnitudes, depicted between the HDPE material-based piping system and the steel material-based one, are equal to: ⁠. Also, the phase-shift induced between the latter two cases is equal to: ⁠. These results, in turn, imply the ratios between the relative attenuation of positive and negative pressure-surges and the phase-shift: ⁠.

Concerning the operating zones of the hydraulic machine involved in this case, Figure 8(b) and 8(c) indicate that the machine operates in the normal pumping zone (zone I) during ⁠; and, subsequently, in the dissipation zone (zone IV) up to ⁠. Thereafter, the machine operates in the turbine zone (zone III).

Third, Figure 8(a) and Table 2 show that the LDPE material-based piping system case leads to less important magnitudes of positive and negative pressure-head surges and a more important wave oscillation period compared with the case utilizing a steel pipe. Specifically, the relative attenuations of positive and negative pressure-surge magnitudes, depicted between the LDPE and steel setup-based piping systems, are equal to: and ⁠, respectively. In addition, the phase-shift induced between the two latter cases is equal to: ⁠. These results involve the ratios between the relative attenuation of positive and negative pressure-surges and the phase-shift: and ⁠, respectively.

Incidentally, it should be pointed out here that the LDPE setup-based piping system leads to more important attenuations of positive and negative pressure-surge magnitudes, and more expansion of pressure-wave oscillation period, than the one employing an HDPE material. Interestingly, the LDPE setup provides a larger ratio than HDPE (⁠ and ⁠). This implies that the LDPE setup provides a more important attenuation accompanied with a less important expansion of pressure-wave signal than the HDPE setup-based piping system. Thereupon, it may be concluded that the LDPE material-based piping system case allows a better trade-off between the attenuations of positive and negative pressure-surge magnitudes and the expansion of the pressure-wave oscillation period compared with the case using an HDPE material.

Regarding the operating zones of the hydraulic machine involved in this case (i.e., LDPE piping system), Figure 8(b) and 8(c) indicate that the machine operates in the normal pumping zone (zone I) during and, subsequently, in the dissipation zone (zone IV) up to ⁠. Thereafter, the machine operates in the turbine zone (zone III).

Ultimately, this study suggests that the use of a plastic pipe-wall material-based pumping system provides a large attenuation effect of the first pressure-head rise and drop, associated with an expansion effect of the period of pressure-wave oscillations, for a pump failure-initiated transient scenario. In addition, the pressure attenuation and period expansion effects are pronounced when using an LDPE pipe-wall material compared with an HDPE material. Specifically, results indicate that the LDPE material allows a better trade-off between the two last effects than HDPE.

Although the present work addressed a particular procedure for valve closure, future investigations on the dependency of the transient pressure-wave behavior upon the check valve setting modes may represent an extension to this study, in particular the determination of the optimal valve closure procedure.

The authors declare that there is no conflict of interest.

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