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A stepwise method is one of the efficient approaches in the calibration of transient flow in viscoelastic pipes. However, there is the lack of comprehensive research on the calibration method for retarded time and creep compliance. Therefore, the sensitivity of the order of magnitude of the creep parameters in the presence of interactions to transient flow pressure damping and phase in the case of the two-element Kelvin–Voigt model is investigated. Also, the calibration methods of retarded time and creep compliance in the stepwise method of transient flow parameter calibration for viscoelastic pipes are proposed. The results indicate that the creep compliances do not affect the phase of the pressure fluctuations when the selected retarded times are greater than the order of magnitude 10−1. For the first Kelvin–Voigt element, when the order of magnitude of the retarded time exceeds 10−2, an increase in creep compliance results in an increase in the degree of damping of pressure fluctuations. When the order of magnitude of the retarded time is less than 10−2, the rule is reversed. For the second Kelvin–Voigt element, an increase in creep compliance results in an increase in the degree of damping of transient flow pressure fluctuations independent of the retarded time.

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  • The calibration of creep parameters in a stepwise method based on the results of sensitivity analysis is presented.

  • The rule of two-element Kelvin–Voigt model creep parameters on transient flow pressure fluctuations is clarified.

  • A calibration reference order of magnitude for the retarded time is presented.

calibration method, creep parameters, transient flow, viscoelastic pipes
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Plastic pipes are used extensively in water supply systems and play a significant role in the transportation of fluids. These pipes are made of viscoelastic materials that exhibit both viscous and elastic mechanical behaviors, distinguishing them from elastic pipes (Kamil et al. 2020). When the pump abruptly ceases operation or the valve closes rapidly, a surge in pipe pressure occurs, resulting in a water hammer phenomenon that compromises the integrity of the piping system (Kamil & Margarida 2023). Thus, the calculation of transient flow is vital to prevent the damage of water hammer. In the simulation of transient flow in viscoelastic pipes, the creep parameters significantly affect the pressure fluctuation. However, the influence of the creep parameters on transient pressure fluctuations does not abide by a single rule. This makes accurate parameter calibration challenging when simulating transient flows in viscoelastic pipes. Therefore, it is necessary to investigate the influence of the creep parameters on the transient flow pressure variability in viscoelastic pipes to facilitate the design and operation of viscoelastic pipe systems.

Calibration of the parameters for transient flows in viscoelastic pipes is a complex task because of the presence of numerous variables (especially for creep parameters) and their effects on pressure fluctuations. Covas et al. (2004) employed the least-squares error method to calibrate creep parameters and examined the influence of the number of Kelvin–Voigt (KV) elements on the accuracy of calibration results. The findings suggest that incorporating three or more KV elements has a negligible impact on the precision of parameter calibration. Pezzinga (2014, 2023) and Pezzinga et al. (2016)) utilized experimental data from high-density polyethylene (HDPE) pipes to compare the simulation results of one-dimensional (1D) elastic and viscoelastic models. The 1D viscoelastic model exhibited superior accuracy in simulating peak pressure damping compared with the 1D elastic model. A micro-genetic algorithm was employed to calibrate the parameters of both 1D and quasi-two-dimensional (quasi-2D) friction models under variations of pipe lengths and flow rates. It was found that there is a linear correlation between the retarded time and the period in the case of the single-element KV model. Kamil et al. (2020) revealed the significant role of unsteady friction effects in plastic pipes during transient flow; however, these effects may be obscured during the calibration of the creep parameters. Fathi-Moghadam & Kiani (2020) designed a transient flow experiment for HDPE pipe networks, where a genetic algorithm and the sequential quadratic program method were employed to calibrate the creep parameters. Sun et al. (2022b) compared the applicability of a quasi-steady friction model and a modified Brunone model at different water temperatures. The results suggest that, particularly in high-water-temperature scenarios, the quasi-steady friction model can accurately simulate the pressure fluctuation in viscoelastic pipes. Pan et al. (2021) utilized the frequency response function to identify the characteristics of viscoelastic pipes by employing the transient wave analysis method. Jalil et al. (2021) investigated the impact of the number of KV elements on creep parameter calibration. They found that, for pipes exceeding 1,000 m, the highest level of accuracy was achieved when using the KV model with three or four KV components. Moreover, they observed that employing a KV model with two elements is capable of simulating transient flow accuracy for pipes longer than 720 m. Monteiro et al. (2023) proposed a model based on a quasi-2D model capable of dealing with the viscoelastic behavior of pipes, which makes it possible to calculate the energy dissipation rate in elastic and viscoelastic pipes directly. They discovered that the viscoelastic behavior accelerates the decay of pressure oscillations. Tjuatja et al. (2023) summarized that there are two methods for calibrating pressure fluctuations of transient flow in viscoelastic pipes, including an optimal algorithm based on the inverse transient analysis (ITA) and a stepwise method. Calibration methods using algorithms can be speedy and save effort but tend to get trapped in local optima and cannot intervene in the calibration process. The stepwise method can intervene in the calibration process to make it easier to adjust, but performing calibration requires a lot of calibration experience in repeated trial and error calculations, and the calibration process is very time-consuming.

Numerous researchers have addressed transient flow pressure evolution in viscoelastic pipes at different water temperatures, Reynolds numbers, and other related variables. Covas et al. (2005) conducted transient flow experiments on HDPE pipes and recorded the evolution of stress and strain to analyze the impact of pipe wall viscoelasticity on pressure fluctuation. Soares et al. (2008) executed transient flow experiments in polyvinyl chloride pipes and concluded that creep effects are influenced not only by pipe and temperature conditions but also by axial and circumferential constraints. Wahba (2017) utilized a dimensional analysis method to develop a quasi-2D model for transient flow in viscoelastic pipes under laminar flow and determined the dimensionless parameters affecting the quasi-2D model using magnitude analysis. The results revealed that, as the pipe diameter and wave speed increase, the impact of viscoelasticity on pressure damping becomes more pronounced. Kumar & Kumar (2020) involved transient flow experiments on a steel pipe, fiberglass-reinforced plastic (GRP), and a composite material consisting of both. Research has established that GRP pipes exhibit superior effectiveness in reducing the water hammer pressure. Abdeldayem et al. (2021) investigated the accuracy of different unsteady-state models based on engineering practices and concluded that the modified Brunone model is relatively suitable for simulating pressure fluctuations in transient flows. Ulugmurad et al. (2021) developed the energy consumption calculation formula for the unsteady process of a water hammer by examining the impact of friction on the maximum pressure, and they validated its feasibility experimentally. Kai et al. (2022) employed a quasi-2D model to investigate the variations in frictional and viscoelastic terms at different Reynolds numbers, revealing that both the work of the friction term and that of the viscoelastic term increase with increasing Reynolds number. Kamil et al. (2023) further extended the model development for quasi-steady-state and unsteady-state assumptions of hydraulic resistance and derived explicit analytical formulas for wall shear stresses.

As previously mentioned, previous studies have examined the calibration methods for transient flow pressure fluctuations in viscoelastic pipes and the relationship between the creep parameters of the single-element KV model and the transient pressure fluctuations; however, there is a lack of comprehensive research on the calibration method for retarded time and creep compliance in the stepwise method and the effect of the compliance parameters on the transient pressure fluctuations in the two-element KV mode. In this study, the influence rule of creep parameters on peak damping and phase offset of transient flow pressure fluctuations in the two-element KV mode are analyzed using one-way analysis of variance (ANOVA) and multiparameter sensitivity analysis. Based on the influence rule of the creep parameters and pressure fluctuations, the proposed stepwise method to calibrate the transient flow pressure fluctuation of the viscoelastic pipeline and the specific calibration method of the creep parameters of viscoelastic pipelines is given. The efficiency and accuracy of the stepwise method are improved and the applicability of the method is widened. The proposed stepwise method is utilized to calibrate the transient flow pressure fluctuations of polypropylene random (PPR) pipes and HDPE pipes with different water temperatures. The recommended order of magnitude of the retarded time in the calibration of transient flow pressure fluctuations of viscoelastic pipes is also proposed based on the rule obtained from the sensitivity analysis. The accurate calibration of creep parameters facilitates the design and operation of viscoelastic piping systems.

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Governing equations

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The governing equations for transient flow in viscoelastic pipes consisting of continuity and momentum equations are as follows (Wylie & Streeter 1993; Covas et al. 2005):
(1)
(2)
where P is the pressure, g is gravitational acceleration, V is the flow velocity, A is the pipe's cross-sectional area, a is the pressure wave speed, x is the distance along the pipe axis, t is the time, D is the diameter of the pipe, ρ is the fluid density, is the wall shear stress, and is the retarded strain.
The wall shear stress using the Darcy–Weisbach equation is expressed as (Zhao & Ghidaoui 2003)
(3)
where f is the frictional resistance coefficient.

KV model

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The KV model comprises a spring and a viscous damper, as depicted in Figure 1. The creep relaxation behavior of viscoelastic pipes can be mathematically described as (Keramat & Haghighi 2014; Gong et al. 2016)
(4)
where J0 is the instantaneous creep compliance, J0 = 1/E0, E0 is the instantaneous elastic modulus, Jk is the creep compliance of the kth element, is the retardation time of the kth element, and N is the number of elements.
Figure 1
Generalized KV model.
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Generalized KV model.

Figure 1
Generalized KV model.
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Generalized KV model.

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Numerical scheme

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The method of characteristics is employed to solve the governing equations in Equations (1) and (2). The resulting ordinary differential equation is
(5)
The relationship between the retarded strain and time is considered (Covas et al. 2005) as follows:
(6)
The retarded strain is given by (Kamil et al. 2016)
(7)
where is the constraint coefficient, is the retarded strain of the kth element, and E is wall thickness.
(8)
Integrating along characteristic lines, the algebraic equations are derived as follows:
(9)
(10)
By substituting Equation (7) into Equations (9) and (10), the pressure and velocity at (n + 1) are expressed as
(11)
(12)
with
(13)
where CR and CS can be obtained as
(14)
(15)

Valve closing curve based on ITA

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By employing the ITA methodology, the valve closure curve can be derived by utilizing the known values of extremity pressure and the preceding time level of the extremity pressure. The flow velocity at the downstream valve can be expressed as
(16)
where is the flow velocity and dT = dx/a is the ratio of the pipe length to the product of the flow velocity and grid number and m is the total grid number.
The valve closing curve is
(17)
(18)
where CV is the value of the valve closing, P0 is the initial pressure, V0 is the initial flow velocity, and is the valve closing.
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Experimental setup

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The experimental setup for investigating the transient flow in viscoelastic pipes is shown in Figure 2. It is composed of PPR pipes, HDPE pipes, a variable-frequency pump, a temperature gauge, a flow transducer, high-precision pressure transducers, a fast-closing solenoid valve, and a thermostatic water tank. The experimental pipes with a diameter of DN20 with a wall thickness of 3 mm are utilized. The frictional resistance coefficient, f, is measured to be 0.04. Furthermore, the total length of the pipe system is 37.8 m. The size of water tank is 1.2 m × 1.2 m × 1.2 m. Variations in water temperature are realized using an internal tank heater, and a thermometer is used to monitor the real-time temperature of the fluid in the tank and main pipe. The pump is a Leo YS3-90S-2 variable-frequency pump with a 57 m H2O head and 4 m3/h flow rate. The pressures at the front and extremity of the pipes are measured using three high-precision pressure transducers. A HELM HM90-H1-2-V2-F2 high-precision pressure sensors are the range of −10 to 120 m, the intrinsic frequency is 500 kHz to 1 MHz, and the integrated accuracy is 0.25% full scale. The flow meter used is a Mike STLD25031111GCB, and the measuring range is 0–2 m3/h. A Danforth quick-closing valve with a closing time of 0.4 s was used. USB-1252 series data acquisition was performed using a 16-channel acquisition card from Beijing Smacq with a maximum sampling frequency of 500 kS/s. The experimental parameters for all cases are listed in Table 1.
Table 1

Experimental data for different cases

CasePipe materialsWater temperature (°C)Initial pressure head (m)Flow rate (m3/h)Water density (kg/m³)Reynolds number
Case 1 PPR 24 62.6 2.59 997.02 71,937 
Case 2 PPR 30 62.3 2.57 995.18 81,829 
Case 3 PPR 40 63.1 2.61 991.94 96,269 
Case 4 HDPE 24 58.7 2.40 997.02 75,826 
Case 5 HDPE 30 59.2 2.42 995.18 86,252 
Case 6 HDPE 40 58.6 2.39 991.94 101,473 
CasePipe materialsWater temperature (°C)Initial pressure head (m)Flow rate (m3/h)Water density (kg/m³)Reynolds number
Case 1 PPR 24 62.6 2.59 997.02 71,937 
Case 2 PPR 30 62.3 2.57 995.18 81,829 
Case 3 PPR 40 63.1 2.61 991.94 96,269 
Case 4 HDPE 24 58.7 2.40 997.02 75,826 
Case 5 HDPE 30 59.2 2.42 995.18 86,252 
Case 6 HDPE 40 58.6 2.39 991.94 101,473 
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Figure 2
Schematic diagram of the experimental setup.
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Schematic diagram of the experimental setup.

Figure 2
Schematic diagram of the experimental setup.
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Schematic diagram of the experimental setup.

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Creep tests

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Creep tests were conducted under controlled experimental conditions, with a temperature of 25 °C and a relative humidity of 75%. The duration of the experiment was set to 1 h while a constant load of 10 N was maintained. Curve fitting analysis was performed on the experimental data obtained from the creep tests, as shown in Figure 3.
Figure 3
Curve fitting of creep tests of PPR and HDPE pipes. (a) PPR pipes; (b) HDPE pipes.
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Curve fitting of creep tests of PPR and HDPE pipes. (a) PPR pipes; (b) HDPE pipes.

Figure 3
Curve fitting of creep tests of PPR and HDPE pipes. (a) PPR pipes; (b) HDPE pipes.
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Curve fitting of creep tests of PPR and HDPE pipes. (a) PPR pipes; (b) HDPE pipes.

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Calibration method

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Transient flow pressure fluctuations in viscoelastic pipes with instantaneous valve closures can be calibrated using the stepwise method (Tjuatja et al. 2023). However, the retarded time and creep compliance in the stepwise method are subject to repeated trial and error calculations requiring knowledge of the law of influence of each parameter. The specific calibration rules for the retarded time and creep compliance in stepwise methods have not been comprehensively investigated. Consequently, the calibration sequence for creep compliance and retarded time is also important in the stepwise calibration method. In this study, the stepwise method was used for the transient flow calibration of viscoelastic pipes based on sensitivity analysis of the relationship between creep parameters and transient pressure fluctuations. The proposed method improves the efficiency of the stepwise calibration method and extends the scope of the calibration. The specific procedure is shown in Figure 4.
Figure 4
Stepwise method for transient flow in viscoelastic pipes.
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Stepwise method for transient flow in viscoelastic pipes.

Figure 4
Stepwise method for transient flow in viscoelastic pipes.
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Stepwise method for transient flow in viscoelastic pipes.

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First, after inputting the first basic parameter, the wave speed is calculated based on the experimental pressure, which is then utilized for subsequent calibration.

Second, the first value of the valve closing time is determined by analyzing the first peak of the experimental pressure. Furthermore, the valve closing parameters are calibrated based on the experimental data of the first peak pressure and wave.

Third, the creep parameters measured by the creep test were used as reference starting values for calibrating the creep parameters. The retardation time is determined using creep tests and experimental pressure fluctuations. The creep compliance is determined by analyzing the peak damping and phase offset of the experimental pressure curves. The creep compliance of the first KV element is determined from the phase offset of the late fluctuations of the pressure fluctuations, and then the creep compliance of the second KV element is determined using the late peak damping of the pressure fluctuations.

Finally, the wave speed, creep compliance, and valve closing parameters are adjusted sequentially, and the optimal calibration results are selected to output the parameters.

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The two-element KV model meets the transient flow pressure fluctuation calibration requirements for viscoelastic pipes. Therefore, a two-element KV model was adopted in this study. As shown in Table 2, the wave speed obtained from the calibration exhibits a close resemblance to those calculated using the steady-state pressures attained before and after the first peak, with a discrepancy of no more than 3.5%. Therefore, the wave speed was determined by measuring the times taken to reach steady-state pressure before and after the first peak, which can serve as an appropriate first value for calibrating the wave speed during the parameter calibration of viscoelastic pipes.

Table 2

Results of wave speed (m/s)

Water temperature (°C)MaterialsCalculatedCalibration
24 PPR 458.1–432.8 470 
HDPE 401.5–368 400 
30 PPR 438.1–411.4 445 
HDPE 393.2–355.9 390 
40 PPR 426.9–390.2 410 
HDPE 381.7–346.9 370 
Water temperature (°C)MaterialsCalculatedCalibration
24 PPR 458.1–432.8 470 
HDPE 401.5–368 400 
30 PPR 438.1–411.4 445 
HDPE 393.2–355.9 390 
40 PPR 426.9–390.2 410 
HDPE 381.7–346.9 370 
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The closing curve of the instantaneous closing valve can be determined using ITA (Meniconi et al. 2012a, 2012b). The experiment involved an indirect closing valve of 0.4 s. An attempt was made to determine whether the valve curve calculated by ITA in the case of indirect valve closing can accurately simulate pressure.

The calibration accuracy of the two-stage valve closing curve and the reverse calculation valve closing curve model are analyzed in this section to assess their accuracy. The shape of the two-stage closing curve is mainly affected by the inflection point and moment point of valve closure. Moreover, by calibrating the time of the first peak of the experimental pressure, it is possible to obtain a more accurate valve closing time. Mean absolute percentage error (MAPE) reflects the accuracy of the peak pressure decay simulation. The evaluation is based on MAPE values, which are used to determine the level of calibration achieved by both methods (Sun et al. 2022b).
(19)
where is the ith simulated peak and valley pressure, Hi,exp is the ith experimental peak and valley pressure, and k is the number of pressure extrema.
The calibration results for PPR pipes at 24 °C are shown in Figure 5, which illustrates the valve closure curves calculated using ITA and the two-stage valve closure curves. When different water temperatures were simulated, altering the valve closing curves affected only the amplitude and shape of the first wave peak. As illustrated in Figure 5(b), the MAPE value decreased with increasing water temperature, indicating the enhanced accuracy of the two calibration results. However, at 40 °C, although the MAPE value for the calculated valve closing curve was lower, the simulation results for the two-stage valve closing curve exhibited greater stability. That's because when employing ITA to calculate the valve closure curve, the inverse valveclosure curve model exhibited significant reliance on the wave speed and creep parameters because of their essential rolein the calculation process.
Figure 5
Pressure head for different valve closing schemes. (a) Simulation of pressure at 24 °C; (b) MAPE at different water temperatures.
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Pressure head for different valve closing schemes. (a) Simulation of pressure at 24 °C; (b) MAPE at different water temperatures.

Figure 5
Pressure head for different valve closing schemes. (a) Simulation of pressure at 24 °C; (b) MAPE at different water temperatures.
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Pressure head for different valve closing schemes. (a) Simulation of pressure at 24 °C; (b) MAPE at different water temperatures.

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The pressure variations of experimental data and calibration results of PPR pipes and HDPE pipes at temperatures of 24 and 30 °C are compared in Figure 6. The first peak pressure of the pressure wave decreases with increasing water temperature. The peak pressure of the PPR pipes is higher than that of the HDPE pipes at the same water temperature while exhibiting a smaller fluctuation period. The phase offset of the pressure in the PPR pipes exhibits less change compared with that in the HDPE pipes with increasing water temperature. These findings indicate that the water temperature has a greater impact on pressure fluctuations in HDPE pipes.
Figure 6
Pressure head at different water temperatures. (a) Simulation of pressure in case 1, (b) simulation of pressure in case 4, (c) simulation of pressure in case 2, and (d) simulation of pressure in case 5.
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Pressure head at different water temperatures. (a) Simulation of pressure in case 1, (b) simulation of pressure in case 4, (c) simulation of pressure in case 2, and (d) simulation of pressure in case 5.

Figure 6
Pressure head at different water temperatures. (a) Simulation of pressure in case 1, (b) simulation of pressure in case 4, (c) simulation of pressure in case 2, and (d) simulation of pressure in case 5.
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Pressure head at different water temperatures. (a) Simulation of pressure in case 1, (b) simulation of pressure in case 4, (c) simulation of pressure in case 2, and (d) simulation of pressure in case 5.

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Figure 6(a) and 6(b) reveals that employing the calibrated creep parameter fitting for calibration results in good agreement between the simulation results and experimental data in terms of phase and pressure damping at later stages because the wave speed is adjusted. The pressure peak differs greatly between the simulation results obtained using the creep test and the experimental data. Figure 6(c) and 6(d) illustrates the pressure curves calibrated using the above method and using the creep test as the initial intrinsic parameter. The calibration results agree well with the experimental pressure curves.

Figure 6 shows that the calibration results of the PPR pipes express the phase deviation later, and there is also a discrepancy in the pressure values, whereas the calibration results of the HDPE pipes perform fine phase calibration later, but there is a significant discrepancy in the pressure values. The accuracy of the results for both pipes improves with increasing water temperature. The accuracy of the quasi-steady friction model increases with increasing water temperature (Sun et al. 2022b).

Accuracy analysis of simulation results

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The root mean square error (RMSE) measures the deviation of the predicted value from the experimental value and is employed as an evaluation metric to assess the accuracy of numerical findings (Sun et al. 2022a). The equation is
(20)
where is the experimental pressure data at computational node M, is the simulation result at computational node M, and m is the total number of computational nodes.
Figure 7 shows the RMSE values of the calibration results for different water temperatures. As the figure reveals, the largest RMSE values in calibration results occur at 24 °C. This can be attributed to the amplified pressure change induced by the water hammer phenomenon at this temperature. At 24 °C, to ensure the accuracy of the first peak, there is a significant disparity in the value of the first trough; however, as the water temperature increases, this effect gradually diminishes. The RMSE decreases with increasing water temperature. The accuracy of the transient flow calibration model for the quasi-steady friction model is enhanced at elevated water temperatures (Sun et al. 2022b).
Figure 7
RMSE values at different water temperatures.
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RMSE values at different water temperatures.

Figure 7
RMSE values at different water temperatures.
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RMSE values at different water temperatures.

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Process of sensitivity analysis

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Pezzinga et al. (2016) investigated the creep parameters of the single-element KV model on transient flow pressure fluctuations. In this paper, the rule of influence of creep parameters on transient pressure fluctuations in the two-element KV model is investigated. The influence of creep parameters on transient pressure fluctuations in the two-element KV model was investigated based on transient pressure fluctuations in PPR pipe at 24 °C. The two-element KV model has four parameters including τ1, J1, τ2, and J2. The effects of retarded time and creep compliance on the transient pressure fluctuations are investigated separately. A one-way ANOVA is employed to calculate the significance of the creep compliance and retarded time on the transient pressure fluctuations separately. However, with the correspondence between the retarded time and creep compliance, the second KV element may affect the relationship between the first KV element and the transient pressure fluctuations. Therefore, when the influence rule of the first KV element on transient pressure fluctuations is investigated further, the influence of the retarded time of the second KV element on the influence rule is considered. In investigating the relationship between the first KV element and transient pressure fluctuations, τ2 is varied for different orders of magnitude, and J2 is fixed. Then, τ1 and J1 are varied, and their effects on pressure damping and phase offset are analyzed. The influence of the first KV element is considered equally in research on the second KV element. However, it is necessary to ensure that the value of τ2 is greater than that of τ1. The specific sensitivity analysis process in the following parts is outlined in Figure 8.
Figure 8
Sensitivity analysis process diagram.
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Sensitivity analysis process diagram.

Figure 8
Sensitivity analysis process diagram.
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Sensitivity analysis process diagram.

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Single-parameter sensitivity analysis

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One-way ANOVA is a method for determining whether one factor has a significant effect on the results of an experiment. Given that creep parameters affect the phase and damping of the pressure wave, ANOVA (Olha et al. 2022) is used to investigate further the impact of retarded time, τ1 and τ2, on pressure fluctuations in the two-element KV model. However, a direct correspondence exists between creep compliance and retarded time. Consequently, in the sensitivity analysis, the values for the creep parameter are determined as J1 = 0.1 and J2 = 1.5 GPa−1, respectively. The initial values of retarded time are determined to be τ1 = 0.1 and τ2 = 1 s. To enhance the accuracy of the sensitivity analysis, other parameters are determined based on the experimental conditions. Each case was simulated three times to mitigate potential errors. The time discrepancy between the pressure peaks serves as an indicator of the phase offset, whereas the pressure difference between the peaks is utilized to quantify the degree of damping.

Consulting the F-distribution table reveals that F0.01(dfA,dfe) = F0.01(3,8) = 7.59, F0.05(3,8) = 4.07. Here, dfA is the degree of freedom of factor A, and dfe is the degree of freedom of the error. In Figure 9, ‘*’ represents the significance of the effect of retarded time on pressure damping and phase offset. To investigate the effects of τ1 and τ2, the same method is used to quantify their effects on pressure by calculating the peak difference and time difference between peaks 1–2, 1–3, and 1–4. The calculated significance values are shown in Figure 9. The effects of τ1 and τ2 on the phase offset of the pressure fluctuations are less significant than their effects on the pressure damping.
Figure 9
Sensitivity analysis of τ1 and τ2 for peak phase offset and peak pressure damping.
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Sensitivity analysis of τ1 and τ2 for peak phase offset and peak pressure damping.

Figure 9
Sensitivity analysis of τ1 and τ2 for peak phase offset and peak pressure damping.
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Sensitivity analysis of τ1 and τ2 for peak phase offset and peak pressure damping.

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The effect of creep compliance on the phase offset of pressure fluctuations and the change in pressure damping with a defined retarded time is also analyzed using ANOVA. The values for retarded time are determined to be τ1 = 0.0225 s and τ2 = 1.1 s. The initial values of the creep parameters are set as J1 = 0.1 GPa−1 and J2 = 0.8 GPa−1, respectively. These findings suggest that J1 has a highly significant influence on both damping and phase, whereas J2 primarily affects damping, with a consequential impact on phase. To enhance the precision of this conclusion, a reanalysis the simulations were reanalyzed using experimental data (Covas et al. 2005), yielding consistent outcomes. The calculated significance is presented in Figure 10, where it can be observed that the influence of J2 on the phase under pressure exhibits a lower magnitude than its impact on damping.
Figure 10
Sensitivity analysis of J1 and J2 for peak phase offset and peak pressure damping. (a) Significance for J1; (b) Significance for J2.
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Sensitivity analysis of J1 and J2 for peak phase offset and peak pressure damping. (a) Significance for J1; (b) Significance for J2.

Figure 10
Sensitivity analysis of J1 and J2 for peak phase offset and peak pressure damping. (a) Significance for J1; (b) Significance for J2.
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Sensitivity analysis of J1 and J2 for peak phase offset and peak pressure damping. (a) Significance for J1; (b) Significance for J2.

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Multiparameter sensitivity analysis

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Because the retarded time and creep compliance correspond to each other, an interaction between them is possible. The two-element KV model (τ1 = 0.0225 s, J1 = 0.1 GPa−1 and τ2 = 1.1 s, J2 = 1.5 GPa−1) was selected for calculation in this study. The relationship between τ1 and τ2 results in variations in the impacts of J1 and J2 on the pressure obtained. The pressure curves were investigated by systematically varying the creep compliance (Δ = 0.5 GPa−1) while altering the orders of magnitude of τ1 and τ2, followed by simulating multiple tests with different retarded times.

The impact of creep compliance on the pressure is determined by its corresponding retarded time. The rules governing the influences of J1 and J2 on pressure exhibit variations when there is a significant variation in the values of τ. When τ2 = 1.1 s and J2 = 1.5 GPa−1, the pressure fluctuation attenuation affected by the different J1 values is investigated by changing the order of magnitude of τ1. When τ1 = 0.0011 s and J1 = 2 GPa−1, the pressure fluctuation attenuation influenced by the different J2 values is investigated by varying the order of magnitude of τ2. To facilitate the analysis of the effects of creep parameters on pressure damping, a local transient analysis is employed to define the contribution of pressure damping (Duan et al. 2017).
(21)
where superscript T denotes the initial results, superscript M denotes the simulated pressure results, and subscript p denotes the point at the pressure peak.
Figure 11 shows the contribution of the pressure damping for different orders of magnitude of τ. The variation of the creep parameters does not affect the time of appearance of the first pressure peak of the pressure fluctuation. The effect on the value of the first pressure peak is also minor, with a maximum of no more than 10%. As Figure 11(a) shows, when the order of magnitude of τ1 is less than 10−1, increased creep compliance leads to enhanced pressure damping. Furthermore, when the creep compliance is varied, the transient pressure fluctuations are offset in phase. In contrast, Figure 11(b) reveals that, when the order of magnitude of τ1 is more than 10−1, the creep compliance does not affect the phase offset. Figure 11(c) and 11(d) shows that τ2 has the same variation rule as τ1. However, unlike the rule of τ1, the contribution of the pressure damping decreases as J2 increases. This indicates that J2 also affects pressure damping; however, as J2 increases, the degree of pressure damping decreases. However, when the order of magnitude of the retarded time is closer to 10−1, the effect of creep compliance on pressure damping is greater than that of others.
Figure 11
Contribution of the pressure damping at different orders of magnitude. (a) τ1 (0–10−1), (b) τ1 (>10−1), (c) τ2 (0–10−1), and (d) τ2 (>10−1).
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Contribution of the pressure damping at different orders of magnitude. (a) τ1 (0–10−1), (b) τ1 (>10−1), (c) τ2 (0–10−1), and (d) τ2 (>10−1).

Figure 11
Contribution of the pressure damping at different orders of magnitude. (a) τ1 (0–10−1), (b) τ1 (>10−1), (c) τ2 (0–10−1), and (d) τ2 (>10−1).
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Contribution of the pressure damping at different orders of magnitude. (a) τ1 (0–10−1), (b) τ1 (>10−1), (c) τ2 (0–10−1), and (d) τ2 (>10−1).

Close modal
Figure 12 shows the influence of the creep parameters on the variation of transient pressure fluctuations. Because the effects of the creep parameters on the first peak of the transient pressure fluctuations are minor, the second peak was examined in this study. The phase offset and pressure damping shown in Figure 12 were produced by the difference in the timing and peak value of the appearance of the second peak as the creep compliance varied from 1 to 2 GPa−1. The reason for the results shown in Figure 12 is the requirement to ensure that τ1 is greater than τ2 in the two-element KV model. Figure 12(a) and 12(c) depicts the effects of the creep parameters on the phase of the transient pressure fluctuations. The creep compliance variation has no effect on the phase when the order of magnitude of τ1 exceeds 10−1. The rules are consistent when the order of magnitude of τ2 exceeds 10−1. Figure 12(b) and 12(d) demonstrates the effects of the creep parameters on the pressure damping of the transient pressure fluctuations. The effects of τ1 and τ2 on the pressure damping of transient pressure fluctuations exhibit the phenomenon of radiating from a line to both sides. The effect of creep compliance on the damping of transient pressure fluctuations is maximized when τ1 and τ2 are on the order of 10−1. The effect of creep compliance on the damping of transient pressure fluctuations diminishes when the retarded time is less than a 10−1 order of magnitude. An increase in the retarded time also leads to a decreasing effect of creep compliance on the damping of transient pressure fluctuations when the retarded time is greater than a 10−1 order of magnitude.
Figure 12
Influence rules of the creep parameters of the two-element KV model. (a) τ1, J1, (b) τ1, J1, (c) τ2, J2, and (d) τ2, J2.
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Influence rules of the creep parameters of the two-element KV model. (a) τ1, J1, (b) τ1, J1, (c) τ2, J2, and (d) τ2, J2.

Figure 12
Influence rules of the creep parameters of the two-element KV model. (a) τ1, J1, (b) τ1, J1, (c) τ2, J2, and (d) τ2, J2.
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Influence rules of the creep parameters of the two-element KV model. (a) τ1, J1, (b) τ1, J1, (c) τ2, J2, and (d) τ2, J2.

Close modal

In the calibration process, the selection of the retarded time is crucial because it simplifies the calibration process and improves calibration efficiency. In this paper, a reference order of magnitude for the calibration of retarded times in the case of a two-element KV model is proposed, with a τ1 order of magnitude equal to or lesser than 10−2 and a τ2 order of magnitude equal to or greater than 1. This is because the calibration creep parameters can be determined rapidly using the phase and pressure damping of the experimental pressure if the retarded time and creep compliance of the two-element KV model do not have the same rule of influence on the transient pressure. Moreover, τ1 and J1 influence the phase and pressure damping of the transient pressure fluctuations, whereas τ2 and J2 only affect the pressure damping of the transient pressure. Therefore, the stepwise calibration method proposed for the calibration of creep parameters is first based on the pressure fluctuations and creep test to determine the order of magnitude of the retarded time and then uses the experimental pressure fluctuation phase and pressure damping variations to determine the creep compliance.

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The transient flow model of viscoelastic pipes was established using a two-element KV model and a quasi-steady friction model. Based on ANOVA and multiparameter sensitivity analysis, the rules of creep parameters on pressure damping and phase offset in a two-element KV model were investigated. The calibration method of the creep parameters is systematically studied based on the relationship between creep parameters and transient pressure fluctuation. A stepwise method is proposed to complement the calibration methods of retarded time and creep compliance, which improves the calibration efficiency and expands the scope of calibration. The transient flow pressure fluctuations of PPR and HDPE pipes were also calibrated using the proposed calibration method with accurate results. The main findings and conclusions are as follows.

  • A stepwise method for the calibration parameter of transient flow applicable to viscoelasticity is developed based on the impact relationship of the creep parameters on pressure fluctuations. The proposed stepwise method is proved to be capable of accurate simulation of pressure fluctuations of transient flows in PPR pipes and HDPE pipes at different water temperatures.

  • One-way ANOVA was used to analyze the effects of creep parameters on transient pressure fluctuations. Creep parameters have a significant effect on pressure damping.

  • The rule of influence of creep compliance on transient pressure fluctuations is only related to the corresponding retarded time.

  • In the two-element KV model, the order of magnitude of the retarded time is greater than 10−1, and the creep compliance has no effect on the phase of transient pressure fluctuations. The order of magnitude of the retarded time is 10−1, and the creep compliance has the maximum effect on the degree of damping of the transient pressure fluctuations.

  • For the two-element KV model, when the order of magnitude of τ1 is equal to or greater than 10−2, the degree of transient pressure fluctuation damping rises with increasing J1. When the order of magnitude of τ1 is less than 10−2, the influence rule of transient pressure fluctuation on the pressure damping is the opposite with the increase in J1. The order of magnitude of τ2 does not alter the rule of the influence of J2 on pressure damping.

  • A more significant increase in J1 has a greater effect on the degree of pressure damping, whereas a more significant increase in J2 has less effect on the degree of pressure damping.

  • A reference order of magnitude for the calibration of the retarded time in the transient flow pressure calibration of viscoelastic pipes for the two-element KV model was determined that τ1 is equal to or less than 10−2 and τ2 is equal to or less than 1.

In the water hammer analysis of transient flow in viscoelastic pipes, there is no viscoelastic pipe model in the current engineering software. This is because there are no accurate creep parameters that can be applied to the model for simulation. Based on our sensitivity analysis of creep parameters, the creep parameters of viscoelastic pipes can be accurately derived and extended to engineering water hammer analysis. Accurate creep parameters can also benefit the optimal design of equipment parameters for water hammer protection against transient flows in viscoelastic pipes and the operation of viscoelastic pipes.

In this study, the order of magnitude of the retarded time resulted in distinct creep compliance rules concerning pressure fluctuation. The outcome resulted from the integration of the two-element KV model, and it is important that future research studies investigate the impact of creep parameters on pressure fluctuations in the context of multiple-element KV models. The solution for the extension of the stepwise method to multiple-element KV models can no longer be carried out in terms of the sensitivity of the creep parameters because it is very difficult to study the sensitivity for more than four parameters and with interactions between the parameters. Therefore, it is necessary to find the relationship between the creep parameters and the creep curve, which in turn can be analyzed for the relationship between the creep parameters and the transient flow pressure fluctuations with multiple-element KV models. In the investigation of creep parameters using multiple-element KV models, it is essential to analyze the impact of the creep parameter selection on the creep curve using creep tests.

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This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 51808102 and 51978202), the Young Innovative Talents Support Program of Harbin University of Commerce (2020CX07), and the Natural Science Fund of Heilongjiang Province (LH2020E028).

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

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The authors declare there is no conflict.

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