## Abstract

Numerous studies on the pressure fluctuations and cavity volume variations of a transient cavitation flow in viscoelastic pipes are available in the literature. However, the effect of water temperature on the cavity volume and energy conversion has been studied less often. This paper employs the discrete vapor cavity model using quasi-steady friction and quasi-two-dimensional friction models to calculate the cavity volume for different water temperatures and investigates the effects of water temperature on the appearance of the first cavitation at the downstream valve, and on the pressure damping in a tank-piping-valve system using an integrated energy analysis approach. The results show that the differences between the pressure and energy variations of the transient cavitation flow simulated using different models were minimal under different water temperature conditions. Moreover, as the water temperature increased, the appearance time of the cavity is postponed, and the volume of the cavity decreases. The energy dissipation increases continuously with an increase in the volume of the cavitation and water temperature in viscoelastic pipes. This study provides valuable insights into the variation pattern of the cavity and the effect of vapor cavities on the rise and decay of the pipeline pressure in different situations.

## HIGHLIGHTS

This study investigated the effect of different water temperatures on the calculated results of transient cavitation flow in viscoelastic pipes.

The energy variations during transient flow and transient cavitation flow in viscoelastic pipes were compared.

The concept of the conversion ratio from kinetic energy to elastic energy was proposed, where a smaller conversion ratio may result in a larger energy dissipation.

## NOMENCLATURE

*The following symbols are used in this paper:*

*A*area of the pipe (m

^{2})*A*_{k}characteristic constant

*a*wave speed (m/s)

*B*constant value

*B*_{k}characteristic constant

*C*_{k}characteristic constant

*C*_{M}characteristic constant

*C*_{P}characteristic constant

*D*pipe diameter (m)

*D*_{f}energy of frictional effect

*F*characteristic constant

*f*Darcy–Weisbach friction factor

*G*instantaneous kinetic energy

*g*acceleration of gravity (m/s

^{2})*H*pressure head (m)

*J*_{0}the instantaneous elastic creep

*J*_{k}the creep compliance of the spring of

*k*-th element*M*the elastic energy of the total internal energy

*m*the number of grid points

*N*the number of K-V elements

the experimentally measured pressure at the valve (N)

the pressure at the valve calculated by the model (N)

*Q*flow rate (m

^{3}/s)*Q*_{i}the downstream flow rate of the cavity (m

^{3}/s)*Q*_{ui}the upstream flow rate of the cavity (m

^{3}/s)the radial flow per unit of the pipe length

*r*radial distance along the pipe from the pipe axis (m)

*x*the distance in the axial direction of the pipe (m)

*t*the time coordinate (s)

*u*axial velocity

*V*average velocity in section (m/s)

*V**the volume of cavity (m

^{3})*V*_{i}the downstream flow velocity of the cavity (m/s)

*V*_{ui}the upstream flow velocity of the cavity (m/s)

*v*the radial velocity

*W*_{R}the power on the pipe wall

*W*_{L}the power from the ends of the pipeline

## GREEK SYMBOLS

## SUPERSCRIPT

## SUBSCRIPTS

## ABBREVIATION

## INTRODUCTION

Water hammer with column separation is a common hydraulic transient phenomenon caused by sudden power failures or the improper manual operation of a pressurized pipeline (Tijsseling 1996; Bergant *et al.* 2006), which can cause a rapid increase in pressure fluctuations. Consequently, the pressure can propagate through an entire pipeline, posing a threat to the safe operation of hydraulic machinery and piping systems (Hadj-Taleb & Hadj-Taleb 2009). Moreover, the viscoelastic properties of pipes can affect the pressure of transient cavitation flow, thereby contributing to the safe operation of piping systems. The viscoelastic properties of pipes can be greatly influenced by the water temperature, directly impacting the transient cavitation flow in viscoelastic pipes.

Various numerical models have been developed for the simulation of transient cavitation flow – such as the discrete cavity and continuous cavity models – among which the discrete cavity model is the more widely used (Simpson & Bergant 1994). Commonly used discrete cavity models include the discrete vapor cavity model (DVCM) and the discrete gas cavity model (DGCM). The classical DVCM describes the essential characteristics of transient cavitation flow and is widely used by researchers (Bergant & Simpson 1999). However, the DVCM combined with the quasi-steady friction model generates virtual pressure oscillations when simulating pressure fluctuations. Consequently, some researchers have used the unsteady friction model instead of the quasi-steady friction model to solve this problem and avoid unrealistic pressure peaks. Soares *et al.* (2017) combined an unsteady friction model with cavitation models to simulate transient cavitation flow in a horizontal copper pipe. The results showed excellent agreement between the numerical results and experimental data, the unsteady friction model also having an attenuating effect on the pressure generated after cavity collapse. Pezzinga & Santoro (2017) and Santoro *et al.* (2018) combined the DVCM with one-dimensional (1D) steady-state and quasi-two-dimensional (quasi-2D) friction models in elastic pipes and found that the quasi-2D friction model could better reproduce the water hammer pressure profile. Based on the experimental and simulation results, the DVCM was more suitable for low-cavitation phenomena or the first stage of cavitation. By comparing the solution of the water hammer equation considering unsteady friction and viscoelasticity with the classical numerical solution, the introduction of viscoelasticity and unsteady friction improved the accuracy of the simulation (Seck 2020). Additionally, Soares *et al.* (2009) combined the DVCM with 1D unsteady friction models to investigate cavitation in polyethylene pipes. They found that the effect of unsteady friction on the pressure was small in viscoelastic pipes, but viscoelasticity had a considerable effect on the results.

Additionally, water temperature not only directly affects the saturation vapor pressure of a liquid but can also influence the intrinsic parameters of the viscoelastic pipe (Ghidaoui *et al.* 2005). Several studies have investigated this topic. Saidani *et al.* (2022) simulated the pressure fluctuations in a copper pipe system in a temperature range of 4–95 °C, and found that the higher the temperature, the higher the cavity volume in the copper pipe. Mousavifard (2022) discussed the effect of water temperature on *k-w* model parameters and transient flow profiles in a viscoelastic pipe. The results showed that the higher the temperature, the lower the velocity gradient near the wall, and the slower the propagation of turbulence to the inner layer. Hadj-Taieeb & Hadj-Taieeb (2009) noted a viscoelastic effect on pressure attenuation and dispersion in their simulations. They pointed out that the higher the temperature, the more evident the pipe creep effect, and the shorter the cavity duration.

However, the aforementioned studies were based on collecting transient pressure signals at specific locations, which could reflect the pipeline system properties. However, these studies did not provide comprehensive information on the entire pipe (Pan *et al.* 2020). An energy analysis method was introduced to study and evaluate the state of pipeline systems. Duan *et al.* (2010) used an integrated energy analysis method to analyze the attenuation of pressure by viscoelastic effects and unsteady friction. The results showed that the viscoelastic properties of a pipe could affect the peaks of later pressure fluctuations (Duan *et al.* 2017). Additionally, the viscoelastic effect proved to be more important in long pipe systems compared with the effect of unsteady friction, while unsteady friction dominated energy dissipation in pipes of small diameter and shorter pipe systems (Duan *et al.* 2012). Lee (2013) employed an integrated energy analysis approach to evaluate the accuracy of a transmission line model. Kashif *et al.* (2023) employed an energy analysis approach to illustrate that pressure vessels greatly affected water hammer protection. Meniconi *et al.* (2014) developed the relationship between pressure dissipation and kinetic and elastic energies in viscoelastic pipes using integrated energy analysis. Andrade *et al.* (2023) explained the energy variations of transient flow in viscoelastic pipes. However, this requires further investigation as there was no clear explanation of the effect of water temperature on pressure attenuation and cavity volume in transient cavitation flow in viscoelastic pipes.

Consequently, this study investigated the effects of 1D quasi-steady friction and quasi-2D models on the numerical simulation of transient cavitation flow in viscoelastic pipes at different water temperatures, the impact of water temperature on the cavity volume and energy variations in the entire pipe being analyzed. This study provides valuable insights into the influence of water temperature on the transient cavitation flow in viscoelastic pipes.

## METHODS: MATHEMATICAL MODEL

### Governing equations

*H*denotes the piezometric head,

*Q*denotes the flow rate,

*x*denotes the distance in the axial direction of the pipe,

*r*denotes the radial distance along the pipe from the pipe axis,

*t*denotes the time coordinate,

*A*denotes the area of the pipe,

*a*denotes the wave speed,

*g*denotes gravitational acceleration, denotes the pipe wall stress,

*u*denotes the axial velocity,

*v*denotes the radial velocity, and denotes the retarded strain.

*et al.*2004b):where

*D*denotes the pipe diameter. The wall shear stress can be calculated using the Darcy–Weisbach equation:where

*f*denotes the Darcy–Weisbach friction factor, and

*V*(

*t*) denotes the average cross-sectional flow velocity.

### Numerical discretization of the retarded strain

*et al.*2004a, 2005):where

*J*denotes the creep compliance of the spring of the

_{k}*k-*th element, denotes the retarded time of the

*k-*th element, and

*N*denotes the number of K-V elements.

### Discrete vapor cavity model

*i*can be calculated from the upstream and downstream flow rates, as shown in Figure 2:

*V*denotes the cavity volume at grid point

_{i}**i*,

*Q*denotes the upstream flow rate of the cavity at grid point

_{ui}*i*,

*Q*denotes the downstream flow rate of the cavity,

_{i}*V*denotes the upstream flow velocity of the cavity,

_{ui}*V*denotes the downstream flow velocity of the cavity, , , and denote the known values at time , , , and denote the unknown values at time ,

_{i}*H*denotes the steam pressure head, and denotes the weight coefficient of calculating the cavity volume, taking the value range of 0.5 ∼ 1.

_{v}### Numerical scheme

#### Numerical solution of the 1D model

#### Numerical solution of quasi-2D model

*et al.*2022):where and denote the weighting factors, denotes the radial flux, the subscripts

*i*and

*j*denote the axial and radial step indices, respectively, and and denote the coordinates of the boundary and middle points of the reaches in the radial direction, respectively.

*i*, substituting Equation (11) into Equations (21) and (22), the unknown parameters and at time can be obtained using Thomas algorithm (Vardy & Hwang 1991). The wall shear stress at time can then be calculated as follows:where denotes the turbulent viscosity.

## RESULTS AND DISCUSSION

### Experimental data

*et al.*(1979) were in a typical tank-pipe-valve system, as shown in Figure 5, comprising a low-density polyethylene pipe of total length of 43.1 m, an inner diameter of 45.8 mm, and a wall thickness of 4.2 mm. The experiments were conducted by heating the fluid in the tank using a heating- and temperature-control device. The valve closure time was 12 ms and three high-precision pressure sensors were installed upstream, midstream, and downstream of the pipe. The experimental parameters were as listed in Tables 1 and 2.

Temperature (°C) . | Initial pressure (×10^{5} Pa)
. | Initial mean velocity (m/s) . | Kinematic viscosity (×10^{−6} m^{2}/s)
. | Reynolds number . | Bulk modulus (×10^{9} Pa)
. | Density (kg/m³) . |
---|---|---|---|---|---|---|

13.8 | 2.99 | 1.27 | 1.17 | 45155 | 2.14 | 999.3 |

25 | 2.99 | 1.27 | 0.892 | 59229 | 2.24 | 997.1 |

31 | 2.99 | 1.27 | 0.784 | 67389 | 2.27 | 995.3 |

38.5 | 2.99 | 1.27 | 0.675 | 78270 | 2.295 | 992.6 |

Temperature (°C) . | Initial pressure (×10^{5} Pa)
. | Initial mean velocity (m/s) . | Kinematic viscosity (×10^{−6} m^{2}/s)
. | Reynolds number . | Bulk modulus (×10^{9} Pa)
. | Density (kg/m³) . |
---|---|---|---|---|---|---|

13.8 | 2.99 | 1.27 | 1.17 | 45155 | 2.14 | 999.3 |

25 | 2.99 | 1.27 | 0.892 | 59229 | 2.24 | 997.1 |

31 | 2.99 | 1.27 | 0.784 | 67389 | 2.27 | 995.3 |

38.5 | 2.99 | 1.27 | 0.675 | 78270 | 2.295 | 992.6 |

Temperature (°C) . | Initial pressure (×10^{5} Pa)
. | Initial mean velocity (m/s) . | Kinematic viscosity (×10^{−6} m^{2}/s)
. | Reynolds number . | Bulk modulus (×10^{9} Pa)
. | Density (kg/m³) . |
---|---|---|---|---|---|---|

13.8 | 1.06 | 0.49 | 1.17 | 17 422 | 2.14 | 999.3 |

25 | 1.07 | 0.55 | 0.892 | 25 650 | 2.24 | 997.1 |

31 | 1.07 | 0.57 | 0.784 | 30 245 | 2.27 | 995.3 |

38.5 | 1.06 | 0.56 | 0.675 | 34 513 | 2.295 | 992.6 |

Temperature (°C) . | Initial pressure (×10^{5} Pa)
. | Initial mean velocity (m/s) . | Kinematic viscosity (×10^{−6} m^{2}/s)
. | Reynolds number . | Bulk modulus (×10^{9} Pa)
. | Density (kg/m³) . |
---|---|---|---|---|---|---|

13.8 | 1.06 | 0.49 | 1.17 | 17 422 | 2.14 | 999.3 |

25 | 1.07 | 0.55 | 0.892 | 25 650 | 2.24 | 997.1 |

31 | 1.07 | 0.57 | 0.784 | 30 245 | 2.27 | 995.3 |

38.5 | 1.06 | 0.56 | 0.675 | 34 513 | 2.295 | 992.6 |

### Parameters selection of DVCM

*t*

_{cav}) was introduced to evaluate the accuracy of the simulation results. According to the literature’ suggestions (Simpson & Bergant 1994), for the 1D-DVCM and 2D-DVCM, axial grid numbers of 50, 100, 200, and 300 were taken into account separately. In the 2D-DVCM, the radial grid numbers ranged from 20 to 100, at intervals of 20. The root mean square error (RMSE) was used to reflect the deviation between the simulated and experimental results. A smaller RMSE value indicates better consistency between the peak pressures of the numerical results and the experimental data. Consequently, the RMSE calculated using each grid number was analyzed, as shown in Figures 6 and 7, to quantify the difference between the calculated and experimental pressure values at the valve.

Figure 6 shows the RMSE calculated using the 2D-DVCM model for different *N _{x}* and

*N*values at different water temperatures. When

_{r}*N*remains constant, an increase in the value of

_{x}*N*results in a decrease in the RMSE. However, the difference is negligible as the variations in the RMSE values are extremely small, especially for the (water temperature) cases of 13.8 and 25 °C. For the

_{r}*N*value, at 13.8 °C, the RMSE values increase considerably with increasing

_{x}*N*, while at other temperatures, the RMSE values exhibit the same trend. This trend is also evident in the 1D-DVCM model.

_{x}Figure 7 shows the RMSE values obtained using the 1D-DVCM model for different grid numbers at various temperatures. Similarly, for the 1D-DVCM, as the *N _{x}* value increases at different temperatures, the RMSE values gradually increase.

Additionally, the influence of the weighting coefficient of the cavity volume on pressure fluctuations (Ψ) with a range of values between 0.5 and 1 was investigated, through the results of the simulations using different Ψ values being compared. It is evident that the differences between simulation results are slight, but the result is the most accurate when Ψ = 1. Consequently, this study adopted a weighting coefficient Ψ = 1, an axial grid number of 50, and a radial grid number for the 2D-DVCM of 80, respectively.

### Comparison of pressure fluctuations calculated using different friction models

*t*

_{cav}can be defined to quantitatively analyze the consistency between the simulation results and the experimental data in terms of the cavity formation time, as follows (Sun

*et al.*2022):. Where

*t*

_{exp}and

*t*

_{mod}denote the formation times of the

*i*th cavity in the experimental and numerical results, respectively.

*t*

_{cav}value computed using both models is less than 3%. Moreover, as the temperature increases, the accuracy of all three models in simulating the cavitation formation time also improves. At 38.5 °C, the difference in the cavitation formation times calculated using the 2D-DVCM model is less than 1%, demonstrating that the friction model is necessary to simulate the formation time of the cavity and that the numerical simulation results of the 2D friction model are better than those of the 1D friction model.

### Effect of water temperature on the cavity volume

Temperature (°C) . | Time of cavity appearance (s) . | Cavity duration (s) . | Maximum volume of cavity (10^{−5} m³)
. | Percentage of maximum cavity (%) . | Appearance time of maximum volume of cavity (s) . |
---|---|---|---|---|---|

13.8 | 0.392 1.439 | 0.546 0.18 | 4.913 | 3.56 | 0.922 |

25 | 0.493 | 0.495 | 3.447 | 2.50 | 0.926 |

31 | 0.552 | 0.467 | 2.641 | 1.91 | 0.981 |

38.5 | 0.706 | 0.38 | 1.135 | 0.82 | 1.076 |

Temperature (°C) . | Time of cavity appearance (s) . | Cavity duration (s) . | Maximum volume of cavity (10^{−5} m³)
. | Percentage of maximum cavity (%) . | Appearance time of maximum volume of cavity (s) . |
---|---|---|---|---|---|

13.8 | 0.392 1.439 | 0.546 0.18 | 4.913 | 3.56 | 0.922 |

25 | 0.493 | 0.495 | 3.447 | 2.50 | 0.926 |

31 | 0.552 | 0.467 | 2.641 | 1.91 | 0.981 |

38.5 | 0.706 | 0.38 | 1.135 | 0.82 | 1.076 |

## ENERGY ANALYSIS

*et al.*2017; Pan

*et al.*2022):where

*M*denotes the elastic energy,

*G*denotes the instantaneous kinetic energy, denotes the energy of the frictional effect, denotes the energy of the viscoelastic behavior, and denotes the power from both ends of the pipeline, respectively.

*et al.*2022):

### Analysis of kinetic-elastic energy variations of different friction models

### Kinetic and elastic energy variations in transient flow

*t*= 0) and decays toward the origin in the direction of the arrows. From Figure 12(a)–12(d), it is evident that the variation in water temperature does not affect the shape of the energy graph. However, water temperature influences the inclination angle of the zigzag waveform. The discussions in detail are as follows.

At the beginning of the transient flow (point A), the kinetic energy reaches its maximum value. The elastic potential energy is not zero but is not easily distinguishable (as is evident in Figure 12(d)). Subsequently, kinetic energy is converted into elastic potential energy. To better evaluate the energy variations, the concept of conversion ratio can be introduced in the following analysis: The conversion ratio is the ratio of the maximum elastic energy to the maximum kinetic energy. Additionally, the conversion ratio is reflected in the inclination angle of the zigzag waveform shown in the graph, the inclination angle being positively proportional to the conversion ratio. A smaller conversion ratio indicates greater energy dissipation, which is consistent with the results of a study conducted by Pan *et al.* (2022).

The conversion ratio decreases over time, however, the decrease is not significant, as shown in Figure 12(c). Moreover, the water temperature has a major influence on the conversion ratio – that is, as the water temperature increases, the conversion ratio decreases – primarily due to the increase in the pipeline's viscoelastic properties at higher temperatures, which allows for better energy storage and dissipation, resulting in increased energy dissipation and decreased energy retention in the system.

### Variations of kinetic and elastic energies of transient cavitation flow

### Analysis of the effect of cavity volume on energy variation

The conversion ratios in the transient flow were 31.69, 22.29, 15.96, and 12.33%, respectively, as the water temperature increasing. Similarly, when cavities were formed, the conversion ratios were 22.43, 19.53, 17.24, and 14.29% separately, as the water temperature increasing. The conversion ratios of the kinetic and elastic energies at different water temperatures indicated that as the temperature increased, the conversion ratios decreased.

Furthermore, by comparing the initial kinetic energy at point A in Figures 12 and 13, it is evident that a higher initial flow velocity is capable of generating a greater initial kinetic energy, with temperature changes having little effect on the initial kinetic energy. Additionally, elastic energy is the energy produced by the coupling between the fluid and pipeline. Therefore, the higher the initial flow velocity, the greater the initial elastic energy, as reflected in the red circle section of Figure 12(d) and the corresponding parts in Figures 12 and 13.

### Comparative analysis of energy terms of frictional effect and viscoelastic behavior

From the above analysis, as the water temperature increases, the conversion ratios of the kinetic and elastic energies decrease considerably and the energy dissipation increases. The volume of the cavity also has a major influence on the conversion ratio of the energy and energy dissipation. The larger the volume of the cavity, the greater the energy dissipation in the cavity, and the faster the energy decay. Moreover, the variations between the energies in the cavity formation and collapse phases are complex and intense.

## CONCLUSIONS

This study investigated the effect of different water temperatures on the calculated results of transient cavitation flow in viscoelastic pipes. The effects of water temperature on the volume and formation time of the cavities were examined. By employing an integrated energy analysis method, energy equations for 1D and quasi-2D transient cavitation flows in viscoelastic pipes were developed, and the energy variations during transient flow and transient cavitation flow in viscoelastic pipes were compared. Moreover, this study explored the effects of water temperature and cavity volume on kinetic energy, elastic energy, frictional energy, and viscoelastic behavior, respectively. The conclusions were as follows.

As the water temperature increased, the maximum volume and duration of cavitation at downstream valve gradually decreased. Additionally, the formation of the cavities and the time of maximum cavity volume occurred later.

The concept of the conversion ratio from kinetic energy to elastic energy was proposed, where a smaller conversion proportion could result in greater energy dissipation. The results indicated that an increase in the water temperature resulted in a decrease in the conversion ratio and an increase in energy dissipation.

The presence of cavities increased the energy dissipation for transient cavitation flow in viscoelastic pipes. Additionally, larger cavity volumes resulted in greater energy dissipation. During the cavity collapse phase, the energy variations became more complex and chaotic.

The differences between the 1D-DVCM and 2D-DVCM simulations in terms of the pressure peak, phase, and cavity appearance time were small. Consequently, it was reasonable to use a one-dimensional quasi-steady friction model to replace the quasi-2D friction model.

## ACKNOWLEDGEMENTS

This work was financially supported by the Fundamental Research Funds for the Central Universities (2572021BJ02), the National Natural Science Foundation of China (51808102, 51978202), Young Innovative Talents Support Program of Harbin University of Commerce (2020CX07), and the Natural Science Fund in Heilongjiang Province (LH2020E028). The authors would like to thank Professor Kamil Urbanowicz from the West Pomeranian University of Technology for providing the experimental data.

## DATA AVAILABILITY STATEMENT

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

## CONFLICT OF INTEREST

The authors declare there is no conflict.