## Abstract

Valves are installed at the end of each branch pipeline in a tree-type long distance gravitational water supply system to regulate flow. However, the sequential closing of all valves may cause a tremendous superposed pressure rise, even larger than the pressure rise under simultaneous valve closure. In this paper, the effects of sequential valve closure on the superposed maximum water hammer pressure rise in a pipeline were investigated. By using the wave superposition principle, a sequential valve closure formula leading to maximum water hammer was proposed and verified using numerical simulation based on a practical project. In addition, the superposed maximum pressure rises in the pipeline were compared under single, simultaneous and sequential valve closure, respectively. The results show that the sequential valve closure formula agrees well with the numerical results and the pressure rise in the pipeline under the sequential closing was the largest. Moreover, compared with the superposed maximum pressure rises at the main pipeline, the effect of sequential valve closure on superposed maximum pressure rise at the branch pipeline is more sensitive.

## INTRODUCTION

The severe shortage of water resources is a crucial issue all over the world for the accelerated construction of urbanization and the concentration of the population (Fan *et al.* 2019; Xiao *et al.* 2019). Through the construction of long distance water supply systems, transporting water resources from an area rich in water resources to the area with shortage of water is the most effective approach (Wang *et al.* 2018). The systems are mainly divided into pumping station pressurization and gravity flow. Compared with pumping station pressurization, because it relies on the topographical drop to supply water, the gravity flow has favorable economics. However, safety along the pipeline is a very serious challenge. In long distance gravitational water supply systems, each branch pipeline forms a branching junction with the main pipeline, and valves are installed at the end of each branch pipeline to regulate flow (Zhang *et al.* 2018). When a valve is illegally closed, a positive water hammer wave is generated at the valve and propagates to the pipeline, and the wave may cause tremendous fluctuations of pressure along the pipeline. Then, when the maximum pressure exceeds the pressure standard of pipeline, a pipe burst will be caused, resulting in enormous damage to the pipeline system. Therefore, it is necessary to investigate how to effectively prevent illegal closure of the valves of a long distance gravitational water supply system.

In a long distance gravitational water supply system, it is generally accepted that valves are components that can effectively regulate flow (Miao *et al.* 2017; Lai *et al.* 2018). In previous studies, the model design of the valves has been intensively investigated (Bostan *et al.* 2018). For instance, Liao *et al.* proposed a plane-sealed large flow directional valve with different throttle windows (Liao *et al.* 2017). Xu *et al.* introduced the structure and working principle of a contra-push check valve (Xu *et al.* 2011), and simulated the steady and transient characteristics of the check valve. In addition, there is a large number of published papers about the closing law of the valves in water supply systems (Zhang 2019). For example, Wang *et al.* (2017) proposed an improved two-segment closing law for a gravitational water supply system; the results showed that the closing law can effectively reduce the water hammer pressure caused by valve closing. Wan & Li (2016) studied the transient pressure, pump speed, and flow for various time differences and the valve opening process for the series pump–valve startup process. However, the above studies have focused on discussing the influence of closing law of a single valve on the water hammer pressure of water supply system, ignoring the impact of closing all valves between multiple branch pipelines.

Traditionally, the water hammer wave caused by valve closing propagates in a single pipeline without considering the effect of wave superposition and interference. Nevertheless, a long distance gravitational water supply system has many branch pipelines and water intakes. According to the propagation and superposition of the wave, the water hammer waves generated by closing valves are mutually strengthened or weakened by propagation and superposition along the pipelines. Typically, simultaneous closing of all valves is considered as the largest water hammer pressure rise case. However, as all branches have different distances from the water intakes, the time when the water hammer waves caused by the simultaneous closing of the valves arrive at a fixed section is different. Then, the water hammer waves cannot be strengthened when they are superposed to the same section. Hence, at a certain section, the superposed maximum pressure rise generated by the simultaneous closing of the valves may not be the largest. Therefore, motivated by the above discussion, this paper focuses on the closing law of valves and its influence on water hammer pressure in the tree-type long distance gravitational water supply system.

## MATHEMATICAL MODEL

### Transient flow model

*et al.*1993; Chaudhry 2014; Moghaddas

*et al.*2017). where

*H*is piezometric head (m),

*Q*is discharge (m

^{3}/s),

*D*is pipe internal diameter (m),

*A*is cross-sectional area (m

^{2}), is gravitational acceleration (m/s

^{2}),

*a*is wave speed (m/s),

*t*is computational time (s),

*x*is axial distance (m),

*f*is friction coefficient, is pipe slope.

### Valve model

Valves are important components for regulating flow in fluid delivery systems such as petroleum, chemical, power plants and long distance water supply systems. Generally, the discharge rate of a valve is determined by the pressure difference between the inlet and outlet, the valve opening and diameter of the valve. The pressure and discharge at the inlet and outlet can be solved through the characteristic lines on both sides of the valve.

^{3}/s), is the area of the valve opening (m

^{2}), is the losses at the valve (m).

## INFLUENCE OF MULTI-VALVE CLOSING LAW ON WATER HAMMER WAVE

_{1}and S

_{2}with the same frequency are generated by the closing valve. The vibration equations can be written as: where and denote the vibration equations of S

_{1}and S

_{2}; and are amplitudes of vibration equations; and and are the phases of the vibration equation.

*P*, the vibration equations can be written as: where and are the vibration equations of S

_{1}and S

_{2}at point

*P*, respectively; and are amplitudes of vibration equations, respectively; is wavelength; and and are the distances from S

_{1}and S

_{2}to point

*P*, respectively.

*P*can be written as: where and is the phase difference of the vibration equation.

When the water hammer waves propagate in the same pipe, firstly, the frequency of propagation period is the same because the distance between two water hammer waves is equal. Secondly, in a fixed pipeline, the propagation direction of the water hammer wave is unidirectional. Therefore, the wave is either a positive or a negative, which spreads in the same direction. Thirdly, the phase difference of the waves at the same pipe in the same direction or reverse propagation is also constant. Therefore, the conditions of wave interference can be satisfied when the water hammer waves are propagating in a pipeline.

If , in which , the superposition water hammer wave is strengthened. , , , are the maximum values of amplitude and vibration intensity of S_{1} and S_{2} at point *P*, respectively.

If , in which , the superposition water hammer wave is weakened. , , , are the minimum values of amplitude and vibration intensity of S_{1} and S_{2} at point *P*, respectively.

According to the prior works, the superposed pressure produced by simultaneous valves closure is the largest. However, the long distance gravitational water supply system has many tree-type branches. All branches have different distances from the water intakes, and the time when the waves propagate to the same section is also different. Then the water hammer waves cannot be strengthened. Therefore, at the same section, the superposed pressure generated by the simultaneous valves closure is not the largest. According to the propagation, superposition and interference conditions of wave, when the waves can propagate to a fixed section at the same time, the amplitude of the superposition water hammer wave can reach the maximum value.

The tree-type long distance gravitational flow layout is shown in Figure 1(b). The system consists of *N* branch pipelines and *N − 1* water intakes. Valves are installed at the end of each branch to regulate the flow.

It is assumed that the closing law of each valve is the same. The maximum water hammer pressure rise at the *j*th water intake generated by the closing valve of the *i*th valve is , in which , . And the time to reach the maximum water hammer pressure is . According to the propagation of the wave, under the same type of pipeline, the farther the water intake is from the valve, the later the moment when the maximum pressure occurs. Therefore, the valve close to the water intake needs to be closed late. In other words, the farther the position of the valve is from the position of the maximum pressure, the earlier the valve needs to be closed.

*j*th water intake is: where is the superposed maximum pressure rise at the

*j*th water intake under the sequential valves closure. is the superposed maximum pressure rise at the

*j*th water intake under the single valves closure. is the superposed maximum pressure rise at the

*j*th water intake under the simultaneous valves closure.

## CASE STUDY AND ANALYSIS

Figure 2(a) shows the layout of a tree-type long distance gravitational water supply system. The parameters of the system are shown in Table 1. The system takes water from the upstream reservoir, conveying water to downstream water plants by gravity in a pipe, with the total design water discharge of 43.3 m^{3}. The design water level of the upstream reservoir is 70.0 m. The system is composed of one main pipeline, six branch pipelines and five water intakes, and a valve is installed at the end of the branch. The elevation of the main pipeline centerline and positions of the water intakes are shown in Figure 2(b). According to the mathematical model, steady flow of the system is calculated by method of characteristic (MOC). The piezometric head of water intakes are 68.76 m, 68.54 m, 51.93 m, 49.86 m and 40.26 m separately.

Pipe number | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | M9 | M10 | M11 |
---|---|---|---|---|---|---|---|---|---|---|---|

Length (km) | 2.142 | 2.393 | 1.207 | 8.850 | 25.634 | 4.910 | 3.068 | 6.410 | 39.610 | 1.193 | 4.451 |

Diameter (m) | 6.0 | 2.8 | 6.0 | 3.5 | 3.4 | 2.4 | 3.4 | 0.6 | 3.2 | 1.2 | 3.2 |

Discharge (m^{3}/s) | 43.3 | 7.6 | 35.7 | 18.3 | 17.4 | 3.2 | 14.2 | 5.1 | 9.1 | 0.8 | 8.3 |

Pipe number | M1 | M2 | M3 | M4 | M5 | M6 | M7 | M8 | M9 | M10 | M11 |
---|---|---|---|---|---|---|---|---|---|---|---|

Length (km) | 2.142 | 2.393 | 1.207 | 8.850 | 25.634 | 4.910 | 3.068 | 6.410 | 39.610 | 1.193 | 4.451 |

Diameter (m) | 6.0 | 2.8 | 6.0 | 3.5 | 3.4 | 2.4 | 3.4 | 0.6 | 3.2 | 1.2 | 3.2 |

Discharge (m^{3}/s) | 43.3 | 7.6 | 35.7 | 18.3 | 17.4 | 3.2 | 14.2 | 5.1 | 9.1 | 0.8 | 8.3 |

### Water hammer pressure under different closing laws of multi-valves

#### Single valve closure

In this section, the effects of single valve closure on the maximum water hammer rise in the 5# water intake are studied. Each valve is closed within 100 s. The transition processes of the system are calculated and analyzed through MOC. Table 2 shows parameters of the time to reach the maximum pressure at the 5# water intake.

Valve | 1# | 2# | 3# | 4# | 5# | 6# |
---|---|---|---|---|---|---|

The maximum pressure (m) | 42.64 | 47.50 | 50.99 | 59.19 | 44.17 | 142.48 |

(m) | 2.38 | 7.24 | 10.73 | 18.93 | 3.91 | 102.22 |

(s) | 99.26 | 91.27 | 72.31 | 75.08 | 28.09 | 83.60 |

Valve | 1# | 2# | 3# | 4# | 5# | 6# |
---|---|---|---|---|---|---|

The maximum pressure (m) | 42.64 | 47.50 | 50.99 | 59.19 | 44.17 | 142.48 |

(m) | 2.38 | 7.24 | 10.73 | 18.93 | 3.91 | 102.22 |

(s) | 99.26 | 91.27 | 72.31 | 75.08 | 28.09 | 83.60 |

From Table 2, both the maximum pressure rise at the 5# water intake and the time of maximum water hammer pressure are different. On the one hand, the reason is that the greater the distance of the valve from the water intake, the longer it takes for the wave caused by the valve closing to propagate to the water intake.

On the other hand, the greater the distance of the valve from the water intake, the greater the attenuation of the wave during the propagation process. The distance from the 1# valve to the 5# water intake is the farthest, at 71.903 km, and the maximum water hammer pressure rise at the 5# water intake generated by the 1# valve is the smallest. And the distance from the 5# and 6# valve to the 5# water intake is the closest, at 1.193 km and 4.451 km, respectively, and the pipeline discharge of the 6# valve is much larger than the discharge of the 5# pipeline (8.3 m^{3}/s ≫ 0.8 m^{3}/s). Therefore, the maximum pressure rise generated by the 6# valve at the 5# water intake is the largest, at 102.22 m. In addition, according to Table 1, the water velocity of M8 is highest because the diameter is the smallest and the discharge is higher. Therefore, from Table 2, the maximum pressure rise generated by the 4# valve of the M8 at the 5# water intake is the second largest.

#### Simultaneous valve closure

This is the same as the single valve closure law. Valves of branches are simultaneously closed within 100 s. Figure 3 presents the variations of water hammer pressure at the 5# water intake generated by the simultaneous valve closure.

Figure 3 shows that the superposed maximum pressure at the 5# water intake is 280.42 m. Compared with the piezometric head of 5# water intake, the superposed maximum pressure rise at 5# water intake under the simultaneous valve closure is 240.16 m (). From Table 2, the superposed maximum pressure rise at the 5# water intake under the single valves closure is 145.41 m . By comparing the superposed maximum pressure rise at the 5# water intake, it can be believed that the superposed maximum rise under simultaneous valve closure is much larger than that under single valve closure. The reason is that the superposition of water hammer waves is completely ignored under the single valve closure. The result is consistent with the results of previous studies. It shows that the mathematical model and calculation method established in this paper are reasonable.

### Effect of sequential valve closure on superposed water hammer pressure of main pipeline

The mathematical model and calculation method established in this paper are verified in the section above. Therefore, these mathematical models can be used to calculate the sequential valve closure. From Table 2, when the valves are closed separately, the time to reach the maximum pressure at 5# water intake has the following relationship. . Based on Equation (13), the sequential closing times of valves can be written as: , , , , , , . When valves are sequentially closed within 100 s, the relationship between the opening of each valve and the time is shown in Figure 4(a). Figure 4(b) presents the variations of the superposed pressure at the 5# water intake under the sequential valve closure.

It can been seen from Figure 4(b), when the valves are closed according to Equation (13), the superposed maximum pressure at the 5# water intake is 287.81 m. Compared with the piezometric head of 5# water intake, the superposed maximum pressure rise is 247.55 m ().

The comparison of the superposed maximum pressure rise at the 5# water intake under the single, simultaneous and sequential valve closure is shown in Figure 4(c). According to the comparison, the superposed maximum pressure rise under the sequential closing is the largest followed by the superposed maximum pressure rise under the simultaneous closing. Then, the superposed maximum pressure rise under the single valve closure is the smallest. Based on the above analyses, it can be believed that the superposed maximum pressure rise in the main pipeline under sequential closing of Equation (13) is the largest.

### Effect of sequential valve closure on superposed water hammer pressure of branch pipelines

In the section above, the effects of sequential valve closure on water hammer pressure of the main pipeline are investigated. The results show that the superposed maximum pressure rise at the main pipeline under sequential valve closure is the largest. According to the propagation of waves, the water hammer wave generated by the valve closing propagates on the main pipeline, as well as on branch pipelines. Therefore, the laws of the closing valves have an important effect on superposed pressure of the branch pipelines.

In this section, the transition processes of the branch pipeline of M8 are calculated and analyzed. The piezometric head at the end valve of branch M8 is 68.07 m. When the valves are closed within 100 s, the variations of pressure at the end valve of branch M8 under single and simultaneous valve closure are shown in Figure 5. Table 3 presents parameters of the time to reach the maximum pressure under the single valves closure.

Valves | 1# | 2# | 3# | 4# | 5# | 6# |
---|---|---|---|---|---|---|

The maximum pressure (m) | 71.15 | 76.92 | 82.42 | 117.44 | 69.87 | 120.38 |

(m) | 3.08 | 8.85 | 14.35 | 49.37 | 1.80 | 52.31 |

(s) | 65.84 | 68.41 | 39.85 | 28.90 | 69.56 | 94.31 |

Valves | 1# | 2# | 3# | 4# | 5# | 6# |
---|---|---|---|---|---|---|

The maximum pressure (m) | 71.15 | 76.92 | 82.42 | 117.44 | 69.87 | 120.38 |

(m) | 3.08 | 8.85 | 14.35 | 49.37 | 1.80 | 52.31 |

(s) | 65.84 | 68.41 | 39.85 | 28.90 | 69.56 | 94.31 |

From Figure 5(a), compared with the piezometric head of the end valve of branch M8, the superposed maximum water hammer pressure rise is 129.76 m . And the water hammer pressure rise generated by the 6# valve accounts for 40.31% of the total pressure rise. In addition, according to Figure 5(b), when all valves are simultaneously closed, the superposed maximum water hammer pressure rise is 143.06 m ().

From Table 3, when valves are closed separately, the time to reach the maximum pressure at the end valve of branch M8 has the following relationship. . Based on Equation (13), the sequential closing times of valves can be written as , , , , , and . When valves are sequentially closed within 100 s, the relationship between the opening of each valve and the time is shown in Figure 6(a). Figure 6(b) presents variations of the superposed water hammer pressure at the end valve of branch M8 under the sequential valves closure.

It can been seen from Figure 6(b), when the valves are closed according to Equation (13), the superposed maximum pressure at the end valve of branch M8 is 287.81 m. Compared with the piezometric head, the superposed maximum pressure rise is 281.36 m ().

The comparison of the superposed maximum pressure rise at the end valve of branch M8 by the single, simultaneous and the sequential valve closure is shown in Figure 6(c). According to the comparison, the superposed maximum pressure rise under the sequential closing is the largest, followed by the superposed maximum pressure under the simultaneous closing. Then, the superposed maximum pressure rise under the single valve closure is the smallest. Based on the above analyses, it can be believed that the superposed maximum pressure rise at branch pipeline under sequential closing according to Equation (13) is the largest.

## DISCUSSION

In order to investigate the effect of valve closing law on the superposed water hammer pressure of a tree-type long distance gravitational water supply system, the formula of sequential valves closure is derived. The formula is verified using numerical simulation based on a practical project. In this section, the effect of sequential valves closure on superposed water hammer pressure of the main pipe and branch pipe is discussed.

The superposed maximum water hammer pressure rise at the main pipe and branch pipe under the single and simultaneous valves closure are compared separately. The variations of superposed pressure of the main pipe and branch pipe are shown in Figures 4 and 5, respectively. From Figure 4, when valves are closed within 100 s, the superposed maximum pressure rise at 5# water intake of the main pipe under single and simultaneous valves closure is 145.41 m and 240.16 m, respectively. It can be believed that the superposed maximum pressure rise at the main pipe under simultaneous closing is larger than that under single valve closure. Figure 5 also shows that, compared with single valve closure, the superposed maximum pressure rise at the branch pipe under simultaneous closing is the largest. The reason is that the superposition of water hammer waves is completely ignored under the single valve closure. These results are consistent with the results of previous studies.

The superposed maximum water hammer pressure rise at the main pipe and branch pipe under the single, simultaneous and the sequential valve closure are compared separately. The comparison of the results is shown in Figures 4 and 6. Figure 4 indicates that, when the formula of sequential closing () is adopted, the superposed maximum pressure rise at 5# water intake is the largest compared with the single and simultaneous valve closure. Figure 6 also indicates that, when valves are closed within 100 s, the superposed maximum pressure rise at the end valves of branch M8 under single, simultaneous and sequential valve closure is 129.76 m, 143.06 m, and 281.36 m respectively.

The reason is that, in previous works, the propagation and superposition of the waves are ignored. The previous works indicate that the superposition maximum pressure rise produced by simultaneous valve closure is the largest compared with the single valve closure. However, every branch has different distances from the water intakes in tree-type long distance gravitational flow, the time when the waves caused by simultaneous valve closure propagate to the same section is different. Then, because the waves cannot be strengthened, the superposition maximum pressure rise is not the largest. Therefore, when the formula of sequential valves closure is used, which is established based on the propagation and superposition of the waves, the superposed maximum water hammer pressure rise at the main pipe and branch pipe is the largest.

The effect of the sequential valve closure on the superposed maximum pressure rise at the main pipeline and the branch pipeline are compared. The superposed maximum pressure rise generated by sequential valve closure at the main pipeline and the end valve of branch M8 are shown in Figures 4(b) and 6(b), respectively. The superposed maximum pressure rise at the main pipeline under the sequential closing is 2.98% larger than the simultaneous closing. The superposed pressure rise at the end valve of branch M8 under the sequential closing is 34.48% larger than the simultaneous closing. Compared with the superposed maximum pressure rise at the main pipeline, it can be believed that the effect of sequential valve closure on the superposed maximum pressure rise at the branch pipeline is more sensitive.

## CONCLUSIONS

For a tree-type long distance gravitational water supply system, in this paper, the sequential valve closure formula was proposed based on the propagation and superposition of waves. The formula was verified using numerical simulation. The results showed that the formula agrees well with the numerical results. In addition, the superposed maximum water hammer pressure rise at the main pipe and branch pipe were compared under the single, simultaneous and sequential valve closure. Based on the results, the main conclusions were drawn as follows. First, compared with the superposed maximum pressure rise under the single and simultaneous valve closure, the maximum pressure rises at main pipeline and the branch pipeline under the sequential valve closure were the largest. Second, compared with the superposed maximum pressure rise at the main pipeline, it can be believed that the effect of sequential valves closure on the superposed maximum pressure rise at the branch pipeline is more sensitive.

This paper proposes that the sequential valve closure is the most unfavorable condition of the tree-type long distance gravitational water supply system, which is more dangerous than conventionally simultaneous valve closure. In actual operation, it is necessary to avoid this valve regulation. Besides, it must be pointed out that the water hammer waves caused by valves are only investigated from the propagation and superposition of the wave in this paper, and other factors are not considered. Therefore, future research is needed on the impact of other factors on the superposed water hammer pressure.

## ACKNOWLEDGEMENTS

This study was supported by the National Natural Science Foundation of China (Grant Nos 51879087 and 51839008), the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. 2019B71114 and SJKY19_0492), and the fifth ‘333 Project’ of Jiangsu Province (Grant No. BRA2018061).