Simulation of water hammer in long distance water transmission pipeline based on Flowmaster

Long distance water transportation projects are widely used in balancing water resources and other fields. However, the opening and closing of valves often causes significant pressure changes in the pipeline, whether it is positive or negative pressure, which can cause damage to the pipeline, thereby affecting the actual water transportation project. Therefore, based on the method of characteristics, this paper studies the influence of pipeline diameter, length, friction coefficient and valve closing time on the transient flow in the pipeline. The results indicated that the pressure pulsation inside the pipe is mainly concentrated at 0.4hz and its harmonics. In addition, the frequency with high amplitude mainly ranges from 2.5s to 10s, and the closer it is to the valve, the higher the amplitude at the node.


Introduction
Water resources are one of the most precious resources [1], and in China, the per capita ownership of water resources is lower than the world average [2].In addition, China's water resources have a serious imbalance in spatial and temporal distribution, and climate change and human production and life can also have a certain impact on water resources.Therefore, there is a serious shortage of water in certain regions of China, and the lack of water has become a bottleneck in the development of some cities.Therefore, it is necessary to adopt long-distance water transportation to optimize water resource allocation [3][4][5][6].
In long-distance water transmission projects, due to the long route of the water pipeline, the pressure at the water intake point is generally not sufficient to support the water loss along the pipeline, resulting in water transmission to the end point.It is often necessary to set up a pump station to pressurize the water transmission.However, due to the shutdown of the pump station and the opening and closing of valves, water hammer may occur during operation [6], which can damage pipelines, valves, and water pumps.Excessive positive water hammer often leads to the inability of the pipeline structure to withstand internal pressure, results in internal to external rupture.Excessive negative water hammer often results in the pipeline structure being unable to withstand external pressure and being compressed by external atmospheric pressure.However, there is little research on the water hammer phenomenon caused by valve opening and closing by using method of characteristics (MOC).Therefore, this article mainly studies the impact of water hammer caused by valve closure on long-distance water pipelines by using Flowmaster.Some parameters of the pipeline, such as length, diameter, friction coefficient was studied for improving the safety and stability of long-distance water transportation.The simulation model and governing equations are described in Section 2. The Results and discussion are described in Section 3. Section 4 concludes the paper.

Model of long-distance water pipeline and numerical method
Where g, H, V, x, t, D, a, f denote gravity acceleration, instantaneous water head, instantaneous flow velocity, pipeline axis distance, wave velocity, pipeline diameter, and friction coefficient respectively.

Method of characteristics
The method of characteristics is used to transform equation ( 1) and ( 2) into ordinary differential equation, and the positive and negative characteristic line equations are obtained as follows:  3) and ( 4) are the characteristic line equations for transient calculation of transient flow in pressurized pipelines, and their significance is explained on the s-t plane shown in Figure 2.
The points L and R in the figure represent the two points that have been given at location s and time t.Their pressure P and velocity V are known.By using the  + curve of point L and the  − curve of point R, the pressure P and velocity V parameters at the intersection point P can be solved using equations ( 3) and (4).

The influence of valve closing time
This study uses Flowmaster to simulate pipeline flow under different linear valve closure times.The linear closing time of the valve is taken as 0.3s, 0.5s, 0.7s and 1s, respectively.Observe the changes in the pressure at the front of the valve and the flow rate upstream of the straight pipe, and analyze the impact of different valve closure times on it.The results are shown in Figure 3.
As shown in Figure 3(a), when the valve is linearly closed, the flow rate does not change significantly during valve closure.With the end of the valve closing state, the upstream flow rate of the straight pipe suddenly decreases and generates negative flow rate in a very small amount of time.In addition, it is also observed that the flow rate value will have a regular change, which is a clear fluctuation phenomenon and the amplitude of the fluctuation gradually decreases.This indicates that the energy possessed by the water hammer is gradually decreasing with the transmission of the water hammer wave.Figure 3(b) indicate the pressure change at the front end of the valve with different valve closing time.During the process of closing the valve, the pressure at the front end of the valve sharply increased to 30 bar, resulting in a water hammer wave.In addition, we also found that the shorter the valve closing time, the greater the amplitude of the water hammer wave pressure formed.

The influence of frictional coefficient
This study uses Flowmaster to simulate pipeline flow under different friction coefficient of pipe.The friction coefficient of pipe is taken as 0.025mm, 0.040mm, 0.055mm respectively.Observe the changes in the pressure at the front of the valve and the flow rate upstream of the straight pipe, and analyze the impact of friction coefficient of pipe on it.The results are shown in Figure 4.
From Figure 4(a), it can be seen that as the friction coefficient increases, the amplitude of the flow rate begins to decrease which indicate that the increase in friction coefficient also increases the energy consumption of water hammer waves.In addition, after the valve is closed, the flow rate upstream of the straight pipe changes because the transmission of water hammer takes time.The pressure in the front section of the valve also reflects the same pattern as the flow rate in the front section of the straight pipe.
The pressure fluctuation at the pipe (node 2, 3 and 4) was analyzed in the time and timefrequency domains through continuous wavelet transform.The pressure pulsation inside the pipe is mainly concentrated at 0.4hz and its harmonics.In addition, the frequency with high amplitude mainly ranges from 2.5s to 10s, and the closer it is to the valve, the higher the amplitude at the node.

The influence of pipeline diameter and length
This study uses Flowmaster to simulate pipeline flow under different diameter and length of pipe.The diameter of pipe is taken as 0.2m, 0.4m, 0.6m respectively.The length of pipe is taken as 200m, 300m, 400m respectively.Observe the changes in the pressure at the front of the valve and the flow rate upstream of the straight pipe, and analyze the impact of friction coefficient of pipe on it.The results are shown in Figures 6 and 7.
As shown in Figure 6(a), as the diameter of the pipeline increases, the peak flow rate of the upstream pipeline decreases.Interestingly, the duration of the peak flow rate does increase.As the diameter of the pipeline increases, the pressure at the front end of the valve gradually decreases as seen in Figure 6(b), which indicate that increasing the diameter of the pipeline will reduce the pressure in front of the valve.Besides that, the upstream pipeline flow gradually decreases as the pipeline length increases, and as the length of the pipeline increases, the time required for the flow rate to change also increases.In the end, the pressure change at the front end of the valve occurs when the pipeline length is greater than 200m, and the peak pressure pulsation at the front end of the valve decreases as the pipeline length increases.

Conclusion
Based on the method of characteristics, this paper uses Flowmaster to study the influence of pipeline length, diameter, friction coefficient and valve closing time on water hammer in longdistance water transmission project.The following conclusions have been drawn： (1) The energy possessed by the water hammer is gradually decreasing with the transmission of the water hammer wave.the shorter the valve closing time, the greater the amplitude of the water hammer wave pressure formed.
(2) The pressure pulsation inside the pipe is mainly concentrated at 0.4hz and its harmonics.In addition, the frequency with high amplitude mainly ranges from 2.5s to 10s, and the closer it is to the valve, the higher the amplitude at the node.
(3) As the diameter of the pipeline increases, the pressure at the front end of the valve gradually decreases.
(4) The upstream pipeline flow gradually decreases as the pipeline length increases, and as the length of the pipeline increases, the time required for the flow rate to change also increases.

2. 1
Physical model Model of long-distance water pipeline as shown in Figure 1.Component 1 and component 2 are the upstream and downstream reservoirs respectively.The water level of the upstream reservoir is 10m, and the water level of the downstream reservoir is 1m.Components 3, 4, and 7 are incompressible circular tubes.Component 5 is a ball valve, Component 6 is a ball valve controller.The calculation time is 20 s, and the calculation time step is 0.02 s.The required data mainly includes the pressure in front of the valve at node 4 and the upstream flow rate at node 1.

Figure 1 .
Figure 1.Model of long-distance water pipeline.

Figure 2 .
Figure 2. Characteristic lines on the s-t plane.

Figure 3 .
Figure 3.Effect of valve linear closing time.
(a) Flow rate change upstream of straight pipe (b) Pressure change at the front end of the valve.

Figure 4 .
Figure 4. Effect of friction coefficient of pipe.

Figure 5 .
Figure 5. Pressure fluctuation at the pipe node 2, 3 and 4 in the time frequency domain.
(a) Flow rate change under different diameter.(b) Pressure change under different diameter.