Study on impedance size optimization of a one-way surge tank in a long-distance water supply system

The size of the impedance hole of the one-way surge tankwill affect its protection against water hammer. In this paper, a mathematical model of one-way surge tank with impedance is established based on the characteristic linemethod. Thepressure-reducingpenetration formulaof theone-waysurge tank including impedance loss and the calculation formula of make-up water flow are deduced. Based on these formulas, the influence of impedance hole diameter ratio (ratio of impedance hole of one-way surge tank to diameter of water pipeline) on the water hammer protection effect is analyzed, and the reasonable value range of impedance hole diameter ratio is given. The correctness of the theoretical analysis results is verified by an engineering example. The results show that the pressure and flow relationship derived from the formula are consistent with the numerical simulation results. The penetration pressure drop of the one-way surge tank is inversely proportional to the impedance size. When the impedance hole diameter ratio is less than 0.2, the penetration pressure drop will cause serious negative pressure. The make-up water flow is proportional to the size of the impedance hole, and the make-up water volume should be reduced while ensuring that the pipeline has no negative pressure.


INTRODUCTION
Long-distance water supply projects can redistribute water resources and improve the utilization rate of water resources. In order to overcome the influence of topography, most long-distance water supply projects need to transfer water from low terrain to high terrain by means of pumping station pressurization (Ren et al. ).
When the water pump is suddenly cut off, the pressure after the pump drops sharply, and the pressure drop wave is quickly transmitted downstream. Where the initial internal pressure is small, the pipeline pressure may drop to the vaporization pressure, and liquid column separation occurs, which in turn induces the bridging of water hammer. The huge pressure generated by bridging the water hammer may cause severe damage to the entire water supply system, leading to the interruption of water supply and heavy losses (Besharat et al. ). Therefore, it is necessary to study the water hammer protection measures of the pumping station water supply system.
The current protection methods used to solve the pump-failure water hammer include air tanks, surge tanks, one-way surge tanks, air valves, etc. Compared with other protection measures, the one-way surge tank has the advantages of low cost, no terrain restriction, and can effectively protect negative water hammer, so it is widely used in engineering. Bergant et al. () introduced the one-way surge tank as a water hammer control device, studied the protection performance of the one-way surge tank, and considered that it can effectively control the surge in the system. Stephenson () proposed that a one-way surge tank can be used for negative pressure protection in low-head pipelines. Liu et al.
() put forward the protection scheme of using a oneway surge tank to prevent water column separation, and verified the rationality of the scheme through experiments. According to different parameters of one-way surge tank, Wang et al. () analyzed the influence of its volume and initial water level on the hydraulic transient process of pumping accidents. Liu et al. () analyzed the main factors affecting the water hammer protection effect of the one-way surge tank. Xu & Zhang () proposed to arrange a one-way surge tank at the high point of the pipeline, which can not only effectively protect the pipeline from negative pressure, but also reduce the height of the one-way surge tank and save investment. Zhang et al. () conducted an in-depth analysis on the installation principle and location of one-way surge tanks in long-distance water delivery projects. Zhou et al. () found that setting a one-way surge tank in the urban pressurized water supply system could solve the possible water hammer problem.
Most of the above-mentioned studies on one-way surge tanks have focused on how to set up the scheme of oneway surge tanks and the impact of changes in the volume parameters of the one-way surge tank on the protection effect of water hammer, and rarely consider the impact of impedance on the protection effect of the one-way surge tank. However, in the actual operation of the water supply system, the influence of impedance cannot be ignored. The selection of impedance hole size of one-way surge tanks is mostly based on numerical simulation or engineering experience, which has a certain degree of blindness. In order to simplify the design process of the impedance hole size, this paper establishes the mathematical model of the impedance one-way surge tank based on the characteristic line method, analyzes the relationship between impedance hole size of one-way surge tanks and pressure drop after pump and make-up water flow rate, and introduces the selection principle and specific range of impedance hole of one-way surge tanks, which can provide a reference for designers to quickly determine the size of impedance hole. At the same time, the method is verified by studying the actual cases introduced in this paper. Figure 1 is a schematic diagram of an impedance-type oneway surge tank. Assuming that the boundary nodes of the water inlet and outlet pipes are numbered 1 and 2, the control equation of the one-way surge tank is:

MATHEMATICAL MODEL
Flow continuity equation: (1) Water head balance equation: Relationship between flow rate and water level: Compatibility equation of pressure pipeline: where H st is the water level of the one-way surge tank (m); A st is the cross-sectional area of the one-way surge tank

SELECTION OF IMPEDANCE HOLE SIZE
The head loss value of the impedance hole of the one-way surge tank can be calculated by Equations (5) and (6).
where g is the acceleration due to gravity (m/s 2 ); A is the pipeline area (m 2 ); η is the ratio of the impedance hole diameter to the pipe diameter; φ is the flow coefficient; and H ω is the head loss through the impedance hole of the one-way surge tank (m).
It can be seen from Equations (5) and (6) that the impedance loss of a one-way surge tank is mainly determined by the size of the impedance hole and the make-up water flow.
When the pump is out of power, the larger the impedance hole size, the smaller the head loss. A strong capacity of the one-way surge tank to replenish water to the pipeline allows the pressure drop behind the pump and along the pipeline to be better controlled. If the impedance hole is too small, the head loss of the water flowing through the impedance hole is too large, which will lead to too little pressure through the one-way surge tank and it is easy to generate negative pressure. On the other hand, if the water in the one-way surge tank has difficulty flowing into the pipeline, the insufficient water supply capacity will lead to the failure of effectively protecting the pressure drop behind the pump. However, when the impedance hole size of the one-way surge tank is too large, the ability of water flowing out of the one-way surge tank is enhanced. This increases the water-level drop speed of the one-way surge tank, and the one-way surge tank makes up too much water to the pipeline, causing the minimum pressure at the bottom of the one-way surge tank to decrease, or even leak out.
From the above analysis, it can be seen that the size of the impedance hole in one-way surge tanks is affected by many factors, such as water hammer pressure, system pressure, height of one-way surge tank, area of one-way surge tank, etc. Therefore, it is necessary to consider the size of impedance holes of one-way surge tanks. The following two conditions should be met when selecting the impedance hole size: 1. The minimum impedance hole of the one-way surge tank should ensure that the pipeline under the influence of water hammer pressure drop does not have negative pressure.
2. The maximum impedance hole of the one-way surge tank should meet the pressure of the pump station system after the valve is quickly closed behind the pump, and the Since the pressure drop wave transmitted to this place is an instantaneous pressure drop, the pressure drop relationship can be approximately calculated according to the direct water hammer calculation formula, and the two satisfy the relationship: where Q 0 is the flow rate during stable operation (m 3 /s); ΔH A is the pressure drop value transmitted to the one-way surge tank (m); and Q p1 is the flow rate when the pressure is reduced to the one-way surge tank (m 3 /s).
Due to the existence of the one-way surge tank, part of the pressure drop wave is reflected by the one-way surge tank as a positive pressure wave and propagates to the front of the one-way surge tank, and part of it penetrates the one-way surge tank and continues to propagate to the pipeline behind the one-way surge tank. In the instantaneous depressurization process, the forward reflected positive pressure wave and backward propagating negative pressure wave can be approximately calculated using a direct water hammer formula, which satisfies the relationship: where ΔH B is the pressure drop value that penetrates the one-way surge tank (m); Q p2 is the flow behind the oneway surge tank (m 3 /s); and Q st is the make-up water flow from the one-way surge tank to the pipeline (m 3 /s).
It can be obtained by simultaneous Equations (6)- (10): Using the Taylor expansion gives the first-order approximation: The meaning of the third term on the right side of Equation (12) is the pressure drop loss caused by impe- is defined as impedance loss ratio, its value is: where If the flow coefficient is 1, the relationship between impedance loss ratio δ and diameter ratio η under different pressure drop differences, as shown in Figure 3.
It can be seen from Figure 3 that when the diameter ratio of impedance hole of the one-way surge tank is less than 0.2, the pressure drop loss ratio of the one-way surge tank will increase sharply, and when the diameter ratio is 0.1, it will increase to more than 1.5 times, which means the protection effect of the one-way surge tank is reduced by more than half. When the diameter ratio is 0.3 or above, the pressure drop loss of the one-way surge tank is not very different. Therefore, it is suggested that the diameter of the impedance hole of the one-way surge tank should be at least greater than 0.2 times the pipe diameter.
Relationship between impedance hole size and make-up water flow When a power failure occurs to the pump, the pump head drops instantaneously, and when the negative pressure water hammer wave is transmitted to the one-way surge tank, the one-way surge tank begins to replenish water.
According to the direct water hammer formula, the water supplement volume of the one-way surge tank in the prime minister process is: where L 1 is the distance from the one-way surge tank to the downstream outlet pool (m).
Equation (14) does not take into account the drop in the water level during the replenishment process of the one-way surge tank. This calculation is only the effective replenishment volume of the one-way surge tank during the main process. Since the water body in the one-way surge tank has inertia, when the size of the impedance hole is large, the water body inertia is too large, which causes excessive water supply. This causes the pressure at the bottom of the one-way surge tank to be too low, which is not conducive to the protection of water hammer in the pipeline.
The actual make-up water volume of the one-way surge tank can be used as follows: where ΔH is the drop height of the one-way surge tank water level (m).
The impedance diameter ratio of the one-way surge tank also has an upper limit due to the influence of make-up water speed. The minimum size of the impedance hole can be determined according to the first wave of pressure drop after the pump, but the influence on the water supplement speed is related to the situation of the entire pipeline. Numerical simulation can be used to determine the pressure change of the entire pipeline. In order to avoid excessive water supply, the size of impedance hole should not be too large.

CASE STUDY AND ANALYSIS
In the water supply system of a pump station, the piping system adopts a single pipe arrangement, and its designed

Without protection
When the water pump is powered off, the pressure behind the pump drops rapidly. If there is no water hammer protection measures in the pipeline behind the pump, serious negative pressure may appear along the pipeline, which will lead to pipeline damage. The power loss of the pump is calculated without any protective measures, and the specific results are shown in Figure 5.
It can be seen from Figure 5(a) that when the pump loses power, the pressure after the pump begins to drop, and when t > 2.6 s, the pressure after the pump drops to negative pressure. When t ¼ 16 s, the pressure after the pump reaches the minimum value À2.14 m. It can be seen from Figure 5(b) that the maximum pipeline pressure meets the maximum pipeline pressure control standard (21.54 m < 28 m), but the minimum pressure in most places along the pipeline is negative. In order to ensure the safe and stable operation of the water supply system, it is necessary to set up reasonable protective measures to protect the negative pressure.

One-way surge tank protection
The project intends to adopt a one-way surge tank as a water It can be seen from Table 1     When the impedance hole size ratio is 0.2 or 0.3, the water head loss of the impedance hole is too large due to the small impedance hole, which makes it difficult to flow into the pipeline in the one-way surge tank, resulting in an insufficient water supply capacity for the pipeline. Under the influence of the first wave of water hammer pressure reduction, the water supply system has different degrees of negative pressure. When the impedance hole size ratio is 0.4-1.0, the pressure drop along the pipeline is controlled due to the small pressure drop caused by the impedance, which has a negative pressure protection effect. When the impedance hole size ratio increases from 0.2 to 0.5, there is a large change in the system pressure and water level in the surge tank change; however, when the impedance hole size ratio increases from 0.5 to 1.0, the change range is small, that is, when the impedance hole reaches a certain value, increasing its size has little effect on the bottom pressure and the water-level change range in one-way surge tanks.  the water body continues to drop, which will cause head loss.
Therefore, when the impedance hole size is greater than 0.5, the pressure at the bottom of the one-way surge tank will continue to decrease after the first wave of penetration pressure drop until the water is replenished. At this time, the bottom pressure of the one-way surge tank is mainly determined by head loss. In the process of changing from 0.5 to 1.0, the bottom pressure of the one-way surge tank shows a decreasing trend, but the difference between different impedance ratios is not large.
Due to the friction h ω in the pipeline, the pressure transmitted to downstream is 12.79 À H ω À h ω . Since the water pressure line along the downstream direction of the pipeline is gradually reduced, negative pressure may occur in the downstream where the initial water pressure line is less than 12.79 À H ω À h ω .
For a one-way surge tank with a small impedance hole size, H ω is too large, the make-up water flow is small, and the influence of h ω is small, so negative pressure is likely to occur. In addition, when the impedance size ratio changes from 0.2 to 0.4, the impedance loss changes in order of magnitude. Therefore, increasing the impedance size can effectively reduce the negative pressure, and, as shown in Figure 8, the pressure at each point of the pipeline changes regularly. The larger the impedance hole size, the greater the minimum pressure corresponding to each point; however, when the size of the impedance hole continues to increase to 0.5, there is no significant difference in impedance loss. The increase of the impedance hole will cause the flow in the pipeline to increase, leading to an increase of h ω , but the impact of h ω on the pipeline is small, so it is better protected against the negative pressure of the water supply system. In the process of impedance size ratio from 0.6 to 1.0, the pressure in the pipeline shows a trend of first rising, then falling, and then rising.
The first wave of rise is due to the pressure rise caused by more water filling in the pipeline, and the larger the size of the impedance hole, the more water is added and the longer the pressure rise distance. Until all the water has replenished, the pressure shows a downward trend under the influence of friction in the pipeline. At a location close to the lower reservoir, the pressure will continue to rise to the water pressure in the lower reservoir because the lower reservoir has a protective effect on nearby pipelines.

CONCLUSION
According to the impact of the one-way surge tank impedance on the water supply system of the pumping station, the impedance size of the one-way surge tank is optimized.
The selection principle of impedance hole size was ana- In practical engineering, the protection effect of water hammer is also affected by the location, height, area, and other factors of the one-way surge tank, and so the optimal choice of the impedance hole size of the one-way surge tank in practical engineering needs further study.
ACKNOWLEDGEMENT This paper was supported by the National Natural Science Foundation of China (grant numbers 51839008 and 51879087).

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