https://orcid.org/0000-0001-5613-8744
Abstract
The protection of negative pressures generated by the hydraulic transient process in the water supply system is crucial to safe and stable operation. In this study, a mathematical model of a pipeline hydraulic transient system with multi-undulating terrain was established based on the method of characteristics (MOC). The generation and development of water hammer negative pressures were analyzed, and double one-way surge tank protection schemes were proposed. It proved that the first surge tank should be located at the initial negative pressure point, and the second surge tank should be located at the second-highest point rather than the highest point. Additionally, compared with the theoretical minimum height, there was an optimization margin for the total surge tank height, which was reduced by 21% in this study. Meanwhile, the applicability of protection schemes and the influence of the one-way surge tank number on the total height were analyzed. The total height reduced with the increase of one-way surge tank number and tended to a minimum value. By comprehensively considering the engineering investment and negative pressure protection effect, the optimal surge tank number could be determined. This research represents an advance in negative pressure protection in multi-undulating terrain and provides support for further engineering studies.
HIGHLIGHTS
Double one-way surge tank schemes can fully protect negative pressures in water supply systems with multi-undulating terrain. The optimal locations are revealed.
Compared with the theoretical minimum surge tank height, the total surge tank height can be further reduced.
The total height reduces with the increase of surge tank number and tends to a minimum value.
INTRODUCTION
Long-distance water supply project is a significant way to solve the problem of uneven distribution of water resources (Zhenghui et al. 2018). With the increasing number of projects, the safety of water supply systems has been paid more and more attention (Zhang et al. 2009; Zhao et al. 2022). When transient processes such as pump power failure occur, water hammer waves will be generated, which create pipeline pressure fluctuations and potential safety hazards (Moghaddas 2018; Kim 2022). The internal pressure may exceed the pipeline pressure standard under the influence of the positive water hammer wave. The internal pressure may reduce to vaporization pressure under the influence of the negative water hammer wave (Yazdi et al. 2019). Therefore, it is critical to study the transient process of the water supply system and put forward water hammer protection schemes.
Water supply pipelines are generally laid according to the topographic terrain. Pumps in pipelines are used to overcome the topographic drop for water delivery. The number of long-distance pipelines constructed in complex terrain has increased in recent years. In some mountainous areas, the pipeline system has undulating terrain. The initial internal pressure of pipelines is low at high points of undulating terrain (Carmona-Paredes et al. 2019). When a pump power failure occurs, the water hammer will produce depressurization waves, which may reduce the lower initial pressures to negative vaporization pressures (Feng et al. 2021). The liquid column separation and cavities collapse are developed, resulting in huge pressures that may damage the pumps, valves, and pipeline system. Some studies of water hammer protection focused on the terrain of the pipeline system. For example, simulations of column separation and cavity collapse were carried out in complex terrain, and preventive measures were proposed to effectively control the water hammer pressure and reduce the occurrence of incidents (Liu et al. 2017; Arefi et al. 2021). According to special terrains in water supply projects, water hammer negative pressure was analyzed and protective schemes were proposed and compared (Lin et al. 2019; Lyu et al. 2022). It is necessary to enhance the understanding of pipeline negative pressure in undulating terrain.
In addition, common water hammer protection devices include air vessels, surge tanks, one-way surge tanks, and air valves (Stephenson 2002; Wan et al. 2014). Appropriate protection measures can be selected according to the characteristics of different projects. Also, joint protection schemes with multi devices were proposed for long-distance water supply systems (Kim et al. 2014; Sgk et al. 2014; Miao et al. 2017; Wang et al. 2019). A one-way surge tank is a useful measure to protect against negative water hammer waves in long-distance water supply projects (Wang et al. 2017; Shi et al. 2021). It has the advantages of a simple structure, easy to manage, and has fewerinstallation limitations. A one-way check valve is set on the connecting pipe at the bottom of the one-way surge tank. Under normal operation conditions, the pressure in the one-way surge tank is less than that in the pipeline, so the one-way surge tank will not operate. When a pump power failure occurs, under the influence of a negative water hammer wave, the pressure in the pipeline will be lower than the pressure in a one-way surge tank. Then the one-way check valve will open to replenish water, which eliminates negative pressure and ensures the safety of the pipeline system. One-way surge tank protection schemes were applied to protect the pipeline from negative pressure. The influence of surge tank height (Chen et al. 2021a, 2021b) and location (Zhang et al. 2011) were significant on the protection effect of the water hammer.
However, water hammer protection of pipeline systems is more complex in multi-undulating terrain. Existing research about multi-undulating terrain water hammer protection was not complete enough. Studies on multi-one-way surge tank protection schemes were less when negative pressures appeared under pump power failure conditions. With the increase of the emphasis on water supply safety, the characteristics of transient processes and protection schemes in multi-undulating terrain need to be further studied.
Primary flowchart of the research procedure.
METHODOLOGY
Mathematical model and governing equation
The basic water hammer governing equations derived from the momentum and continuity theorem are as follows (Geng et al. 2017; Li et al. 2019):
is the pipeline diameter.
is the pipeline area. The subscripts A and B represent the parameters of two adjacent nodes, and the subscript p represents the unknown quantity. The calculation starts from 
. The Q and H of each node are calculated when 
, and then the Q and H of each node are derived when 
. In this way, the solution process continues until the required time is reached.One-way surge tank model
and 
represent the flow rates of the front and rear nodes, respectively, and 
is the flow out rate of the one-way surge tank. 
is the initial water level of the one-way surge tank. When a pump power failure occurs, the pressure at node K of the pipeline is lower than the head pressure in the surge tank, and the one-way check valve opens to replenish water in the pipeline (
). Then the pressure in the pipeline continues to rise, and when the pressure at node K of the pipeline is greater than or equal to the head pressure in the surge tank, the check valve closes to stop replenishing water (
) (Yiran et al. 2023). Accordingly, the mathematical model of a one-way surge tank is shown in Figure 2. 
Mathematical model of one-way surge tanks.
The equations of the surge tank calculation node are as follows:
and 
are the flow rate in the pipeline, and 
is the replenishing flow rate of one-way surge tank.
is the head at point P, 
is the water level of the surge tank, and 
is the impedance head loss coefficient of a one-way check valve.
is the water level of the surge tank, 
is the sectional area of the surge tank. The compatibility equations are derived as follows:
Boundary conditions of the mathematical model
. The reservoir node auxiliary equations are as follows:
When the rotating speed n and flow rate q of the pump were determined, the pump head and shaft torque can be calculated through the pump characteristic curve. Then, the flow rate function 
and torque function 
were obtained by the similarity theory and transformation. When the pump power failure occurs, the shaft torque of the pump is 0. Combined with the pump head balance equation 
, through the derivation of the pump rotation equations and compatibility equations, the mathematical model Equation (11) of the pump node is obtained.
is the valve flow rate, 
is the valve flow rate coefficient, 
is the valve flow area, 
is the valve loss, 
is the valve flow resistance coefficient.RESULTS
Water hammer negative pressure analysis without protection
of high points were 11.46, 13.85, and 16.79 m, respectively. The section behind point C had no obvious high points and undulations along the pipeline, and the overall pipeline elevation was decreasing. The water levels of the inlet reservoir and outlet reservoir were 6.81 and 5.25 m, and elevations at the beginning and end of the pipeline were relatively close. The pumping station was equipped with two single-stage double-suction pumps, the head of the pumps was 26.68 m, the rated flow rate was 2.4 m3/s and the rated speed was 740 r/min. The total flow rate of the water supply system was 4.8 m3/s and the total pipeline length was about 26 km. According to the projection of head loss along the pipeline, the section behind the high point C belonged to gravity flow under constant flow conditions, while the section before point C belonged to the pumping station pressurization section. The positive overpressure problem in the gravity flow section will be further investigated in future research. This study paid more attention to negative pressure protection, which was generally considered in the pumping station pressurization section and high points (Chen et al. 2021a, 2021b). The terrain of the pressurization section was multi-undulating, which was used as the protective facility layout area and the negative pressure protection study area. 
Schematic diagram of the pipeline system with multi-undulating terrain: (a) schematic diagram of the overall pipeline system and (b) elevation of the pump station pressurization section.
Schematic diagram of the pipeline system with multi-undulating terrain: (a) schematic diagram of the overall pipeline system and (b) elevation of the pump station pressurization section.
of the whole pipeline was 878 m/s, according to the water hammer formula 
, the theoretical value of instantaneous pressure drop 
behind the pump was 26.8 m. Through the calculation of the water supply system model with multi-undulating terrain, the instantaneous pressure behind the pump 
and the discharge behind the pump 
was obtained after pump power failure as shown in Figure 4(a). It could be seen that the discharge 
immediately dropped from 4.8 to 3.86 m3/s within 2 s after pump power failure, which led to the velocity decrease from 1.528 to 1.229 m/s. The calculated instantaneous pressure drop value behind the pump was 
, which was close to the theoretical value. It proved the accuracy of the calculation results and also clarified that water hammer pressure drops under pump power failure conditions needed more attention. 
Water hammer analysis under unprotected power failure conditions: (a) instantaneous pressure and discharge behind the pump; (b) minimum pressure envelope; and (c) maximum pressure envelope.
Water hammer analysis under unprotected power failure conditions: (a) instantaneous pressure and discharge behind the pump; (b) minimum pressure envelope; and (c) maximum pressure envelope.
The minimum internal pressure 
and maximum internal pressure 
envelope curves along the pipeline under the unprotected power failure condition are shown in Figure 4(b) and 4(c). The abscissa was the distance from the inlet reservoir. It could be seen that negative pressure appeared in most of the pipeline. Taking −10 m as the maximum negative pressure limit. When the pressure was lower than −10 m, the cavity would occur. The specific pressures lower than −10 m of pipeline were theoretical calculation values to show the severity of negative pressure. Without any protective measures, there were no water replenishment measures in the pipeline system when the pump power failure occurred. Pressure drops in the pressurization section would inevitably be generated after the pump, and the depressurization wave started to transmit. The negative pressures at point B and point C of the pumping station pressurization section were close to or greater than −10 m. Furthermore, in the absence of water replenishment facilities, negative pressure had already been observed in the pressurization section before the high point, and a significant pressure drop would inevitably occur in the gravity flow section and the rest of the pipeline system after the high point. Negative pressure, especially lower than −10 m, was more likely to occur at higher elevations. Therefore, there were more negative pressure values greater than −10 m at the end of the gravity flow section, with the maximum 
reaching −18.8 m. Negative pressures would lead to serious safety accidents like water hammer pipe bursts, so it was necessary to propose further protective measures in multi-undulating terrain pipeline system.
Protection schemes of double one-way surge tanks
It is efficient and economical to use one-way surge tanks to protect against negative pressure under pump power failure conditions in the water supply system. The total height of one-way surge tanks 
is a key factor in protection capacity. The higher the height is, the stronger the protection capacity, and the longer the protection distance, while it is uneconomical. On the other hand, when the height is low, the protective capacity will be insufficient. Generally, the theoretical minimum 
is calculated based on the first wave pressure drop behind the pump, which is, the sum of the total surge tank height and the pressure at the end of the pipeline should be greater than or equal to the instantaneous pressure drop behind the pump. In this study, the instantaneous pressure drop was 27.3 m and the pressure at the end of the pipeline was 4 m. Thus, the minimum theoretical 
was 24 m. Considering the negative pressure at multi-high points in the pipeline system with multi-undulating terrain, it was difficult to protect the negative pressure only by a single one-way surge tank. Therefore, double one-way surge tank protection schemes were proposed as shown in Table 1. The rationalized heights of each surge tank were permuted and combined into multiple schemes. The protective effects of each scheme with the same total height and total number were calculated, and the scheme with the best protection effect was selected and presented in Table 1. The height of each surge tank H was selected after optimization. The pipeline elevation 
of each location was given. In this study, the water supply pipeline diameter was 2 m. Generally, the impedance hole diameter of one-way surge tank is selected as 20%-50% of the water supply pipeline diameter. Therefore, the initial sectional area and impedance hole diameter of one-way surge tanks were not taken as variables, and were defined as 100 m2 and 1 m respectively.
Double one-way surge tank protection schemes
| No. | Surge tank location |  (m) |  (m) |  (m) |
|---|---|---|---|---|
| S1 | P | 7 | 24 | 10.38 |
| B | 17 | 13.85 | ||
| S2 | P | 12 | 24 | 10.38 |
| C | 12 | 16.79 | ||
| S3 | A | 14 | 24 | 11.46 |
| B | 10 | 13.85 | ||
| S4 | A | 16 | 24 | 11.46 |
| C | 8 | 16.79 | ||
| S5 | P | 4 | 19 | 10.38 |
| B | 15 | 13.85 | ||
| S6 | P | 8.5 | 19 | 10.38 |
| C | 10.5 | 16.79 |
| No. | Surge tank location | (m) | (m) | (m) |
|---|---|---|---|---|
| S1 | P | 7 | 24 | 10.38 |
| B | 17 | 13.85 | ||
| S2 | P | 12 | 24 | 10.38 |
| C | 12 | 16.79 | ||
| S3 | A | 14 | 24 | 11.46 |
| B | 10 | 13.85 | ||
| S4 | A | 16 | 24 | 11.46 |
| C | 8 | 16.79 | ||
| S5 | P | 4 | 19 | 10.38 |
| B | 15 | 13.85 | ||
| S6 | P | 8.5 | 19 | 10.38 |
| C | 10.5 | 16.79 |
The variable of S1–S4 schemes was surge tank location. The pumping station pressurization section with multi-undulating terrain was the main pipeline for negative pressure protection. Setting one-way surge tanks before point C could cut down the pressure drop wave generated by pump power failure and prevent it from propagating backwards, which provided protection for the gravity flow section. Considering the conservative principle, the first surge tank location selected point P behind the pump, and the second surge tank location selected the second-highest point B or the highest point C. In this way, the forward projection distance of the one-way surge tank was not considered. The first surge tank was located at the initial point of negative pressure protection, so as to make full use of the surge tank protection capacity to replenish water backwards and reflect water hammer waves. Considering economic principle, the first surge tank location selected point A which was a high point and close to the inlet reservoir. The second surge tank location selected the second-highest point B or the highest point C. In this way, the forward projection distance of the one-way surge tank was fully considered, and all surge tanks were set at the high points of multi-undulating terrain.
The variable of S5–S6 schemes was the total height 
. Based on the theoretical 
and schemes S1–S2, schemes with a lower 
were proposed, which explored 
optimization margin for negative pressure protection.
Effects of one-way surge tank location on negative pressure protection
was fixed to the theoretical value of 24 m. The influence of different locations of the first surge tank and second surge tank on the water hammer was calculated. Figure 5(a) shows the minimum internal pressure envelope under different one-way surge tank protection schemes in pump power failure conditions. Figure 5(b) illustrates the minimum pressure at the highest point C after pump power failure. It could be seen that surge tank locations would have a great impact on negative pressure protection along the pipeline. Compared with the unprotected pipeline system, the setting of double one-way surge tanks could lift the pipeline pressure under pump power failure conditions as a whole. S1 and S2 schemes met the negative pressure protection requirements along the pipeline, and the pipeline minimum pressure envelopes were greater than 0 m. However, there were still some negative pressure areas at the end of the pipeline in the S3 and S4 schemes. 
Pressure under different one-way surge tanks location schemes: (a) minimum internal pressure envelope and (b) minimum pressure at the highest point along the pipeline.
Pressure under different one-way surge tanks location schemes: (a) minimum internal pressure envelope and (b) minimum pressure at the highest point along the pipeline.
Under scheme S1, the minimum internal pressure of the multi-undulating section was 2.95 m. The minimum internal pressure of the gravity flow section was 1.7 m. The minimum internal pressure at the highest point C was 3.3 m. Under scheme S2, the minimum pressure of the multi-undulating section appeared at the highest point C, which was 2.12 m, and the minimum pressure of the gravity flow section was 1.78 m. These two schemes ignored the forward protection distance of the first surge tank and the high points but ensured that the location of the surge tank was at the initial negative pressure point so that the water hammer negative pressure caused by pump power failure along the pipeline could be eliminated.
S3 and S4 schemes made full use of the forward projection distance of the one-way surge tanks, and all surge tanks located at the high points of the pipeline system. However, due to the limitation of the actual forward replenishment protection capacity, when the total height was fixed, the first surge tank height should be higher and the second surge tank height should be lower, which weakened the backward protection distance. As a result, there was still negative pressure at the end of the pipeline system.
of one-way surge tanks in different schemes are shown in Figure 6. The changing trend of 
with the time after a power failure was similar under different schemes. The water replenishment of double one-way surge tanks after power failure was large in S1 and S2. However, due to the gravity flow behind the highest point C and the second surge tank location, the second surge tank would always replenish water backwards in the S2 scheme before the end valve was completely closed. Consequently, there was more water loss in S2, which also explained that the minimum internal pressure at the highest point in S2 was smaller than that in S1. Moreover, the water replenishment flow rates of S3 and S4 were significantly smaller than those of S1 and S2, which further revealed that S3 and S4 schemes had insufficient protection capacity in multi-undulating terrain pipeline system. 
Total water replenishing flow rate of one-way surge tanks in different schemes.
In general, for the pipeline system with multi-undulating terrain, the double one-way surge tank schemes could fully protect water hammer negative pressure of pump power failure. The effect of surge tank location was crucial. When selecting a one-way surge tank location, the forward protection effect should not be considered. The first surge tank should be directly set at the initial negative pressure point to strengthen the protection capacity of the negative pressure water hammer wave. In addition, as shown in Figure 5(a), both S1 and S2 were able to effectively protect the pipeline from negative pressure when the total height of surge tanks was the same, indicating that setting the second surge tank at the highest or second-highest point could all meet the negative pressure protection requirements, which was the basis of the protection scheme selection. As shown in Figure 5(b), it could be found that the minimum pressure at the highest point in the scheme of setting the highest point surge tank (S2) was lower than that in the scheme of setting the second-highest point surge tank (S1), which indicated that scheme S1 had a better negative pressure protection effect on the highest point. Moreover, there was a long-distance gravity flow section behind the highest point. Through Figure 6 and theoretical analysis, considering the gravity flow characteristics caused by undulating terrain and the unidirectional water replenishment of a one-way surge tank, the instantaneous water replenishment was greater and water replenishment would always be going on in the scheme of setting the highest point surge tank (S2), which might increase the risk of empty surge tank of water and reduce the negative pressure protection effect of the highest point and pipeline system. Hence, considering the effect of negative pressure protection at the highest point, the second surge tank location could be set at the second-highest point along the multi-undulating pipeline rather than the highest point.
Effects of one-way surge tank height on negative pressure protection
were calculated and compared with S1 and S2. The surge tank locations of S5 and S6 were the same as S1 and S2. The effect of 
on water hammer negative pressure was studied. Figure 7(a) shows the minimum internal pressure envelope under different 
in pump power failure conditions. Figure 7(b) illustrates the minimum pressure at the highest point C after pump power failure. It could be seen that S5 and S6 could also increase the negative pressure to more than 0 m, which was effective for negative pressure protection. There was no negative pressure at the highest point of the terrain during the whole propagation process of the water hammer wave. 
Pressure under different one-way surge tank total height schemes: (a) minimum internal pressure envelope and (b) minimum pressure at the highest point along the pipeline.
Pressure under different one-way surge tank total height schemes: (a) minimum internal pressure envelope and (b) minimum pressure at the highest point along the pipeline.
of S1 and S2, 
of the double one-way surge tanks in S5 and S6 decreased by 21%. It showed that, compared with the theoretical minimum height of the surge tank, there were still margins for total height optimization in double one-way surge tank schemes with multi-undulating terrain. Therefore, in multi-undulating terrain pipeline system, the double one-way surge tanks protection scheme was conductive to further decrease the total height of surge tanks. The protection effect was improved and the project cost was reduced. 
Comparison of surge tank heights in different protection schemes.
DISCUSSION
The water hammer negative pressure protection is more significant and complex in multi-undulating terrain pipeline systems. The applicability and generalizability of the present double one-way surge tanks research results are worth discussing. On the one hand, the water supply project system of this present research had a low head and high flow rate, which represented a common working condition of the water supply pipeline. Therefore, the results might be more applicable to similar working conditions. On the other hand, the elevation difference between the highest point and the second-highest point was not very large in the present multi-undulating terrain research section. So the effect of a one-way surge tank location on negative pressure protection was proposed. However, when the elevation difference between the highest point and the second-highest point was large enough, the schemes of setting second-highest point surge tank might no longer meet the negative pressure protection requirements. At this time, setting a one-way surge tank at the second-highest point might make it difficult to eliminate negative pressure at the highest point. The surge tank location selections of the present results might be unreasonable and the surge tank locations of protection schemes would be further reconsidered.
Considering the total height 
of one-way surge tanks, the present results showed that the height of the optimized double surge tanks scheme could be less than the theoretical minimum value. The reasons for the theoretical height optimization margin were discussed. Due to the water gravity flow, negative pressure is always generated at the high point of the multi-undulating terrain pipeline in pump power failure conditions. In order to ensure the pipeline flow continuity, water replenishment of the surge tanks arranged at high points will also be strengthened. The protection distance before and after the one-way surge tank at the high points is extended with the increase of the undulating terrain slope angle. Moreover, in practical engineering, there are many attenuation factors of water hammer waves to weaken the pressure drop, such as pipeline friction, and the attenuation mechanism is complex. However, the attenuation mechanism of the water hammer wave propagating downstream was not considered in the theoretical total height calculation. Accordingly, the surge tanks 
can be further reduced in double surge tank protection schemes with multi-undulating terrain.
Furthermore, the influence of the one-way surge tanks number on 
was discussed. The present research demonstrated double surge tank schemes to protect the negative pressure in multi-undulating terrain. When the surge tank number is increased, the protection effect and 
needs to be further studied. As shown in Table 2, more optimized schemes with multi-surge tanks were proposed. Except for the single surge tank scheme, multi-surge tank schemes could protect water hammer negative pressure along undulating pipeline, and the lowest pressure 
was greater than 0 m. 
reduced with the increase of one-way surge tank number, but the reduction range gradually decreased and tended to a minimum value. Therefore, in an undulating terrain pipeline system, it was necessary to comprehensively consider the engineering investment and negative pressure protection effect to select the optimal surge tank number.
Negative protection effect of optimized schemes with different number and total height of surge tanks
| No. | Surge tank location | Surge tank number |  (m) |  (m) |
|---|---|---|---|---|
| S7 | P | 1 | 24 | −2.40 |
| S5 | P | 2 | 19 | 0.21 |
| B | ||||
| S8 | P | 3 | 18 | 0.16 |
| A | ||||
| B | ||||
| S9 | P | 4 | 17 | 0.09 |
| A | ||||
| B | ||||
| C |
| No. | Surge tank location | Surge tank number | (m) | (m) |
|---|---|---|---|---|
| S7 | P | 1 | 24 | −2.40 |
| S5 | P | 2 | 19 | 0.21 |
| B | ||||
| S8 | P | 3 | 18 | 0.16 |
| A | ||||
| B | ||||
| S9 | P | 4 | 17 | 0.09 |
| A | ||||
| B | ||||
| C |
CONCLUSIONS
The negative pressure protection of the water supply system is critical to the safety operation under pump power failure conditions. In multi-undulating terrain, the pipeline with obvious high points is more complex and the protection measures need to be further investigated. In this study, by establishing a mathematical model and equations about the transient process, multi-one-way surge tank protection schemes were proposed. The effects of surge tank location and height on negative pressure protection were revealed. The main conclusions are as follows:
- (1)
Double one-way surge tank schemes could fully protect water hammer negative pressure of pump power failure. Selecting a one-way surge tank location should not consider the forward protection effect. The first surge tank should be located at the initial negative pressure point. The second surge tank should be located at the second-highest point along the multi-undulating pipeline rather than the highest point.
- (2)
Compared with the theoretical minimum surge tank height, there was still optimization margin for total surge tank height in double one-way surge tank schemes. Under the premise that there was no negative pressure along the pipeline, the surge tank total height in this study was reduced by 21%.
- (3)
The applicability of one-way surge tank protection and the reasons for the theoretical height optimization margin were discussed. The influence of the one-way surge tank number on total height was analyzed. Total height reduced with the increase of one-way surge tank number and tended to a minimum value. The optimal surge tank number could be selected by comprehensively considering the engineering investment and negative pressure protection effect.
This study enhances the understanding of negative pressure protection in water supply systems with multi-undulating terrain, and provides a technical guide for the design of one-way surge tank protection schemes.
ACKNOWLEDGEMENTS
This study was supported by the National Natural Science Foundation of China (Grant Nos 51879087 and 51839008).
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.
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