## 4 Result and Discussion  
### One-Stage Linear Shut-Off Valve Water Hammer Change Rule  
#### 4.1.1 Change Rule of Water Hammer Pressure Along the Pipeline  
Figure 4 illustrates the variation of water hammer pressure along the pipeline under single-stage linear valve closure time conditions. Given that the main pipeline is shared among the units prior to branching at the junction, the water hammer pressure value in the shared segment remains constant. Thus, only the water hammer pressure situation along the shared main pipeline segment is presented in Figure 4a, while the subsequent \begin{table}
\begin{tabular}{c c c c} \hline
**Closure Type** & **Condition** & **Large Unit (s)** & **Small Unit (s)** \\ \hline  & T1-1 & 3 & 3 \\ Single-Stage Linear & T1-2 & 6 & 4 \\ Valve Closure & T1-3 & 9 & 7 \\  & T1-4 & 12 & 10 \\ \hline Two-Stage Linear & T2-1 & 3 & 2 \\ Valve Closure (Fast & T2-2 & 4.5 & 3.5 \\ Closure at 80\%) & T2-3 & 6 & 5 \\ \hline Two-Stage Linear & T3-1 & 3 & 2 \\ Valve Closure (Fast & T3-2 & 4.5 & 3.5 \\ Closure at 60\%) & T3-3 & 6 & 5 \\ \hline \end{tabular}
\end{table} 
Table 1: One-phase and two-phase linear shutoff condition settings. Figure 4b-d depict different sections of the pipeline under various unit conditions, each exhibiting differing water hammer pressures. Figure 4 illustrates the variation in the water hammer pressure along the pipeline from the upstream pipeline to the valves of different units. To better represent the pressure changes in the latter section of the pipeline, Figure 4a shows the pressure variation along the entire pipeline, whereas Figure 4b-d focuses on the water hammer pressure variations from the 710 m point of the pipeline to the respective unit valve endpoints. It can be observed that the closer the head along the pipeline is to the terminal equipment, the greater the fluctuation in the water hammer pressure, which tends to increase. The maximum water hammer pressures recorded along the pipeline ranged from 5 to 3932 KPa, 5 to 4123 KPa, 5 to 4735 KPa, and 5 to 11,149 KPa for conditions T1-4, T1-3, T1-2, and T1-1, respectively. When the valve closing time was short (i.e., rapid closure), a high transient peak pressure was generated at the initial moment of closure. Once the valve was fully closed, it remained in the closed state for the remainder of the time. Valves can generate excessive hammer pressures, mainly because of the conversion of liquid kinetic energy into pressure energy during the closing process, resulting in an increase in pressure within the pipeline. Longer pipelines may accumulate more reflected energy under specific boundary conditions; however, this must be analyzed in conjunction with actual reflection and superposition scenarios. The results of this study suggested potential strategies for mitigating the water hammer effect, such as optimizing the valve closing time and installing damping devices, to enhance the safety and reliability of pipeline operations [39]. Consequently, this study identified a phenomenon in which closing the valves compressed the fluid in front, creating a high-pressure zone that significantly increased the water hammer pressure at the terminal end. It was also determined that the maximum water hammer pressure occurred near the valve, situated between 1300 and 1450 m along the pipeline. It is also noteworthy that during the valve closure time, which occurs near the water hammer wave phase (at 3 s), the pipeline under condition T1-1 exhibits a significant  
Figure 4: Variations in pipeline pressure along the line for different units at different times: (**a**) Unit D1; (**b**) Unit D2; (**c**) Unit X1; (**d**) Unit X2. negative pressure of \(-98\) kPa. Figure 5 illustrates the air volume distribution along the D1 unit pipeline at the moment when the maximum water hammer extreme occurred under the T1-1 condition. According to the software simulation results, the air volume within the pipeline reached its peak value at \(t=8.76\).