## 4 Result and Discussion
### Two-Stage Linear Shut-Off Valve Water Hammer Change Rules  
#### 4.2.1 Change Rule of Water Hammer Pressure Along Pipelines  
Figure 7 illustrates the variation patterns of the water hammer pressure along the pipeline under two-stage linear valve closure conditions, revealing that the water hammer pressure variation patterns of the four units are generally consistent, with the maximum water hammer pressure gradually increasing along the pipeline. Specifically, the T2-1 condition ranges from 5 to 5544 kPa, whereas the T3-1 condition ranges from 5 to 4857 kPa. Figure 6: Pressure wave variations at the valve end for different units at four different times: (**a**) Unit D1; (**b**) Unit D2; (**c**) Unit X1; (**d**) Unit X2. A comparison between the T2-1 and T3-1 closure conditions indicates that the maximum water hammer pressure in the latter is reduced by 12.4% compared with the former, demonstrating that the valve closure speed has a significant impact on the water hammer pressure, as different closure speeds lead to varying flow state changes. Notably, when the closure speed is excessively rapid, the oscillation of the water hammer waves within the pipeline becomes more intense. Furthermore, the T2-1 condition exhibits negative pressure in the pipeline, indicating the occurrence of flow interruption, which suggests that a fast closure at 80% is more likely to induce negative pressure in the pipeline than a fast closure at 60%. Additionally, under a closure time of 6 s for the T2-1 condition, it is important to note that regardless of the speed of closure, the maximum water hammer pressure within the unit's pipeline exceeds the pressure limit of 5 MPa for the steel pipe. The maximum water hammer pressures corresponding to each pipeline are presented in Table 2. Within the same timeframe, a faster closure speed yields a higher water hammer peak with recorded pressures of 4264 KPA for T3-2, 4724 KPa for T2-2, 3985 KPa for T3-3, and 4303 KPa for T2-3. Specifically, under identical closure-time conditions, the maximum water hammer pressure observed in the pipeline under the 80% fast closure scenario is greater than that observed under the 60% fast closure scenario, with increases ranging from 7.4% to 9.7%. This phenomenon can be preliminarily attributed to the rapid conversion of the kinetic energy of the fluid into pressure energy resulting from the high closure speed, which consequently increases the peak pressure within the pipeline. However, as the closure time increases, the influence of the fast closure rate on the maximum water hammer pressure exhibits a diminishing trend. The results of this study are consistent with the conclusions obtained by previous researchers, who illustrated various cases of emergency closure and its effect on the water hammer pressure through simulation and analysis and verified the effectiveness of the fast valve under different operating conditions [43]. Furthermore, it was determined that when maintaining the same unit parameters, the position of the unit exerts a certain influence on the maximum water hammer pressure  
Figure 7: Water hammer pressure along the pipeline during two-stage linear closure (6 s for large units and 4 s for small units): (**a**) Unit D1; (**b**) Unit D2; (**c**) Unit X1; (**d**) Unit X2. within the pipeline. In this study, the peak water hammer pressure recorded in the D1 unit was slightly higher than that in the D2 unit, with a range of variation between 0.07% and 0.18%. Similarly, the X1 unit showed a greater maximum water hammer pressure than that in the X2 unit, with differences ranging from 0.17% to 0.7%. This suggests that the closer a unit is to the terminus of the main water supply pipeline, the greater the resulting peak water hammer pressure, which can be preliminarily attributed to the increased fluid kinetic energy at the valve closure point, influenced by the act of valve closure, although the observed differences under identical unit parameters are not statistically significant.