## 4 Result and Discussion
### One-Stage Linear Shut-Off Valve Water Hammer Change Rule
#### 4.1.2 Variation Characteristics of Water Hammer Pressure near the End Valve  
Figure 6 shows the variation in the valve pressure at the pipeline end over time under single-stage linear valve closure. The figure shows the pressure fluctuations at the end valves of each unit under different operating conditions, with the pressure variation patterns being generally consistent across all four units. Over time, these fluctuations gradually stabilized and reached a steady state at approximately 30 s. As can be observed from Figure 6a-d, under the four-valve closure conditions (T1-1, T1-2, T1-3, and T1-4), the water hammer pressure ranges at the valve ends of the different units are as follows: \(-98\) to 11,149 kPa, 2544 to 4735 kPa, 2845 to 4125 kPa, and 3104 to 3932 kPa. Under the T1-1 condition, the minimum value (\(-98\) kPa) occurred at the ends of all units, while the maximum value (11,149 kPa) was observed in the X2 unit (Figure 6d). For the T1-2 condition, the minimum value (2544 kPa) appeared in the X2 unit (Figure 6d), and the maximum value (4735 kPa) occurred in the X1 unit (Figure 6c). In the T1-3 condition, the minimum value (2845 kPa) was found in the X1 unit (Figure 6c), while the maximum value  
Figure 5: Variation diagram of air/steam volume of unit D1 under T1-1 condition.  
(4125 kPa) was observed in the D1 unit (Figure 6a). Similarly, under the T1-4 condition, the minimum value (3104 kPa) occurred in the X1 unit (Figure 6c), and the maximum value (3932 kPa) was observed in the D1 unit (Figure 6a). Moreover, as the valve closure time increases, the extremes of the water hammer pressure gradually decrease, and the frequency of the peak occurrences decreases. This is primarily because the conversion of fluid kinetic energy into pressure energy occurs more gradually when the valve closure speed is lower. As a result, the peak values of the transient water hammer pressure and the amplitude of multiple wave superpositions are reduced, leading to fewer peak pressures and reflections within the pipeline. This phenomenon is consistent with that observed in previous studies that effectively revealed the transient flow characteristics of pressurized pipelines through CFD analyses. During valve closure, the flow velocity underwent three distinct phases: an initial slight change, followed by a sharp decrease, and finally, a gradual reduction [42].  
Additionally, a comparison of the end valve pressures of the large units D1 and D2 with those of the small units X1 and X2 showed that the pressure wave patterns at the end valves of the small units X1 and X2 are more densely packed. This indicates that the initial pressure readings have a more chaotic flow oscillation at the small unit valves. This phenomenon can be preliminarily attributed to the smaller diameters of the branch pipelines associated with the small units. The flow velocity becomes turbulent as the main water supply pipeline's flow enters the small unit branch lines, consequently leading to excessive oscillation frequencies of the water pressure within the pipeline when the valve suddenly closes.