The gas characteristics of an air vessel is one of the key parameters that determines the protective effect on water hammer pressure. Because of the limitation of the ideal gas state equation applied for a small-volume vessel, the Van der Waals (VDW) equation and Redlich–Kwong (R–K) equation are proposed to numerically simulate the pressure oscillation. The R–K polytropic equation is derived under the assumption that the volume occupied by the air molecules themselves could be ignored. The effects of cohesion pressure under real gas equations are analyzed by using the method of characteristics under different vessel diameters. The results show that cohesion pressure has a significant effect on the small volume vessel. During the first phase of the transient period, the minimum pressure and water depth calculated by a real gas model are obviously lower than that calculated by an ideal gas model. Because VDW cohesion pressure has a stronger influence on the air vessel pressure compared to R–K air cohesion pressure, the amplitude of head oscillation in the vessel calculated by the R–K equation becomes larger. The numerical results of real gas equations can provide a higher safe-depth margin of the water depth required in the small-volume vessel, resulting in the safe operation of the practical pumping pipeline system.

In the long-distance and high-drop gravitational water supply systems, the water level difference between the upstream and downstream is large. Thus, it is necessary to ensure energy dissipation and pressure head reduction to reduce the pipeline pressure head. The energy dissipation box is a new type of energy dissipation and pressure head reduction device, which is widely used in the gravitational flow transition systems. At present, there is still a dearth of systematic knowledge about the performance of energy dissipation boxes. In this paper, a relationship between the location of the energy dissipation box and the pressure head amplitude is established, a theoretical optimal location equation of the energy dissipation box is derived, and numerical simulations using an engineering example are carried out for verification. The protective effects of an energy dissipation box placed at the theoretical optimal location and an upstream location are compared. The results indicate that for the same valve action time, the optimal position allows effectively reducing the total volume of energy dissipation box. The oscillation amplitudes of the water level in the box and the pressure head behind the box are markedly reduced. Under the condition that the water level oscillation of the energy dissipation box is almost the same, the optimal location offers better pressure head reduction protection performance than the upstream location.

In this study, the water hammer pressure due to the sudden closure of the partially-open valve was investigated experimentally and analytically. Because the partially-open valve could produce local non-uniform flow, a supplementary Joukowsky's water hammer equation was derived based on the assumption of the local non-uniform flow and the kinetic energy equation. A physical model was set up to measure the maximum water hammer pressure of the first positive wave due to the sudden closure of partially-open valve under different conditions, including different water heads, flow velocities, pipe diameters and valve types. The results showed that Joukowsky's equation obtained by the momentum theorem in the uniform flow field was applicable to the uniform flow field with the valve fully open. The experimental results of the partially-open valve-closure water hammer pressure were 3.5%~21% larger than Joukowsky's equation, which consisted with the theoretical analysis of the supplementary Joukowsky's water hammer equation. This phenomenon had repeatability and was unrelated with the water head, the inlet flow velocity, the pipe diameter and the valve type. This study could provide guidance for water hammer protection in hydropower stations and pump stations.