Solving water column separation and cavity collapse for pipelines by semi-analytical method

For pressurized water supply systems, excessive installation of air valves will inevitably increase the risk of secondary water hammer especially when the air valve fails. Therefore, reducing redundant air valves is of positive significance whether from the maintenance cost on apparatus or from the possible adverse impact caused by the malfunction. Compared to traditional analytical and numerical methods, the semi-analytical method adopted in this paper could not only ensure the clear and intuitive physical meaning of the research results but also expand the application scope of the analytical method, which was also conducive to the further study of key parameters. In this research, it aimed to find out the primary relationship between the function of air valves and the geometry characteristic of the system. Taking a simplified reservoir-pipe-reservoir system with one air valve installed at the elevated point for example, the research initially employed the basic theory of fluid transients to analyze the water hammer wave propagation process. Since the gas inhaled through the air valve separated the water column as the depressurized pressure arrived at the elevated point, the system was divided into 2 subsystems at the point according to their different wave propagation processes. The semi-analytical formulas of the target parameters such as the duration time of cavity growth and collapse, maximum air pocket volume and extreme pressure spike， were firstly proposed in a frictionless condition. Based on the formulas, it studied the key factors affecting the protective effect of the air valve against the water hammer. The semi-analytical solution indicated that the relative length of the downstream pipe section and the relative elevation of the high point played a leading role in the process of cavity growth and collapse. The effect of friction was later taken into the consideration of the semi-analytical expressions serving to reveal its influences on the system. Numerical simulations established on the method of characteristics, which had been proved to a credible and effective numerical method, were then conducted and compared with the semi-analytical solutions to validate the corresponding expressions with and without friction. The outcomes of the 2 approaches presented a consistent variation tendency with the principle variable. However, deviations still existed particularly when the targeted point had a relatively low elevation. The reasons for above deviations were discussed which probably stemmed from some hypotheses during the derivation of semi-analytic formulas, mainly including the omission of gas storage in the upstream section, linear and averaging treatment of the flow variation process in the derivation step. It could also be speculated from the results that the length and roughness of the downstream pipeline determined the maximum pressure spike and the specific pipeline elevation. The semi-analytical formula proposed in this paper was applicable to the containing-one-air-valve pressurized water supply system, and it required to be discussed in the future research for the complex system containing multiple water hammer protection devices with mutual influence. In spite of the limitations, the semi-analytical formulas still reflected the key factors correctly of the air valve as the protection device against water hammer. The findings are helpful to understand the action mechanism of the air valve in the hydraulic transition and provide references for the research of water hammer protection.