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# Mathematical Model of Small-Volume Air Vessel Based on Real Gas Equation

by 1,*, 1, 2 and 1,*
1
College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China
2
College of Water Conservancy and Civil Engineering, Xinjiang Agricultural University, Urumqi 830052, China
*
Authors to whom correspondence should be addressed.
Water 2020, 12(2), 530; https://doi.org/10.3390/w12020530
Received: 22 January 2020 / Revised: 9 February 2020 / Accepted: 10 February 2020 / Published: 13 February 2020
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. View Full-Text
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Ni, W.; Zhang, J.; Shi, L.; Wang, T.; Zhang, X.; Chen, S. Mathematical Model of Small-Volume Air Vessel Based on Real Gas Equation. Water 2020, 12, 530.