Analytical models for critical heights of vortexing in flat and spherical bottom tanks with siphon and bottom drains

Critical height or critical submergence is liquid level at which air-core vortex extends from the free surface into drain hole when a liquid is drained from a container/tank. Extensive analytical and experimental studies have been reported on critical height of bath tub vortex, for liquid draining downward from flat bottom propellant tanks. Rockets making use of liquid propellants mostly employ spherical bottom propellant tanks as well as siphon or upward drain flow. Keeping in view of such practical applications, analytical models are developed for critical height, considering the effects of siphon drain and the shape of tank bottom. Additional design parameters influencing the behavior for each case are identified. Appropriate governing equations and a solution methodology are developed pertinent to the system considered, to predict the critical height for siphon drain and spherical bottom tank independently as well as for both combined. The results indicate that the critical height for spherical bottom tank is higher than for flat bottom tank, due to higher local flow velocity. Siphon geometry can be designed for critical height much less than normal drain from flat bottom tank. These observations are in accordance with the results published in the literature. This paper reports the analytical models and solution methodology to predict the critical height for vortexing for normal draining from spherical tank bottom, siphon draining from both flat and spherical bottom tanks.