| A comprehensive, non-equilibrium, two-domain (liquid and vapor), physics based, mathematical model is developed to investigate the onset and growth of the natural circulation and thermal stratification inside cryogenic propellant storage tanks due to heat transfer from the surroundings. A two-dimensional (planar) model is incorporated for the liquid domain while a lumped, thermodynamic model is utilized for the vapor domain. The mathematical model in the liquid domain consists of the conservation of mass, momentum, and energy equations and incorporates the Boussinesq approximation (constant fluid density except in the buoyancy term of the momentum equation). In addition, the vapor is assumed to behave like an ideal gas with uniform thermodynamic properties. Furthermore, the time-dependent nature of the heat leaks from the surroundings to the propellant (due to imperfect tank insulation) is considered. Also, heterogeneous nucleation, although not significant in the temperature range of study, has been included. The transport of mass and energy between the liquid and vapor domains leads to transient ullage vapor temperatures and pressures. (The latter of which affects the saturation temperature of the liquid at the liquid-vapor interface.) This coupling between the two domains is accomplished through an energy balance (based on a micro-layer concept) at the interface. The resulting governing, non-linear, partial differential equations (which include a Poisson's equation for determining the pressure distribution) in the liquid domain are solved by an implicit, finite-differencing technique utilizing a non-uniform (stretched) mesh (in both directions) for predicting the velocity and temperature fields. (The accuracy of the numerical scheme is validated by comparing the model's results to a benchmark numerical case as well as to available experimental data.) The mass, temperature, and pressure of the vapor is determined by using a simple explicit finite-differencing technique. With the model at hand, the effects of variable fluid transport/thermo-physical properties, levels of initial sub-cooling, operating pressure, and initial liquid aspect ratio on the natural circulation patterns and thermal stratification are numerically investigated. Liquid oxygen (LOx) is the primary working fluid in the study. However, a simulation with liquid nitrogen (LN2) as the propellant is also carried out for comparison purposes. |
