A mathematical model describing an unsteady leak of compressed air from an open underwater storage tank

Abstract

A new mathematical model is developed here to predict the discharge dynamics of compressed air from a massive leak occurring at the top of a submerged storage tank with an original bottom opening. The model represents a transient leak due to variations in the air volume and air pressure remaining inside the tank using a mass balance equation and the compressible Bernoulli equation. A first-order-non-linear ordinary differential equation is solved in terms of the air height inside the reservoir for different tank geometries: a right circular cylinder and a remaining combined with a hemisphere of the same radius to form a bell. The total discharge time is obtained by analytical integration. The model is then applied to a case study in which the amount of air is fixed, and the investigated parameters are the immersion depth (ranging from 75 to 200 m) and the radius of the storage tanks (5 or 7.5 m) on the discharge time. Finally, a representative steady rate is given based on the average discharge rate in order to rapidly estimate the discharge time without further knowledge of the storage tank configuration. The immersion depth has no influence in the discharge time of a cylindrical tank, and a wide tank causes a longer discharge than a narrow one. It has also been shown that, in the range of initial air height considered (close to ten meters), the variation of density can be neglected in the model for the cylindrical geometry, and for the bell geometry provided that the ratio radius-to-initial height remains low.