Designing the shape of a large scientific balloon

Abstract

Large scientific balloons are used to carry out research in the upper atmosphere. By Archimedes' principle, a balloon in equilibrium at a fixed altitude must displace an amount of air equal to its weight. The system that we model here includes the balloon film, reinforcing caps, load tapes and payload. We consider two mathematical models for the design shape of a balloon. One is a design in the shape of an ellipsoid-on-cone. The equilibrium condition gives rise to a cubic equation whose solution gives the unique shape for a given set of design parameters. A second model is the natural-shape design, a system of nonlinear ordinary differential equations that is commonly used for modeling the shape of large scientific balloons. In both models, caps are included as an added thickness, thus the resulting film weight density is discontinuous. The discontinuities in the natural-shape model are handled with a parallel shooting method. We present numerical solutions for various design parameters and consider issues related to the efficient design of a balloon.