A simple model of the World Trade Center fireball dynamics

An analytical model of the initial expansion of a fireball is presented. The model is based on an exact solution of the low Mach number combustion equations in the form initially proposed by the authors. The equations consist of the conservation of mass, momentum, and energy with an isobaric equation of state. The heat release rate is a prescribed spherically symmetric function characterized by a flame expansion velocity, a flame brush thickness that increases with time, and a heat release rate per unit surface area. The introduction of a prescribed heat release rate obviates the need for an explicit turbulence model. Thus, the inviscid forms of the conservation equations can be used in the analysis. The velocity field is decomposed into a spherically symmetric expansion field and a solenoidal component determined by the buoyancy induced vorticity field. The expansion field together with the induced pressure rise and temperature fields are spherically symmetric. However, the buoyancy forces induce vorticity where the temperature changes rapidly and break the spherical symmetry of the velocity field. The solution is used to study the initial expansion of the fireballs generated in the attack on the World Trade Center south tower. Video images are used to estimate the expansion rate of the fireball. This information, when combined with the analysis, leads to an estimate of the fuel consumed in the fireball that is independent of any assumptions about either the initial fuel distribution or the state of the building following the crash.