The Multiphase Buoyant Plume Solution of the Dusty Gas Model

Starting from the balance equations of mass, momentum and energy we formulate an integral 1D model for a poly-disperse mixture injected in the atmosphere. We write all the equations, either in their most general formulation or in the more simplified, taking particular care in considering all the underlying hypothesis in order to make clear when it is possible and appropriate to use them. Moreover, we put all the equations in a non-dimensional form, making explicit all the dimensionless parameters that drive the dynamics of these phenomena. In particular, we find parameters to measure: the goodness of the Boussinesq approximation, the injected mass flow, the column stability and his eventual collapse, and the importance of the atmospheric stratification, the initial kinetic energy and the gravitational potential energy. We show that setting to zero some of these parameters, it is possible to recover some of the existing jet and plume models for single-phase flows. Moreover, we write a simplified set of equations for which it is possible to find analytical solutions that can be used to describe also the dynamics of multiphase "weak-plumes".