  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".
