A modified kinematic model for the discharge of a granular-like material from long vertical cylinders

The rapid and uncontrolled discharge of a large-sized granular-like materia l from a vertical cylinder is modelled assuming a modified kinematic relati onship exists between granular pressure, speed and density. Discharge is dr iven by the initial Janssen pressure, and a Beverloo-type equation is deriv ed for the initial discharge. A shock occurs at the bottom of the cylinder, and another upwards travelling shock separates static and moving material. The initial discharge is nonconstant, with the constant density and discha rge case violating the 'entropy' condition. Two sets of characteristics are found : one travelling upwards, associated with the motion of voids; and t he other travelling downwards, associated with work performed by the partic le pressure. Contrary to hopper models, a low density of solids is predicte d about the cylinder exit. The modified kinematic model allows density wave s to travel either up or down through the cylinder, but the frequency and s peed of the waves is not fixed uniquely by the model. The waves exhibit a s aw-tooth behaviour, with a continuously increasing magnitude in Bur at the orifice, interspersed with discontinuous decreases.