A differential equation for the flow rate during silo discharge: Beyond   the Beverloo rule
  We present a differential equation for the flow rate of granular materials during the discharge of a silo. This is based in the energy balance of the variable mass system in contrast with the traditional derivations based on heuristic postulates such as the free fall arch. We show that this new equation is consistent with the well known Beverloo rule, providing an independent estimate for the universal Beverloo prefactor. We also find an analytic expression for the pressure under discharging conditions. 
Marcos A. Madrid, Jos\'e R. Darias and Luis A. Pugnaloni
cond-mat.soft