Hydrodynamic model for a vibrofluidized granular bed

Equations relating the energy flux, energy dissipation rate, and pressure within a three-dimensional vibrofluidized bed are derived and solved numerically, using only observable system properties, such as particle number, size, mass and coefficient of restitution, to give the granular temperature and packing fraction distributions within the bed. These are compared with results obtained from positron emission particle tracking experiments and the two are found to be in good agreement, without using fitting parameters, except at high altitudes when using a modified heat law including a packing fraction gradient term. Criteria for the onset of the Knudsen regime are proposed and the resulting temperature profiles are found to agree more closely with the experimental distributions. The model is then used to predict the scaling relationship between the height of the centre of mass and mean weighted bed temperature with the number of particles in the system and the excitation level.