Cut-off model and exact general solutions for fragmentation with mass loss

Abstract

Exact solutions are obtained for a linear rate equation for the evolution of the particle mass distribution during fragmentation with mass loss. These involve general power-law dependences on the particle mass for the fragmentation rate, the daughter-mass distribution, and the mass-loss rate. Consumption of bridges joining two or more otherwise disconnected regions leads to fragmentation such as might occur during the combustion of porous charcoal particles. Exact results for mass-loss rates proportional to the particle mass are relevant to random mass-removal processes such as percolation theory. For pure fragmentation without mass loss, a mass cut-off below which no fragmentation occurs is introduced to avoid the unbounded fragmentation rate for small particles in the `shattering' regime, in which the fragmentation rate becomes unbounded for particle masses approaching zero. This cut-off model predicts that, for monodisperse initial conditions in the strong shattering regime, the number of particles whose masses exceed the cut-off mass remains near unity for the duration of the fragmentation process, and then drops quickly to zero. The associated asymptotic behaviour excludes the scaling solution in this regime, but includes the scaling solution otherwise.