An Exact Solution of the Fragmentation Equation

Abstract

The accuracy of the self-similarity assumption employed in the study of the grinding equation is examined in detail. This is made possible by obtaining an exact solution for any homogeneous breakup function, thereby enabling the asymptotic limit as time proceeds to be examined carefully. For the Randolph-Ranjan model of breakup, we have obtained some explicit results and these have been employed to highlight the limitations of current self-similar solutions. In particular, we note that use of a self-similar solution which depends only on the zeroth and first moment of the distribution cannot give any detailed information on the higher moments. Nevertheless, at times very soon after the start of grinding, self-similarity does lead to useful and practical asymptotic results for size distributions, since it appears that higher moments are then of less importance. Thus the reason for the success of similarity is explained and the rate of approach to this condition is given. We have also introduced a new model of breakup which assumes that the minimum particle size in a given breakup process is always a fixed fraction of the initial size. This has the advantage of eliminating the divergence of the total number of particles produced per grinding action while still allowing the equation to be dealt with analytically. Finally, we discuss some steady state grinding distributions that arise from the new model.