On the Evolution of Particle Size Distribution in Pipe Flow of Dispersions Undergoing Breakage

The problem of a flowing dispersion in a pipe with particles undergoing breakage and radial diffusion is considered in this note. This problem has been addressed in an earlier work [Kostoglou and Karabelas, AIChE J., 2004, 50, 1746] where it is solved analytically for the case of radially uniform parameters, while the case of turbulent flow (with radially dependent parameters) was treated by other researchers [Nere and Ramkrishna, Ind. Eng. Chem. Res. 2005, 44, 1187]. Here, simplified versions of the extended mathematical problem in the limits of small and large diffusion influence are presented. The conditions under which self-similar particle size distributions can be developed are determined. Finally, a closed-form solution for the total particle number concentration for linear breakage rate with radial dependence and radial uniform velocity and diffusivity is derived.