Similarity Solutions for the Population Balance Equation Describing Particle Fragmentation

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Abstract
Similarity solutions are obtained to the population balance equation describing particulate systems undergoing fragmentation. The solutions are restricted only by the requirement that the breakage rate of particles of volume v be of the form φ(v) = Avb and the breakage distribution function of the form ω(u, v) = ƒ(v/u)/u, where u is the volume of the fragmenting particle. Under such conditions, the moments of the distribution asymptotically approach the form Ni(t) α t (1−i)/b , and the particle size distribution function is shown to obey a first-order linear ordinary integro-diflerential equation. For the case ƒ(v/u) = γ(v/u)γ−2, analytical solutions to the above equation were obtained. Complete solutions as well as asymptotic behavior are given. The results are potentially applicable to a wide range of particle fragmentation problems, including char/ash fragmentation during pulverized coal combustion, explosively generated aerosol formation, ore comminution, powder crushing and grinding, floe breakage, and crystallization kinetics.