Similarity Solutions for the Population Balance Equation Describing Particle Fragmentation

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, a...

[1]  S. Friedlander,et al.  The self-preserving particle size distribution for coagulation by brownian motion☆ , 1966 .

[2]  John H. Seinfeld,et al.  Coagulation and growth of a multicomponent aerosol , 1978 .

[3]  L. G. Austin,et al.  The variation of breakage parameters with ball and powder loading in dry ball milling , 1980 .

[4]  A. F. Sarofim,et al.  Fragmentation behavior of single coal particles in a fluidized bed , 1985 .

[5]  Doraiswami Ramkrishna.  Drop-breakage in agitated liquid—liquid dispersions , 1974 .

[6]  John H. Seinfeld,et al.  Dynamics of aerosol coagulation and condensation , 1976 .

[7]  L. Spielman,et al.  Floc breakage in agitated suspensions: Theory and data processing strategy , 1982 .

[8]  S. Friedlander.  THEORETICAL CONSIDERATIONS FOR THE PARTICLE SIZE SPECTRUM OF STRATOSPHERIC AEROSOL , 1961 .

[9]  L. Spielman,et al.  Floc breakage in agitated suspensions: Effect of agitation rate , 1983 .

[10]  Adel F. Sarofim,et al.  The Physical Transformation of the Mineral Matter in Pulverized Coal Under Simulated Combustion Conditions , 1977 .

[11]  Leonard G. Austin,et al.  Introduction to the mathematical description of grinding as a rate process , 1971 .

[12]  Alan D. Randolph,et al.  Effect of a material-flow model in prediction of particle-size distributions in open- and closed-circuit mills , 1977 .

[13]  Doraiswami Ramkrishna,et al.  Breakage functions for droplets in agitated liquid-liquid dispersions , 1984 .

[14]  I. Lifshitz,et al.  The kinetics of precipitation from supersaturated solid solutions , 1961 .

[15]  P. C. Kapur.  A similarity solution to an integro-differential equation describing batch grinding , 1970 .

[16]  Dynamics of source-reinforced, coagulating, and condensing aerosols , 1978 .

[17]  K. J. Reid.  A solution to the batch grinding equation , 1965 .

[18]  S. Friedlander.  SIMILARITY CONSIDERATIONS FOR THE PARTICLE-SIZE SPECTRUM OF A COAGULATING, SEDIMENTING AEROSOL , 1960 .

[19]  A. Sarofim,et al.  Comparison of comminution data with analytical solutions of the fragmentation equation , 1985 .