Analytical solution for a population balance equation with aggregation and fragmentation

Lage (2002) and Patil and Andrews (1998), denoted here as LPA, have derived a particular solution for the population balance equation (PBE) with simultaneous aggregation(coalescence) and fragmentation (breakage), but for the special case where the total number of particles is constant. The more general reversible case, when either fragmentation or coalescence can dominate, has numerous applications, and is thus of considerable importance (McCoy and Madras, 1998; McCoy and Madras, 2002; Sterling and McCoy, 2001, McCoy, 2001). We wish to comment that a more general solution is available when the number of particles is not constant. The discrete-distribution solution with arbitrary ratio of aggregation and fragmentation rate coefficients was presented by Blatz and Tobolsky (1948) and Family et al. (1986), and later extended by Browarzik and Kehlen (1997) to continuous distributions. Aizenman and Bak (1979) showed that the exponential distribution is of the form of Flory’s (1963) most probable distribution. Cohen (1992) presented a combinatorial approach to derive the similarity solution, and Vigil and Ziff (1989) provided moment solutions that focused on the evolution to steady state distributions. Spicer and Pratsinis (1996) numerically simulated the evolution to a steady-state similarity solution for different forms of the breakage stoichiometry and rate coefficients. McCoy and Madras (1998, 2002) based their solution on moment analysis, which was validated by substitution into the PBE. LPA have suggested applying a straightforward Laplace transform method to solve the PBE, which we explore in this comment