A new analytical solution for solving the population balance equation in the continuum-slip regime

Abstract

A new analytical solution is first proposed to solve the population balance equation due to Brownian coagulation in the continuum-slip regime. An assumption for a novel variable g (g=m0m2/m12, where m0, m1 and m2 are the first three moments, respectively) is successfully applied in executing a separate variable method for ordinary differential equations of the Taylor expansion method of moments. The sectional method is selected as a reference to verify the new solution. The accuracy between the new solution and Lee et al. analytical solution (Lee et al., 1997, Journal of Colloid and Interface Science, 188, 486–492) is mainly compared. The geometric standard deviation of number distribution for the new analytical solution is revealed to be limited to 1.6583. Within the valid range of the geometric standard deviation, the new analytical solution is confirmed to solve the population balance equation undergoing Brownian coagulation with the very nearly same accuracy as Lee et al. analytical solution. For the total particle number concentration, the new solution usually yields higher accuracy. The new solution and Lee et al. analytical solution approximately become one solution as the Knudsen number is smaller than 0.1000. The new solution has the potential to become a competitive analytical solution for solving population balance equation regarding its accuracy and very straightforward derivation.