A new approximation approach for analytically solving the population balance equation due to thermophoretic coagulation

Abstract

The analytical solution of the population balance equation (PBE) for coagulation by differential settling velocities remains a challenging issue due to the absolute value in the collision kernel. A geometric mean approximation approach is first proposed in this study to handle the integration of the absolute value and convert the moment equation into an integrable form. Using this approach along with the log-normal method of moments, the PBE is first analytically solved for thermophoretic coagulation in the low Knudsen number limit. The present analytical solution is validated by comparing the size distribution parameters with the sectional method. The maximum relative errors in this study are quite acceptable as compared with previous analytical solutions for Brownian coagulation. Moreover, the self-preserving size distribution predicted by the present analytical solution compares well with the sectional method and previous numerical studies on thermophoretic coagulation. Based on the analytical solution, the characteristics of the size distribution evolution during thermophoretic coagulation are investigated theoretically. The results show that the coagulation process is faster with a larger initial geometric standard deviation and the time required to reach the self-preserving distribution increases with the difference between the initial and asymptotic geometric standard deviations.