Analytical solution for a three-dimensional non-homogeneous bivariate population balance equation-a special case

There has been a dramatic increase in the number of research publications using the population balance equation (PBE). The PBE allows the prediction of the spatial distribution of the dispersed phase size for an accurate estimation of the flow fields in multiphase flows. A few recent studies have proposed new efficient numerical methods to solve non-homogeneous multivariate PBE and implemented the same in computational fluid dynamics (CFD) codes. However, these codes are generally benchmarked against other numerical methods and applied without verification. To address this gap, an analytical solution for a three-dimensional non-homogeneous bivariate PBE is presented here for the first time. The method of manufactured solutions (MMS) has been used to construct a solution of the non-homogeneous PBE containing breakage and coalescence terms, and an additional source term appearing as a result of this method. The analytical solution presented in this work can be used for the rigorous verification of computer codes written to solve the non-homogeneous bivariate PBE. Quantification of the errors due to different numerical schemes will also become possible with the availability of this analytical solution for the PBE.