A methodology for solving the population balance equation based on an embedded reduced order representation

Abstract

Polydispersed particles are frequently described in terms of a property distribution whose evolution is governed by the population balance equation (PBE). In the present article, we develop a numerical solution scheme for the univariate, spatially inhomogeneous PBE that allows for the economical resolution of the particle size distribution at a moderate accuracy and is robust with respect to growth-dominated applications. The method combines the notion of characteristic curves in particle size space with a constrained Galerkin projection and is informed by a training step for the identification of representative shape functions. This step is based on the proper orthogonal decomposition (POD) of snapshot matrices containing adaptive grid solutions of the PBE obtained in simplified flow configurations. In order to assess the viability, accuracy and convergence properties of the combined PBE-POD approach, we analyze the pure growth and dispersion in a laminar plane jet of a polysized particle population.