Improved numerical inversion methods for the recovery of bivariate distributions of polymer properties from 2D probability generating function domains

Abstract

The 2D probability generating function technique is a powerfulmethod for modeling bivariate distributions of polymer properties. It isbased on the transformation of bivariate population balance equationsusing 2D probability generating functions (pgf) and a posteriori recovery of the distribution from the transform domain by numerical inversion. A key step of this method is the inversion of the pgf transforms. Available numerical inversion methods yield excellent results for pgf transforms of distributions with independent dimensions of similar orders of magnitude.However, numerical problems are found for 2D distributions in which the independent dimensions have very different range of values, such as the molecular weight distribution-branching distribution in branched polymers. In this work, two new 2D pgf inversion methods are developed,which regard the pgf as a complex variable. The superior accuracy ofthese new methods allows constructing a 2D inversion technique suitablefor any type of bivariate distribution.This enhances the capabilities of the 2D pgf modeling technique for simulation and optimization of polymer processes. An application example of the technique in a polymeric system of industrial interest is presented.