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

Abstract

Please cite this article as: Brandolin, Adriana., Balbueno, Ayslane Assini., & Asteasuain, Mariano., Improved numerical inversion methods for the recovery of bivariate distributions of polymer properties from 2D probability generating function domains.Computers and Chemical Engineering http://dx. GRAPHICAL ABSTRACTIn this work two 2D pgf inversion methods are developed, for which the pgf is regarded as a complex variable. These methods provide an outstanding accuracy in the inversion, thus allowing extending the 2D pgf technique for modeling bivariate distributions without restrictions in the range of values of its independent domains HIGHLIGHTS  Advanced modeling of polymer processes.  Prediction of bivariate (2D) distributions of polymer molecular properties.  Improvement of the pgf modeling method.  Development of new pgf inversion methods that use complex and/or real pgf.  Accurate modeling of 2D distributions with domains of different orders of magnitude. AbstractThe 2D probability-generating function technique is a powerful method for modeling bivariate distributions of polymer properties. It is based on the transformation of bivariate population balance equations using 2D probability generating functions (pgf) followed by a recovery of the distributions 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 with similar orders of magnitude, for example bivariate molecular weight distributions in copolymerization systems.However, numerical problems are found for 2D distributions in which the independent dimensions have very different ranges of values, such as the molecular weight distributionbranching 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 of these innovative methods makes them suitable for recovering 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.

Acrylate Network Formation by Free‐Radical Polymerization Modeled Using Random Graphs

“…Many 1D and 2D PBEs were solved numerically by being transformed into partial differential equations in the generating function domain; see, for example, refs. . The recent developments in approximation with radial basis functions gave access to modeling with multidimensional PBEs .…”

“…accessible computational power on the other resulted in a gradual shift of the focus to MC simulations, [34,35] population balance equation (PBE), [36][37][38][39][40][41] lattice modeling, [42] and molecular dynamics, [43] methodologies offering more flexibility than analytical models. Leiza and co-workers [44,45] were working on elaborate MC simulations for crosslink polymerization.…”

“…Notoriously, the analytical models are hard to custom‐tailor to the specifics of the real‐world chemical systems since the latter typically feature a mixture of monomers with functional groups of various kinds and substitution effects. This inability on one hand and increasingly accessible computational power on the other resulted in a gradual shift of the focus to MC simulations, population balance equation (PBE), lattice modeling, and molecular dynamics, methodologies offering more flexibility than analytical models. Leiza and co‐workers were working on elaborate MC simulations for crosslink polymerization.…”

LDPE Production in Tubular Reactors: Comprehensive Model for the Prediction of the Joint Molecular Weight-Short (Long) Chain Branching Distributions

“…The model is an extension of a previous comprehensive steady-state model . The bivariate distributions are predicted using the 2D probability generating functions technique (2D pgf), − without a priori knowledge of the shape of the distributions. It is easy to implement, in spite of the complexities of both the kinetic mechanism and the reactor configuration.…”

Solution of population balance equations by logarithmic shape preserving interpolation on finite elements

“…The generating function method [37,38] relies on solving the PBE in the Laplace space and then inverting the solution to re-obtain the desired distribution. This inversion step is numerical in most cases and represents the main drawback of the generating functions approach, given the related numerical complexities.…”