Improved numerical inversion methods for the recovery of bivariate distributions of polymer properties from 2D probability generating function domains |
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| Affiliation: | 1. Planta Piloto de Ingeniería Química (PLAPIQUI), Universidad Nacional del Sur • CONICET, C. La Carrindanga km 7, Bahía Blanca, 8000, Argentina;2. Federal University of Viçosa, Centro, Viçosa, Minas Gerais, Brazil;1. Mechanical and Industrial Engineering, The University of Iowa, Iowa City, IA 52242, United States;2. Aerospace Engineering, San Diego State University, San Diego, CA 92115, United States;1. Department of Internal Medicine, University of Tennessee Health Science Center, Memphis, Tennessee;2. Division of Cardiovascular Medicine, Department of Internal Medicine, St. Luke''s University Health Network, Bethlehem, Pennsylvania;3. Division of Cardiovascular Medicine, Department of Internal Medicine, Lehigh Valley Health Network, Allentown, Pennsylvania;4. Department of Cardiovascular Medicine, John Ochsner Heart and Vascular Institute, Ochsner Clinical School—The University of Queensland School of Medicine, New Orleans, Louisiana;1. Institute of Structural Mechanics, Bauhaus-Universität Weimar, Marienstr. 15, D-99423 Weimar, Germany;2. Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran;3. Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China;1. School of Environment and Civil Engineering, Dongguan University of Technology, Dongguan, Guangdong 523808, PR China;2. School of Chemical Engineering, South China University of Technology, Guangzhou 510640, PR China |
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| Abstract: | The 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 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 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. |
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| Keywords: | Modeling Polymerization Bivariate distribution 2D probability generating function |
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