Modeling of Bivariate Distributions of Polymer Properties: Speeding up Simulations By Using Parallel Computing and 2D Probability Generating Functions

PINTOS ESTEBAN; ASTEASUAIN MARIANO; FORTUNATTI CECILIA

Congreso; 2018 AIChE Annual Meeting; 2018

Polymers are used in most areas of our daily life. The applications of polymers depend on their processing and end-use properties, which are greatly determined by the microstructure of the polymer chains. One key molecular property is the molecular weight distribution (MWD). The MWD affects both the processing properties (i.e. melting point temperature and flow properties of melted polymers) as well as the mechanical properties (tensile strength, impact strength, etc.). Other important molecular properties are the copolymer composition distribution (CCD), short-chain branching distribution (SCBD), long-chain branching distribution (LCBD), and particle size distribution (PSD). The prediction of the distributions of polymer properties as a function of the process conditions is a challenging task. Moreover, knowledge of joint (multivariate) distributions is highly desired in some cases, which makes the calculation more complex.Several methods have been proposed for the prediction of the univariate MWD.1â??3 However, the resulting system of equations is very large in most cases. Extension to multivariate distributions that join the MWD with the CCD, SCBD, LCBD or the PSD is even more challenging, so a reduced number of approaches have been reported.4 One of the major drawbacks of these large models is not only the complexity of their formulation but also the efficiency in terms of CPU time and memory requirement.A computational technique that has the potential for solving large models for the calculation of multivariate distributions of polymer properties is parallel computing. As the improvement of single-processor computing speed has slowed down, researchers are realizing the importance of parallel computing. Most modern computers possess more than one CPU (typically between 4 and 8). However, the majority of the methods reported in the literature for the prediction of multivariate distributions are designed for serial computation using only 1 CPU. Harnessing the power of the multiple readily available CPUs allows many computations to be completed more quickly.Unfortunately, not all problems can be parallelized. In order to use parallel computing to solve a particular problem, this problem needs to be broken down into independent parts, so that each part can be solved or calculated concurrently with the other ones. The probability generating function (pgf) technique is a powerful method for modeling distributions of polymer properties, which is particularly suitable to be used in a parallel environment. The pgf method has been used extensively for modeling univariate and multivariate distributions of polymer properties.5,6 This modeling technique is based on solving the pgf transform equations of the population balances describing a polymer process, and the subsequent inversion of the resulting pgf transform in order to recover the desired distribution. A key step of the technique involves choosing an appropriate inversion method.The development of an algorithm designed for parallel computing usually takes more effort and time compared with that for serial computing. Besides, the use of an appropriate programming language is of critical importance to ease this process. This work employs a novel programming language called Julia7. Julia is an open-source, modern high-level dynamic programming language designed to achieve high-performance in technical applications, numerical analysis, and scientific computing. The language combines several modern programming language features, such as â??multiple dispatchâ??, â??Just-In-time compilationâ?? and a â??Low-level Virtual Machine (LLVM) compiler infrastructureâ?? among others. It also has native parallel programming support. The code parallelization is quite straightforward compared to other languages where parallelization is not always native and the use of another application or libraries is needed.The model presented in this work, implemented in Julia, uses the 2D pgf technique for the prediction of the bivariate MWD-CCD in the nitroxide mediated copolymerization of styrene and a-methyl styrene. Three different pgf inversion methods were tested: the 2D IFG method, 2D PAP Method and 2D PAP-IFG method.8 These methods were compared for accuracy and computation times. The computation times were obtained for different CPU configurations (serial using one CPU core and parallel using 2, 4 and 8 cores).The 2D IFG method showed very good accuracy in all cases, the 2D Pap-IFG method yielded good accuracy, and the 2D Papoulis method showed the lowest level of accuracy of the three. Parallel execution of the algorithm in the Julia programming language allowed a significant reduction in CPU time compared to the serial execution. The results showed that the use of the pgf technique in combination with parallel computing technologies allows performing an accurate calculation of bivariate distributions in short times.