(167g) Computuationally-Efficient High-Fidellity Modelling of the High-Pressure Polymerization of Ethylene in Tubular Reactors Using Parallel Computing.

Low-density polyethylene (LDPE) is produced by high-pressure radical polymerization in tubular or autoclave reactors. Its high content of branches gives LDPE useful final properties, such as low density, high ductility, flexibility, strength, and barrier properties.[1] LDPE is known as a versatile material. It is possible to obtain different grades with different final properties by changing the process conditions. Each LDPE grade may have molecules with different amount and type of branches and also different molecular weights. Short-chain branches (SCB) influence the morphology of the crystalline structure of the polymer, while long-chain branches (LCB) have a remarkable effect on melt rheology.[2] For this reason, is crucial to obtain detailed information about the characteristics of these branches.The molecular structure of LDPE is usually characterized in terms of the molecular weight distribution (MWD), the bivariate molecular weight-short-chain branching distribution (MWD-SCBD), the bivariate molecular weight-long-chain branching distribution (MWD-LCBD), the number of branches every 1000 carbon atoms (LCB/1000C) and the branching index (g). These properties are usually difficult to measure experimentally, so several mathematical models have been developed to predict them as a function of the process conditions.

Mathematical models reported in the literature may be divided into two classes: deterministic and stochastic. The deterministic approach is based on polymerization kinetics. It involves the derivation of population balance equations (PBEs) of the polymeric species. On the other hand, stochastic approaches involve the use of probabilistic theories for the reconstruction of the polymerization system. Recently, we presented a deterministic model of the LDPE polymerization in high-pressure tubular reactors predicted the bivariate MWD-SCBD and MWD-LCBD, as well as the LCB/1000C and g distributions.[3] An update of this model was recently developed,[4] in which the mentioned molecular information is used as input for the calculation of rheological properties of the polymer.

The rheological behavior of the polymer is higly sensitive to the architecture of the molecules. Even subtle changes in the latter may significantly modify the rheological properties, which in turn affect the processing behavior.[5] In the particular case of LDPE the processability and the final end-use properties are highly influenced by the presence of LCBs, even in small concentration, and by the presence of high molecular weight tails in the MWDs.[6] For this reason, it is extremely important to reduce as much as possible the errors associated with the prediction of the MWD-LCBD and other related properties.

The aforementioned models of the LDPE reactor [3, 4] use the 2D probability generating function (2D pgf) technique to predicti the MWD-LCBD. This technique consists of two steps: in the first one, the infinite system of PBEs is transformed into the pgf domain. In this way, a smaller system of equations is obtained in which the dependent variable is the pgf transform of the distribution.[7] In the second step, the pgf transform obtained from the numerical solution of the pgf balances is inverted to recover the desired distribution.[8] Several inversion methods were developed to recover 2D distributions.[9] Particularly, the 2D IFG method was specially developed to overcome the numerical problems found when recovering bivariate distributions in which the distributed variables had very different range of values. This is the case of the MWD-LCBD in which the molecular weight dimension usually varies from 1 to 10⁵ and the branching dimension between 1 and 20. However, this inversion method results in a very large system of equations for systems of high molecular weight, which is the case of LDPE. Therefore, a model employing this inversion technique may be very time-consuming, or even infeasible to solve.

The efficiency concerning computation times, data processing, CPU and RAM memory requirements of high-fidelity mathematical models. A computational technique that has the potential to speed up the execution of mathematical models is parallel computing. To solve a particular problem using parallel computing, the problem needs to be divided into independent sections, so that each section can be solved concurrently with the others. Not all the methods proposed for calculating multivariate distributions meet this condition. The 2D pgf techinique meets this requirement, because it yields a system formed by subblocks of equations that are independent of each other.

Besides, developing an algorithm designed for parallel computing generally requires more effort and time than one designed for serial computing. The use of the appropriate programming language is very important to facilitate this process. In this work, the detailed high-pressure tubular reactor mathematical model was programmed in Julia, which is an open-source programming language easy to use and that performs very fast calculations.

The mathematical model of the tubular reactor developed in Julia allowed predicting the MWD-LCBD in a very efficient manner. The application of the 2D pgf technique with the 2D IFG inversion method results in a mathematical model consisting of 385 009 independent blocks of 12 pgf equations each. Therefore, the total number of pgf equations is 4 620 108 differential equations. Solving a model of this size, using an implicit DAE solver as required by the stiff system of equations, is infeasible on a regular PC. On the other hand, the parallelized implementation of the solution of the independent blocks in Julia, using a desktop PC equipped with an Intel Core i7-9700 CPU processor and 32 GB RAM, demanded a very short time. In this way, the bivariate MWD-LCBD of several samples of LDPE produced in a high-pressure tubular reactor were computed on a very fine grid of molecular weights and number of branches with high-precission, as well as the LCB/1000C and g distributions. This allowed to accurately compute the rheological properties of the LDPE samples, which was verified by comparison with experimental data.

References

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3.Dietrich, M.L., et al., LDPE Production in Tubular Reactors: Comprehensive Model for the Prediction of the Joint Molecular Weight-Short (Long) Chain Branching Distributions. Industrial Engineering Chemistry Research, 2019. 58(11): p. 4412-4424.

4.Dietrich, M.L., et al., Modelling Low-Density Polyethylene (LDPE) Production in Tubular Reactors: Connecting Polymerization Conditions with Polymer Microstructure and Rheological Behavior. Macromolecular Reaction Engineering. 2022. In press.

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