(576g) Optimization Methods for Polymerization Processes with Detailed Microstructural Quality Indices

A polymerization process is expected to produce polymer grades with different properties. The end-use properties of polymers are directly determined by their microstructural indices¹. The key indices are molecular weight distribution (MWD) and chemical composition distribution (CCD). They can describe detailed information on polymers, in contrast to traditional experimental indices, such as melt index (MI), average molecular weight and polydispersity index (PDI). Generally, there are two different categories of methods to obtain these indices¹. One category is based on solving equations derived from material balances. This method takes advantage of gradient-based optimization algorithms to develop optimal operating policies. But to construct complete microstructural information, millions of equations are involved and impractical to solve. The other category is based on Monte Carlo simulation. This method is essentially “math-free” and does not need to solve complex differential-algebraic equations. However, since Monte Carlo simulations are computationally expensive, integrating the Monte Carlo model into optimization may become computationally intractable.

Model-based optimization provides a systematic way to develop polymer products faster, at lower cost and with higher quality. It also allows us to move away from a reliance on multiple rounds of lab testing and toward computer modeling and testing in an efficient way. Our work focuses on the model-based optimization of polymerization processes, with microstructural indices into consideration. The first part of our work is dynamic optimization for grade transitions based on MWD. The optimization goal is to switch from one grade to another grade with newly specified MWD in minimum time during operation. Ideally, the complete population balance model should be included in the optimization problem. However, this model consists of a huge set of differential equations. Even if polymer chain length is truncated to a reasonable value, the model is still too large to solve by existing optimization methodologies. In this talk, we develop an alternative representation of MWD using orthogonal collocation methods². In this method, the population balance equations of polymer chains are strictly satisfied only at the collocation points in the chain length domain. The resulting solution is used to reconstruct a piecewise polynomial representation of the entire MWD. This MWD collocation method has been demonstrated on an industrial high-density polyethylene (HDPE) slurry process with a continuous stirred-tank reactor. Different dynamic optimization formulations, considering transition time and off-grade production, are proposed to determine optimal trajectories of manipulated variables³. Then a direct transcription optimization approach is applied to solve these problems after fully discretizing state and control variables.

The second part of our work deals with optimization of a Monte-Carlo-simulation-based model, with detailed microstructural quality indices. Here an adaptive simulation algorithm is proposed to reduce computational cost, based on error estimation of the Monte Carlo model. The parallel computing on the graphics processing unit⁴ is utilized to further accelerate the Monte Carlo simulation. Considering the uncertainties in the Monte Carlo simulation, a derivative-free algorithm based on trust region method is employed for solution⁵. A successive boundary shrinkage formulation is developed to improve the convergence of problem solving. The above-mentioned methods are successfully integrated and implemented on a copolymerization process to maximize the co-monomer conversion ratio with specified CCD. Numerical experiments have been applied to validate the high efficiency and good performance of the proposed methods⁶.

Keywords: Model-based optimization, Molecular weight distribution, Chemical composition distribution

References:

[1] Soares, J.B.P., McKenna, T.F.L., Polyolefin Reaction Engineering, 2012. Weinheim: Wiley-vch Verlag GmbH & Co. KGaA.

[2] Ma Y., Chen X., Biegler L.T., Dynamic optimization of polymer grade transitions with molecular weight distribution models. 2017 SIAM Annual Meeting, July 10-14, 2017, Pittsburgh, USA.

[3] Eason J.P., Ma Y., Chen X., Biegler L.T. Dynamic Optimization of Polymerization Processes with Detailed Molecular Weight Distributions. 2017 AICHE Annual Meeting, Oct. 29 - Nov. 3, 2017, Minneapolis, USA.

[4] Ma Y., Shao Z., Chen X., Biegler L.T., A Parallel Function Evaluation Approach for Solution to Large-scale Equation-oriented Models, Computers & Chemical Engineering, 2016. Vol 93, pp 309–322.

[5] Eason J.P., Biegler L.T., A trust region filter method for glass box/black box optimization[J]. AICHE Journal, 2016. Vol 62, No 9. pp 3124-3136.

[6] Ma Y., Chen X., Biegler L.T., Monte-Carlo-simulation-based optimization for copolymerization processes with embedded chemical composition distribution, Computers & Chemical Engineering, 2018. Vol 109, pp 261-275.