Dynamic optimization can be used to find fast and economical grade transitions in polymerization processes. The high degree of nonlinearity and large size of these systems makes this a very challenging problem. Many end-use polymer properties on the molecular weight distribution (MWD). Conventionally, a Flory-Schulz distribution with a small-scale moment model has been used to represent the MWD, but it is valid only under steady state or pseudo steady state assumptions. Ideally, the full population balance model should be included in the optimization process so that an operator can control from one desired MWD to another desired MWD in minimum time. However, population balance models consist of a countable system of differential equations. Even if chain length is truncated at a suitable value, these problems are too large for existing dynamic optimization methodologies. In this talk, we discuss alternative representations of the MWD using orthogonal collocation methods1,2
. In the collocation method, the population balance equations of polymer chains are satisfied exactly at the collocation points in the chain length domain. The resulting solution is used to reconstruct a piecewise polynomial representation of the entire MWD. There are several outstanding challenges compared to typical applications of collocation. First, the inherently discrete MWD is being approximated by smooth functions, and the collocation points must be chosen as integers. In addition, reactions are driven by finite difference relations rather than derivatives, which introduces numerical challenges1
. We investigate the impacts of these potential sources of error and discuss potential adaptive approaches to regulate and reduce the error.
The MWD collocation method is demonstrated on an industrial high-density polyethylene (HDPE) slurry process with a continuous stirred-tank reactor. A detailed mathematical model is presented to track the behavior of the entire plant with liquid and vapor recycles3. We will develop and discuss the steady state optimization of the process to produce a product of a specified MWD. A Monte-Carlo simulation method is used to verify the accuracy of the collocation approach. In addition, a dynamic grade transition problem will be presented, where a direct transcription optimization approach is applied to solve this problem after fully discretizing state and control variables in time.
- Canu, P., and Ray, W.H. Discrete weighted residual methods applied to polymerization reactions. Computers and Chemical Engineering, 1991. Vol 15, issue 8, pp 549-564
- Pontes, K.V., Embirucu, M., Maciel, R., Hartwich, A., Marquardt, W. Optimal process operation for the production of linear polyethylene resins with tailored molecular weight distribution. AIChE Journal, 2011. Vol 57, No 8. pp 2149-2163.
- Zhang, C., Shao, Z., Chen, X., Yao, Z., Gu, X. and Biegler, L. T. Kinetic parameter estimation of HDPE slurry process from molecular weight distribution: Estimability analysis and multistep methodology. AIChE Journal, 2014. 60: 3442â3459.Â