Large-Scale Parameter Estimation in Low-Density Polyethylene Tubular Reactors

We propose a simultaneous approach for the solution of the associated DAE-constrained parameter estimation problem. Parameter estimation is an essential task in the development and on-line update of first-principles models for low-density polyethylene tubular reactors, consisting of nonlinear and stiff differential -algebraic equations (DAE). Our approach discretizes the reactor model equations in space, leading to a large-scale nonlinear program (NLP) that can be solved efficiently with state-of-the-art general-purpose NLP solvers. In doing so, more efficient estimation strategies can be considered, enabling the solution of challenging estimation problems including multiple data and large parameters sets. This approach is efficient in handling advanced regression problems such as the errors-in-variables-measured (EVM) formulation. The methodology is fast, robust, and reliable and can be used both for off-line and on-line purposes. Moreover, substantial improvements on the reactor model predictions have been obtained over previous approaches, making the model amenable for real-time optimization and control tasks. 1. Introduction The chemical industry faces a challenging and competitive market where high-quality commodity polymers play a central role. These polymers are produced in high-throughput and flexible continuous processes which produce a multitude of polymer grades with tight quality specifications. Stringent performance conditions coupled with high operating costs have motivated the application of advanced real-time optimization and control schemes. These applications require the development of comprehensive first-principles models with accurate predictive capabilities. In all of these developments, a considerable amount of time is spent in discriminating among candidate models and determining the associated parameters. Low-density polyethylene (LDPE) grades are produced in processes with high-pressure, multizone tubular reactors. Reacting in the gas phase at high temperature (130-300 °C) and pressure (1500-3000 atm), ethylene and a comonomer are copolymerized through a free-radical mechanism 1 in the presence of complex mixtures of peroxide initiators. A typical tubular reactor can be described as a jacketed, multizone device with a predefined sequence of reaction and cooling zones. Different configurations of monomer, comonomer, and initiator mixtures enter in feed and multiple sidestreams and are selected to maximize the reactor productivity and obtain the desired polymer properties. The total reactor length ranges between 0.5 and 2 km, while its internal diameter does not exceed 70-80 mm. A schematic representation of a typical tubular reactor is presented in Figure 1. The final end-use properties of the different LDPE grades are mainly correlated to the polymer density and macromolecular properties. Different additives or chain-transfer agents (CTAs) are added to the axial feed streams to control the polymer melt index. In general, the required polymer properties are enforced through complex recipes that try to keep the reactor under strict operating conditions. Mathematical modeling of industrial LDPE tubular reactors is a fundamental but difficult task that motivates a huge amount of research effort. A number of comprehensive steady-state tubular reactor models are available in the literature. 2-5 These large-scale models differ in the mechanisms postulated to describe the polymerization kinetics, 6 the prediction approach of the final polymer properties, 7 the prediction methods of the reacting mixture physical properties, 8 assumptions regarding the flow regime, different approaches for taking into account the reactor variability, 9,10 and, finally, the kinetic and transport parameters used for model validation. 4 A common observation in all these studies is the lack of a consistent database of parameters that can be used for model development. Therefore, it is often necessary to re-estimate these parameters using experimental data from the particular laboratory or industrial reactor under study, a complicated and time-consuming task. 5,8