Computers and Chemical Engineering 33 (2009) 379–390 Contents lists available at ScienceDirect Computers and Chemical Engineering journal homepage: www.elsevier.com/locate/compchemeng Optimization-based strategies for the operation of low-density polyethylene tubular reactors: Moving horizon estimation Victor M. Zavala ∗ , Lorenz T. Biegler Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA article info Article history: Received 20 May 2008 Received in revised form 12 September 2008 Accepted 8 October 2008 Available online 22 October 2008 Keywords: Moving horizon estimation LDPE Distributed reactors Partial differential equations Discretization Uncertainty Large-scale Nonlinear programming abstract We present a moving horizon estimation (MHE) application for multi-zone low-density (LDPE) polyethy- lene tubular reactors. The strategy incorporates a first-principles dynamic model comprised of large sets of nonlinear partial, differential and algebraic equations (PDAEs). It was found that limited temperature measurements distributed along the reactor are sufficient to infer all the model states in space and time and to track uncertain time-varying phenomena such as fouling. A full discretization strategy and a state- of-the-art nonlinear programming (NLP) solver are used to enable the computational feasibility of the approach. It is demonstrated that the MHE estimator exhibits fast performance and is well suited for applications of industrial interest. Published by Elsevier Ltd. 1. Motivation and background information Low-density polyethylene (LDPE) is an important commodity polymer in today’s economy due to its high flexibility and rela- tively low-cost (Knuuttila, Lehtinen, & Nummila-Pakarinen, 2004). LDPE is mostly produced in tubular reactors by free-radical poly- merization of ethylene at supercritical conditions (2000–3000 atm and 150–350 ◦ C). A typical tubular reactor and corresponding tem- perature profiles for the reactor core and jackets are sketched in Fig. 1. This type of reactor consists of long pipes (1–3 km) with small inner diameters (5–10 cm) and thick reactor walls (2–5 cm) which are divided into several reaction and cooling zones. Each zone is equipped with a jacket cooling system used to remove the large amounts of heat produced by polymerization. Multiple side streams containing monomer, comonomer, chain transfer agent (CTA) and initiators can be fed along the reactor to control the temperature profile and the resulting polymer properties. The large heat trans- fer areas and low degrees of back-mixing resulting in these units permit the high throughput production of LDPE resins with unique processability and end-use properties. ∗ Corresponding author. Tel.: +1 412 268 2238. E-mail address: vzavala@andrew.cmu.edu (V.M. Zavala). Despite the multiple benefits offered by LDPE tubular reactors, there exist several factors limiting their performance. The first issue arises due to their distributed and multivariable nature which gives rise to complex interactions along the pipe. The most common approach to cope with this complexity is to find feasible operat- ing conditions able to produce a particular grade by trial and error and/or experience. The resulting operating recipes are enforced strictly through an appropriate regulatory control system. While these recipes work well in many cases, they tend to be rather con- servative and need to be constantly adapted for each new grade incorporated into the product portfolio. A second important prob- lem arising in LDPE reactors is the persistent and slow deposition of polymer on the inner reactor walls (Buchelli et al., 2005a,b; Lacunza, Ugrin, Brandolin, & Capiati, 1998). The resulting fouling layer is highly insulating and severely decreases the heat-transfer rate to the cooling jacket. Since the polymerization reactions are highly exothermic, the production rate needs to be dropped pro- gressively in order to keep the temperature profile within safe limits and avoid thermal runaway. The impact of fouling on the overall profitability of high-throughput LDPE reactors is extremely large. The potential economic benefits and high operational complex- ity of LDPE reactors have motivated research efforts in many areas. Extensive experimental studies have been performed in order to understand the fundamental interactions between the reactor 0098-1354/$ – see front matter. Published by Elsevier Ltd. doi:10.1016/j.compchemeng.2008.10.008