Adriana Brandolin

Large-scale dynamic optimization for grade transitions in a low density polyethylene plant

This paper presents the optimal control policy of an industrial low-density polyethylene (LDPE) plant. Based on a dynamic model of the whole plant, optimal feed profiles are determined to minimize the transient states generated during the switching between different steady states. The industrial process under study produces LDPE by high-pressure polymerization of ethylene in a tubular reactor using oxygen and organic peroxides as initiators. The plant produces polyethylene of different grades that require continuous changes from one steady state to another, in order to switch among the different final products. These changes generate disturbances that keep the product out of specifications during the transient states, with a consequent economic loss. The plant model consists of two parts; the first one corresponds to the tubular reactor. Here, the partial differential equations corresponding to the mass and energy dynamic balances are discretized along the distance coordinate by using finite differences. The resulting ordinary differential equations include the energy balance and individual mass balances for oxygen, peroxides, ethylene, butane, free radicals and polymer. Although, methane is also present in the plant, in the reactor model it is considered as a nonreacting impurity along with the other impurities coming from the rest of the process. The second part of the model corresponds to the rest of the plant. Here we considered four components: ethylene, butane, methane and impurities. An interesting aspect of this process is the presence of many time delays that are incorporated in the optimization model. The resulting differential algebraic equation (DAE) plant model includes over five hundred equations. The dynamic optimization problem is solved using a simultaneous nonlinear programming (NLP) approach. The continuous state and control variables are discretized, by applying orthogonal collocation on finite elements. The resulting NLP is solved with a reduced space Interior Point Algorithm, which is applied directly to the NLP. In addition, a new mesh refinement strategy is applied to this model to confirm that no further improvement can be found in the optimal control profiles. The paper studies two cases of switching among different polymer grades, determining the optimal profiles of butane fed to the plant, in order to minimize the time to reach the steady state operation corresponding to the desired new product quality. The results are also compared with simpler model where reactor was considered as a black-box with the conversion level taken as constant data for each polymer grade. As a result, the dynamic model we developed and the solution methodology used is a flexible and practical tool to help process engineers for taking decisions during the plant operation.

High Pressure Polymerization of Ethylene. an Improved Mathematical Model for Industrial Tubular Reactors

ABSTRACTThis work presents an exhaustive mathematical model for the high pressure polymerization of ethylene in tubular reactors of configuration similar to that encountered in the industry. Multiple injection of monomer, mixtures of initiators and chain transfer agents are considered together with realistic flux configurations. Typical heat transfer coefficients are estimated from industrial plant data. The effects of pressure pulse on the reactor behavior are also analyzed. Instantaneous temperature profiles produced by such pressure pulse were recovered from stationary simulations showing a very good agreement with the corresponding experimental data. The model features are demonstrated by predictions of temperature, concentrations of reactants and products and molecular properties as a function of reactor length. Also, appropriate predictive capabilities are disclosed by comparison of model simulation results and experimental data The generation of a high temperature initiator, derived from oxygen, is...

Recovery of molecular weight distributions from transformed domains. Part I. Application of pgf to mass balances describing reactions involving free radicals

We present a general framework for the application of a transform technique, probability generating functions (pgf), to mass balances that describe free radical reactions, in particular synthesis or modification of polyolefins. Contributions of specific reactions to the mass balances are identified and transformed separately, so that a modular approach is possible for the construction of the pgf balance equations for different free radical processes. This simplifies the transformation step hopefully making the method useful to more people. Three examples taken from the literature are transformed using this modular method showing its ease of use. In Part II of this work, the resulting transforms are inverted to recover the complete molecular weight distribution.

Recovery of molecular weight distributions from transformed domains. Part II. Application of numerical inversion methods

This work is a part of a study aiming at developing tools for the prediction of complete molecular weight distributions (MWDs) of polymers at the exit of a reactor. This reactor may be a synthesis reactor or one used to modify a preexisting resin. In this work, we analyse the suitability of three methods for the numerical inversion of probability generating functions (pgfs), those due to Papoulis, de Hoog and Garbow. The three methods have been proposed in the literature for the inversion of Laplace transforms. We show how to adapt them to the problem at hand, and apply them to two situations. The first one is the recovery of experimentally measured MWDs, through a process that consists of finding the pgf of the distribution, numerically inverting it and comparing the result with the known MWD. The second one is to solve the pgf balances of polymerisation systems with known MWDs, and comparing those MWD with the ones that result from the inversion with the three methods. We discuss the relative advantages of each inversion method and propose guidelines for their proper use with unknown MWD functions.

Dynamic simulation and optimisation of tubular polymerisation reactors in gPROMS

Abstract In this paper a dynamic model of the high-pressure polymerisation of ethylene in tubular reactors is introduced and a dynamic optimisation problem is formulated for studying start-up strategies. The optimisation objectives proposed are to maximise outlet conversion and optimise the time necessary for its stabilisation while keeping product molecular properties between commercial ranges. Results are shown giving the time responses for temperature, number-average molecular weight and conversion along the reactor axial distance, which were obtained using different control variable profiles. Improving in reactor productivity is achieved. The interface gOPT of the gPROMS simulator was used to resolve the optimisation problem and to perform the simulations.

Large-scale dynamic optimization of a low density polyethylene plant

Abstract This paper presents the optimal control policy of an industrial low-density polyethylene (LDPE) plant. Based on a dynamic model of the whole plant, optimal feed profiles are determined to minimize the transient states generated during the switching between different steady states. This industrial process produces LDPE by high-pressure polymerization of ethylene in a tubular reactor. The plant produces different final products. The model consists of two parts, the first one corresponds to the reactor and the second to the rest of the plant. The process has many time delays that are also incorporated into the optimization model. The resulting differential algebraic equation (DAE) plant model includes over 500 equations. The continuous state and control variables are discretized by applying orthogonal collocation on finite elements. The resulting NLP is solved with a reduced space interior point algorithm. The paper studies two cases of switching among different polymer grades determining the optimal butane flow rates, in order to minimize the time to reach the steady state operation corresponding to the desired new product quality.

Peroxide modification of polyethylene. Prediction of molecular weight distributions by probability generating functions

We present a mathematical model able to describe the complete molecular weight distributions of polyethylene during reactive modification by organic peroxides. The method is applicable to batch processes, such as modification in a press or a plug-flow extruder, and in its present form is valid up to the gel point. We apply probability generating function definitions to the mass balances of radical and polymer species in the reacting medium. Three different probability generating functions are applied, each one directly applicable either to the number, weight or chromatographic distributions. These generating functions are numerically inverted to obtain the corresponding calculated molecular weight distribution. Two different inversion methods are used, and their relative performances analyzed. Predictions are compared with qualitative experimental data obtained in a press. Model predictions on molecular weight distributions are in agreement with experimental trends.

EFFECT OF MULTIPLE FEEDINGS IN THE OPERATION OF A HIGH-PRESSURE POLYMERIZATION REACTOR FOR ETHYLENE POLYMERIZATION

A thorough model of the high pressure polymerization of ethylene in tubular reactors previously developed by the authors is used to analyze operating conditions for a specific industrial reactor. The objective is to maximize productivity while keeping product quality within desired parameters. The variables under analysis are: position, rate and temperature of lateral reactor feeds, reactor jacket configuration, temperature and flow rate of the refrigerant. Separating the initiator injections provides an increase in conversion, but also in polydispersity. However, this undesirable effect in polydispersity can be sorted out by manipulating the modifier flow rate properly. Besides, the refrigerating cost can be significantly reduced by means of a proper jacket configuration and an optimized refrigerant flow rate. In this way we show the usefulness of the mathematical model as a predictive tool -in comparison with pilot or industrial scale trials- especially when processes are carried out under severe operating conditions.

Prediction of the full molecular weight distribution in RAFT polymerization using probability generating functions

Abstract In this work, a model for the RAFT polymerization following the slow fragmentation approach was developed in order to obtain the full molecular weight distribution (MWD) using probability generating functions (pgf). A combination of univariate and bivariate pgf is applied to deal with the univariate chain length distributions of macroradical, dormant and dead polymer chains, and the bivariate distribution of the two arms intermediate adduct. This allows rigorous modeling of the polymerization system without simplifying assumptions. For comparison purposes, the population balances were solved by direct integration of the resulting equations. Our results show that the pgf technique allows obtaining an accurate solution efficiently in terms of computational time. What is more, the model provides a detailed characterization of the polymer that could be of great help for grasp the process fundamentals.

Comprehensive Mathematical Modeling of Controlled Radical Copolymerization in Tubular Reactors

Abstract In this work a comprehensive mathematical model of the nitroxide mediated polymerization in tubular reactors is developed. The model is able to predict average molecular properties, such as the average molecular weights and copolymer composition. Besides, detailed calculation of the copolymer microstructure is included. The model is able to predict the bidimensional molecular weight distribution of the copolymer, the sequence length distribution, the global molecular weight distribution and the copolymer composition distribution. In particular, styrene-α methyl styrene and styrene-methyl metha-crylate copolymerizations are studied. Model outputs are consistent with known features of the system. The detailed information on the copolymer molecular structure provided by the model makes it a valuable tool for the process design.