Modeling of “living” free‐radical polymerization processes. I. Batch,semibatch, and continuous tank reactors

A comprehensive mathematical model is developed for “living” free‐radical polymerization carried out in tank reactors and provides a tool for the study of process development and design issues. The model is validated using experimental data for nitroxide‐mediated styrene polymerization and atom transfer radical copolymerization of styrene and n‐butyl acrylate. Simulations show that the presence of reversible capping reactions between growing and dormant polymer chains should boost initiation efficiency when using free nitroxide in conjunction with conventional initiator and also increase the effectiveness of thermal initiation. A study shows the effects of the value of the capping equilibrium constant and capping reaction rate constants for both nitroxide‐mediated styrene polymerization (using alkoxyamine as polymer chain seeds) and atom transfer radical polymerization of n‐butyl acrylate (using methyl 2‐bromopropionate as chain extension seeds). Also the effect of introducing additional conventional initiator into atom transfer radical polymerization of n‐butyl acrylate is studied. It is found that the characteristics of long chain growth are determined by the fast exchange of radicals between growing and dormant polymer chains. Polymerization results in batch, semibatch, and a series of continuous tank reactors are analyzed. The simulations also show that a semibatch reactor is most flexible for the preparation of polymers with controlled architecture. For continuous tank reactors, the residence time distribution has a significant effect on the development of chain architecture. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 86: 1630–1662, 2002     

悬浮聚合反应的数学模型

以苯乙烯悬浮聚合反应为例,从机理分析出发,建立了聚合过程的动态模型,其中通过引入若干个动态校正系数,较充分地考虑了凝胶效应的影响,根据实验结果,对部分参数进行了辨识,提高了模型的精度,并适用整个反应过程,为先进控制规律的研究提供了必要基础.     

The influence of reactor type and operating conditions on the molecular weight distribution in vinyl acetate polymerization

The vinyl acetate polymerization system was investigated with respect to the breadth of the molecular weight distribution (MWD) in batch, continuous segregated, and continuous micromixed reactors. Models were developed employing a complex kinetic scheme including polymer transfer and terminal double bond polymerization, without neglecting initiation and termination steps. Inclusion of a gel effect for terminal double bond polymerization gave better agreement with experimental molecular weight data in suspension polymerization. Simulation results showed the MWD order in the three reactor types is not fixed, but a function of reactant concentrations and the importance of chain branching. In some cases changing the initiator type and concentration will change the MWD order.     

Application of optimal temperature trajectory to batch PMMA polymerization reactor

A mathematical model is developed for the polymerization of methyl methacrylate (MMA) in a batch reactor. The model includes chain transfers to the monomer and solvent and termination by both combination and disproportionation and also takes into account the density change of the reactor contents and the gel effect. The usual pseudo-steady-state assumption is relaxed here. The validity of the proposed model is tested by an isothermal experiment of batch PMMA polymerization. Indeed, the experimental results show that the proposed model can describe the real polymerization system very well in view of both monomer conversion and average molecular weights. The optimal control theory is applied together with Pontryagin's minimum principle to calculate the optimal temperature trajectory for a batch polymerization reactor system which would lead to a polymer product having the desired properties set a priori. The performance index of the control system is composed of three factors—the desired monomer conversion and number- and weight-average molecular weights. The desired values of number- and weight-average molecular weights are obtained at a specified monomer conversion within acceptable error ranges. Control experiments are conducted to track the optimal temperature trajectory obtained from the model and the results are found to be in good agreement with the desired values. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 69: 59–68, 1998     

Simulation of batch and continuous reactors with co-immobilized yeast and β-galactosidase

A reaction-diffusion model was used to simulate a co-immobilized system utilizing the numerical method of orthogonal collocation. The production of ethanol from deproteinized whey using β-galactosidase co-immobilized with Saccharomyces cerevisiae in calcium alginate gel beads was chosen as a model system. Calculated concentrations of lactose, glucose, galactose and ethanol were compared with experimental data for a batch reactor and a continuous horizontal packed-bed reactor. The mathematical model has been used to analyse the influence of internal and external mass transfer for the continuous reactor. The external mass transfer was shown to be of minor importance. The introduction of baffles decreased the backmixing in the horizontal packed-bed reactor. Internal mass transfer was found to be the main cause of the reduction in the apparent reaction rate. Thus, much of the expected increase in reaction rate is diminished by mass transfer hindrance when the cell concentration is increased.     

Kinetics of high conversion polymerization of vinyl acetate. Effects of mixing and reactor type on polymer properties

In the present article the kinetics of polymerization of vinyl acetate in suspension up to high conversion was studied. The molecular weight distribution and the side chain branching of polyvinyl acetate produced were examined with respect to micro and macro mixing as well as to reactor type. The following results were achieved: the time–activity curves of the polymerization can be described up to high conversions considering the exponential increase in viscosity of the polymerizing system and combining the viscosity with rate constants of the polymerization. The change of volume of the polymerizing system has no significant influence on kinetics. The narrowest molecular weight distribution of the poly(vinyl acetate) produced was achieved when polymerizing in the homogeneous continuous stirred tank reactor while the broadest molecular weight distribution was observed in the segregated continuous stirred tank reactor. The batch reactor and the flow tube reactor produce polymers with molecular weight distributions lying in between. Considering the side chain branching, another order was found. The batch reactor and the tube reactor show the lowest side chain branching, the homogeneous continuous stirred tank reactor shows a larger one and the segregated continuous stirred tank reactor shows the largest. Possible reasons for the different behavior of the different reactors are discussed. The degree of segregation was determined by experiments.     

The computer simulation of batch emulsion polymerization reactors through a detailed mathematical model

A computational algorithm for the detailed simulation of a batch emulsion polymerization reactor is discussed. The model is applied to the polymerization of methyl methacrylate, and the model predictions are shown to be in good agreement with experimental data. Further computations show the influence of reactor operating conditions on the polymer product and the reactor performance.     

Temperature control of highly exothermic batch polymerization reactors

In this article a highly exothermic batch polymerization reactor is considered. The reactor is simplified as a mixing tank with the internal heat generation treated as a disturbance. A fuzzy-hybrid-PID-feedback (FH-PID) control structure is developed in which the output of fuzzy hybrid portion is used to adjust the set point of a PID controller to compensate for the effect of the major disturbance, the heat of reaction. In this way, the hybrid portion of the controller does not influence the stability of the original PID control system. A fuzzy model was constructed to estimate the heat of reaction inside the fuzzy hybrid block. The fuzzy parameters of the hybrid portion do not depend on the process model and can be estimated from the transient response obtained with a conventional PID controller. This FH-PID control strategy has been applied to the temperature control of batch solution and batch inverse emulsion polymerizations of acrylamide in a 1 gallon pilot scale reactor. The results show that this fuzzy hybrid—PID-feedback control strategy improves the control performance of the batch polymerization reactor.     

Modeling gel effect in branched polymer systems: Free‐radical solution homopolymerization of vinyl acetate

In this work a comprehensive mathematical framework is developed for modeling gel effect in branched polymer systems with application in the solution polymerization of vinyl acetate. This model is based on sound principles such as the free‐volume theory for polymer chains diffusion. The model predictions for monomer conversion and number‐ and weight‐average molecular weights were found to be in good agreement with published data in the literature. Moreover, the joint molecular‐weight distribution–long chain branching distribution is calculated by direct numerical integration of a large system of nonlinear ordinary integral‐differential equations describing the mass conservation of macromolecular species in a batch reactor. This allows studying the effect of process conditions such as initiator and solvent concentration on the product quality. It is believed that this work might contribute to a more rational design of polymerization reactors. © 2008 Wiley Periodicals, Inc. J Appl Polym Sci, 2009     

Über die Auswahl von Polymerisationsreaktoren

Choice of polymerization reactors . Industrial syntheses of polymers always yield a highly complex product whose adequate characterization often requires application-oriented tests in addition to chemical and physical parameters. In contrast to the synthesis of low-molecular mass substances subsequent corrections, e. g. by distillation or crystallization, are usually impossible. Extremely detailed reaction control is of paramount importance. Polyreactions can be classified according to kinetic aspects (monomer linkage with and without termination reaction, linkage of polymers). Apart from homogeneous bulk polymerization, there are the heterogeneous processes of precipitation-, bead-, and emulsion-polymerization. The polymerization reactors can be assigned to the known ideal types (batch reactor, continuous plug flow reactor, series of stirred tanks, and continuous stirred tank reactor). High viscosity often thwarts thorough mixing in bulk polymerization and segregation may occur. The article surveys the variants of the reactor types used for various polyreactions and polymerization processes in industry and examines the reasons for this choice. Most commonly used is the batch reactor, followed by the cascade of stirred tanks. Continuous polymer-linkage (polycondensation, polyaddition) is frequently performed in a series of several different kinds of reactors.     

Development of a unified framework for calculating molecular weight distribution in diffusion controlled free radical bulk homo-polymerization

In the present work, two different approaches to model diffusion controlled free radical polymerization, namely the free volume model and the entanglement theory are compared. These approaches are applied to methyl methacrylate bulk polymerization in a batch reactor to calculate the conversion, total radical concentration, the number and weight average molecular weights as well as the entire molecular weight distribution as a function of the polymerization time and the process conditions. All the diffusion-controlled phenomena were taken into account, including gel, glass and cage effects as well as residual termination. The molecular weight distribution is calculated by direct numerical integration of a large system of non-linear ordinary differential equations describing the conservation of the mass of macromolecular species in the batch reactor. Model predictions are in good agreement with available experimental data for conversion, number and weight average molecular weights as well as the entire molecular weight distribution, thus justifying the ability of these models to describe the main issues of the diffusion-controlled free radical polymerization.     

Sensitivity analysis of a batch polymerization reactor

The sensitivity of model output variables for a batch polymerization reactor to uncertainties in the kinetic parameters and initial conditions is studied. Differential equations that describe the time variation of sensitivity coefficients for the batch reactor are derived. Numerical integration of the sensitivity equations reveals that the system output responses are very sensitive to parameter variations especially when the polymerization exhibits an autoacceleration of the reaction rate.     

Modeling of Emulsion Copolymerization of 2‐Hydroxyethyl Methacrylate and Styrene

A mathematical model is developed to describe the reaction behavior of emulsion copolymerization systems where significant polymerization occurs in both the latex particle and aqueous phases. 2‐Hydroxyethyl methacrylate (HEMA) and styrene system was used to illustrate the development of a batch reactor. The model considers the gel and glass effects and monomer transfer. The principal model parameters are taken from the literature. Copolymerization reactivity ratios were estimated using a nonlinear least‐squares procedure. Model predictions have been compared with experimental data on monomer conversion.     

Algorithms for Using Some Models of Gel and Glass Effects in Free-Radical Polymerization of Methyl Methacrylate

The aim of this article is the modeling and simulation of the batch bulk polymerization of methyl methacrylate (MMA). Simple dependencies between the propagation and termination rate constants and monomer conversion for the gel and glass effects are proposed. The empirical parameters in these relations are determined from conversion and molecular-weight experimental data obtained under various reaction conditions (initiator concentration and temperature). Sometimes, different relations were necessary to express the variation of the kinetic constants on conversion subintervals. Thus, these models have been used continuously or discontinuously, as a function of their results. An algorithm for using this model was also established.

The analysis of this model has two aims: (1) to get a good agreement between simulation and experiment; (2) to provide a simple model to be used under different reactor conditions (batch, semibatch, or continuous) or which can be easily handled in polymer engineering studies, such as sensitivity analysis, optimal control, and so forth.     

Continuous emulsion polymerization of styrene in a single Couette–Taylor vortex flow reactor

The effect of the geometrical and operational parameters on the mixing characteristics of a Couette–Taylor vortex flow reactor (CTVFR) were investigated and were correlated with the same parameters by using the tank‐in‐series model. Continuous emulsion polymerization of styrene was conducted at 50°C in a CTVFR to clarify the effects on kinetic behavior and reactor performance of operational parameters such as rotational speed of inner cylinder (Taylor number), reactor mean residence time, and emulsifier and initiator concentrations in the feed streams. It was found that steady‐state monomer conversion and particle number could be freely varied only by varying the Taylor number. In order to explain the observed kinetic behavior of this polymerization system, a mathematical model was developed by combining the empirical correlation of the mixing characteristics of a CTVFR and a previously proposed kinetic model for the continuous emulsion polymerization of styrene in continuous stirred tank reactors connected in series (CSTRs). On the basis of these experimental results, it was concluded that a CTVFR is suitable for the first reactor (prereactor) of a continuous emulsion polymerization reactor system. © 2001 John Wiley & Sons, Inc. J Appl Polym Sci 80: 1931–1942, 2001     

A radial cell model for a tubular polymerization reactor

A cell model for the prediction of temperature and concentration gradients in a nonisothermal tubular polymerization reactor at steady state is presented. Both radial and longitudinal gradients are considered. The complete molecular weight distribution is calculated as well as the leading moments of the distribution. The model is easily reduced to predict the performance of a plug-flow tubular reactor, batch reactor, and continuous stirred tank reactor (CSTR). The specific polymerization mechanism application consists of free-radical initiation, propagation, and combination termination.     

A Laboratory Reactor for Gas‐Phase Olefin Polymerization

A new reactor system for gas‐phase ethylene/α‐olefin polymerization is described. Good gas‐phase temperature control at high polymerization rates was achieved with the 2‐L semi‐batch reactor. Ethylene/1‐hexene and ethylene polymerization results showing the effects of operating conditions on temperature profiles are presented. Good gas‐phase temperature control is required to obtain reliable activity profiles. A gas‐sampling and analysis system, which allows relatively rapid (< 3 min) and accurate determination of ethylene/1‐hexene contents in the gas‐phase of the reactor, is also described. Rapid and reliable hydrogen contents were also measured with this relatively inexpensive system.     

Dynamic On‐Line Reoptimization Control of a Batch MMA Polymerization Reactor Using Hybrid Neural Network Models

A hybrid neural network model based on‐line reoptimization control strategy is developed for a batch polymerization reactor. To address the difficulties in batch polymerization reactor modeling, the hybrid neural network model contains a simplified mechanistic model covering material balance assuming perfect temperature control, and recurrent neural networks modeling the residuals of the simplified mechanistic model due to imperfect temperature control. This hybrid neural network model is used to calculate the optimal control policy. A difficulty in the optimal control of batch polymerization reactors is that the optimization effort can be seriously hampered by unknown disturbances such as reactive impurities and reactor fouling. With the presence of an unknown amount of reactive impurities, the off‐line calculated optimal control profile will be no longer optimal. To address this issue, a strategy combining on‐line reactive impurity estimation and on‐line reoptimization is proposed in this paper. The amount of reactive impurities is estimated on‐line during the early stage of a batch by using a neural network based inverse model. Based on the estimated amount of reactive impurities, on‐line reoptimization is then applied to calculate the optimal reactor temperature profile for the remaining time period of the batch reactor operation. This approach is illustrated on the optimization control of a simulated batch methyl methacrylate polymerization process.     

尼龙66连续缩聚过程模拟

为加深对聚己二酰己二胺(尼龙66)缩聚过程的理解,指导尼龙66生产过程的优化和新工艺开发,本研究基于聚合反应特性以及反应器流体力学特征,耦合聚合反应动力学模型、水-聚合物体系的汽液平衡模型以及传质模型,建立了尼龙66连续缩聚过程管式反应器和后缩聚反应器的数学模型,实现了尼龙66连续聚合过程的模拟。聚合物分子量及水含量的模拟值符合工业生产数据,验证了模型的可靠性,模拟分析了聚合工艺参数对聚合物分子量及水含量的影响,结果表明减少醋酸含量、提高反应器温度、降低管式反应器和后缩聚反应器压力均有利于快速提升聚合物分子量。较优的工艺条件为后缩聚器温度280~285℃,卧管式反应器压力1.600~1.750MPa。建立的模型准确性好,可用于考察工艺参数的影响,指导工业应用。     

Experimental investigation of vinyl chloride polymerization at high conversion—reactor dynamics

A series of suspension polymerizations of vinyl chloride monomer (VCM) was carried out in a 5-L pilot plant reactor over the temperature range, 40–70°C. The reactor pressure and monomer conversion were monitored simultaneously every 7–8 min. The critical conversion X_f, at which the liquid monomer phase is consumed, was considered to occur when the reactor pressure fell to 98% of the vapor pressure of VCM for suspension at the polymerization temperature. The reactor model predictions of pressure are in excellent agreement with the experimental data over the entire conversion and temperature ranges studied. The mechanism of reactor pressure development for VCM suspension polymerization is discussed herein in some detail. For isothermal batch polymerization, the reactor pressure falls in two stages due to the effect of polymer particle morphology on pressure drop. The first stage is due to the volume increase of the vapor phase as a result of volume shrinkage due to conversion of monomer to polymer. The monomer phase is not yet consumed at this stage, but it is trapped in the interstices between primary particles creating a mass transfer resistance; therefore, the reactor pressure drops slowly. The second stage is due to both the volume increase of the vapor phase and to the monomer in the vapor phase diffusing into the polymer phase because of the subsaturation condition with respect to monomer in the polymer phase. The reactor pressure drops dramatically with an increase in monomer conversion at this stage. The present model can be used to predict reactor dynamics during suspension polymerization under varying temperature and pressure conditions.