Polymerization of melamine and formaldehyde in homogeneous continuous-flow stirred-tank reactors using functional group approach: Part B: Molecular weight distribution

Polymerization of melamine and formaldehyde in homogeneous continuous-flow stirred-tank reactors (HCSTRs) is reversible and leads to formation of branched polymer due to the hexafunctional nature of melamine. The reversed reaction of branched molecules depends upon the chain structure, and herein a simple model is presented to account for this. The functional group analysis of part A of this series has been extended and the molecular weight distribution (MWD) relations for HCSTRs have been solved. The MWD of the polymer is extremely sharp and the molecules are more branched compared to those for batch reactors and can be explained as follows. A given reacted bond can react with water as well as free formaldehyde through the reversed reaction, and the rate constant for the latter step is much larger. Consequently, the chain growth is limited as long as there is free formaldehyde in the reaction mass. Lower conversion of melamine and formaldehyde in HCSTRs is used to explain the behavior observed.     

An analytical solution of the molecular weight distribution of reversible step growth polymerization in homogeneous continuous flow stirred tank reactors

Mole balance for the molecular weight distribution in homogeneous continuos-flow stirred tank reactors (HCSTRs) for reversible step-growth polymerization has been written. The relation for the moment generating function G is found to be a nonlinear ordinary differential equation and has been solved analytically. The solution of the MWD of the polymer formed is shown to be valid even if the condensation product is removed. At equilibrium, the solution reduces to the Flory distribution. The computations show that the polydispersity of the polymer first increases with the residence time θ of the reactor, but, for large θ, it reduces to the equilibrium value after undergoing a maximum.     

Molecular weight distribution of polyethylene terephthalate in homogeneous,continuous-flow-stirred tank reactors

Poly(ethylene terephthalate) (PET) formation in homogeneous, continuous-flow-stirred tank reactors (HCSTRs) operating at steady state has been simulated. The feed to the reactor is assumed to consist of the monomer bis-(hydroxyethyl) terephthalate and monofunctional compound (MF₁) cetyl alcohol. The overall polymerization is assumed to consist of the polycondensation, reaction with monofunctional compounds, redistribution, and cyclization reactions. At a given time, the reaction mass consists of polyester molecules (P_n), polyester molecules with an ending of molecules of monofunctional compound (MF_n), and cyclic polymers (C_n). A mass balance for each of these species in the reactor gives rise to a set of algebraic equations to be solved simultaneously. The MWD calculations show that the redistribution reaction plays a major role and cannot be ignored, This result is in contrast lo the observation for semi-batch reactors, for which redistribution becomes important when the cyclization reaction is included. For the same residence times of semi-batch and HCSTRs, the latter gives considerably lower-number average molecular weight, N_av, and polydispersity index, ρ. However, for the same conversions, the ρ for CSTR is higher. The concentration of the monofurctional compound, [MF₁]₀, in the feed and the reactor temperature both influence ρ, but the effect is small within the range studied.     

Algorithm for computation of molecular weight distribution and its moments in reversible step-growth polymerization in batch reactors

The differential equations governing the molecular weight distribution (MWD) in step-growth polymerization are coupled and nonlinear and a large number of them must be solved simultaneously to keep the truncation error low. In this work, these equations have been decoupled so that they can be solved sequentially. The solution of these is independent of the truncation error and there is considerable saving of computation time. To demonstrate the efficiency of the algorithm, the formation of polyethylene terephthalate (PET) in batch reactors with ethylene glycol evaporating has been analyzed. The feed to the reactor is taken as polymer with its oligomers present according to the Flory's distribution. The effect of pressure and temperature of the reactor on the progress of polymerization has been modelled and evaluated. The amount of ethylene glycol distilled, the concentrations of the first five oligomers Q₁ to Q₅, the number average chain length, and the polydispersity index of the polymer have been determined. It is shown that the reduced pressure and increased temperature reduce the concentration of the condensation product in the reaction mass, thus pushing the polymerization in the forward direction. Lastly the CPU time on Dec 1090 using this algorithm is only 0.40 s compared to about 10 min for similar computations using other existing methods.     

Computational scheme for the calculation of molecular weight distributions for nylon 6 polymerization in homogeneous,continuous-flow stirred-tank reactors with continuous removal of water

An efficient computational scheme has been established for obtaining the molecular-weight distributions (MWDs) for Nylon 6 polymerization in homogeneous, continuous-flow stirred-tank reactors (HCSTRs) with water vaporization. A three-step procedure is used: the moment equations are solved by a sequential application of Brown's algorithm, the moments so calculated are used to give an empirical Flory-Schultz distribution and, finally, this empirical distribution is used as a starting guess in Brown's algorithm applied to species mass-balance equations to give the exact MWD. The moment-closure approximations normally used for hutch reactors have been found to work equally well with HCSTRs for residence times of industrial significance.     

Simulation of reversible nylon-66 polymerization in homogeneous continuous-flow stirred tank reactors

Experimental data of Ogata¹ has been curve-fitted to obtain the forward and reverse rate constants for nylon-66 polymerization. Its molecular weight distribution (MWD) has been simulated in homogeneous continuous-flow stirred tank reactors (HCSTR) for 11 h of residence time when the reaction mass is very close to equilibrium. The set of algebraic equations have been solved using Brown's algorithm,² which was found to be more efficient compared to the Gauss-Jordon techniques of solution. The MWD thus obtained is compared with our earlier simulation of the molecular weight distribution from batch reactors³ and was found to differ significantly. In HCSTR, the weight fraction distribution does not undergo a maximum and the polydispersity index ρ of the polymer formed is much higher than that obtained from batch reactors. The number and weight average of the polymer formed in HCSTR is found to be significantly lower.     

Polymerization of melamine and formaldehyde in homogeneous continuous-flow stirred-tank reactors using functional group approach: Part A: Conversion of functional groups

A comprehensive kinetic model using the functional group approach has been proposed for the polymerization of melamine and formaldehyde. The kinetic model is consistent with the basic chemistry of polymerization and involves five rate constants which have been estimated using the experimental data of Tomita. Homogeneous continuous-flow stirred-tank reactors (HCSTRs) have been modelled and the mole balance relations for various functional groups have been written. The performance of HCSTRs is governed by algebraic equations and, for any specified residence time, is found by the method of successive substitution using the Brown's algorithm. The computations show that as long as free formaldehyde is present, the reaction mass would consist predominantly of substituted melamine molecules. However, after formaldehyde is completely reacted, larger oligomers are formed in larger concentrations. On comparison of results with batch reactors, it is found that for the same reaction time HCSTRs yield polymer with higher branching.     

Dynamic optimization of a semi-batch reactor for polyurethane production

In this work, the dynamic optimization of a polyurethane copolymerization reactor is addressed. A kinetic-probabilistic model is used to describe the nonlinear step-growth polymerization of a mixture of low- and high-molecular-weight diols, and a low-molecular-weight diisocyanate. The dynamic optimization formulation gives rise to a highly complex and nonlinear differential-algebraic equation (DAE) system. The DAE optimization problem is solved using a simultaneous approach (SDO) wherein the differential and algebraic variables are fully discretized leading to a large-scale nonlinear programming (NLP) problem. The main reactor operation process control objective is the maximization of the molecular weight distribution (MWD) under a desired batch time, subject to a large set of operational constraints, while simultaneously avoiding the formation of polymer network (gel molecule). Typically, polyurethane formation is carried out using batch reactors. However, batch operation leads to attain relatively low MWD values and, if the process is not efficiently operated, there is always the possibility of obtaining a polymer network. In this work, it was found that process operation is greatly enhanced by the semi-batch addition of 1,4-butanediol and diamine, and the manipulation of the reactor temperature profile, allowing to obtain high molecular weights while avoiding the onset of the gelation point.     

Modelling of reversible poly(ethylene terephthalate) reactors

The second stage of batch poly(ethylene terephthalate) (PET) reactor with bis(2-hydroxyethyl) terephthalate (BHET) as the feed has been simulated. In this stage, the overall polymerization is not diffusion limited and is known to be a complex reaction. In this work it has been assumed to consist of polycondensation, reaction with monofunctional compounds (cetyl alcohol), redistribution, and cyclization reactions. The forward and reverse steps of each of these have been modelled in terms of the rate constants involving functional groups and the reacted bonds. The equations for the calculation of the molecular weight distribution (MWD) in batch reactors have been written and solved numerically. The MWD reported in this work is assumed to include the monofunctional products only, and, for the case where ethylene glycol is not removed from the reaction mass, it was found to be unaffected by the choice of the redistribution rate constant (k_r). Since the removal of ethylene glycol is not mass transfer controlled, its concentration in the reaction mass is assumed be given by the vapor–liquid equilibrium existing at the pressure applied on the reactor. In this work, the level of ethylene glycol concentration, y_g (?[G]/[P₁]₀), has been taken as a parameter, and, on application of vacuum, the MWD results were found to vary with k_r with the sensitivity increasing with y_g. It was then shown that the importance of the redistribution reaction is enhanced when the cyclization reaction also occurs. The effect of vacuum on the performance of the reactor has been studied by varying y_g. For y_g less than 0.01, the change in the MWD of the polymer becomes very small. The effects of polymerization temperature and initial concentration of monofunctional compounds on MWD were found to be small.     

Modeling of molecular weight and copolymer composition distributions for ethylene-propylene solution copolymerization

A comprehensive computational fluid dynamics (CFD) model was developed to investigate spatial distributions of molecular weight distribution (MWD) and copolymer composition distribution (CCD) for ethylene-propylene (EPM) copolymers in a bubble column reactor. The CFD approach incorporated Euler–Euler two-fluid model, copolymerization kinetics, and copolymer microstructural distribution model together by user-defined functions for ethylene-propylene heterogeneous copolymerization process. MWD and CCD distributions were calculated by introducing Flory's distribution and Stockmayer's distribution, respectively. CFD model results were validated with literature data. The multiphase hydrodynamics, interphase mass transfer, spatial–temporal variations of MWD and CCD distributions were analyzed. Both distributions are wider at the inlet of reactor for the inefficient mixing, but narrower at the outlet due to fully developed flow and polymerization. This model is beneficial to the improvement of polymer products and process control in industrial EPM reactor.     

Optimal operation of ethylene polymerization reactors for tailored molecular weight distribution

This work presents a comprehensive steady‐state model of the high‐pressure ethylene polymerization in a tubular reactor able to calculate the complete molecular weight distribution (MWD). For this purpose, the probability generating function technique is employed. The model is included in an optimization framework, which is used to determine optimal reactor designs and operating conditions for producing a polymer with tailored MWD. Two application examples are presented. The first one involves maximization of conversion to obtain a given MWD, typical of industrial operation. Excellent agreement between the resulting MWD and the target one is achieved with a conversion about 5% higher than the ones commonly reported for this type of reactor. The second example consists in finding the design and operating conditions necessary to produce a polymer with a bimodal MWD. The optimal design for this case involves a split of the initiator, monomer, and modifier feeds between the main stream and two lateral injections. To the best of our knowledge, this is the first work dealing with the optimization of this process in which a tailored shape for the MWD is included. © 2007 Wiley Periodicals, Inc. J Appl Polym Sci, 2007     

Mass transfer effects in polycondensation reactors wherein functional groups are not equally reactive

Three simplified models of polycondensation reactors are considered in which the condensation product is continuously removed by application of vacuum. Reversible polycondensation reactions of monomers violating the equal reactivity hypothesis have been simulated in these reactors. The effect of various rate and reactor design variables on the molecular weight distribution (MWD) and its moments is studed. It is observed that when the reverse reactions are rapid, the results are fairly sensitive to the level of vacuum applied and to the mass transfer resistance; whereas when the forward reactions predominate, results lie very close to earlier plots for the corresponding irreversible polymerizations. These reactor variables then have relatively small influence on the MWD. Splitting of the MWD curves for odd and even values on n is observed under certain conditions, the effects being more pronounced in the presence of mass transfer than in its absence.     

半间歇聚合反应分子量分布周期操作的迭代学习控制

通过对半间歇聚合反应的引发剂进料实施周期操作,研究了这类操作方式对聚合物分子量分布的影响。研究结果显示,周期操作能改善聚合反应过程,对分子量分布有明显的加宽作用。对性能指标进行改进,以引发剂周期进料的占空比为控制变量,采用基于粒子群优化的迭代学习算法,对分子量分布进行了优化控制。仿真分析表明,在实际对象和模型存在不匹配的情况下,运用迭代粒子群算法,控制输入随着批次学习的进行而逐渐趋于最优解,聚合反应的分子量分布则不断逼近希望的分子量分布。实验结果验证了以周期操作方式对半间歇聚合过程分子量分布进行迭代优化控制的可行性。     

Heterogeneous continuous kinetics modeling of PET depolymerization in supercritical methanol

The kinetics of PET depolymerization in supercritical methanol was investigated. A continuous kinetics model was developed to analyze PET decomposition behavior. This model includes molecular weight distribution (MWD) changes in the polymer by random and specific scissions and secondary reactions of monomer components for complex macromolecular reactions. The changes of MWD and monomers as a function of time were simulated by continuous kinetics. Reactions in two phases, polymer melt phase and supercritical phase, were considered. By comparing simulated and experimental results, values of the rate constants were determined. These results indicated that random scission proceeds predominantly in the heterogeneous phase during the initial stage of PET depolymerization in supercritical methanol and specific scission proceeds predominantly in the homogeneous phase during the final stage. It was also shown that mass transfer influences the depolymerization of PET.     

Modeling of polyethylene terephthalate reactors: 7. MWD considerations

A mathematical model has been developed to compute the molecular weight distribution (MWD) in the polyethylene terephthalate (PET) manufacturing process. Unlike the previous efforts, this model takes into account the influence of side reactions and various interchange reactions on MWD. The process of blending of molten polyester chips has also been simulated with a view to calculate the equilibrium MWD as well as the time required to reach the equilibrium MWD. The1 significance of the results has been discussed in terms of industrial operations involving PET polymerization and PET blending.     

Solution of mass transfer in step growth polymerization in films with bulk in equilibrium

We have analyzed step growth polymerization in a flat film with finite mass transfer resistance. We have shown rigorously that the molecular weight distribution (MWD) at equilibrium is given by the Flory distribution, and under reaction the form of the MWD does not change if the feed is either pure monomer or in equilibrium initially. Extensive computations have shown that it is possible to split the film into growing interfacial and shrinking bulk regions. It is possible to obtain similarity transformations of concentrations of condensation product, and polymer as time invariant profiles. Based on this finding, we have determined a solution for step growth polymerization with finite mass transfer in films. The results lie within 5% of the “exact” numerical computations, for all possible variations of parameters.     

Optimization of the transesterification stage of polyethylene terephthalate reactors

The tranesterification step of the polyethylene terephthalate (PET) formation consists of several side reactions in addition to the main ester interchange, transesterification, and polycondensation reactions. The side reactions considered in this work are acid end group, acetaldehyde, diethylene glycol, water, and vinyl end group formations. The objective function of the batch esterinterchange reactor is assumed to consist of maximizing the conversion and simultaneously minimizing the formation of side products. The control vector iteration procedure has been used to optimize the esterinterchange reactor and the temperature-time profile that gives the best performance has been found. It is found that the reactor should be operated at a high temperature initially to obtain high conversion of dimethyl terephthalate (DMT) first, but then it should be lowered to reduce the formation of side products.     

A multi-thread parallel computation method for dynamic simulation of molecular weight distribution of multisite polymerization

Molecular weight distribution (MWD) is an important quality index of polymer products. Many methods have been proposed to dynamically simulate the MWD of polymerization, but these methods are normally designed for serial computations. In this paper, a multi-thread parallel computation method was proposed for multisite free-radical polymerization. Analysis of the relationship among different subtasks revealed a combined parallel strategy by fully exploiting the parallel feature of the process. A good performance was obtained to accelerate the dynamic simulation of MWD based on Flory method. We theoretically analyzed the speedup ratio (SR) and parallel efficiency (PE). Results showed that software algorithm and hardware configuration exhibited a good match. The efficiency of the proposed parallel method was presented through industrial slurry processes that used high-density polyethylene (HDPE).     

Molecular weight distribution in low-density polyethylene polymerization; impact of scission mechanisms in the case of a tubular reactor

It has been shown that scission kinetics strongly affects the molecular weight distribution (MWD) of low-density polyethylene (ldPE) in a continuous stirred tank reactor (CSTR). The present paper focuses on the effects of different chemical scission mechanisms, linear and topological scission, as well as mechanical scission on MWD in batch and tubular reactors. In contrast to the CSTR, a batch reactor MWD does not show bimodalities or long tails. The tubular reactor was modeled as an industrially representative system with four initiator injection points and a proper ‘cocktail’ of different initiators. Calculated MWD was compared to one experimentally determined with SEC-MALLS for a commercial tubular product and fair agreement was found. Typically, these MWDs are broad, but not bimodal. Sensitivity studies were performed as to scission kinetics and the effect of chain transfer agent (CTA). Both batch and tubular reactor were observed to be less sensitive to scission kinetics than a CSTR. In addition, alternative CTA injection strategies (downstream positions) were tested. These showed interesting behavior leading to very broad and bimodal MWD. The model allows following the MWD broadening along the tube. We conclude that batch and tubular ldPE reactors lead to completely different MWD behavior than a CSTR and that it is possible to manipulate it to a great extent.     

聚酯装置酯化生产过程动态模拟

动态模型是进行生产过程动态优化的基础。本文采用链段法建立了聚酯装置酯化生产过程反应器和工艺塔相互影响的动态模型,分析了基本控制系统作用下过程操作工况的动态阶跃响应特性。模拟结果表明,酯化反应器进料摩尔配比及反应器温度、压力、液位的调整显著影响了酯化过程气相流的变化,且对反应产物中端羧基含量和聚合度等指标的响应比较灵敏;控制系统对稳定酯化生产过程操作起着显著的作用。