Polymerization reactors mathematical modeling

Keywords polymerization kinetics, polymerization reactors, mathematical modelling, molecular weight distribution (MWD), chemical composition distribution (CCD), Ziegler-Natta catalysts, metallocenes, microstructure, isotacticity distribution, mass transfer resistances, heat transfer resistances, effects of multiple site types. 

Using copolymerization theory and well known phase equilibrium laws a mathematical model is reported for predicting conversions in an emulsion polymerization reactor. The model is demonstrated to accurately predict conversions from the head space vapor compositions during copolymerization reactions for two commercial products. However, it appears that for products with compositions lower than the azeotropic compositions the model becomes semi-empirical. 

A considerable amount of work has been published during the past 20 years on a wide variety of emulsion polymerization and latex problems. A list of 11, mostly recent, general reference books is included at the end of this chapter. Areas in which significant advances have been reported include reaction mechanisms and kinetics, latex characterization and analysis, copolymerization and particle morphology control, reactor mathematical modeling, control of adsorbed and bound surface groups, particle size control reactor parameters. Readers who are interested in a more in-depth study of emulsion polymerization will find extensive literature sources. 

Tosun, G., 1992. A mathematical model of mixing and polymerization in a semibatch stirred tank reactor. American Institution of Chemical Engineers Journal, 38, 425 37. 

Although a dynamic mathematical model of the polymerization system has been developed (17) it is not capable of providing the necessary operating policies for the reactor in order to preselect the time-averaged MWD in the product. Hence the flow policies for the reagents were selected empirically and for experimental convenience. 

Although the papers represent the whole range of kinds of polymers and processes, there are common themes which reveal the dominant concerns of polymerization reactor engineers. Fully half the papers are concerned rather closely with devising and testing mathematical models which enable process variables to be predicted and controlled very precisely. Such models are increasingly demanded for optimization and com-... 

In this paper we present a meaningful analysis of the operation of a batch polymerization reactor in its final stages (i.e. high conversion levels) where MWD broadening is relatively unimportant. The ultimate objective is to minimize the residual monomer concentration as fast as possible, using the time-optimal problem formulation. Isothermal as well as nonisothermal policies are derived based on a mathematical model that also takes depropagation into account. The effect of initiator concentration, initiator half-life and activation energy on optimum temperature and time is studied. 

Mathematical Modeling of Bulk and Solution Polymerization in a Tubular Reactor... 

The derivation and development of a mathematical model which is as general as possible and incorporates detailed knowledge from phenomena operative in emulsion polymerization reactors, its testing phase and its application to latex reactor design, simulation, optimization and control are the objectives of this paper and will be described in what follows. 

The analytical predictor, as well as the other dead-time compensation techniques, requires a mathematical model of the process for implementation. The block diagram of the analytical predictor control strategy, applied to the problem of conversion control in an emulsion polymerization, is illustrated in Figure 2(a). In this application, the current measured values of monomer conversion and initiator feed rate are input into the mathematical model which then calculates the value of conversion T units of time in the future assuming no changes in initiator flow or reactor conditions occur during this time. 

Single Phase Polymerization Mathematical Modelling Ziegler-Natta Polymerization Polymerization Processes (Monograph) Emulsion Polymerization Polymerization Reactions and Reactors Continuous Reactors (ed. volume)... 

In theory, by feeding the MWD and experimental rate data into a mathematical model containing a variety of polymerization mechanisms, it should be possible to find the mechanism which explains all the experimental phenomena and to evaluate any unknown rate constants. As pointed out by Zeman (58), as long as there are more independent experimental observations than rate parameters, the solution should, in principle, be unique. This approach involves critical problems in choice of experiments and in experimental as well as computational techniques. We are not aware of its having yet been successfully employed. The converse— namely, predicting MWD from different reactor types on the basis of mathematical models and kinetic data—has been successfully demonstrated, however, as discussed above. The recent series of interesting papers by Hamielec et al. is a case in point. 

The objective was to develop a model for continuous emulsion polymerization of styrene in tubular reactors which predicts the radial and axial profiles of temperature and concentration, and to verify the model using a 240 ft. long, 1/2 in. OD Stainless Steel Tubular reactor. The mathematical model (solved by numerical techniques on a digital computer and based on Smith-Ewart kinetics) accurately predicts the experimental conversion, except at low conversions. Hiqh soap level (1.0%) and low temperature (less than 70°C) permitted the reactor to perform without plugging, giving a uniform latex of 30% solids and up to 90% conversion, with a particle size of about 1000 K and a molecular weight of about 2 X 10 . 

The detailed course of a polymerization is determined by the nature of the particular reaction as well as by the characteristics of the reactor which is used. The design and control of the operation are greatly aided by mathematical modeling of the process. Such models may be based on empirical relations between the independent and dependent operating variables. This is not as satisfactory, however, as a model that is derived from accurate knowledge of the polymerization process and reactor operation, because only the latter tool permits extrapolation to reaction conditions that have not yet been tried. 

One may use an approach discussed earlier in which a comprehensive mathematical model is developed, which relates the reactor hardware to fluid dynamics and polymerization reactions in one framework. However, it is extremely difficult... 

In the case of polyester synthesis from divinyl esters, hydrolysis of the vinyl end group partly took place, resulting in the limitation of the polymer growth.201 A mathematical model showing the kinetics of the polymerization predicts the product composition. On the basis of these data, a batch-stirred reactor was designed to minimize temperature and mass-transfer effects.202 The efficient enzymatic production of polyesters was achieved using this reactor poly(l,4-butylene adipate) with Mn 2 x 104 was synthesized in 1 h at 60 °C. 

Penlidis, A. Macgregor, J.F. Hamielec, A.E. Mathematical-modeling of emulsion polymerization reactors—a population balance approach and its applications. ACS Symp. Ser. 1986, 313, 219-240. 

Mathematical modeling is a powerful tool not only for the development of process understanding but also for that of the advanced reactor controls in polymerization processes. The modeling techniques for polymerization processes are reasonably well developed and several commercial simulation packages are available. The modeling of heterogeneous polymerizations such as precipitation polymerization and emulsion polymerization remains a challenge. In the past decade, excellent... 

Dube, M.A. Soares, J.B.P. Penlidis, A. Hamielec, A.E. Mathematical modeling of multicomponent chain-growth polymerizations in batch, semibatch, and continuous reactors a review. Ind. Eng. Chem. Res. 1997, 36, 966-1015. 

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