The dynamic behavior of continuous polymerization reactors—I: Isothermal solution polymerization in a CSTR

A mathematical model having no adjustable parameters is developed and compared with both dynamic and steady-states” in the neighborhood of unstable middle steady states is thought to be due to a combination of methacrylate with ethyl acetate solvent, in a CSTR, was shown to have ignition and hysteresis behavior characteristic of multiple steady states for sufficiently low solvent concentrations. The observation of “pseudosteady-states” in the neighborhood of unstable middle steady states is thought to be due to a combination of kinetic and hydrodynamic mechanisms.     

The dynamic behaviour of continuous polymerization reactors—III: An experimental study of multiple steady states in solution polymerization

A study involving small scale reactor experiments coupled with detailed non-isothermal reactor modelling has been carried out shows that ‘isoia” ty     

The dynamic behavior of continuous polymerization reactors—II Nonisothermal solution homopolymerization and copolymerization in a CSTR

The possible types of dynamic behavior have been determined for a CSTR in which bulk or solution free radical polymerization is carried out. To illustrate the methodology, the expected behavior of reactors for vinyl acetate or methylmethacrylate homopolymerization and vinyl acetate-methylmethacrylate copolymerization has been explicitly determined. The monomer feed ratios, solvent concentrations, and reactor cooling capacity which allow multiple steady states and limit cycles have been carefully mapped.     

Theoretical investigations of dynamic behavior of isothermal continuous stirred tank biological reactors

The dynamic behavior of an isothermal biological CSTR modelled by idealized cell and substrate balance equations has been investigated theoretically in terms of multiplicity and stability of steady states and existence and stability character of limit cycles. Various types of dynamic behavior have been classified in terms of a Damköhler number and two other system parameters. The predicted types of behavior have been illustrated by numerical computation of cell and substrate concentration trajectories.     

A two-parameter model for partial segregation: Application to flow reactors with pre- and unmixed feed

A two-parameter (TP) mixing model for partial segregation has been applied to continuous stirred tank reactors (CSTR) for cases with premixed and unmixed feed. It was found that in the case of perfectly mixed, homogeneous feed the model predicts similar or better results than earlier one-paramete models. However, a significant advantage of the TP model was found in the case of unmixed feed. Its two parameters are suitable for representing two se situations existing in the case of unmixed feed, namely, imperfect mixing of the feed and partial segregation of the reacting mixture. The single-param models cannot be used to represent both phenomena. Reversible and irreversible reactions were studied, and deviations from a perfectly mixed CSTR were into account by considering the RTD expression of Wolf and Resnick. The parameters of the model were determined from experimental data, when available, from arbitrary values within their permissible range, otherwise. Good agreement was found between experimental and predicted conversions.     

Markov chain models of complex chemical reactions in continuous flow reactors

A stochastic approach, namely, a Markov chain, has been adopted to simulate the dynamics of complex reactions in a flow chemical reactor without either solving directly differential equations governing the performance of the reactor or obtaining a closed form solution to such equations. All calculations have been carried out iteratively with the aid of a computer. The results are in good agreement with the known results obtained from the deterministic approach. The present technique can be applied to a variety of flow reactors with complex chemical reactions, which cannot be handled easily by deterministic approaches.     

Periodic control of continuous stirred tank reactors—II Cases of a nonisothermal single reactor

The optimal periodic control of a single nonisothermal CSTR in which a single reaction, a parallel reaction or a consecutive reaction is taking place is studied. Both the feed rate and the heat transfer between the reactor content and the immersed coil are periodically controlled. Utility consumption and conversion of the raw material into the waste are included in the performance index together with conversion into the desired product. The results indicate a respectable improvement in the reactor performance.     

The performance of continuous cyclic ion-exchange reactors—II: Reactions with intraparticle diffusion controlled kinetics

This work extends the previous experimental and theoretical study of the Cloete-Streat stage-wise solid-liquid contactor to the case of high liquid concentrations where the rate of ion exchange is limited by the kinetics of intraparticle diffusion. In all cases, the reaction studied was neutralizat fo NaOH with a cationic exchange resin. Experiments were carried out in a three-stage system and simulations using literature values of the kinetic parameters and a population-based model of this system gave excellent agreement (within 27%) with the experimental results.It was found that small improvements beyond steady-state performance could be obtained by varying the fraction of solids periodically transferred, keeping flow rates and mean residence times of both phases constant. However, if solids conversion is high (～90%), as was unavoidable in these experiments, the scope for such improvement in performance decreases and any inefficiency due to the method of addition of solids can actually lead to a small decrease in the overall performance when compared to that of steady-state.     

The performance of continuous,cyclic ion-exchange reactors—I: Reactions with film diffusion controlled kinetics

This work is an experimental and theoretical study performed on the Cloete-Streat stage-wise semicontinuous solid-liquid contactor in which counter-cur flow of solids and liquid is achieved by a periodic transfer of solids from stage to stage. The reacting system investigated was the neutralization of NaOH with a cation exchange resin.Part I reports on a range of operating conditions where film diffusion is the controlling resistance for mass transfer and Part II pertains to the case of intraparticle diffusion control.The effect of periodic flow on the performance of the contactor was studied by varying the fraction of solids in a stage periodically transferred keeping flow rates and mean residence times of solid and liquid constant. The system performance improved monotonically with higher values of fractiona transfer up to 0.9; specifically, the residual ion concentration in the liquid leaving the system could be reduced by 35% when compared with the same system operating under conditions approaching a continuous steady-state. Viewed differently, the separation typically required 50% fewer stages than the same separation done under steady-state conditions.The behaviour of the system was simulated by a population-based model for the solid particles and assuming perfect macro-mixing in both phases. The res     

Microporous membrane reactors

Microporous membranes were fabricated from nickel powders of nearly uniform particles having mean diameters varying from 5 to 98 μm. These membranes$\text{?k}\text{aD}$=0.14 ($\text{caseυ}\text{aD}$)^0.5 for v/D = 1270 and 20 ? (v/aD) ≤ 500. This is in agreement with correlations obtained     

Steady state multiplicity depending on coalescence in liquid—liquid continuous stirred reactors with reaction in the dispersed phase

The influence of drop coalescence and breakup on the existence of multiple steady states is studied for a two-phase stirred isothermal reactor where the chemical reaction in the d?ispersed phase obeys the rate expression ? r = kC/(1 + KC)². The random coalescence model developed by Curl was simulated using a modified Spielman and Levenspiel Monte Carlo technique.For certain range of the coalescence rate, Damköhler number, and dimensionless feed concentration, multiple steady states have been investigated.A special case has also been considered wherein the existence of multiple steady states for finite values of the coalescence rate is contrasted to the unique steady state solution for an infinite coalescence rate.     

Periodic control of continuous stirred tank reactors—I: The pi criterion and its applications to isothermal cases

The pi criterion which is useful for examining if the optimal periodic control is proper is briefly introduced in a somewhat generalized form and then it is applied to optimal periodic control problems of CSTR's in isothermal condition.     

Minimum end time policies for batchwise radical chain polymerization—II A two-stage process for styrene polymerization

For styrene polymerization, a two-stage process is considered. The first stage, initiated by initiator, is operated along the best isothermal policy for a predetermined number average molecular weight and monomer conversion at end of the stage as described in the previous paper. The second stage is operated at higher temperature level at which thermal initiation polymerization is significant. A mathematical model is proposed for describing the changes of conversion and number average molecular weight with time in the second stage based on the experimental data obtained. All parameters involved in the model are calculable using the kinetic constants of the initiator initiation system and the thermal initiation system. Experimental verification on the optimal temperature policies for the second stage shows promising. Further calculations indicate that the optimal two-stage process can be significantly better than the optimal process of initiator initiation or thermal initiation polymerization.     

A heterogeneous one-dimensional model for non-adiabatic fixed bed catalytic reactors

A heterogeneous one dimensional model which takes separately into account the heat transfer through the solid and fluid phases is introduced. The response of this model is compared with that of the heterogeneous two-dimensional model, and a very good agreement, especially at mild conditions, is obtained. Its performance is better than previous one dimensional models and the deviations are explained in tenns of the operative conditions and of the value of dimensionless groups. Conditions at which the different one and two-dimensional models are recommended to be used are presented.     

Modeling of fixed bed catalytic reactors

The simulation of a fixed-bed catalytic reactor requires the selection of a model, which is a set of balance equations that describes the reactor, as well as correlations for the model parameters involved. In this work general criteria, leading to a better choice of a model that fulfills the objectives of the simulation, are established. Different ways in which the parameters can be obtained are analyzed, and the numerical methods for solving the model equations are discussed.     

Transient response of three-phase slurry reactors

Analytical time domain solutions are provided for the transient response of three-phase slurry reactors, with diffusion and first-order irreversible ch input of the reactant in the inlet gas stream; and (b) when the catalyst particles are suddenly introduced into the reactor which has otherwise reached obtained for the general case of continuous slurry reactors but can be easily reduced for application to semi-batch slurry reactors in which there is n and are expected to be useful for evaluating the rate and equilibrium constants from dynamic response experiments.Asymptotic solutions have also been obtained for the case of large Thiele parameter (φ ≥ 3) and some other simpler expressions have been derived as only for gaseous reactants of moderate or high solubility in the liquid phase and are not applicable to those with low solubility.     

The bifurcation behavior of tubular reactors

Methods for studying the bifurcation behavior of tubular reactors have been developed. This involves the application of static and Hopf bifurcation theory for PDE's and the very precise determination of steady state profiles. Practical computational methods for carrying out this analysis are discussed in some detail. For the special case of a first order, irreversible reaction in a tubular reactor with axial dispersion, the bifurcation behavior is classified and summarized in parameter space plots. In particular the influence of the Lewis and Peclet numbers is investigated. It is shown that oscillations due to interaction of dispersion and reaction effects should not exist in fixed bed reactors and moreover, should only occur in very short “empty” tubular reactors. The parameter study not only brings together previously published examples of multiple and periodic solutions but also reveals a hitherto undiscovered wealth of bifurcation structures. Sixteen of these structures, which come about by combinations of as many as four bifurcations to multiple steady states and four bifurcations to periodic solutions, are illustrated with numerical examples. Although the analysis is based on the pseudohomogeneous axial dispersion model, it can readily be applied to other reaction diffusion equations such as the general two phase models for fixed bed reactors.     

Free radical polymerization engineering—I: A new method for modeling free radical homogeneous polymerization reactions

This paper deals with a new appraisal of free radical polymerization modeling. Classical properties of the distribution of polymerization degrees and methods for studying them thanks to the z-transform are recalled. When simple mechanisms are involved, the kinetics may be represented by a “detailed analytic” model relying on macromolecule balances. This is no longer possible when complex processes are occurring, e.g. transfer to polymer, β-scission, terminal double bond propagation… In order to deal with these more complex cases, a “tendency model” is proposed, relying on balances of quantities such as free radicals, macromolecules and the moments of the distribution of polymerization degrees. The quality of the polymer is described by chemical characters such as double bonds, long and short branching points, terminal double bonds… for which kinetic equations are established. Equations are given for calculating the moments of free radicals and those of instantaneously produced macromolecules via various processes. The simplicity and the usefulness of the method are illustrated by several examples and a comparison is made, when possible, with the detailed approach. Finally, equations are given, making it possible to calculate the quality of polymer produced in any kind of reactor and for different states of segregation.     

Multiplicity of conversion in a cascade of imperfectly stirred tank reactors

The multiple steady states of conversion in the model of non-ideal CSTR's in series have been studied for irreversible, homogeneous exothermic reaction of first order. The influences of the parameters p and n and those of some kinetic parameters on the occurrence of multiple conversions for a given residence time are presented. The existence of steady-state multiplicity of conversion in each CSTR is shown to be independent of n and depen largely on the values of E/RT₀ (activation energy), b/T₀ (heat of reaction), and p. For certain parametric values, more than three steady states can exist in such a model.