# Multiplicity features of reacting systems. Dependence of the steady-states of a CSTR on the residence time

TL;DR: Singularity theory with a distinguished parameter, as developed by Golubitsky and Schaeffer, is a very useful tool for predicting the influence of changes in a control or design variable on the steady-state of lumped-parameter systems as mentioned in this paper.

Abstract: Singularity theory with a distinguished parameter, as developed by Golubitsky and Schaeffer, is a very useful tool for predicting the influence of changes in a control or design variable on the steady-states of lumped-parameter systems. The theory is used to construct various bifurcation diagrams describing the influence of changes in the residence time on the temperature in a CSTR in which several reactions occur simultaneously. The number of different bifurcation diagrams increases very rapidly with increasing number of reactions. The predictions of this local theory provide important theoretical guidance in the global analysis of the multiplicity features.

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TL;DR: In this paper, a review brings attention to much of the recent literature describing these developments and their applications, particularly regarding complex and chaotic oscillations, as well as reported experimental observations which, in many cases, have provided support for prior theory and in many others, have led to new theoretical developments.

Abstract: Regarded less than two decades ago as the province of theoreticians and understood mainly in terms of a single-step, exothermic, homogeneous reaction in a well-stirred vessel, the description of steady-state multiplicities and intrinsic dynamics of chemically reacting systems has become increasingly experimental, broad, rich and complex. Through the past decade or so researchers, investigating systems which involve varying degrees of physical and chemical complexity, have (1) made use of catastrophe and singularity theories to produce a systematic representation and new understanding of the steady state and of instabilities and self-sustained oscillations in certain instances, and (2) reported experimental observations which, in many cases, have provided support for prior theory and, in many others, have led to new theoretical developments, particularly regarding complex and chaotic oscillations. This review brings attention to much of the recent literature describing these developments and their applications.

172 citations

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TL;DR: In this paper, a systematic, efficient scheme is presented for finding parameter values corresponding to a specific number of solutions. But this scheme is not suitable for large-scale systems, where many chemical reactions occur simultaneously with a large number of parameters.

Abstract: Mathematical models of lumped-parameter systems in which many chemical reactions occur simultaneously contain a large number of parameters, so that a p Theoretical guidance is needed to determine all the multiplicity features and the corresponding parameter regions. A systematic, efficient scheme is presented for finding parameter values corresponding to a specific number of solutions. A new scheme is developed for bifurcation diagrams, which describe the dependence of a state variable on a slowly changing operating variable. Some general predictions are made abou systems. Bounds on the values of the bifurcation or state variable may create bifurcation diagrams which cannot be found close to the highest order sin of solutions even when an isola variety does not exist. Several examples illustrate the application of the mathematical techniques.

157 citations

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TL;DR: In this paper, a technique based on the bifurcation theory and the shooting algorithm is developed, which permits fast and efficient tracking of bifurbation in these systems, and it is predicted that the homogeneous ignition temperature is higher with surface reaction than without surface reaction.

Abstract: Bifurcation analysis of ignition and extinction of combustion in stagnation-point flow was carried out for conditions when both homogeneous and heterogeneous reactions can occur. A technique based on the bifurcation theory and the shooting algorithm is developed, which permits fast and efficient tracking of bifurcation in these systems. The influences of the different parameters on the ignition and extinction behavior were investigated by assuming (1) only catalytic surface (heterogeneous) reaction, (2) only homogeneous reaction on a hot inert surface, and (3) both surface and homogeneous reactions. The coupling effects of the homogeneous and heterogeneous reactions are clearly demonstrated. It is shown that the heterogeneous reaction dominates the system behavior at the lower temperature while both homogeneous and heterogeneous reactions play important role at higher temperature. It is predicted that the homogeneous ignition temperature is higher with surface reaction than without surface reaction. However, homogeneous-heterogeneous reactions expand the stabilized operating regions with high reaction rates compared with either heterogeneous or homogeneous reaction alone. The numerical results for propane and for methane oxidation on platinum foil also show good qualitative agreement with the experimental results of Part I of this article, which include the two types of ignition, extinction and autothermal behavior of homogeneous-heterogeneous reactions.

109 citations

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TL;DR: In this paper, the effect of process parameters such as the cooling jacket flow rate, heat transfer coefficient, heat of reaction, and cooling jacket feed temperature on the steady-state multiplicity of the three-state CSTR model is investigated.

Abstract: Research on exothermic reactor operation has been based mostly on the classic two-state continuous stirred tank reactor model, implicitly assuming that the cooling jacket temperature dynamics are negligible In this case, the cooling jacket temperature is the manipulated input instead of the cooling jacket flow rate for feedback control of reactor temperature Adding a cooling jacket energy balance results in much more complex behavior than a simple lag effect A stabilizing inner-loop cascade controller is assumed in the two-state CSTR model, because the three-state model incorporating cooling jacket temperature dynamics may be open-loop unstable when the two-state model is open-loop stable The influence of design parameters on the multiplicity behavior of a three-state model is considered Elementary catastrophe theory is used to study the effect of process parameters such as the cooling jacket flow rate, heat-transfer coefficient, heat of reaction, and cooling jacket feed temperature on the steady-state multiplicity of the three-state model This multiplicity analysis is particularly relevant for control because the primary bifurcation parameter is the cooling jacket flow rate, the manipulated input for feedback control in the three-state model This multiplicity analysis guides improvements in process design and/or operation to eliminate difficult operating regions associated with steady-state multiplicities; the presence of multiple steady states results in safety and operation problems due to ignition/extinction phenomena Reactor scale-up affects the presence of these infeasible reactor operating regions Certain design parameter changes that remove multiplicities in the two-state model cannot remove multiplicities in the three-state model

108 citations

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TL;DR: In this paper, the Damkohler number (Da), the ratio of a characteristic residence time to a characteristic reaction time, is used to describe the effect of kinetics on solution multiplicity.

Abstract: A model is developed to describe kinetic effects in reactive distillation. A Damkohler number (Da), the ratio of a characteristic residence time to a characteristic reaction time, is the key parameter. A comprehensive picture of solutions is found with arc length continuation methods that trace solution branches as functions of Da, reboil ratio, or reflux ratio. For methyl tert-butyl ether (MTBE) synthesis in the Jacobs–Krishna column configuration, the model reproduces known solution multiplicities in the limit of reaction equilibrium. In addition to these previously known equilibrium solutions, we report new results concerning the effect of kinetics on solution multiplicity. In particular, isolated solutions branches are found at several values of Da; the isola shrinks and disappears below a critical value of Da. Results for tert-amyl methyl ether (TAME) synthesis in a model for the test column of Mohl et al. [Chem. Eng. Sci. 54 (1999) 1029] are in sharp contrast; multiplicities present in the kinetic regime disappear above a critical value of Da for all values of reflux ratio.

90 citations

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TL;DR: In this paper, the authors apply the theory of singularities of differentiable mappings (SOMD) to study the effect of imperfections in a system subject to bifurcation.

Abstract: : This paper applies the theory of singularities of differentiable mappings - specifically the unfolding theorem - to study the effect of imperfections in a system subject to bifurcation. In a number of special cases we have classified (up to a suitable equivalence) all the possible perturbations of the bifurcation equations by a finite number of imperfection parameters. These cases include both bifurcation from a double eigenvalue and from a simple eigenvalue degenerate in the sense of Crandall-Rabinowitz.

299 citations

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TL;DR: In this article, the dynamic behavior of the continuous stirred tank reactor is analyzed and classified for a variable reactor residence time, and plots are given to show the influence of system parameters on the reactor behavior.

Abstract: The dynamic behavior of the continuous stirred tank reactor is analysed and classified for a variable reactor residence time. Although earlier work, treating the bifurcation to limit cycles and steady states with changing Damkohler number, yields a complete description of the problem, the evolution of multiple steady states and limit cycles is much more bizarre as the reactor residence time varies. In addition, it is the reactor residence time which is most easily varied experimentally so that the present results are more readily compared with experiment. Plots are given to show the influence of system parameters on the reactor behavior.

241 citations

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TL;DR: In this paper, the maximal number of steady-state solutions of a lumped parameter system in which several chemical reactions occur simultaneously is determined, and the method can predict also the different types of diagrams describing the dependence of a state variable of the reactor on a design or operating variable.

Abstract: A new, powerful mathematical technique enables a systematic determination of the maximal number of steady-state solutions of lumped parameter systems in which several chemical reactions occur simultaneously. The method can predict also the different types of diagrams describing the dependence of a state variable of the reactor on a design or operating variable. The technique is applied to several reaction networks giving new results and insight. For example, it is proven that when N independent, parallel exothermic reactions with equal and high activation energies occur in a CSTR there exist N ! distinct regions of parameters in each of which 2 N + 1 steady-state solutions exist.

150 citations