# Global analysis of the multiplicity features of multi-reaction lumped-parameter systems

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.

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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 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: The literature on the stirred tank reactor with consecutive reactions is reviewed and a census of multiplicity diagrams, bifurcation diagrams and phase portraits is given in this article, where the location of the organizing centre, introduced by Balakotaiah and Luss, and some features of the butterfly singularity are discussed with reference to variations in the ratio of activation energies.

Abstract: The literature on the stirred tank reactor with consecutive reactions is reviewed and a census of multiplicity diagrams, bifurcation diagrams and phase portraits is given. The location of the organizing centre, introduced by Balakotaiah and Luss, and some features of the butterfly singularity are discu particularly with reference to variations in the ratio of activation energies. The forms that the regions of attraction of the various steady states ma take in the three-dimensional state space are also discussed.

82 citations

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TL;DR: In this paper, a simplified model of fire growth in a compartment and a preliminary analysis of the dynamics exhibited, in terms of both transient simulations and quasi-steady evolution manifolds, for variations in controlling parameters.

Abstract: Flashover is a phenomenon whereby a room fire undergoes a rapid increase in size and intensity. Such a transition is suggestive of a nonlinear process. We therefore seek to apply modern geometrical and computational techniques of nonlinear dynamics to a simplified model of fire growth to investigate flashover and other instabilities occuring in compartment fires. We present here a simplified model of fire growth in a compartment and conduct a preliminary analysis of the dynamics exhibited, in terms of both transient simulations and quasi-steady evolution manifolds, for variations in controlling parameters.

68 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 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

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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.

135 citations

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TL;DR: In this article, a catastrophe theory-implicit function theorem approach was proposed for the determination of regions of unique and multiple solutions to certain nonlinear equations via a catastrophe theoretic approach, which yields exact, explicit bounds for all n ≥ 0.

Abstract: The development of a simple, generalized technique for the exact determination of regions of unique and multiple solutions to certain nonlinear equations via a catastrophe theory-implicit function theorem approach, is presented The application of this technique to the nth order chemical reaction in the nonadiabatic and adiabatic CSTR yields exact, explicit bounds for all n ≥ 0 To our knowledge, this is the first report of exact, explicit bounds for these systems, except for n = 0, 1 for the adiabatic CSTR, and n = 1 for the nonadiabatic CSTR For the nonadiabatic CSTR, these bounds show that the higher the reaction order, the smaller the region in parameter space for which multiplicity can occur for all γ and x2c, (dimensionless activation energy and coolant temperature, respectively) This behavior is similar to that reported by Van den Bosch and Luss[1] for the adiabatic CSTR The zeroth order reaction in the nonadiabatic CSTR exhibits more complex behavior and assumes characteristics of both high and low reaction orders insofar as increasing and/or decreasing the uniqueness space, in comparison to all other n > 0 An exact implicit bound between regions of uniqueness and multiplicity is also derived for the nth order reaction in a catalyst particle with an intraparticle concentration gradient and uniform temperature, and is fully demonstrated for the first order reaction In addition, explicit criteria, sufficient for uniqueness and multiplicity of the catalyst particle steady state, stronger than those of Van den Bosch and Luss, are also developed by combining the present technique with bounds suggested by these authors

55 citations

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TL;DR: In this paper, a simple calculation of steady states is suggested based on solution of a single transcendental equation for a continuous flow stirred tank reactor with a consecutive first order exothermic reaction.

Abstract: Heat and mass balances are written in dimensionless form describing transient behaviour of a continuous flow stirred tank reactor with a consecutive first order exothermic reaction. A simple calculation of steady states is suggested based on solution of a single transcendental equation. In the parametric plane Damkohler number Da — adiabatic temperature rise B regions involving information on both multiplicity and stability of the steady states are drawn. In view of this analysis a systematic classification of the steady state behaviour may be performed. A projection of the three-dimensional phase space on a particular plane is adopted for visualization of trajectories in the phase space. The region of occurrence of limit cycles is examined in detail. Analogously to a two-dimensional case an asymptotically stable steady state exist with a corresponding orbitally stable limit cycle. The procedures for evaluation of the shape of separatrices are proposed.

32 citations