Liapunov Stability of Parabolic Distributed Systems: The Catalyst Particle Problem

Murphy, Eoin; Crandall, Edward D.

doi:10.1115/1.3424985

Cited by 8 publications

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“…For the problem of stability and uniqueness of the steady state of a chemical reaction occurring in a catalyst particle with slab geometry, several sufficient conditions for stability have been developed for both cases of unity and nonunity Lewis numbers in terms of the system parameters and steady state profiles, the system parameters and the catalyst center temperature, and the system parameters alone. These conditions are shown to be stronger than the previously reported results (Murphy and Crandall, 1970). Sufficient conditions for uniqueness have also been developed.…”

mentioning

confidence: 67%

“…The net result is that this is equivalent to Case I11 in which a new state variable u3 was used and two parameters were optimized, that is, the same Liapunov function is obtained when the parameters are set to their optimal values. Condition (26) is shown in Appendix D to be less conservative than Condition (25), the result of Murphy and Crandall (1970). This is due to the introduction and optimization of k2 to obtain the least conservative result.…”

Section: Stability Conditions For Catalyst Particlementioning

confidence: 91%

“…A limited version in which A2 is a diagonal positive definite matrix was presented by Murphy and Crandall (1970). The difficulty in the application of this theorem is usually found in the search for the weighting matrix S ( x ) .…”

Section: Ll Apu Nov Stab I Llty Analysismentioning

confidence: 99%

“…This problem has been studied by Wei (1965), Berger and Lapidus (1968), and Murphy and Crandall (1970) who modeled the catalyst particle as a slab and applied the Liapunov functional technique. This problem has also been considered by Kuo and Amundson (1967) using the comparison theorem and Sturm's oscillation theorem for the case of equal mass and heat diffusion (unity Lewis number).…”

Section: Scopementioning

confidence: 99%

“…The uniqueness criterion presented here for the more general case of the nonunity Lewis number, Condition (35), is a new result previously not available in the literature. Also, the stability criterion for nonunity Lewis number presented here, Condition (26), is less conservative than the stability criterion previously developed by Murphy and Crandall (1970), that is, Condition (25).The relationship between the degree of conservatism of the sufficient conditions for stability and the number of March, 1973 …”

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confidence: 91%

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Stability and uniqueness of a catalyst particle problem: Parameter optimization in Liapunov functions

1973

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In the analysis of stability and uniqueness of steady state through the Liapunov functional technique, the Liapunov functional is not unique and different forms can lead to significantly different results. A method is proposed in which the state vector and/or the parameters in the weighting matrix S ( x ) are optimized to obtain less conservative results than those reported previously. For the problem of stability and uniqueness of the steady state of a chemical reaction occurring in a catalyst particle with slab geometry, several sufficient conditions for stability have been developed for both cases of unity and nonunity Lewis numbers in terms of the system parameters and steady state profiles, the system parameters and the catalyst center temperature, and the system parameters alone. These conditions are shown to be stronger than the previously reported results (Murphy and Crandall, 1970). Sufficient conditions for uniqueness have also been developed. For unity Lewis number our uniqueness condition is stronger than the prevfously reported result (Luss and Amundson, 1967), and for non-unity Lewis number our uniqueness result is new. CHING-TIEN LlOU SCOPEUnder certain conditions multiple steady states are possible for a chemical reaction occurring in a catalyst particle with slab geometry (Weisz and Hicks, 1962;Amundson and Raymond, 1965). When this occurs at least one of the steady states is shown to be unstable (Gavalas, 1968). Since many industrially important reactors are packed with catalysts, the study of the stability and uniqueness plays an important role in chemical reactor design and operation.The desirable results are criteria in terms of design and operating parameters, which will predict a single stable steady state and also predict which one of steady states is stable when multiple steady states are possible.This problem has been studied by Wei (1965), Berger and Lapidus (1968), and Murphy and Crandall (1970) who modeled the catalyst particle as a slab and applied the Liapunov functional technique. This problem has also been considered by Kuo and Amundson (1967) using the comparison theorem and Sturm's oscillation theorem for the case of equal mass and heat diffusion (unity Lewis number). Lee and Luss (1968) used Galerkin's method to determine the local stability character of the steady states of spherical catalyst particles. Conditions for existence of a single steady state, that is, uniqueness conditions, have been developed by Gavalas (1966), Amundson (1963, andLuss (1968) by using topological methods.One advantage of the Liapunov functional approach is that the Liapunov functional is not unique, and different forms can lead to significantly less conservative results than those currently available. In this paper this is accomplished for the problem of the catalyst particle with slab geometry by a judicious choice of state variables and optimizing the parameters in the Liapunov function. CONCLUSIONS AND SIGNIFICANCEThe Liapunov stability technique has been applied to a class of distributed para...

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mentioning

confidence: 67%

Section: Stability Conditions For Catalyst Particlementioning

confidence: 91%

Section: Ll Apu Nov Stab I Llty Analysismentioning

confidence: 99%

Section: Scopementioning

confidence: 99%

mentioning

confidence: 91%

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Stability and uniqueness of a catalyst particle problem: Parameter optimization in Liapunov functions

1973

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Local stability of tubular reactors

Clough¹,

Ramirez²

1972

AIChE Journal

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Liapunov analysis techniques employing a general quadratic functional are used to derive stability conditions for tubular reactor systems. The adiabatic tubular reactor without axial dispersion is shown to be locally stable, which excludes the possibility of multiple steady states, and the reactor with axial dispersion is proven locally stable if a condition involving only system parameters and steady state values is satisfied. Peclet numbers for heat and mass transfer are not specified equal for the latter proof.Results of simulation studies are used to confirm the validity of the derived stability condition, and it is shown that the parametric region of multiplicity is quite well defined. For the nonlinear equations, single steady state cases appear to possess nonuniform stability.This paper investigates the stability of tubular reactors using both theoretical and computational techniques. Liapunov's direct method employing a functional based on a general quadratic norm is used as the theoretical method. The theoretical analysis yields through straightforward algebraic manipulation stability results without the need for a solution of the partial differential equations describing the system. Dynamic simulations were computationally carried out by finite differencing the describing equations and using the quasilinearization technique.In the literature much attention has recently been devoted to stability studies of distributed parameter systems. Murphy and Crandall (19) applied Liapunov stability theory to the catalyst particle problem using a theorem somewhat similar to that to be presented here. Their analysis yielded stability conditions depending on system parameters and steady state values. Han and Meyer ( 1 5 ) used bounded-input-bounded-output concepts to analyze the stability of a nondispersive, catalytic bed reactor. They obtained a stability condition based on discretized nonlinear equations which was different from previous results based on linearized equations. Other work has been done on nondispersive models, and an example is the recent article by Agnew and Narsimhan (1). that is, positive away from a given steady state and zero only at the steady state is specified, and stability is ensured if it can be shown that the time derivative of this function is negative definite. This time derivative involves time derivatives of the state variables, and these are conveniently available in the system differential equations.Liapunov's first method, according to Bertram and Kalman ( 7 ) , "deals with stability questions via an explicit representation of the solutions of a differential equation," whereas Aris ( 4 ) is more specific in stating, "His so-called first method shows how the process of linearization about the steady state may be justified and exploited."The difficulty in the application of the second method is usually found in the search for the Liapunov function. An inverse formulation involving first the specification of a negative definite Liapunov function derivative has been widely used for line...

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To the editor: Stability of tubular reactors

Soliman

McGreavy

1972

AIChE Journal

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Liapunov Stability of Parabolic Distributed Systems: The Catalyst Particle Problem

Cited by 8 publications

References 0 publications

Stability and uniqueness of a catalyst particle problem: Parameter optimization in Liapunov functions

Stability and uniqueness of a catalyst particle problem: Parameter optimization in Liapunov functions

Local stability of tubular reactors

To the editor: Stability of tubular reactors

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