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

Abstract

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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“…In anticipation, this vector is chosen to possess two elements expressing excursion from the steady-state temperature and concentration profile in terms of dimensionless variables defined in Eq. [4]. Consequently, the elements of the perturbation vector are w~(% w -= 0(% 8) -0~ (8) [10a] and w.,(T, w -= F(~, 8) -F~ (8) [10b]…”

“…It then follows that any positive descending function in the [0,1] domain is acceptable: e-M~; M > 0, Ne-M~; M > 0, N > 0; M cos "M2, (1 -w M > 0 are obvious candidates. The choice of an exponential function is perhaps the most convenient, as indicated by Weigand et al (4,5), but there is no physical significance attached to the sl (w and s2 (w functions.…”

“…The excitation mode, either forward (i) or reverse (ii) polarization, is determinant for the EL characteristics of the ZnO/electrolyte junction. II-VI compounds are electroluminescent (EL) materials whose properties have been extensively studied in solidstate junctions called light emitting diodes (LED's) (1)(2)(3)(4). Most of these devices are based on metal-semiconductor (MS) or metal-insulator-semiconductor (MIS) junctions since self-compensation effects usually impede the realization of p-n homojunctions.…”

“…Berger and Lapidus (3) considered the stability of distributed systems with single and multiple equilibrium profiles using the adiabatic catalyst particle and an empty tubular reactor with axial diffusion as working examples. Liou, Lim, and Weigand (4,5) refined the stability-of-a-catalyst-particle problem in terms of parametrically optimized Liapunov functions and extended the technique to nonadiabatic tubular reactors with axial mixing and recycle. A general theoretical framework for the stability analysis of distributedparameter-type chemical reaction systems has also been presented (6).…”