For a chemical reaction in a porous catalyst particle in the presence of heat and mass diffusion. the stabilitv of steadv state solutions is analysed by extending the two methods of Liapunov to a function space with a suitable metric.
1. INTROD~OTI~N
WHEN a first-order reaction takes place inside a porous catalyst in the presence of heat and mass diffusion, multiple steady-state solutions are sometimes obtained [l--3]. Some of these solutions are not stable with respect to perturbations. This stability problem can not be handled by the two Liapunov methods since the state of the system is not described by a finite number of parameters (YD Y29 *.* y.) such as concentrations and temperatures at different points; it is described, instead, by a function, y(x), representing the temperature and concentration profile inside the catalyst. This is sometimes referred to as a "distributed parameter system." The dimension of the problem is no longer n, but a non-denumerable infinity, and the phaseplane is hard to imagine. The system has an itinite number of degrees of freedom, and a stable system must have a restoring force for each degree of freedom.
An extension of the Liapunov methods to this problem requires us to define a Liapunov functional, u[y(x)], that has the function y(x) as argument and a non-negative number as its value [4]. This functional, u, describes the distance or metric from the function y(x) to the steady-state solution y*(x). If t, has the property of ever-decreasing in a neighborhood of y*(x), then the steady-state solution is stable. When the partial differential equations of the chemical reaction with heat and mass transfer are linearized, one obtains a Sturm-Liouville equation and the stability problems can be examined in detail.