An important feature of feedback systems is their ability to maintain stability for a given set of conditions. In reaction engineering such systems include, among others, autothermal reactors and reactors with recycle. In the former case the feedback is provided by the recycle of heat produced in the system, while in the latter case - by recycling both heat and reacted mass. The set of conditions is determined by the steady-state parameters, defined in turn by the state of feed and the initial conditions. It follows from the control engineering theory that a dynamic feedback system may be either stable or unstable, depending on the values of these parameters. The quantities of crucial importance are the feedback ratio and the rate of generation of heat and mass. Thus, the problem arises of deriving an efficient method for analysing the stability of the systems discussed. If we deal with tubular reactors described by partial differential balance equations with boundary conditions, the problem of the stability of the solutions to these equations is by no means trivial. It is further complicated by the fact that the solutions may not be unique, leading to multiple states of a reactor [1-3]. The present work concerns an adiabatic, homogeneous tubular recycle reactor with plug flow (Fig. 1). A single reaction is assumed of any rate expression. The work is an extension and generalization of papers [1] and [5], and presents a more detailed analysis of the mathematical model discussed. It consists of two parts. The first, based on Lapunov's method, provides an analytical condition for the local asymptotic stability of the reactor (inequality (60)). By tabulating this inequality the regions of stable and unstable operation are determined. A representative bifurcation diagram, which includes multiple states, is given in Fig. 2. The second part of the paper deals with the analysis of the global asymptotic stability of the reactor. A method for examining global stability is closely related to the previous analysis: if a given stationary point is locally asymptotically stable, it is necessary to know the depth of this stability, i.e., to evaluate the maximum deviation of a state variable from the steady state, which still enables the asymptotic return of the system to this state. Based on Popov's method [6, 12] an analytical condition is derived for the global asymptotic stability of the reactor (inequality (75)). By tabulating this inequality the regions of initial conditions are obtained which define both stable and unstable solutions to the balance equations. In the present analysis Lapunov's second method has not been used since, unlike Popov's method, it requires the definition of the function in full. As the choice of this function is usually fortuitous, only partial stability region for the system analysed may be obtained. Popov's method, if only it may be employed, is free of these restrictions and yields a maximum region of stable solutions of the model. It should be stressed that the problem of stability of recycle reactors has been dealt with in the literature for a long time (cf. [7-11]). A simple method for examining stability has, however, been lacking; the methods proposed so far have required the use of tedious numerical iterative procedures and concerned exclusively local stability. The methods developed in the present paper are straightforward, applicable to single reactions of any rate expression and, most important, concern the global stability of a steady state analysed.
