Stability of the steady state

To pursue this question we shall examine the stability of certain steady state solutions of Che above equaclons by the well known technique of linearized stability analysis, which gives a necessary (but noc sufficient) condition for the stability of Che steady state. 

3 ( 0) denote a solution of the steady state balance equaclons, and consider small perturbations about these values, writing 

On subsciCuLlng (12.49) into uhe dynamical equations we may expand each term in powers of the perturbations and retain only terms of the zeroth and first orders. The terms of order zero can then be eliminated by subtracting the steady state equations, and what remains is a set of linear partial differential equations in the perturbations. Thus equations (12.46) and (12.47) yield the following pair of linearized perturbation equations 

If attention is now limited to solutions separable in the space and time variables, we find 

The eigenvalue problem defined by equations (12.56) and (12.37) has been studied by Lee and Luss l79j and, more recently, in considerable detail by Villadsen and Michelsen When - I it is easy to show 

---

The main theoretical methods have in connnon the detemiination of the stability of steady-state or other... 

Despite the very restricted circumstances In which these equations properly describe the dynamical behavior, they are the starting point for almost all the extensive literature on the stability of steady states in catalyst pellets. It is therefore Interesting to examine the case of a binary mixture at the opposite limit, where bulk diffusion controls, to see what form the dynamical equations should take in a coarsely porous pellet. 

As a first indication, the stability of steady-state can be assessed by combining equations (7.2.7) and (7.2.13) into... 

Fig. 11. The effect of reactor volume V on the stability of steady states A, V = 0.93 liter B, V = 2.08 liters C, V = 4.48 liters. One steady state (45). (Copyright by Pergamon Press. Reprinted with permission.)...

These properties characterize both the type and the stability of steady-state points in system (5). If the steady-state point is unique, it is stable. If there are several steady states, then at least one of them is unstable. Stable and unstable steady states alternate. 

The stability of steady states is analyzed [139] like the investigation performed for the three-step mechanism. In stable steady state, the inequality dg(0o)/d0o > df(0o)/ddo is fulfilled. In the unstable steady state, the sign of this inequality reverses. It can easily been shown that the unique steady state is always stable. If there are three steady states, the outer are stable and the middle is unstable. It can be suggested that the addition to the three-step adsorption mechanism of the impact step that is linear with respect to the intermediate does not produce any essential changes in the phase pattern of the system. The only difference is that at k. x = k 2 = 0 the dynamic model corresponding to the two-route mechanism can have only one boundary steady state (60 = 0, 9C0 = 1). 

In Section VI we have touched upon the subject of the stability of steady-state wave propagation and pointed out the signs of a monotonic instability in the low-velocity autowave process. Here we shall consider qualitatively another... 

The stability of steady states and cycles implies the restoration of steady states and cycles, respectively, following a small perturbation. One type of bifurcation is period-doubling bifurcation, in which a stable cycle of period n becomes unstable and a new stable cycle of period In is generated with a new parameter. Equation (13.4), for example, produces successive period-doubling as a increases. For 3 [Pg.633]

Comparison between the average lifetime of foam bubbles in a cylindrical and conical vessels for solutions of alkyl glycosides has been carried out by Waltermo et al. [120]. In all cases the results obtained about the lifetimes are close, t = 10-40 s. A more precise characteristic of the stability of steady-state foams has been proposed in [94-97] the retention time (rt). It represents the average time of gas retention in the whole solution+foam system. This characteristic is determined by the slope of the linear segment in the dependence of the total gas volume used in foam formation versus its volumetric rate. 

The explanation of the change in the stability of steady-state foams in the homologous series of fatty alcohols and acids is based on that correlation. A detailed discussion of the stability and the related to it other properties of steady-state foams can be found in [113,123]. 

Horak and Jiracek (1972) observed three steady states and indicated that the stability of steady states depends on the ratio of the reactor volume to the amount of catalyst. 

A few words on the stability of steady states of polymerization. This question arises immediately as soon as the multiplicity of steady-state conditions spears. It is well known that three solutions are possible in the flow of reactants. The general theory of thermal instability of reactors has been developed in detail in Refe. [16-20,30,31], and the theory of kinetic instability caused by peculiarities of the kinetic schenK (self-acceleration. gel-effect, etc. in Refs. [37-40]). The instability of steady states of poly-nKiization plug reactors of a hydrodynamic nature is more interesting for the present paper. It can be assumed that the state corresponding to the negative slopes of the P(Q) curve are unstable if P = const is maintained [30, 33, 34]. At Q = const, all states are stable and realizable. The analysis of this problem in zero-dimensional formulation [41], for a reactor determined by only one value of T, p, q and a complex variable hydrodynamic resistance has shown that the slope of the curve is not an exhaustive stability criterion. 

Before plunging into the meat of our discussions, we will review some basic but necessary ideas. Much of this material will be familiar to many readers, and we encourage you to move quickly through it or to skip it completely if appropriate. If you have not encountered these concepts before, you will find it worthwhile to invest some time here and perhaps to take a look at some of the more detailed references that we shall mention. We begin with a review of chemical kinetics. We then consider how to determine the stability of steady states in an open system using analytical and graphical techniques. Finally, we look at some of the methods used to represent data in nonlinear dynamics. 

The simplest definition of the stability of steady states is as follows ... 

More complex oscillations have been found when the full TWC microkinetic model (Eqs. 1-31 in Table 1) has been used in the computations, cf. Fig. 4. The complex spatiotemporal pattern of oxidation intermediate C2H2 (Fig. 4, right) illustrates that the oscillations result from the composition of two periodic processes with different time constants. For another set of parameters the coexistence of doubly periodic oscillations with stable and apparently unstable steady states has been found (cf. Fig. 5). Even if LSODE stiff integrator (Hindmarsh, 1983) has been succesfully employed in the solution of approx. 10 ODEs, in some cases the unstable steady state has been stabilised by the implicit integrator, particularly when the default value for maximum time-step (/imax) has been used (cf. Fig. 5 right and Fig. 3 bottom). Hence it is necessary to give care to the control of the step size used, otherwise false conclusions on the stability of steady states can be reached. 

Another factor to keep in mind is the stirring rate. Although one generally assumes that mixing is essentially perfect in the CSTR, recent experiments by ROUX et. al. [50] on the chlorite-iodide reaction have shown marked dependence of the stability of steady states on the rate of stirring even at such "high" speeds as 700 rpm. Similar effects are to be expected for oscillatory states. 

---