Locally stable steady-state values

Figure 16. The stability of a steady state, as determined by a rate balance plot [287]. Left panel The rate of synthesis and consumption vcon of a substrate S. Right panel The (net) flux difference vnet vco . The steady state value S° is locally stable After transient perturbation...

The local stability of a steady-state can be ascertained by an examination of the eigen values for the local representation of the system (Savageau, 1976) these eigen values also can be obtained by a single command from within ESSYNS (Voit et al., 1989). If the real parts of all the eigen values are negative, then the steady-state is locally stable, and the system will return to the steady-state following small perturbations. This condition also is required for the consistency of any model that represents a stable-steady state. 

Here, the second subscript denotes the steady state value. The roots of the quadratic characteristic equation (eigenvalues) of the matrix A determine the stability of the equations the system will converge exponentially to the steady state if all roots have a negative real part and, therefore, is asymptotically stable. It will show a limit cycle if the roots are imaginary with zero real parts. It is unstable if any of the roots has a positive real part. Since the perturbations will decay asymptotically if and only if all the eigenvalues of the matrix A have a negative real part, it follows that the necessary and sufficient conditions for local stability are ... 

The packed bed breakthrough method for investigation of mass transfer phenomena in sorbent systems can in many instances offer certain advantages not found in other experimental methods. The method is especially useful when the adsorption isotherms for the principal sorbate exhibit favorable curvature (convex toward loading axis). In such a case, there is the potential for a portion of the sorption front to approach a stable wave form (shape of the front invariant with time). Given the existence of a stable or "steady-state" mass transfer zone (MTZ) and a detailed knowledge of the equilibrium loading characteristics within that zone, one can extract local values of the effective mass transfer resistance at any concentration in the zone. 

Not a single steady-state point in kinetic equations cannot be asymptotically stable in Z) if it does not coincide with a point of G minimum. Indeed, let us denote this steady-state point as Na and assume that it is not the point of G minimum. Then in any vicinity of Na there exist points N for which G(N) < G(N0) (otherwise N0 would be a point of local minimum). But a solution of the kinetic equations whose initial values are such values of N, since G(N) < G(N0), at t - oo cannot tend to N0 G(N) can only diminish with time. Consequently, NQ is not an asymptotically stable rest point in D. In its vicinity in D there exists such N points that, coming from these points, solutions for kinetic equations do not tend to Na at t - oo. 

The plots of Weisz and Hicks (1962) are reproduced in Figure 7.6. The nature of the curves at high values of /3m suggests multiple solutions. In other words, the reaction can occur at three steady states, two stable and one unstable. We shall not be concerned with this aspect of effectiveness factors, but it is instructive to note that e given by one of the solutions in the multiple steady-state region can be orders of magnitude higher than unity. Instabilities of this kind are essentially local in nature, and are briefly considered in Chapter 12. The reactor as a whole can also exhibit multiple steady states, a feature that is briefly treated in Chapter 13. 

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