Global stability criteria

These equations have immediate implications for the signs of the /indices. Namely, by the global stability criterion [31, 49] q = 8p/0N > 0, so that any addition of electrons to the system, dN > 0, raises the global chemical potential dp(dN > 0) > 0. Assume that both initial and final states correspond to the global equilibrium, so that dji = djia = djip =. .. > 0. Moreover in a stable molecular system all hardness eigenvalues are positive h > 0, a = 1,..., m. Hence it immediately follows from the EE equations (23) that, for stable systems,... 

The local equilibrium assumption was the basis on which the Brussels school developed a global thermodynamic theory. Use of this assumption makes possible the macroscopic evaluation of entropy production and entropy flow terms with macroscopic thermodynamic methods. The assumption states that "there exists within each small mass element of the medium a state of local equilibrium for which the local entropy, s, is the same function of the local macroscopic variables as at equilibrium state" (Glansdorff and Prigogine, 1971, p. 14). In other words, each small element of a system may be treated as a state near equilibrium but need not necessarily be at equilibrium. This does not mean that the system as a whole need be near equilibrium thus, neighboring local elements may differ in parameters (temperatures, chemical affinities, etc.) which are reflected in the function describing their local entropy. The additional assumption is made that the sum of the criteria of local stability for each element corresponds to the global stability criterion for the whole system. 

Plug Flow With a significant amount of axial dispersion, Equation (3), describing the normal bed temperature profile, must be modified to account for this dispersion. The effect of this modification is that the ultimate vertical asymptote in temperature is moved forward in the extended bed. Dispersion enhances the tendency of a reactor to run away. However, with the type of dispersion that occurs in a trickle bed, by variations in velocity from point to point, the profile retains its vertical asymptote. The solution of Equation (3) plus dispersion is almost identical with Equation (5), but with a different value of SD. Since SD drops out in the ultimate stability criterion, axial dispersion cannot be of any particular significance in the development of local hot spots. It affects the global stability of the normal part of the reactor, but it has little influence on the way disturbances grow, relative to the normal regions. 

The general iterative scheme is illustrated by the flowchart in Fig. 3. Note the iteration loop on the determination of the coupled fields < >/ and u. The equilibrium potential is determined first. Then Eqs. (46) and (47) are solved alternately until convergence of the global electric and mass fluxes I and U. An estimate of the velocity is given (zero velocity when the routine enters the loop) an estimate of l is determined by solving (46) the velocity field is then updated by solving Eqs. (47). The stabilization criterion for the fluxes is generally set to 10. ... 

The minimization of the right hand side of (9.3), over all periodic functions, shows that this right hand side will be negative for n/L > k and therefore the nonlinear KS equation (6.10) is globally stable for an initial condition with a wavenumber satisfying the linear stability criterion. 

When the second derivative is positive, the mixed system is stable and does not form separate phases. When the second derivative is negative, the system does form separate phases. To decide whether the system is stable, look at whether the free energy function is concave upward (]d F)/ dx ) > 0) or concave downward ( d F) j(dx ) < 0). This stability criterion can be applied to models for G(x) (when pressure is constant) or F(x) (when volume is constant) to predict phase boundaries. W e return to questions of local stabilities and small fluctuations on page 477. First, we consider global stabilities. 

With such an understanding on system complexity in mind, the DBS model is composed of two simple force balance equations, respectively, for small or large bubble classes, and one mass conservation equation as well as the stability condition serving as a variational criterion and a closure for conservative equations. For a given operating condition of the global system, six structure parameters for small and large bubble classes (their respective diameters dg, dL, volume fraction... 

Theoretical ionization energies are in good agreement with the experimental values. For all the molecules, the HOMO-LUMO gap is larger for the most stable isomers. This confirms previous results that claim that the stability of aromatic hydrocarbons depends on the HOMO-LUMO gap. The principle of maximum hardness establishes that the system would be more stable if the global hardness, related to the HOMO-LUMO gap, is a maximum. As shown in Table 61, the HOMO-LUMO gap correlates well with the expected stability of these molecules and the energy difference between the HOMO and HOMO-1 for benzo[3]thiophene is smaller than for benzo[c]thiophene (Figure 27). Therefore, it is possible to use hardness as a criterion of stability. 

If a network has been shown to have a unique steady state, either by application of the uniqueness criterion of Section 7.5 or by an individual inspection, there are still three possibilities for its global behaviour a) the state of the network tends into the steady state as t—> i.e., global and asymptotic stability of the... 

The analytic part of the above proof is evidently not restricted to loop-free networks but may be applied as a criterion of global and asymptotical stability to arbitrary networks as an individual test for the network with respect to conditions 1), 2) and 3). 

Apply conditions 1), 2) and 3) as expressed by (7.72) to (7.76) as a criterion for global and asymptotical stability to the networks throughout this book and show that those networks which have already been identified as globally and asymptotically stable indeed satisfy the conditions whereas those showing multiple steady states or limit cycles violate conditions 2) or 3) or even both. 

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