The Stability Function

There are some special conditions in electrochemical simulations that have an effect on stability. 

This leads, for a given step of length ST, to the solution 

It is at this point that the various Pad6 approximations to the exponential functions come in. One of them is simply the sequence on the right-hand side of (15.39) cut off after the first two terms, producing the propagation equation. 

An interesting case is extrapolation, and we consider the simplest variant, the second-order case. It is a number of steps using BI. First one takes two successive steps with half step size, and subtracts from twice the result of this the result of one whole step. Thus, we can directly write [39]. 

Another way of investigating stability, that at the same time provides information on the behaviour of a given method, is what Gourlay and Morris [277] call the symbol of the algorithm, also called the symbol of the method [514] or, more logically perhaps, the stability function [286]. It is developed from Pade approximations to the general solution of the diffusion equation. Equation (14.6) can be semidiscretised to the system of odes as 

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The tube-current stabilizer is usually put in the grounded return output circuit of the high-voltage transformer. The stabilizer functions by properly adjusting the a-c heating current through the filament (x-ray tube cathode), and in this way regulating the electron emission. 

The fundamental aspects of the structure and stability of carbanions were discussed in Chapter 6 of Part A. In the present chapter we relate the properties and reactivity of carbanions stabilized by carbonyl and other EWG substituents to their application as nucleophiles in synthesis. As discussed in Section 6.3 of Part A, there is a fundamental relationship between the stabilizing functional group and the acidity of the C-H groups, as illustrated by the pK data summarized in Table 6.7 in Part A. These pK data provide a basis for assessing the stability and reactivity of carbanions. The acidity of the reactant determines which bases can be used for generation of the anion. Another crucial factor is the distinction between kinetic or thermodynamic control of enolate formation by deprotonation (Part A, Section 6.3), which determines the enolate composition. Fundamental mechanisms of Sw2 alkylation reactions of carbanions are discussed in Section 6.5 of Part A. A review of this material may prove helpful. 

Solution of the above constrained least squares problem requires the repeated computation of the equilibrium surface at each iteration of the parameter search. This can be avoided by using the equilibrium surface defined by the experimental VLE data points rather than the EoS computed ones in the calculation of the stability function. The above minimization problem can be further simplified by satisfying the constraint only at the given experimental data points (Englezos et al. 1989). In this case, the constraint (Equation 14.25) is replaced by... 

In Equation 14.27, cT, oP and ax are the standard deviations of the measurements of T, P and x respectively. All the derivatives are evaluated at the point where the stability function cp has its lowest value. We call the minimization of Equation 14.24 subject to the above constraint simplified Constrained Least Squares (simplified CLS) estimation. 

The set of points over which the minimum of

gradient method for the location of the minimum of the stability function, we advocate the use of direct search. The rationale behind this choice is that first we avoid any local minima and second the computational requirements for a direct search over the interpolated and the given experimental data are rather negligible. Hence, the minimization of Equation 14.24 should be performed subject to the following constraint... 

Using the values (-0.2094, -0.2665) for the parameters ka and kd in the EoS, the stability function, stability function at each isotherm are shown in Figure 14.5. The stability function at 311.5 K is shown in Figure 14.6. As seen,

liquid phases. This is also evident in Fig. 14.7 where the EoS-based VLE calculations at 311.5 K are shown. 

Hgure 14.5 The minima of the stability function at the experimental temperatures for the diethylamine-water system [reprinted from Computers Chemical Engineering with permission from Elsevier Science]. 

Figure 14.6 The stability function calculated with interaction parameters from unconstrained least squares estimation.

Therefore, although the stability function was found to be positive at all the experimental conditions it becomes negative at mole fractions between 0 and the first measured data point. Obviously, if there were additional data available in this region, the simplified constrained LS method that was followed above would have yielded interaction parameters that do not result in prediction of false liquid phase splitting. 

New interaction parameter values were obtained and they are also shown in Table 14.5. These values are used in the EoS to calculate the stability function and the calculated results at 311.5 K are shown in Figure 14.10. 

As seen from the figure, the stability function does not become negative at any pressure when the hydrogen sulfide mole fraction lies anywhere between 0 and 1. The phase diagram calculations at 311.5 K are shown in Figure 14.11. As seen, the correct phase behavior is now predicted by the EoS. 

The improved method guarantees that the EoS will calculate the correct VLE not only at the experimental data but also at any other point that belongs to the same isotherm. The question that arises is what happens at temperatures different than the experimental. As seen in Figure 14.10 the minima of the stability function increase monotonically with respect to temperature. Hence, it is safe to assume that at any temperature between the lowest and the highest one, the EoS predicts the correct behavior of the system. However, below the minimum experimental temperature, it is likely that the EoS will predict erroneous liquid phase separation. 

The above observation provides useful information to the experimenter when investigating systems that exhibit vapor-liquid-liquid equilibrium. In particular, it is desirable to obtain VLE measurements at a temperature near the one where the third phase appears. Then by performing CLS estimation, it is guaranteed that the EoS predicts complete miscibility everywhere in the actual two phase region. It should be noted, however, that in general the minima of the stability function at each temperature might not change monotonically. This is the case with the C02-n-Hexane system where it is risky to interpolate for intermediate temperatures. Hence, VLE data should also be collected at intermediate temperatures too. 

The stabilizing function of macromolecular surfactants in solid-liquid systems is exercised through protective colloid action. To be effective, they must have a strong solution affinity for hydrophobic and hydrophilic entities. In liquid-liquid systems, surfactants are more accurately called emulsifiers. The same stabilizing function is exercised in gas-liquid disperse systems where the surfactants are called foam stabilizers. 

From the range of methods for determining stability of a given algorithm such as EX, CN, B1 or BDF, etc., this chapter restricts itself to the heuristic, the Neumann and the matrix methods, as well as a fourth that makes use of the stability function. 

The similarity of the stability function to the conventional stability constant is evident from a comparison of equations 6 and 4. Equation 6 does not require knowledge of the concentration of the individual ligand or complex species in the equilibrium mixture. All of the complexes of a given stoichiometry are grouped together Tn... 

When equation 8 is satisfied for alt ligand species, the stability function adopts constant, limiting values, given by ... 

These constant, limiting values of the stability function have been previously discussed by MacCarthy (20), and have been given the name CLASP values (conditional, limiting average stability products). For reasons which will become evident later, we will refer to them here as upper CLASP values, or CLASP-w. In practice, eqns. 7, 8 and 9 may be applicable when at least one of the following conditions is satisfied ... 

We will refer to these limiting values of the stability function as lower CLASP values, or CLASP-7. In practice, equations 10 and 11 apply when both of the following experimental conditions are satisfied ... 

Stability Surfaces. In order to provide an overall picture of multiligand complexatlon behavior, the stability function should be plotted for all values of metal-multiligand solution composition. This is easily achieved by representing the solution composition as a point on a two-dimensional grid, one axis of... 

The stability surfaces depicted in Figure 3 are quite simple in shape. As mentioned previously (during discussion of Stability Profiles), a situation may be reached where some of the ligand species are almost totally complexed while the others are largely uncomplexed. This can lead to a temporary constancy of the stability function over an intermediate range of ratios,... 

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