The rigorous approach

Based on the ideas discussed in Section 6.2, various rigorous extensions to slab-like systems have been proposed [245-251]. Because these systems are 

As before the main step consists of rewriting the (rf-like) charge distribution f) r) corresponding to Eq. (6.33) as a sum of three eontributions ef., Eq. (6.7), (6.10)1 which are then considered separately. This procedure is sketched in Appendix F.3.1. One finally obtains 

In terms of computer time, evaluation of the Fourier contribution to the total configurational potential energy, requires significantly more effort than 

for both the ionic and the dipolar systems, the actual use of the rigorously derived Ewald summation for slab systems loads to a substantial increase in computer time. One way of dealing with this problem would be to employ precalculated tables [252] for potential energies (and forces) on a three-dimensional spatial grid amended by a suitable interpolation scheme. Another strategy is to employ approximate methods such as the one presented in the subsequent Section 6.3.2. 

---

We leave it to the reader to pursue the rigorous approach to probability theory in appropriate texts. 

Certainly the effort of following a checklist cannot be avoided. A less rigorous alternative is to call several manufacturers, have their representatives stop by, visit their plants or customer locations and witness demonstrations of their equipment, and then decide which seems to best meet one s needs. Both this approach, which seems to be the standard in industry, and the more rigorous approach to overall planning suggested here, will yield a system. But the rigorous approach is most likely to yield a successful system and it is often an approach that is easier to sell to supervision and management. 

A more basic difficulty and one not yet adequately resolved is that encountered in the use of artificial models to represent molecules. From a rigorous point of view the entire behavior of a molecular encounter is determined by the force field surrounding each molecule. By representing molecular force fields by artificial models we avoid the impossible mathematical problem involved in the rigorous approach. The result, however, is to introduce an entirely new set of molecular parameters which remain as yet unpredictable from simpler molecular properties. In the case of the hard sphere model we have introduced the molecular diameter additional parameters which were somewhat concealed in the discussion, namely, the two accommodation coefficients, one for velocity transfers between molecules in collision and the other for collision between molecules and surfaces. 

The rigorous approach to a kinetic-theory derivation of the fluid-dynamical conservation equations, which begins with the Liouville equation and involves a number of subtle assumptions, will be omitted here because of its complexity. The same result will be obtained in a simpler manner from a physical derivation of the Boltzmann equation, followed by the identification of the hydrodynamic variables and the development of the equations of change. For additional details the reader may consult [1] and [2]. 

The disadvantage of the rigorous approach to this type of problem is that the mathematics are very difficult for any case except the most simple e.g., this first order reaction). Instead, it is common to consider the somewhat vague concept of the reaction layer . This is an approach which gives a physical idea of the processes involved as well as allowing rate coefficients to be derived for more complicated kinetic mechanisms. The reaction layer is a hypothetical layer surrounding the electrode within which all the HA molecules produced by reaction (9) reach the electrode and are reduced. Its thickness p depends on the reverse rate coefficient, ky, (p = Suppose the applied potential is only sufficient to discharge... 

When more than two components are present, the efficiencies of each are not necessarily the same. The rigorous approach to handling multicomponent mixtures, outlined by Taylor and Krishna, uses the Maxwell-Stefan diffusional equations. Chan and Fair used the rigorous approach to compare multicomponent system separations with those predicted by the use of the equivalent pseudobinary systems. They found that if the dominant pair of components present in the mixture is used to determine efficiency for all of the components, the separation determined is quite close to that resulting from rigorous multicomponent procedures. 

It has been pointed out above that the Wheeler-Ono approach (see Sec. III.l) to the idealized mathematically plane surface problem is the rigorous approach, though actual numerical calculations based on the general equations are not practical. On the other hand, the Frenkel-Halsey-Hill method (see Sec. III.4) is essentially a very approximate solution of this same problem resulting in a simple and surprisingly successful isotherm equation, Eq. (38), for 0 not too small. This method can be applied to capillary condensation (see Sec. III.5) and is capable of accounting for isotherm types II to V (1,55,75). 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

This is the simplified, two-region potential, on which the augmented-plane wave (APW) method by Slater [226,227] is essentially based. It stands for the rigorous approach which correctly describes both the spherical, atomic-like region close to the nucleus and also the flat region between the nuclei, by treating the two regions differently, in the spirit of the above muffin-tin idea. 

The Rigorous Approach is for vessel where L/D > 5 or the vessel is supported near the top or bottom of the vessel. In such cases the simplified approach may not be adequate. In this case the vessel is divided into two parts the upper and lower part. The division between the upper and lower part is the line of support. 

There is one very recent example of application of the rigorous approach outlined here. It deals with the deactivation in the solid acid alkylation process for the production of high octane gasoline [Martinis and Froment, 2006]. The kinetic modeling of the reaction between 1-butene and i-butane on a Y-zeolite catalyst was expressed in terms of elementary steps and the kinetics were written in terms of single events, discussed in Chapters 1 and 2. It was found that the... 

The reversible e stage takes place in ed process for the SMFR and FR cases, so the influence of the rate constant of the d stage on is described here in terms of the conventional theory of the ec reactions as a kinetic correction A i/2 [225]. The rigorous approach to the assessment of solvent effects is based on finding the... 

The rigorous approach to the correction factors, detailed below, resulted in equations which involved the mercury fill volume in the sample cell, the compressibilities of the mercury, glass and sample and the volume of the sample. Figure 4.9 shows the unconected intrusion curve for nylon samples together with partial and full corrections. 

The rigorous approach to this problem would call for use of Eq. 8.13.1, a rather cumbersome task. Empirically, the commonly used expression for the estimation of the enthalpy of vaporization at some temperature from its value at temperature T2 is that of Watson ... 

---