Heterogeneous error structure

Figure 1.18. Mean residual pattern characteristic of a heterogeneous, or relative, error structure in the experimental data.

It took some time to adopt a similar view of other heterogeneous elimination and substitution reactions. Most efficient experimental tools have been found in stereochemical studies, correlation of structure effects on rates and measurement of deuterium kinetic isotope effects. The usual kinetic studies were not of much help due to the complex nature of catalytic reactions and relatively large experimental error. The progress has been made possible also by the studies of surface acid—base properties of the solids and their meaning for catalysis (for a detailed treatment see ref. 5). 

Besides the hypothesis of spatially homogeneous processes in this stochastic formulation, the particle model introduces a structural heterogeneity in the media through the scarcity of particles when their number is low. In fact, the number of differential equations in the stochastic formulation for the state probability keeps track of all of the particles in the system, and therefore it accounts for the particle scarcity. The presence of several differential equations in the stochastic formulation is at the origin of the uncertainty, or stochastic error, in the process. The deterministic version of the model is unable to deal with the stochastic error, but as stated in Section 9.3.4, that is reduced to zero when the number of particles is very large. Only in this last case can the set of Kolmogorov differential equations be adequately approximated by the deterministic formulation, involving a set of differential equations of fixed size for the states of the process. 

The modelling of gas permeation has been applied by several authors in the qualitative characterisation of porous structures of ceramic membranes [132-138]. Concerning the difficult case of gas transport analysis in microporous membranes, we have to notice the extensive works of A.B. Shelekhin et al. on glass membranes [139,14] as well as those more recent of R.S.A. de Lange et al. on sol-gel derived molecular sieve membranes [137,138]. The influence of errors in measured variables on the reliability of membrane structural parameters have been discussed in [136]. The accuracy of experimental data and the mutual relation between the resistance to gas flow of the separation layer and of the support are the limitations for the application of the permeation method. The interpretation of flux data must be further considered in heterogeneous media due to the effects of pore size distribution and pore connectivity. This can be conveniently done in terms of structure factors [5]. Furthermore the adsorption of gas is often considered as negligible in simple kinetic theories. Application of flow methods should always be critically examined with this in mind. 

In the Dubinin-Stoeckli (DS) method, a Gaussian pore size distribution is assumed for 7(B) in Eq. (39), based on the premise that for heterogeneous carbons, the original DR equation holds only for those carbons that have a narrow distribution of micropore sizes. This assumption enables Eq. (39) to be integrated into an analytical form involving the error function [119] that relates the structure parameter B to the relative pressure A = -RT ln(P/Po)-The structure parameter B is proportional to the square of the pore halfwidth, for carbon adsorbents that have slit-shaped micropores. 

In order to meet the ever-increasing danands for enantiopure compounds, heterogene-ons, homogeneous and enzymatic catalysis evolved independently in the past. Although all three approaches have yielded industrially viable processes, the latter two are the most widely used and can be regarded as complanentary in many respects. Despite the progress in structural, computational and mechanistic smdies, however, to date there is no universal recipe for the optimization of catalytic processes. Thus, a trial-and-error approach remains predominant in catalyst discovery and optimization. 

Hy some estimates (Shifrin, 1951 van de Hulst, 1957 Heller, 1963 Kerker and Farone, 1963 Moore et al., 1968 Kerker, 1969), the results of the soft pajticle approximation are qualitatively true within, at le.a,st, 0.8 < m < 1.5 if Q 1 when m < 1.15 they are valid quantitatively with a slight error. Hence, Rayleigh-Debye and van de Hulst s approximations are very fruitful in studying the heterogeneous structures of polymer and biological origin. 

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