Disordered Pore Media

Amorphous catalysts do not have the regular or structured morphology dealt with above. Yet, a more representative model of the structure is required for an accurate prediction of their performance, in particular when the structure is modified during its application, e.g., through pore blockage by poisons carried by the feed or by coke formed by the process itself. A pore medium can evidently be considered as a network of channels — preferably 3-dimensional — with a size distribution, but the disorder also has to be included. The problem is two-fold a structure has to be generated and the operation and performance of such a structure has to be described. Two frequently used methods are briefly described here the Monte-Carlo generation and simulation and the Effective Medium Approximation. 

The calculations are repeated a large number of times for the same overall blockage probability and average pore size and the calculated set of values of De is averaged. Both Knudsen diffusivities and molecular diffusivities can be obtained in this way. They can be compared with values derived from a continuum model. 

The objective of EMA is to construct a small size network that permits the derivation of the relation between diffusivity and blockage without considering the complete network. 

More information on percolation theory and EMA can be found in Kirkpatrick [1973] Burganos and Sotirchos [1987] Zhang and Seaton [1992] Sahimi [1994] Keil [1996] Beyne and Froment [2001]. 

An early example is the work of Pereira and Beeckman [1989] and of Hegedus and Pereira [1990] who optimized the porous structure, more particularly the micropore diameter and the mean porosity of a titania-supported vanadia catalyst for the reduction of NOx by a mixture of NH3 and O2. Another example is given for hydrodemetallation by Keil and Rieckmann [1994]. The support of such a catalyst is manufactured by pelletizing powder of the support 

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We are not going to deal with all these examples of application of percolation theory to catalysis in this paper. Although the physics of these problems are different the basic numerical and mathematical techniques are very similar. For the deactivation problem discussed here, for example, one starts with a three-dimensional network representation of the catalyst porous structure. Systematic procedures of how to map any disordered porous medium onto an equivalent random network of pore bodies and throats have been developed and detailed accounts can be found in a number of publications ( 8). For the purposes of this discussion it suffices to say that the success of the mapping techniques strongly depends on the availability of quality structural data, such as mercury porosimetry, BET and direct microscopic observations. Of equal importance, however, is the correct interpretation of this data. It serves no purpose to perform careful mercury porosimetry and BET experiments and then use the wrong model (like the bundle of pores) for data analysis and interpretation. 

Percolation phenomena deal with the effect of clustering and coimectivity of microscopic elements in a disordered medium [129], Percolation theory represents a random composite material as a network or lattice structure of two or more distinct types of microscopic elements or phase domains, the so-called percolation sites. These elements represent mutually exclusive physical properties, e.g., electrically conducting vs. isolating phase domains, pore space vs. solid matrix, atoms with spin up vs. spin down states. Here, we will refer to black and white elements for definiteness. The network onto which black and white elements of the composite medium are distributed could be continuous (continuum percolation) or discrete (discrete or lattice percolation) it could be a disordered or regular network. With a probability p a randomly chosen percolation site will be... 

The theory developed must be apphed with care if the adsorbate is disintegrated into separate molecules occupying fixed sites in the adsorbent. This may happen in extremely microporous media (like those formed by cavities in crystalline structures) or on structured heterogeneous surfaces. The reason for such a limitation is that adsorbate is described by means of a thermodynamic equation of state. This assumes that adsorbate behaves more or less like a continuous medium and that interactions between its molecules are the same as in a bulk phase. On the contrary, a pore in a microporous medium may often contain only few molecules of the adsorbate, so that statistical fluctuations from a thermodynamic equihbrium state become significant. It may be expected that errors associated with these fluctuations cancel out in disordered media. However, a more ordered adsorbate might possess some properties introduced by the order. In this case, the above-described localized approach, the semi-analytical approach by Aranovich and Donohue [111], or direct molecular simulations become more suitable. 

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