Dirac-delta function, catalyst

The effects of non-uniform distribution of the catalytic material within the support in the performance of catalyst pellets started receiving attention in the late 60 s (cf 1-4). These, as well as later studies, both theoretical and experimental, demonstrated that non-uniformly distributed catalysts can offer superior conversion, selectivity, durability, and thermal sensitivity characteristics over those wherein the activity is uniform. Work in this area has been reviewed by Gavriilidis et al. (5). Recently, Wu et al. (6) showed that for any catalyst performance index (i.e. conversion, selectivity or yield) and for the most general case of an arbitrary number of reactions, following arbitrary kinetics, occurring in a non-isothermal pellet, with finite external mass and heat transfer resistances, the optimal catalyst distribution remains a Dirac-delta function. 

Catalyst attached to membrane pore surface. The final distribution of the catalyst in the membrane pores can significantly impact the reactor performance. The optimal form of the catalyst distribution for maximizing conversion was studied mathematically by Keller et al. [1984]. They determined that the optimal distribution of the catalyst concentration is of the Dirac delta function. 

Dirac delta function porosity of catalyst particle parameter for Stockmayer distribution... 

A non-uniform and precisely controlled position of the catalyst within the membrane pores can impact the reactor performance positively [17]. A modeling study on the first-order reaction in a catalytic membrane indicates that a Dirac-delta function of the concentration of catalyst in the membrane, placed at the feed side, allows the highest conversions. In other words, it is better to promote the reaction as close to the membrane as possible (on its surface), letting the rest of the membrane work as a mere separator of some of the reaction products [24]. The membrane structure is critical to the preparation of non-uniform catalytic membranes. If a sufficiently homogeneous membrane structure is present, simple impregnation may be sufficient to obtain a controlled, non-uniform distribution of active materials. 

Figure 9.6 Total conversion for inert membrane reactor with catalyst on the feed side (IMRCF), catalytic membrane reactor (CMR) and conventional fixed-bed reactor (FBR) with uniform and Dirac delta catalyst activity distributions as a function of the dimensionless residence time [Yeung et al., 1994]...

Figure 9.7 Effect of location of Dirac delta catalyst activity distribution on toul conversion as a function of dimensionless residence time for (a) IMRCF (b) CMR and (c) FBR [Yeung etal., 1994]...

Figure 5.18. Total conversion as function of the dimensionless residence time 0 for a CMR, PBMR and PBR with uniform and Dirac delta catalyst activity located at the feed side (CMR) or on the surface of catalyst pellets (PBMR, PBR) at identical Thiele modulus (0=5) for membrane and catalyst pellets. From [5.62], with permission from Elsevier Science.

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