Heterogeneous catalysis Effectiveness factor

In the same way as in heterogeneous catalysis, effective diffusion coefficients (for fluxes with respect to the total geometric area) are decreased relative to the values in the melt Dy because of the obstruction due to the crystalline phase at a volume fraction (j) and because of the tortuosity factor xo (which depends on the structure of the solid, values of 1.5 to 3 being common) the relationship is given in Eq. (47). 

Theoretically, the problem has been attacked by various approaches and on different levels. Simple derivations are connected with the theory of extrathermodynamic relationships and consider a single and simple mechanism of interaction to be a sufficient condition (2, 120). Alternative simple derivations depend on a plurality of mechanisms (4, 121, 122) or a complex mechanism of so called cooperative processes (113), or a particular form of temperature dependence (123). Fundamental studies in the framework of statistical mechanics have been done by Riietschi (96), Ritchie and Sager (124), and Thorn (125). Theories of more limited range of application have been advanced for heterogeneous catalysis (4, 5, 46-48, 122) and for solution enthalpies and entropies (126). However, most theories are concerned with reactions in the condensed phase (6, 127) and assume the controlling factors to be solvent effects (13, 21, 56, 109, 116, 128-130), hydrogen bonding (131), steric (13, 116, 132) and electrostatic (37, 133) effects, and the tunnel effect (4,... 

Sachtler WMH. 1997. Factors influencing catalytic action—Ensemble and ligand effects in metal catalysis. In Ertl G, Kndzinger H, Weitkamp J, eds. Handbook of Heterogeneous Catalysis. Volume 3. Weinheim VCH-Wiley. 

Inspection of Fig. 15.3 reveals that while for jo 0.1 nAcm , the effectiveness factor is expected to be close to 1, for a faster reaction with Jo 1 p,A cm , it will drop to about 0.2. This is the case of internal diffusion limitation, well known in heterogeneous catalysis, when the reagent concentration at the outer surface of the catalyst grains is equal to its volume concentration, but drops sharply inside the pores of the catalyst. In this context, it should be pointed out that when the pore size is decreased below about 50 nm, the predominant mechanism of mass transport is Knudsen diffusion [Malek and Coppens, 2003], with the diffusion coefficient being less than the Pick diffusion coefficient and dependent on the porosity and pore stmcture. Moreover, the discrete distribution of the catalytic particles in the CL may also affect the measured current owing to overlap of diffusion zones around closely positioned particles [Antoine et ah, 1998]. 

Case 3 will be of special interest in this paper. It is encountered in all the examples listed above for Equations (1) and (2). Especially in heterogeneous catalysis it shows what has been called the compensation effect (C.E.). This term indicates that an increase in the enthalpy of activation AH frequently has not the expected result of a considerable decrease in the rate constant, because there occurs a simultaneous increase in the entropy of activation AS or of the frequency factor A, which compensates partly or entirely for the change in the exponent (AH /Er, or AE/ET). 

In the case of gel entrapped biocatalysts, or where the biocatalyst has been immobilised in the pores of the carrier, then the reaction is unlikely to occur solely at the surface. Similarly, the consumption of substrate by a microbial film or floe would be expected to occur at some depth into the microbial mass. The situation is more complex than in the case of surface immobilisation since, in this case, transport and reaction occur in parallel. By analogy with the case of heterogeneous catalysis, which is discussed in Chapter 3, the flux of substrate is related to the rate of reaction by the use of an effectiveness factor rj. The rate of reaction is itself expressed in terms of the surface substrate concentration which in many instances will be very close to the bulk substrate concentration. In general, the flux of substrate will be given by ... 

Any real system is known to suffer constantly from the perturbing effects of its environment. One can hardly build a model accounting for all the perturbations. Besides, as a rule, models account for the internal properties of the system only approximately. It is these two factors that are responsible for the discrepancy between real systems and theoretical models. This discrepancy is different for various objects of modem science. For example, for the objects of planetary mechanics this discrepancy can be very small. On the other hand, in chemical kinetics (particularly in heterogeneous catalysis) it cannot be negligible. Strange as it is, taking into consideration such unpredictable discrepancies between theoretical models and real systems can simplify the situation. Perturbations "smooth out some fine details of dynamics. 

It plays the same role as the effectiveness factor in heterogeneous catalysis and is a measure of the film thickness uniformity. It represents the ratio of the total reaction rate on each pair of wafers to that we would obtain if the concentration in the cell formed by the two wafers were equal to the bulk concentration everywhere. Thus, if the surface reaction is the rate controlling step, n = 1, whereas if the diffusion between the wafers controls, n < 1. In the limit of strong diffusion resistance the deposition is confined to a narrow outer band of the wafers and a strongly nonuniform film results. 

In heterogeneous catalysis reactants have to be transported to the catalyst and (if the catalyst is a porous, solid particle) also through the pores of the particle to the active material. In this case all kinds of transport resistance s may play a role, which prevent the catalyst from being fully effective in its industrial application. Furthermore, because appreciable heat effects accompany most reactions, heat has to be removed from the particle or supplied to it in order to keep it in the appropriate temperature range (where the catalyst is really fully effective). Furthermore, heterogeneous catalysis is one of the most complex branches of chemical kinetics. Rarely do we know the compositions, properties or concentrations of the reaction intermediates that exist on the surfaces covered with the catalytically effective material. TTie chemical factors that govern reaction rates under these conditions are less well known than in homogeneous catalysis. Yet solid catalysts display specificities for particular reactions, and selectivity s for desired products, that in most practical cases cannot be equaled in other ways. Thus use of solid catalysts and the proper (mathematical) tools to describe their performance are essential. 

Diffusivity and tortuosity affect resistance to diffusion caused by collision with other molecules (bulk diffusion) or by collision with the walls of the pore (Knudsen diffusion). Actual diffusivity in common porous catalysts is intermediate between the two types. Measurements and correlations of diffusivities of both types are known. Diffusion is expressed per unit cross section and unit thickness of the pellet. Diffusion rate through the pellet then depends on the porosity and a tortuosity factor "C that accounts for increased resistance of crooked and varied-diameter pores. Effective diffusion coefficient is = Aheoi / C. Empirical porosities range from 0.3 to 0.7, tortuosities from 2 to 7. In the absence of other information, Satterfield (Heterogeneous Catalysis in Practice, McGraw-Hill, 1991) recommends taking =0.5 and T = 4. In this area, clearly, precision is not a feature. 

The effects discussed here demonstrate that oscillations in heterogeneous catalysis are very complex phenomena. The closer an experimental system is to an industrial catalytic process, the more factors there are that have to be taken into account in order to truly understand the mechanisms leading to oscillatory reaction rates. Table II shows which factors influence the macroscopically observed oscillatory behavior of a catalytic system in different pressure regimes. 

Figure 12-5 (a) Effectiveness factor plot for nth-order kinetics spherical catalyst particles (from Mass Transfer in Heterogeneous Catalysis, hy C. N. Satterfield, 1970 reprint edition Robert E. Krieger Publishing Co., 1981 reprinted by permission of the author), (b) First-order reaction in different pellet geometries (from R. Aris, Introduction to the Analysis of Chemical Reactors, 1965, p. 131 reprinted by permission of Prentice-Hall, Englewood Cliffs, NJ)... 

Little attention in what follows is given to photocatalytic effects, these having been reviewed in Volume 2 of this series for heterogeneous catalysis in general. Rather, emphasis is placed on catalytic studies that characterize mechanism for partial oxidation, if only to surface species. However, we must note the extensive studies from various research schools for elucidation of factors that control activity for total oxidation over metal oxides. 

The use of microscale reactors is not confined to single-phase systems. Both striated and droplet flows of two-phase liquid mixtures have been studied, as have suspensions of solid particles. It seems that almost any chemistry can be used at the microscale. Effectiveness factors in heterogeneous catalysis will be nearly 1.0 since diffusion distances are so small. As pointed out below, rapid molecular diffusion gives nearly instantaneous cross-channel mixing and may cause significant axial mixing. 

In the general case of immobilized enzymes not only the internal diffusion addressed above, but also diffusion through the film should be taken into account. Similarly to heterogeneous catalysis the catalyst effectiveness factor for slab geometry and low substrate concentrations (first order kinetics) is decribed by eq. (9.173) in a somewhat different form... 

Solvent effects in heterogeneous catalysis are examined in terms of physical or chemical modifications to control the chemo-, regio- and stereoselectivity of a reaction. The main factors affecting selectivity are reactant solubility, polarity, reactivity or acido-basicity of solvents and competitive chemisorption of products and solvents In the special case of molecular sieves, selectivity control of a reaction by competitive adsorption, diffusion or shape selectivity and confinement catalysis are also examined. 

In the previous section, it was pointed out that electrocatalysis is akin to heterogeneous catalysis. The essential differences are the effect of the electric field on the reaction rate and the presence of nonreacting species (ions of electrolyte, solvent), which may also affect the reaction rate. The following sections are concerned with (i) elucidation of the effect of the electric field on the reaction rate (n) role of adsorption which is somewhat more complicated in electrocatalysis by the fact that the adsorbed species are not only reactants, intermediates, or products, but also the solvent or ions of the solution (in) conditions under which a comparison of the electrocatalytic activity of various substrates for a particular reaction should be made (iv) the role of electronic and geometric factors of the electrocatalyst. 

Figure 7.4 Effectiveness factors for power-law kinetics and various geometries (sphere) = 3 (slab). [After C.N. Satterfield, Mass Transfer in Heterogeneous Catalysis, with permission of MET Press, Cambridge, MA, (1970).]...

Heinen, A.W., Papadogianakis, G., Sheldon, R.A., Peters, J.A., and van Bekkum, H. (1999) Factors effecting the hydrogenation of fructose with a water soluble Ru-TPPTS complex. A comparison between homogeneous and heterogeneous catalysis, J. Mol. Catal. A Chem. 142,17-26. 

Catal5dic gas-phase reactions are generally carried out in continuous fixed-bed reactors, which in the ideal case operate without backmixing. The model reactor is the ideal plug flow reactor, the design equation of which is derived from the mass-balance equation. As we have already learnt, in heterogeneous catalysis the effective reaction rate is usually expressed relative to the catalyst mass / cat, which gives Equation (14-1). The left side of this equation is known as the time factor the quotient is proportional to the residence time on the catalyst. 

---