Effectiveness factor, particle definition

Using this definition of the Thiele modulus, the reaction rate measurements for finely divided catalyst particles noted below, and the additional property values cited below, determine the effectiveness factor for 0.5 in. spherical catalyst pellets fabricated from these particles. Comment on the reasons for the discrepancy between the calculated value of rj and the ratio of the observed rate for 0.5 in. pellets to that for fine particles. 

The definition of the particle effectiveness factor 77 involves the intrinsic rate of reaction, ( rA)int> for reaction A - products, at the exterior surface conditions of gas-phase concentration (cAs) and temperature (Ts). Thus, from equation 8.55,... 

Ca relates the concentration difference over the film to procurable quantities and is therefore a so-called observable [4]. A criterion for the absence of extra-particle gradients in the rate data can be derived from the definition of an effectiveness factor for a particle. This should not deviate more than 5% from unity ... 

This equation, when is used in the definition of the effectiveness factor based on the fluid bulk concentration a leads to the combined resistance of fluid and particle... 

According to the above definitions, the effectiveness factor for any of the above shapes can adequately describe simultaneous reaction and diffusion in a catalyst particle. The equation for the effectiveness factor in a slab is the simplest in Table 6.3.1 and will be used for all pellet shapes with the appropriate Thiele modulus ... 

Before this section is concluded, there is one key point that should be emphasized. The ultimate goal of the calculation was to calculate the effectiveness factor rjcat- In the present case, this came down to solving Eq. (4-170) for the concentration distribution in the boundary-layer near the particle surface. Because of all the assumptions that we made in the original problem setup, Eq. (4-170) is simple and easy to solve. However, even if the equation we achieved in the boundary-layer region had been much more complicated so that we could not solve it, the process of setting up the asymptotic framework, by means of nondimensionalization and rescaling, provides most of the important information about rjcat, and would do so even if we had not been able to solve (4-170). If we go back to the definition of rjcat in Eq. (4-191), we see that the effectiveness factor depends on (dc/dr) =. However, we see from fhe rescaling process that... 

Catalyst supports such as silica and alumina have low thermal conductivities so that temperature gradients within catalyst particles are likely in all but the finely ground powders used for infrinsic kinetic studies. There may also be a film resisfance fo heaf fransfer af fhe exfemal surface of the catalyst. Thus the internal temperatures in a catalyst pellet may be substantially different than the bulk gas temperature. The definition of the effectiveness factor, Equation 10.23, is unchanged, but an exothermic reaction can have reaction rates inside the pellet that are higher than would be predicted using the bulk gas temperature. In the absence of a diffusion limitation, rj > 1 would be expected for an exothermic reaction. (The case > 1 is also possible for some isothermal reactions with weird kinetics.) Mass transfer limitations may have a larger... 

A comparison of the expressions before. Equations 5.85 through 5.87 and Equations 5.99 through 5.101, for spherical and flake-formed geometries, shows that the particle geometry is of less importance for the effectiveness factor. Let us recall the definition of the Thiele modulus, as in Equation 5.68 ... 

The last equation required is the definition of the catalyst effectiveness factor ETA. for reaction j. The formula is derived from the definition giverl in the Introduction. Since the particle is assumed to be spherical, the equation is ... 

Figure 9-7 can be used to estimate the effectiveness factor for a wide range of situations, as long as the catalyst particle is essentially isothermal. For example, even though the definition of (t> is based on an nth-order reaction, the behavior of many Langmuir-Hinshelwood and Michaelis-Menten rate equations can be bracketed by two values of n, say 0 and 1 or 0 and 2. This type of rate equation can be treated by using an intramediate order, say 1/2, in Eqns. (9-13) and (9-13a), orEqn. (9-14). 

The quality factor is a dimensionless quantity that depends in part on the stopping power for charged particles, and it accounts for the differences in biological effectiveness found among the types of radiation. By definition it is independent of tissue and biological end point and, therefore, of little use... 

It is clear that the influence of surface geometry upon catalytic activity is extremely complex and many more studies are required before any definitive relationship between catalytic activity and metal particle size can be established. Such studies will require to take cognisance of such factors as the perturbation of surface structure due to the formation of carbidic residues, as noted by Boudart [289] and by Thomson and Webb [95], and by the modification of catalytic properties on adsorption, as noted by Izumi et al. [296—298] and by Groenewegen and Sachtler [299] in studies of the modification of nickel catalysts for enantioselective hydrogenation. Possible effects of the support, as will be discussed in Sect. 6.3, must also be taken into account. 

The diffusion of small particles depends upon many factors. In addition to Brownian motion, we must consider the effect of gravity and the motion of the fluid in which the particles reside. Ordinary diffusion as understood in colloid chemistry must be modified considerably when we deal with turbulence. However, we still retain the usual definition of diffusion, namely that it is the number or mass of particles passing a unit cross section of the fluid in unit-time and unit-concentration gradient. That is, if dw particles (or mass) move through an area / in time dt and dC/dx is the concentration increase in the jc-directior then... 

It will be clear that, to be conceptually sound, any enhancement effect should exclude these factors. This requires careful definition of the flux enhancement due to the particles, for which we proposed [76]... 

Hie selection of a solid catalyst for a given reaction is to a large extent still empirical and based on prior experience or analogy. However, there are now many aspects of this complex situation that are quite well understood. For example we know how the true chemical kinetics, which are an intrinsic property of the catalyst, and all the many aspects of transport of material and heat around the catalytic particles, interact. In other words, the physical characteristics around the catalyst system and their effects on catalyst performance are well known today. The chemist searching for new and better catalysts should always consider these physical factors, for they can be brought under control, and often in this way definite gains can usually be made both in activity and in selectivity. Further, this knowledge enables us to avoid... 

Under this definition, the E function is characterized by an experimental 5 factor, which can be estimated experimentally (Saad et al, 2001). Note that the cumulative B factor applied in the final reconstruction is a composite of experimental and computational causes. The computational B factor is attributable to additional blurring effects such as inaccuracy in determining the orientation of particles, which could also be described by a Gaussian function type. The dampening of the image contrast by the... 

The concept of direct attack mechanisms as direct contact phenomena requiring intimate physical contact between the mineral surface and the organism requires some precise definition. Vanselow (1976) has thrown some light on the possible physical interactions between cells and minerals in studies on the effects of dilution on the rates of oxygen uptake by T. ferrooxidans strains in the presence of synthetic covellite. The dilution of a slurry will lower the rate of copper sulfide oxidation per unit volume, and the relationship between the dilution factor, and the factor by which the oxidation rate is lowered, will depend upon the nature of the physical interaction between the cells and the mineral particles. Three principal situations were postulated, namely, that in which the oxidation was carried out by cells... 

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