Diffusion catalyst effectiveness

Catalyst Effectiveness. Even at steady-state, isothermal conditions, consideration must be given to the possible loss in catalyst activity resulting from gradients. The loss is usually calculated based on the effectiveness factor, which is the diffusion-limited reaction rate within catalyst pores divided by the reaction rate at catalyst surface conditions (50). The effectiveness factor E, in turn, is related to the Thiele modulus,

first-order rate constant, a the internal surface area, and the effective diffusivity. It is desirable for E to be as close as possible to its maximum value of unity. Various formulas have been developed for E, which are particularly usehil for analyzing reactors that are potentially subject to thermal instabilities, such as hot spots and temperature mnaways (1,48,51). 

Physical properties of catalysts also may need to be checked periodically, includiug pellet size, specific surface, porosity, pore size and size distribution, and effective diffusivity. The effectiveness of a porous catalyst is found by measuring conversions with successively smaller pellets until no further change occurs. These topics are touched on by Satterfield (Heterogeneous Cataly.sls in Jndustiial Practice, McGraw-Hill, 1991). 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Illustration 12.3 indicates the use of the effective diffusivity approach for estimating catalyst effectiveness factors when this parameter is determined experimentally or may be estimated. 

ILLUSTRATION 12.3 DETERMINATION OF CATALYST EFFECTIVENESS FACTOR FOR THE CUMENE CRACKING REACTION USING THE EFFECTIVE DIFFUSIVITY APPROACH Use the effective diffusivity approach to evaluate the effectiveness factor for the silica-alumina catalyst pellets considered in Illustration 12.2. 

For situations where the reaction is very slow relative to diffusion, the effectiveness factor for the poisoned catalyst will be unity, and the apparent activation energy of the reaction will be the true activation energy for the intrinsic chemical reaction. As the temperature increases, however, the reaction rate increases much faster than the diffusion rate and one may enter a regime where hT( 1 — a) is larger than 2, so the apparent activation energy will drop to that given by equation 12.3.85 (approximately half the value for the intrinsic reaction). As the temperature increases further, the Thiele modulus [hT( 1 — a)] continues to increase with a concomitant decrease in the effectiveness with which the catalyst surface area is used and in the depth to which the reactants are capable of... 

When the hydrogen pressure is 1 atm, and the temperature is 77 °K, the experimentally observed (apparent) rate constant is 0.159 cm3/ sec-g catalyst. Determine the mean pore radius, the effective diffusivity of hydrogen, and the catalyst effectiveness factor. 

At this point it is instructive to consider the possible presence of intraparticle and external mass and heat transfer limitations using the methods developed in Chapter 12. In order to evaluate the catalyst effectiveness factor we first need to know the combined diffusivity for use... 

From Figure 12.2 it is evident that the catalyst effectiveness factor for isothermal operation will be approximately 0.47. At higher temperatures the effectiveness factor will be smaller because the rate constant will increase more rapidly with temperature than will the combined diffusivity. However, the reactions in question are quite... 

The study of the intra-phase mass transfer in SCR reactors has been addressed by combining the equations for the external field with the differential equations for diffusion and reaction of NO and N H 3 in the intra-porous region and by adopting the Wakao-Smith random pore model to describe the diffusion of NO and NH3 inside the pores [30, 44]. The solution of the model equations confirmed that steep reactant concentration gradients are present near the external catalyst surface under typical industrial conditions so that the internal catalyst effectiveness factor is low [27]. 

Practical solid-catalyzed rate processes also may be influenced by rates of diffusion to the external and internal surfaces. In the latter case the rate equation is modified by inclusion of a catalyst effectiveness to become... 

The data of Table 17.8 exhibit a fairly narrow range of xp, an average of about 4, but there seems to be no pattern to xrn, which is not surprising since the diffusions actually are intermediate between bulk and Knudsen in these cases. In order to be able to calculate the effective diffusivity, it is necessary to know the pore size distribution, the specific surface, the porosity, and bulk diffasivity in the reaction mixture under reaction conditions. Such a calculation is primarily of theoretical interest. Practically it is more useful to simply measure the diffusivity directly, or even better to measure the really pertinent property of catalyst effectiveness as defined next. 

Reactor and Gas Loop Model The number of lumps used is determined by comparing the rigorous steady-state results with models using different numbers of lumps. The rigorous steady-state reactor exit temperature is 500 K with a 460 K inlet reactor temperature. When a 10-lump model is used with the same amount of catalyst, the steady-state exit temperature is 505 K. This occurs because of the numerical diffusion or effective back-mixing that is inherent with a lumped model. A 50-lump reactor model is used in all the simulations. It gives a steady-state reactor exit temperature of 501 K. The differences between the dynamic responses of the 10-lump and 50-lump reactors are shown in the next section. 

The support has an internal pore structure (i.e., pore volume and pore size distribution) that facilitates transport of reactants (products) into (out of) the particle. Low pore volume and small pores limit the accessibility of the internal surface because of increased diffusion resistance. Diffusion of products outward also is decreased, and this may cause product degradation or catalyst fouling within the catalyst particle. As discussed in Sec. 7, the effectiveness factor Tj is the ratio of the actual reaction rate to the rate in the absence of any diffusion limitations. When the rate of reaction greatly exceeds the rate of diffusion, the effectiveness factor is low and the internal volume of the catalyst pellet is not utilized for catalysis. In such cases, expensive catalytic metals are best placed as a shell around the pellet. The rate of diffusion may be increased by optimizing the pore structure to provide larger pores (or macropores) that transport the reactants (products) into (out of) the pellet and smaller pores (micropores) that provide the internal surface area needed for effective catalyst dispersion. Micropores typically have volume-averaged diameters of 50 to... 

The assessment of the internal and external diffusion on the overall reaction rate is necessary before starting kinetic simulation. Criteria for estimating catalyst effectiveness are available for simple irreversible reactions [18]. With some approximation they can be applied in the case of more complex reactions, namely to detect the need of including it in the analysis. 

In relation 8.6 - rA is the reaction rate in kmol/(kg catalyst s), pc density of solid catalyst, R particle radius, I ) i diffusion coefficient in liquid and CAs reactant concentration at the catalyst surface. If CWp 1 there are no diffusion limitations, but if CWP 1 the catalyst effectiveness is severely affected. 

The value 1.693 suggests that the internal diffusion should only slightly affect the catalyst effectiveness. From a Thiele-modulus calculation, the effectiveness should be at least 90%. On the other hand, because of rapid acid consumption on down-... 

The quotient from the effective reaction rate (reff) and that (r) without diffusion control (cG = cs) defines the outer catalyst effectiveness factor, r ( Xt ... 

A special type of fluid-solid catalyzed reaction is obtained when either the reaction rate is so fast that the reactants become completely exhausted at the external catalyst surface (i.e. at very high reaction temperatures) or when the catalyst is nonporous. Then, pore diffusion and effective heat conduction inside the pellet need not be considered. Thus, the problem is reduced to a treatment of the coupled interphase heat and mass transport. 

When the effective reaction rate is controlled by pore diffusion, then the asymptotic solution of the catalyst effectiveness factor as a function of the generalized Thiele modulus can be utilized (cq 108). This (approximate) relationship has been derived in Section 6.2.3.1. It is valid for arbitrary order of reaction and arbitrary pellet shape. 

The analysis of the effects of transport on catalysis has focused on a comparison of the availability of reacting species by diffusion to the rate of reaction on the catalytic sites. High-surface-area catalysts are usually porous. Comparison of transport to reaction rates has usually been based on Knudsen diffusion (by constricted collision with the pore walls) as the dominant mode of transport. DeBoer has noted that for small pores surface diffusion may dominate transport (192). Thiele modulus calculations may therefore not be valid if they are applied to systems where surface diffusion can be significant. This may mean that the direct participation of spillover species in catalysis becomes more important if the catalysts are more microporous. Generalized interpretations of catalyst effectiveness may need to be modified for systems where one of the reactants can spill over and diffuse across the catalyst surface. 

If a transport parameter rc — CS/CL is defined, where Cs is the concentration of C at the catalyst surface, then Peterson134 showed that for gas-solid reactions t)c < rc, where c is the catalyst effectiveness factor for C. For three-phase slurry reactors, Reuther and Puri145 showed that rc could be less than t)C if the reaction order with respect to C is less than unity, the reaction occurs in the liquid phase, and the catalyst is finely divided. The effective diffusivity in the pores of the catalyst particle is considerably less if the pores are filled with liquid than if they are filled with gas. For finely divided catalyst, the Sherwood number for the liquid-solid mass-transfer coefficient based on catalyst particle diameter is two. 

In many processes of interest to the hydrocarbon processing industry the size and shape of the catalyst has been chosen as a compromise between catalyst effectiveness and pressure drop. Hence, with effectiveness factors for the main reaction somewhat below 1, intraparticle pore diffusion is generally a factor to be reckoned with. Its effect is not easily quantified since the processing of a practical feedstock involves the conversion of a large variety of molecules with widely different reaction rates and therefore the translation of catalyst performance data obtained with crushed particles to that of the actual catalyst may be difficult and of questionable validity. 

---