Internal effectiveness factor

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). 

In the equation referred to above, it is assumed that there is 100 percent transmission of the shear rate in the shear stress. However, with the slurry viscosity determined essentially by the properties of the slurry, at high concentrations of slurries there is a shppage factor. Internal motion of particles in the fluids over and around each other can reduce the effective transmission of viscosity efficiencies from 100 percent to as low as 30 percent. 

Treatment of thermal conductivity inside the catalyst can be done similarly to that for pore diffusion. The major difference is that while diffusion can occur in the pore volume only, heat can be conducted in both the fluid and solid phases. For strongly exothermic reactions and catalysts with poor heat conductivity, the internal overheating of the catalyst is a possibility. This can result in an effectiveness factor larger than unity. 

Hurst (19) discusses the similarity in action of the pyrethrins and of DDT as indicated by a dispersant action on the lipids of insect cuticle and internal tissue. He has developed an elaborate theory of contact insecticidal action but provides no experimental data. Hurst believes that the susceptibility to insecticides depends partially on the cuticular permeability, but more fundamentally on the effects on internal tissue receptors which control oxidative metabolism or oxidative enzyme systems. The access of pyrethrins to insects, for example, is facilitated by adsorption and storage in the lipophilic layers of the epicuticle. The epicuticle is to be regarded as a lipoprotein mosaic consisting of alternating patches of lipid and protein receptors which are sites of oxidase activity. Such a condition exists in both the hydrophilic type of cuticle found in larvae of Calliphora and Phormia and in the waxy cuticle of Tenebrio larvae. Hurst explains pyrethrinization as a preliminary narcosis or knockdown phase in which oxidase action is blocked by adsorption of the insecticide on the lipoprotein tissue components, followed by death when further dispersant action of the insecticide results in an irreversible increase in the phenoloxidase activity as a result of the displacement of protective lipids. This increase in phenoloxidase activity is accompanied by the accumulation of toxic quinoid metabolites in the blood and tissues—for example, O-quinones which would block substrate access to normal enzyme systems. The varying degrees of susceptibility shown by different insect species to an insecticide may be explainable not only in terms of differences in cuticle make-up but also as internal factors associated with the stability of oxidase systems. 

Initially the LP-DE effect was ascribed to the high internal pressure generated by the solubilization of the salt in diethyl ether [34]. Today the acceleration is explained in terms of Lewis-acid catalysis by the lithium cation [35]. The contribution of both factors (internal pressure and lithium cation catalysis) has also been invoked [36]. 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

The size of the cataly.st particle influences the observed rate of reaction the smaller the particle, the less time required for the reactants to move to the active catalyst sites and for the products to diffuse out of the particle. Furthermore, with relatively fast reactions in large particles the reactants may never reach the interior of the particle, thus decreasing the catalyst utilization. Catalyst utilization is expressed as the internal effectiveness factor //,. This factor is defined as follows ... 

The internal effectiveness factor is a function of the generalized Thiele modulus (see for instance Krishna and Sie (1994), Trambouze et al. (1988), and Fogler (1986). For a first-order reaction ... 

If diffusion of reactants to the active sites in pores is slower than the chemical reaction, internal mass transfer is at least partly limiting and the reactant concentration decreases along the pores. This reduces the reaction rate compared to the rate at external surface conditions. A measure of the reaction rate decrease is the effectiveness factor, r, which has been defined as ... 

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]. 

All of these steps are rate processes and are temperature dependent. It is important to realize that very large temperature gradients may exist between active sites and the bulk gas phase. Usually, one step is slower than the others, and it is this rate-controlling step. The effectiveness factor is the ratio of the observed rate to that which would be obtained if the whole of the internal surface of the pellet were available to the reagents at the same concentrations as they have at the external surface. Generally, the higher the effectiveness factor, the higher the rate of reaction. 

When a solid acts as a catalyst for a reaction, reactant molecules are converted into product molecules at the fluid-solid interface. To use the catalyst efficiently, we must ensure that fresh reactant molecules are supplied and product molecules removed continuously. Otherwise, chemical equilibrium would be established in the fluid adjacent to the surface, and the desired reaction would proceed no further. Ordinarily, supply and removal of the species in question depend on two physical rate processes in series. These processes involve mass transfer between the bulk fluid and the external surface of the catalyst and transport from the external surface to the internal surfaces of the solid. The concept of effectiveness factors developed in Section 12.3 permits one to average the reaction rate over the pore structure to obtain an expression for the rate in terms of the reactant concentrations and temperatures prevailing at the exterior surface of the catalyst. In some instances, the external surface concentrations do not differ appreciably from those prevailing in the bulk fluid. In other cases, a significant concentration difference arises as a consequence of physical limitations on the rate at which reactant molecules can be transported from the bulk fluid to the exterior surface of the catalyst particle. Here, we discuss... 

Oldigs, B. (1986). Effects of internal factors upon hematological and clinical chemical parameters in the Gottinger miniature pig. Swine Biomed. Res. 2 809-813. 

The expression for the effectiveness factor q in the case of zero-order kinetics, described by the Michaelis-Menten equation (Eq. 8) at high substrate concentration, can also be analytically solved. Two solutions were combined by Kobayashi et al. to give an approximate empirical expression for the effectiveness factor q [9]. A more detailed discussion on the effects of internal and external mass transfer resistance on the enzyme kinetics of a Michaelis-Menten type can be found elsewhere [10,11]. 

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]. 

Here, we consider the general case of a porous catalyst, where the internal diffusion effect is included in the effectiveness factor (//,). 

It is possible to combine the resistances of internal and external mass transfer through an overall effectiveness factor, for isothermal particles and first-order reaction. Two approaches can be applied. The general idea is that the catalyst can be divided into two parts its exterior surface and its interior surface. Therefore, the global reaction rates used here are per unit surface area of catalyst. 

First-order reactions without internal mass transfer limitations A number of reactions carried out at high temperatures are potentially mass-transfer limited. The surface reaction is so fast that the global rate is limited by the transfer of the reactants from the bulk to the exterior surface of the catalyst. Moreover, the reactants do not have the chance to travel within catalyst particles due to the use of nonporous catalysts or veiy fast reaction on the exterior surface of catalyst pellets. Consider a first-order reaction A - B or a general reaction of the form a A - bB - products, which is of first order with respect to A. For the following analysis, a zero expansion factor and an effectiveness factor equal to 1 are considered. 

Calculation of the internal effectiveness factor for spherical pellets and fust order reaction The Thiele modulus is... 

Furthermore, eq. (5.233) is valid for a reaction-limited system, i.e. the controlling mechanism is the reaction step. Furthermore, the effectiveness factor is unity. The situation becomes more complex in the case where internal and external resistances exist as the effectiveness factor and the mass transfer coefficient should be taken into account (see eq. (5.211)) and they are a function of temperature. 

The reaction is first order with respect to hydrogen peroxide and the effectiveness factor t]v is found to be equal to 0.24. This effectiveness factor accounts only for internal transport eifects. Due to the dilute feed of hydrogen peroxide, the operation can be considered isothermal. 

Evaluate the effective rate coefficient, i.e. the product of internal effectiveness factor and intrinsic rate coefficient for this reaction... 

The intraparticle phenomena The next step is the evaluation of the internal effectiveness factor. The unknown parameter is the effective solid-phase diffusion coefficient, which is (eq. (3.602))... 

The resulting conversion under the specified conditions is approximately 25%. Evaluate the rate coefficient of the reaction using the Orcutt—Davisdon Pigford model. Assume that the internal effectiveness factor is unity and the expansion factor zero. 

This study employed conventional diffusion-reaction theory, showing that with diffusion-limited reactions the internal effectiveness factor of a heterogeneous catalyst is inversely related to the Thiele modulus. Using a standard definition of the Thiele modulus [100], the observed reaction rate of an immobilized-enzyme reaction will vary with the square root of the immobilized-enzyme concentration in a diffusion-limited system. In this case, a plot of the reaction rate versus the enzyme loading in the catalyst formulation will be nonlinear. 

In the spatially ID model of the monolith channel, no transverse concentration gradients inside the catalytic washcoat layer are considered, i.e. the influence of internal diffusion is neglected or included in the employed reaction-kinetic parameters. It may lead to the over-prediction of the achieved conversions, particularly with the increasing thickness of the washcoat layer (cfi, e.g., Aris, 1975 Kryl et al., 2005 Tronconi and Beretta, 1999 Zygourakis and Aris, 1983). To overcome this limitation, the effectiveness-factor concept can be used in a limited extent (cf. Section III.D). Despite the drawbacks coming from the fact that internal diffusion effects are implicitly included in the reaction kinetics, the ID plug-flow model is extensively used in automotive industry, thanks to the reasonable combination of physical reliability and short computation times. 

Calculated reaction rates can be in the spatially ID model corrected using the generalized effectiveness factor (rf) approach for non-linear rate laws. The effect of internal diffusion limitations on the apparent reaction rate Reff is then lumped into the parameter evaluated in dependence on Dc>r, 8 and Rj (cf. Aris, 1975 Froment and Bischoff, 1979, 1990 Leclerc and Schweich, 1993). 

The resistance to mass transfer of reactants within catalyst particles results in lower apparent reaction rates, due to a slower supply of reactants to the catalytic reaction sites. Ihe long diffusional paths inside large catalyst particles, often through tortuous pores, result in a high resistance to mass transfer of the reactants and products. The overall effects of these factors involving mass transfer and reaction rates are expressed by the so-called (internal) effectiveness factor f, which is defined by the following equation, excluding the mass transfer resistance of the liquid film on the particle surface [1, 2] ... 

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