Diffusion internal effectiveness factor

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 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))... 

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. 

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... 

When the diffusion of a reactant inside the pellet is not fast enough to compensate for its disappearance by reaction a decreasing concentration profile is established in the pellet. For positive partial reaction orders with respect to the reactant this leads to lower reaction rates at positions away from the external surface and hence to a lower reaction rate when averaged over the complete pellet volume. A measure for the degree of internal diffusion limitations is given by the internal effectiveness factor, t, defined as ... 

The magnitude of the effectiveness factor (ranging from 0 to 1) indicates the relative importance of diffusion and reaction limitations. The internal effectiveness factor is defined as... 

We observe that as the particle diameter becomes very small, c ) decreases, so that the effeetiveness faetor approaches 1 and the reaetion is surface-reac-tion-liimited. On the other hand, when 4, is large ( 30), the internal effectiveness factor T is small (i.e., < 1), and the reaetion is diffusion-limited within... 

Diffusion and Reaction in the Catalyst Pellet In Section 12.2 we showed that the internal effectiveness factor was the ratio of the actual rate of reaction, to the rate that would exist if the entire interior of the pellet were exposed to the reactant concentration at the external surface, Consequently, the actual rate of reaction per imit mass of catalyst can be written... 

Obviously, the internal effectiveness factor, qi, depends on the effective diffusivity, Dg, and kinetic parameters such as the rate coefficient, fcy,p/ but also on the shape of the catalyst particle. 

A measure of the absence of internal (pore diffusion) mass transfer limitations is provided by the internal effectiveness factor, t, which is defined as the ratio of the actual overall rate of reaction to the rate that would be observed if the entire interior surface were exposed to the reactant concentration and temperature existing at the exterior of the catalyst pellet. A value of 1 for rj implies that all of the sites are being utilized to their potential, while a value below, say, 0.5, signals that mass transfer is limiting performance. The value of rj can be related to that of the Thiele modulus, 4>, which is an important dimensionless parameter that roughly expresses a ratio of surface reaction rate to diffusion rate. For the specific case of an nth order irreversible reaction occurring in a porous sphere,... 

It is useful to define an external effectiveness factor along the lines of the effectiveness factor for diffusion within the solid, which may now be more appropriately called the internal effectiveness factor. ... 

The same expression as in Equation 2.61 is derived for the t of a catalyst with flat plate geometry, with the only difference that Dcomb in Equation 2.60 is replaced by of the flat plate having many cylindrical pores. For other particle geometries, such as cylindrical or spherical, the concentration profiles and the correlations for the isothermal internal effectiveness factor tj are different, and the effective mass diffusivity, D, is used in all the analyses. 

For isothermal particles, under conditions when

3, which is in the region with strong pore diffusion effects in Figure 2.10, the isothermal internal effectiveness factor r) will be inversely proportional to tp. Considering an nth order... 

This micromodel could be used to investigate the effect ofzeoHte particle size on its catalytic performance. The influence of crystal size actually represents the impact of species diffusion on the reaction, and could be quantified by the internal effective factors, which are defined as following... 

The influence of the internal effectiveness factor, t, on global rate thus has similarities to that of the external effectiveness factor, fj, in that a) the higher the reaction order, the greater the diffusional effect b) t unity for small values of the Thiele modulus, (/>, and similarly, fj unity for small values of the Damkohler number, Dao and c) at large values of these two moduli, T = l/(/)(for 0 > 3) and fj = 1/Dao. Assuming that external mass transfer limitations have been removed (Cg = Co), the effect of internal (pore) diffusion on the observed kinetics can be determined i.e., for cf) > 3, i] = l/4> and... 

The generalized internal effectiveness factor in the asymptotic region of strong diffusion effects is obtained from ... 

So far, only a single reaction has been considered. While the reactor point effectiveness cannot be expressed explicitly for a reversible reaction, the internal effectiveness factor can readily be obtained analytically using the generalized modulus (see Problem 4.23). For complex multiple reactions, however, it is not possible to obtain analytical expressions for the global rates and one has to solve the conservation equations numerically. The numerical solution of nonlinear, coupled diffusion equations with split boundary conditions is by no means trivial and often presents convergence difficulties. In this section, the same approach is taken as was used for the reactor point effectiveness. This enables the global rates to be obtained in a straightforward manner and the diffusion equations to be solved as an initial value problem (Akella 1983). 

The catalyst surface area per unit volume, g(Ss), can be made to vary with pellet coordinate by choosing an appropriate impregnation method. Hence, this function represents not only the level of dispersion but also activity distribution. A partially impregnated (or equivalently hollow) pellet is a typical example of a pellet with a certain activity distribution. The motivation for making such a pellet becomes obvious if it is recognized that the reactant concentration becomes almost zero at some point in the pellet when the reaction is diffusion-limited. The fraction of the volume of the pellet for which the concentration is zero is not utilized at all. If this fraction is made hollow or inert, then the observed rate on a per pellet basis should be the same as the fully impregnated pellet. Let us examine this further. Suppose that a pellet is hollow or partially impregnated for a distance Li from the center. Consider a diffusion-limited, first-order reaction. The internal effectiveness factor for this hollow pellet is ... 

For diffusion-affected reactions, the generalized internal effectiveness factor for fresh catalyst, tjq, can be used to arrive at the following overall pellet effectiveness for uniform chemical deactivation ... 

While the effect of deactivation alone is rather straightforward to analyze, the combined effect of deactivation and diffusion is not, because of diffusional intrusion into deactivation even when all reactions occur on the same site. Therefore, the selectivity can vary with time regardless of the nature of the active sites. Attention here will be confined to the case in which ail reactions occur on the same sites. It has been shown in Chapter 5 that the internal effectiveness factor for a pellet undergoing deactivation is given by ... 

This equation applies in the asymptotic region of strong diffusion effects, where g is the fraction of external surface activity still remaining active after deactivation, and Tc is the intrinsic rate in terms of surface conditions. This internal effectiveness factor for a deactivated pellet is simply the internal effectiveness factor for a fresh pellet multiplied by g. The restriction placed on the relationship was that the following condition be met ... 

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). 

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. 

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. 

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]. 

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 (//,). 

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