Spherical catalyst pellets diffusion/reaction

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

In problems such as the drying of droplets or diffusion through films around spherical catalyst pellets, it is more convenient to use Eqs. (40b) and (49) in spherical coordinates. Then for steady state diffusion in the radial direction alone, one has in the absence of chemical reactions... 

Villadsen, J. and Michelsen, M. L. (1972) Diffusion and reaction in spherical catalyst pellets steady state and local stability analysis. Chem. Engng Sci. 27, 751. 

The well known Thiele modulus of the reaction. This is defined as the ratio of the intrinsic chemical rate, calculated at bulk fluid phase conditions, to the maximum rate of effective diffusion at the external pellet surface. For spherical catalyst pellets, the Thiele modulus is given by... 

A reaction of the order Vi is carried out in a spherical catalyst pellet. For the given surface temperature and concentration the product ksCAjmii equals 0.2 s 1. The effective diffusion coefficient has been determined as 2 x Iff7 m2 s"1. The diameter of the sphere is 6 mm. Fur-... 

A lack of significant intraphase diffusion effects (i.e., 17 > 0.95) on an irreversible, isothermal, first-order reaction in a spherical catalyst pellet can be assessed by the Weisz-Prater criterion [P. B. Weisz and C. D. Prater, Adv. Catal., 6 (1954) 143] ... 

The irreversible, first-order reaction of gaseous A lo B occurs in spherical catalyst pellets with a radius of 2 mm. For this problem, the molecular diffusivity of A is 1.2 X 10" cm s and the Knudsen diffusivity is 9 X 10 " cm s. The intrinsic first-order rate constant determined from detailed laboratory measurements was found to be 5.0 s . The concentration of A in the surrounding gas is 0.01 mol L . Assume the porosity and the tortuosity of the pellets are 0.5 and 4, respectively. 

P12-6b a first-order heterogeneous irreversible reaction is taking place within a spherical catalyst pellet which is plated with platinum throughout the pellet (see Figure 12-3). The reactant concentration halfway between the external surface and the center of the pellet (i.e. r = F/2) is equal to one-tenth the concentration of the pellet s external surface. The concentration at the external surface is 0.001 g mol/dm, the diameter (2R) is 2 X 10 cm, and the diffusion coefficient is 0.1 cmVs. 

P12-15c Reconsider diffusion and reaction in a spherical catalyst pellet forthe case where the reaction is not isothermal. Show that the energy balance can be written as... 

The effect of pore-mouth poisoning can be obtained by equating the rate of diffusion through the outer, deactivated layer to the rate of reaction on the inner fully active part of the pellet. Figure 11-12 depicts a spherical catalyst pellet at a time when the radius of the unpoisoned central portion is r, corresponding to a thickness of completely poisoned catalyst — r. Consider a first-order reaction where the concentration of reactant at the outer surface is C. The rate of diffusion into one pellet will be... 

Problem 4-3. Reaction-Diffusion in a Spherical Catalyst Pellet. Consider a spherical catalyst pellet. Assume that transport of product is by diffusion, with diffusivity Deff. As in the problem discussed in Section F, we assume that the transport of reactant within the pellet is decoupled from the transport of product. Mass transfer in the gas is sufficiently rapid so that the reactant concentration at the catalyst surface is maintained at c,x,. 

Figure 12.6 Effectiveness factor plot for spherical catalyst pellets based on the effective diffusivity model for a first-order reaction.

This relation is plotted as curve B in Figure 12.11. Smith (68) has shown that the same limiting forms for T are observed using the concept of effective diffusivities and spherical catalyst pellets. Examination of curve B indicates that for fast reactions on catalyst surfaces where the poisoned sites are uniformly distributed over the pore surface, the apparent activity of the catalyst declines much less rapidly than for the case where catalyst effectiveness factors approach unity. Under these circumstances, the catalyst effectiveness factors are considerably less than unity, and the effects of the portion of the poison adsorbed near the closed end of the pore are not as apparent as in the earlier case for small values of hj. With poisoning, the Thiele modulus hp decreases, and the reaction merely penetrates deeper into the pore. 

As a first step towards obtaining the global rate equation r, a rate equation is derived by taking into account the rate of diffusion through the internal pores of the catalyst and the rate of reaction at the active sites (r2 and r,). The derivation of this rate equation for a slabshaped catalyst pellet and a spherical catalyst pellet is presented in the following sections. 

Internal Pore Diffusion and Reaction in a Spherical Catalyst Pellet... 

Figure 9 Spherical model for simultaneous diffusion and reaction R spherical catalyst pellet of radius, dr spherical shell of thickness, r. spherical shell of radius) (Charles and Thomas, 1963).

First, the potential exhibits a maximum or a minimum at a point or axis of symmetry. These locations can be the centerline of a slab, the axis of a cylinder, or the center of a sphere. Figure 1.2a and Figure 1.2b consider two such cases. Figure 1.2a represents a spherical catalyst pellet in which a reactant of external concentration Q diffuses into the sphere and undergoes a reaction. Its concentration diminishes and attains a minimum at the center. Figure 1.2b considers laminar flow in a cylindrical pipe. Here the state variable in question is the axial velocity v, which rises from a value of zero at the wall to a maximum at the centerline before dropping back to zero at the other end of the diameter. Here, again, symmetry considerations dictate that this maximum must be located at the centerline of the conduit. 

Chemical reaction and diffusion in a spherical catalyst pellet... 

Consider the case of a nonisothermal reaction A B occurring in the interior of a spherical catalyst pellet of radius R (Figure 6.4). We wish to compute the effect of internal heat and mass transfer resistance upon the reaction rate and the concentration and temperature profiles within the pellet. If Z)a is the effective binary diffusivity of A within the pellet, and we have first-order kinetics, the concentration profile CA(f) is governed by the mole balance... 

Diffusion effects can be expected in reactions that are very rapid. A great deal of effort has been made to shorten the diffusion path, which increases the efficiency of the catalysts. Pellets are made with all the active ingredients concentrated on a thin peripheral shell and monoliths are made with very thin washcoats containing the noble metals. In order to convert 90% of the CO from the inlet stream at a residence time of no more than 0.01 sec, one needs a first-order kinetic rate constant of about 230 sec-1. When the catalytic activity is distributed uniformly through a porous pellet of 0.15 cm radius with a diffusion coefficient of 0.01 cm2/sec, one obtains a Thiele modulus y> = 22.7. This would yield an effectiveness factor of 0.132 for a spherical geometry, and an apparent kinetic rate constant of 30.3 sec-1 (106). 

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. 

Figure 12.5 contains a series of curves representing the concentration profile in the spherical pellet for different values of the Thiele modulus s. For small values of 0S, (say less than 0.5) the concentration profile is relatively flat and the reactant concentration is reasonably uniform. For large values of (say greater than 5), the reaction is rapid relative to diffusion and the reactant concentration at the center of the catalyst pellet is less than 7% of that at the external surface. Notice that in all cases the concentration gradient approaches zero at the center of the pellet. 

This study was carried out to simulate the 3D temperature field in and around the large steam reforming catalyst particles at the wall of a reformer tube, under various conditions (Dixon et al., 2003). We wanted to use this study with spherical catalyst particles to find an approach to incorporate thermal effects into the pellets, within reasonable constraints of computational effort and realism. This was our first look at the problem of bringing together CFD and heterogeneously catalyzed reactions. To have included species transport in the particles would have required a 3D diffusion-reaction model for each particle to be included in the flow simulation. The computational burden of this approach would have been very large. For the purposes of this first study, therefore, species transport was not incorporated in the model, and diffusion and mass transfer limitations were not directly represented. 

Effectiveness factors for a first-order reaction in a spherical, nonisothermal catalysts pellet. (Reprinted from R B. Weisz and J. S. Hicks, The Behavior of Porous Catalyst Particles in View of Internal Mass and Heat Diffusion Effects, Chem. Eng. Sci., 17 (1962) 265, copyright 1962, with permission from Elsevier Science.)... 

To begin our discussion on the diffusion of reactants from the bulk fluid to the external smface of a catalyst, we shall focus attention on the flow past a single catalyst pellet. Reaction takes place only on the catalyst and not in the fluid surroimding it. The fluid velocity in the vicinity of the spherical pellet will vaiy with position aroimd the sphere. The hydrodynamic boundary layer is usually defined as the distance from a solid object to where the fluid velocity is 99% of the bulk velocity U. Similarly, the mass transfer boundary layer thickness, 8, is defined as the distance from a solid object to where the concentration of the diffusing species reaches 99% of the bulk concentration. 

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