Diffusivity in a Catalyst Pellet

The evaluation of E requires a knowledge both of the diffusivity and the reaction rate constants. Our task in the present illustration is to estimate the diffusivity of a reacting species, given certain physical parameters of the catalyst particle. 

Let us consider the diffusion of oxygen in a silica-alumina cracking catalyst with an average pore radius of 24 A = 24 x 1(L cm and a void fraction e of 0.3 cm /cm. The tortuosity factor is not known, and we consequently use an average value of x = 4. Because pore radius is much smaller than the mean free path, which is of the order of 10 cm at atmospheric pressures, it is suspected that Knudsen diffusion may be operative. We turn to Equation 3.14 and obtain, for a reaction temperature of 420 K, 

This value is more than 10 times lower than the molecular diffusivities of 0.20 cm /s. We conclude therefore that Knudsen diffusion is indeed the operative mode of diffusion. To calculate the effective diffusivity for the entire particle, we draw on Equation 3.13a and obtain 

This is the effective diffusivity to be used in assessing catalyst performance. 

---

The analysis below is given for the ORR, since the agglomerate and embedded models mainly examine the cathode reaction at the anode can be derived in a similar manner. The analysis is basically the same as that of reaction and diffusion in a catalyst pellet. For the analysis, an effectiveness factor is used, which allows for the actual rate of reaction to be written as (see eq 55)... 

The material balance for simultaneous reaction and diffusion in a catalyst pellet can be extended to include more complex reactions. For example, the generalized Thiele modulus for an irreversible reaction of order n is ... 

You saw how the equations governing energy transfer, mass transfer, and fluid flow were similar, and examples were given for one-drmensional problems. Examples included heat conduction, both steady and transient, reaction and diffusion in a catalyst pellet, flow in pipes and between flat plates of Newtonian or non-Newtonian fluids. The last two examples illustrated an adsorption column, in one case with a linear isotherm and slow mass transfer and in the other case with a nonlinear isotherm and fast mass transfer. Specific techniques you demonstrated included parametric solutions when the solution was desired for several values of one parameter, and the use of artificial diffusion to smooth time-dependent solutions which had steep fronts and large gradients. 

The location of a metal sulfide deposit in a catalyst pellet is dependent on the relative rates of reaction and diffusion. The theory describing diffusion and reaction in catalysts was first developed by Thiele (1939) and extended by many others, including Wheeler (1955), Weisz (1962) and Satterfield (1970), and has been discussed in Section IV. Sato et al. (1971) and Tamm et al. (1981) have discussed metal deposit profiles in the context of the Thiele analysis. 

The question remains as to when the various diffusion effects really influence the conversion rate in fluid-solid reactions. Many criteria have been developed in the past for the determination of the absence of diffusion resistance. In using the many criteria no more information is required than the diffusion coefficient DA for fluid phase diffusion and for internal diffusion in a porous pellet, the heat of reaction and the physical properties of the gas and the solid or catalyst, together with an experimental value of the observed global reaction rate (R ) per unit volume or weight of solid or catalyst. For the time being the following criteria are recommended. Note that intraparticle criteria are discussed in much greater detail in Chapter 6. 

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 ... 

Madon and Boudart propose a simple experimental criterion for the absence of artifacts in the measurement of rates of heterogeneous catalytic reactions [R. J. Madon and M. Boudart, Ind. Eng. Chem. Fundam., 21 (1982) 438]. The experiment involves making rate measurements on catalysts in which the concentration of active material has been purposely changed. In the absence of artifacts from transport limitations, the reaction rate is directly proportional to the concentration of active material. In other words, the intrinsic turnover frequency should be independent of the concentration of active material in a catalyst. One way of varying the concentration of active material in a catalyst pellet is to mix inert particles together with active catalyst particles and then pelletize the mixture. Of course, the diffusional characteristics of the inert particles must be the same as the catalyst particles, and the initial particles in the mixture must be much smaller than the final pellet size. If the diluted catalyst pellets contain 50 percent inert powder, then the observed reaction rate should be 50 percent of the rate observed over the undiluted pellets. An intriguing aspect of this experiment is that measurement of the number of active catalytic sites is not involved with this test. However, care should be exercised when the dilution method is used with catalysts having a bimodal pore size distribution. Internal diffusion in the micropores may be important for both the diluted and undiluted catalysts. 

After substituting Equation (12-7) into Equation (12-6) we arrive at the following differential equation describing diffusion with reaction in a catalyst pellet ... 

The aim of this work is to analyze the effect of internal and interfacial diffusion on a catalyst pellet in the presence of an activity decay by simultaneous poisoning and sintering. The study has been applied to a copper based catalyst used in the WGSR CO + H2O = CO2 + H2, for which a Langmuir-Hinshelwood type kinetics has been considered [2]. 

DifTeneniial equation and boundary condilioos describing diffusion and reaction in a catalyst pellet... 

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... 

The strings "1 eft" and "ri ght" specify that we would like to have collocation points at the endpoints of the Interval in addition to the zeros of the orthogonal polynomial. We solve for the concentration profile for the reactiou diffusion problem in a catalyst pellet to illustrate the collocation method. 

A similar model that specifically considers the poison deposition in a catalyst pellet was presented by Olson [5] and Carberry and Gorring [6], Here the poison is assumed to deposit in the catalyst as a moving boundary of a poisoned shell surrounding an unpoisoned core, as in an adsorption situation. These types of models are also often used for noncatalytic heterogeneous reactions, which was discussed in detail in Chapter 4. The pseudo-steady-state assumption is made that the boundary moves rather slowly compared to the poison diffusion or reaction rates. Then, steady-state diffusion results can be used for the shell, and the total mass transfer resistance consists of the usual external interfacial, pore diffusion, and boundary chemical reaction steps in series. 

The concept of diffusion is used whenever one is dealing with transport within a phase as a function of time and position. For example, when a chemical reaction occurs in a catalyst pellet, the reactant has to diffuse through the catalyst and react while it is still diffusing. Thus, in any rational analysis of such a situation, we (chemists or chemical engineers) are concerned with diffusion. As we shall see in Chapter 7, the Thiele modulus, which is central to the analysis of catalytic reactions, is based on the joint use of diffusion and reaction coefficients in a single dimensionless group. 

Let us refine this crude argument by investigating diffusion and reaction in a long narrow cylinder, the walls of which are covered with catalytic material. This reactor can serve as an idealized model of a pore in a catalyst pellet. Let the length of reactor be L, the length coordinate 4 ( = 0 at the reactor entrance), the radius of the reactor p. Reactant A penetrates into the reactor by diffusion only. Its concentration at = 0 is (A)o. Its diffusivity is D. Reaction on the walls proceeds at a rate ... 

Fig. 3. Dimensionless concentration as a function of the dimensionless position in a catalyst pellet during simultaneous reaction and diffusion for first-order kinetics and slab...

If a catalyst pellet (of any shape) has well-structured pores that are of imiform diameter d and length L and the pores are uniformly distributed throughout the volume of the pellet, then the overall rate equation can be derived by accounting for the rate of diffusion and rate of reaction in one single pore within the catalyst pellet. Consider a cylindrical pore of diameter d and length L (Figure 4.24) in a catalyst pellet in contact with a gas stream containing reactant A at concentration Ag- AS is the concentration of A in the gas at the pore mouth on the outer surface of the catalyst pellet. 

Illustration 4.9 Reaction and Diffusion in a Catalyst Particle. The Effectiveness Factor and the Design of Catalyst Pellets... 

Illustration 4.9 considered the model that describes the isothermal diffusion and reaction in a catalyst pellet. Solution of that model yields tiie reactant concentration profile within the pellet, which is then converted by integration... 

Burghardt and Aerts [12] proposed a method for evaluation of the pressure change in an isothermal porous pellet within which a single chemical reaction takes place, accompanied by mass transfer by Knudsen diffusion, bulk diffusion and viscous convective flow of the reacting mixture. The pressure change did also depend on the reaction and on the mixture composition on the pellet surface. It was concluded that the pressure changes in a catalyst pellet under conditions normally encountered in industry are most likely so small that they can be neglected in process simulations. 

Galerldn Finite Element Method In the finite element method, the domain is divided into elements and an expansion is made for the solution on each finite element. In the Galerldn finite element method an additional idea is introduced the Galerldn method is used to solve the equation. The Galerldn method is explained before the finite element basis set is introduced, using the equations for reaction and diffusion in a porous catalyst pellet. 

When a carrier is impregnated with a solution, where the catalyst deposits will depend on the rate of diffusion and the rate of adsorption on the carrier. Many studies have been made of Pt deposition from chloroplatinic acid (HgPtClg) with a variety of acids and salts as coim-pregnants. HCl results in uniform deposition of Pt. Citric or oxalic acid drive the Pt to the interior. HF coimpregnant produces an egg white profile. Photographs show such varied distributions in a single pellet. 

Barrer (19) has developed another widely used nonsteady-state technique for measuring effective diffusivities in porous catalysts. In this approach, an apparatus configuration similar to the steady-state apparatus is used. One side of the pellet is first evacuated and then the increase in the downstream pressure is recorded as a function of time, the upstream pressure being held constant. The pressure drop across the pellet during the experiment is also held relatively constant. There is a time lag before a steady-state flux develops, and effective diffusion coefficients can be determined from either the transient or steady-state data. For the transient analysis, one must allow for accumulation or depletion of material by adsorption if this occurs. 

---