Reaction rates in porous catalyst

An important problem in catalysis is to predict diffusion and reaction rates in porous catalysts when the reaction rate can depend on concentration in a non-linear way.6 The heterogeneous system is modeled as a solid material with pores through which the reactants and products diffuse. We assume for diffusion that all the microscopic details of the porous medium are lumped together into the effective diffusion coefficient De for reactant. 

The overall rate of reaction is equal to the rate of the slowest step in the mechanism. When the diffusion steps (1.2. 6. and 7 in Table 10-2) are very fast compared with the reaction steps (14. and 5), the concentrations in the immediate vicinity of the active sites are indistinguishable from those in the bulk Ouid. In this situation, the transport or diffusion steps do not affect the overall rate of the reaction. In other situations, if the reaction. steps are very fast compared with the diffusion steps, mass transport does affect the reaction rate. In systems where diffusion from the bulk gas or liquid to the catalyst surface or to the mouths of catalyst pores affects the rate, changing the flow conditions past the catalyst should change the overall reaction rate. In porous catalysts, on the other hand, diffusion within the catalyst pores may limit the rate of reaction. Under these circumstances, the overall rate will be unaffected by external flow conditions even though diffusion affects the overall reaction rate. 

The reaction kinetics in porous catalysts are kinetically controlled only if the reaction is sufficiently slow, while the rate of rapid reactions becomes limited by reactant diffusion via pores. Quantitatively, the role of diffusion is usually scrutinized by employing the phenomenological reaction-diffusion equations [1]. For example, in the case of the simplest first-order reaction, one has... 

Bokhoven, C. and van Raayen, W. (1954) Diffusion and reaction rate in porous synthetic ammonia catalysts. /. Phys. Chem., 58,471-476. 

The catalyst activity depends not only on the chemical composition but also on the diffusion properties of the catalyst material and on the size and shape of the catalyst pellets because transport limitations through the gas boundary layer around the pellets and through the porous material reduce the overall reaction rate. The influence of gas film restrictions, which depends on the pellet size and gas velocity, is usually low in sulphuric acid converters. The effective diffusivity in the catalyst depends on the porosity, the pore size distribution, and the tortuosity of the pore system. It may be improved in the design of the carrier by e.g. increasing the porosity or the pore size, but usually such improvements will also lead to a reduction of mechanical strength. The effect of transport restrictions is normally expressed as an effectiveness factor q defined as the ratio between observed reaction rate for a catalyst pellet and the intrinsic reaction rate, i.e. the hypothetical reaction rate if bulk or surface conditions (temperature, pressure, concentrations) prevailed throughout the pellet [11], For particles with the same intrinsic reaction rate and the same pore system, the surface effectiveness factor only depends on an equivalent particle diameter given by... 

The porous structure of either a catalyst or a solid reactant may have a considerable influence on the measured reaction rate, especially if a large proportion of the available surface area is only accessible through narrow pores. The problem of chemical reaction within porous solids was first considered quantitatively by Thiele [1] who developed mathematical models describing chemical reaction and intraparticle diffusion. Wheeler [2] later extended Thiele s work and identified model parameters which could be measured experimentally and used to predict reaction rates in... 

As mentioned earlier, if the rate of a catalytic reaction is proportional to the surface area, then a catalyst with the highest possible area is most desirable and that is generally achieved by its porous structure. However, the reactants have to diffuse into the pores within the catalyst particle, and as a result a concentration gradient appears between the pore mouth and the interior of the catalyst. Consequently, the concentration at the exterior surface of the catalyst particle does not apply to die whole surface area and the pore diffusion limits the overall rate of reaction. The effectiveness factor tjs is used to account for diffusion and reaction in porous catalysts and is defined as... 

The analysis is concerned with a one-dimensional model of electrodes in which reaction rates are distributed unevenly due to diffusion as well as a variation in electrode potential.4,5 The treatment of the problem of a simultaneous variation in electrolyte concentration and potential distribution in the electrode is treated in an analogous manner to that of non-isothermal chemical reactions in porous catalysts.16 The results show that several dimensionless groups or numbers control the electrode behavior. Figure 8 shows a back fed porous anode used in the model. 

In principle, the rate constant k T can contain, among other things, mass transfer rates to the surface of the particles (ks) and diffusion limitation inside the particles. For example, for a first-order reaction at the porous catalyst surface,... 

At steady state, the rate and current density of an electrocatalyst in a MER are uniform. In a CER, however, reactant concentration declines along the reactor and current decreases under potentiostatic control for non-zero-order, single, or multiple reactions. Current nonuniformity in a CER becomes more pronounced with decreasing reduction or increasing oxidation potentials (60-62). With slow diffusive transport in porous catalysts, significantly lower potentials are necessary to reach the same degree of nonuniformity as in the absence of pore diffusion (61). 

Consider the effect of surface diffusion on the effectiveness factor for a first-order, irreversible, gaseous reaction on a porous catalyst. Assume that the intrinsic rates of adsorption and desorption of reactant on the Surface are rapid with respect to the rate of surface diffusion. Hence equilibrium is established between reactant iii the gas in the pore and reactant adsorbed on the surface. Assume further that the equilibrium expression for the concentration is a linear one. Derive an equation for the effectiveness factor for each of the following two cases ... 

In the general case where the active material is dispersed through the pellet and the catalyst is porous, internal diffusion of the species within the pores of the pellet must be included. In fact, for many cases diffusion through catalyst pores represents the main resistance to mass transfer. Therefore, the concentration and temperature profiles inside the catalyst particles are usually not flat and the reaction rates in the solid phase are not constant. As there is a continuous variation in concentration and temperature inside the pellet, differential conservation equations are required to describe the concentration and temperature profiles. These profiles are used with intrinsic rate equations to integrate through the pellet and to obtain the overall rate of reaction for the pellet. The differential equations for the catalyst pellet are two point boundary value differential equations and besides the intrinsic kinetics they require the effective diffusivity and thermal conductivity of the porous pellet. 

Some of the first considerations of the problem of diffusion and reaction in porous catalysts were reported independently by Thiele [E.W. Thiele, Ind. Eng. Chem., 31, 916 (1939)] Damkohler [G. Damkohler, Der Chemie-Ingenieur, 3, 430 (1937)] and Zeldovich [Ya.B. Zeldovich, Acta Phys.-Chim. USSR, 10, 583 (1939)] although the first solution to the mathematical problem was given by Jiittner in 1909 [F. Jiittner, Z. Phys. Chem., 65, 595 (1909)]. Consider the porous catalyst in the form of a flat slab of semi-infinite dimension on the surface, and of half-thickness W as shown in Figure 7.3. The first-order, irreversible reaction A B is catalyzed within the porous matrix with an intrinsic rate (—r). We assume that the mass-transport process is in one direction though the porous structure and may be represented by a normal diffusion-type expression, that there is no net eonveetive transport eontribution, and that the medium is isotropic. For this case, a steady-state mass balance over the differential volume element dz (for unit surface area) (Figure 7.3), yields... 

Consider the following nine examples of diffusion and chemical reaction in porous catalysts where the irreversible kinetic rate law is only a function of the molar density of reactant A. Identify the problems tabulated below that yield analytical solutions for (a) the molar density of reactant A, and (b) the dimensionless correlation between the effectiveness factor and the intrapellet Damkohler number. 

The absorption rate varies with the square root of the rate constant and the molecular diffusivity. Both mass transfer and chemical reaction influence the overall process, but neither can be said to control the rate. Note that the square-root dependence on a rate constant and a diffusivity matches that found for diffusion and reaction in porous catalysts at high values of the Thiele modulus [Eq. (4.34)]. 

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