Porous catalyst systems diffusion

Equation (11-13) also reduces to Equation (11-16) for porous catalyst systems in which the pore radii are very small. Diffusion imder these conditions, known as Knudsen diffusion, occurs when the mean free path of the molecule is greater than the diameter of the catalyst pore. Here the reacting molecules eollide more often with pore walls than with each other, and molecules of different speeies do not affect each other. The flux of species A for Knudsen diffusion (where bulk flow is neglected) is... 

The reaction kinetics approximation is mechanistically correct for systems where the reaction step at pore surfaces or other fluid-solid interfaces is controlling. This may occur in the case of chemisorption on porous catalysts and in affinity adsorbents that involve veiy slow binding steps. In these cases, the mass-transfer parameter k is replaced by a second-order reaction rate constant k. The driving force is written for a constant separation fac tor isotherm (column 4 in Table 16-12). When diffusion steps control the process, it is still possible to describe the system hy its apparent second-order kinetic behavior, since it usually provides a good approximation to a more complex exact form for single transition systems (see Fixed Bed Transitions ). 

Mass transport may constitute another problem. Since many catalysts are porous systems, diffusion of gases in and out of the pores may not be fast enough in comparison to the rate of reaction on the catalytic site. In such cases diffusion limits the rate of the overall process. 

The computer-reconstructed catalyst is represented by a discrete volume phase function in the form of 3D matrix containing information about the phase in each volume element. Another 3D matrix defines the distribution of active catalytic sites. Macroporosity, sizes of supporting articles and the correlation function describing the macropore size distribution are evaluated from the SEM images of porous catalyst (Koci et al., 2006 Kosek et al., 2005). Spatially 3D reaction-diffusion system with low concentrations of reactants and products can be described by mass balances in the form of the following partial differential equations (Koci et al., 2006, 2007a). For gaseous components ... 

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. 

Figure 2 shows a schematic of a porous catalyst slab that is supplied by reactant from the outer surface and in which reaction takes place at the internal catalytic surface. It is known in such systems that, when diffusion of species internally in the structure is slow in comparison to the rate of reaction, a variation in reactant concentration will occur in the catalyst. This variation in concentration changes the rate locally in the electrode. 

When the reaction mixture diffuses into a porous catalyst, simultaneous reaction and diffusion have to be considered when obtaining an expression for the overall conversion rate. The pore system is normally some kind of complex maze and must be approximated to allow mathematical description of mass transport inside the particles, such as a sys-... 

The HDM reaction mechanism of model compounds is found to be unique to the type of porphyrin and independent of the type of catai st employed (7). This makes HDM of VO-TPP useful fw screening and testing of different catalyst systems. Prerequisite fw hydrodemetallisation is the diffusion of these large porphyrins into the catalyst porous texture. Diffusion of these molecules can be limited by geometric exclusion and hydrodynamic drag. When the solute molecular size is significant as compared to the pore size, a restrictive factor should be introduced to account fw the reduction in diffusivity. As a consequence, clarification of detailed HDM reaction kinetics may be obscured by diffusion limitations. 

An important consideration about exact lumping is the following one, which is due to Coxson and Bischoff (1987a). If one considers reactor types other than a batch (or, equivalently, a plug flow) reactor, exact lumping carries over. In other words, the dynamics of the reduced system behave as if they were representative of true intrinsic kinetics. (This, as discussed in Section IV,C, is not true for overall kinetics, which may be regarded as nonexact lumping.) A somewhat similar result was proved by Wei and Kuo (1969) for the case of reactions with diffusion, such as occurs in porous catalysts. 

Although many similarities exist between gas-solid catalytic and gas-solid noncatalytic reactions, the noncatalytic systems, particularly when a porous reactant is converted to a porous product, are more complex. Both occur as the result of a number of series-parallel steps. Mass transfer of reacting gas from the bulk gas to the exterior of the solid and that of gas product from the solid to the bulk gas are involved in each. Diffusion of the reacting gas from the exterior surface into a porous catalyst or porous solid reactant and that of gas product from the pores to the exterior surface are also common to the two types of reactions. Adsorption of reacting gas, surface reaction, and... 

Effective diffusivities in porous catalysts are usually measured under conditions where the pressure is maintained constant by external means. The experimental method is discussed in Sec. 11-2 it is mentioned here because under this condition, and for a binary counterdiffusing system, the ratio is the same regardless of the extent of Knudsen and bulk... 

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 starting point for studying the diffusion with chemical reactions for multicomponent systems in porous catalyst pellets is to derive the mathematical models that describe the system under study. 

All these factors are functions of the concentration of the chemical species, temperature and pressure of the system. At constant diffu-sionai resistance, the increase in the rate of chemical reaction decreases the effectiveness factor while al a constant intrinsic rate of reaction, the increase of the diffusional resistances decreases the effectiveness factor. Elnashaie et al. (1989a) showed that the effect of the diffusional resistances and the intrinsic rate of reactions are not sufficient to explain the behaviour of the effectiveness factor for reversible reactions and that the effect of the equilibrium constant should be introduced. They found that the effectiveness factor increases with the increase of the equilibrium constants and hence the behaviour of the effectiveness factor should be explained by the interaction of the effective diffusivities, intrinsic rates of reaction as well as the equilibrium constants. The equations of the dusty gas model for the steam reforming of methane in the porous catalyst pellet, are solved accurately using the global orthogonal collocation technique given in Appendix B. Kinetics and other physico-chemical parameters for the steam reforming case are summarized in Appendix A. 

Chapter 5 is dedicated to the single particle problem, the main building block of the overall reactor model. Both porous and non-porous catalyst pellets are considered. The modelling of diffusion and chemical reaction in porous catalyst pellets is treated using two degrees of model sophistication, namely the approximate Fickian type description of the diffusion process and the more rigorous formulation based on the Stefan-Maxwell equations for diffusion in multicomponent systems. 

For non-porous catalyst pellets the reactants are chemisorbed on their external surface. However, for porous pellets the main surface area is distributed inside the pores of the catalyst pellets and the reactant molecules diffuse through these pores in order to reach the internal surface of these pellets. This process is usually called intraparticle diffusion of reactant molecules. The molecules are then chemisorbed on the internal surface of the catalyst pellets. The diffusion through the pores is usually described by Fickian diffusion models together with effective diffusivities that include porosity and tortuosity. Tortuosity accounts for the complex porous structure of the pellet. A more rigorous formulation for multicomponent systems is through the use of Stefan-Maxwell equations for multicomponent diffusion. Chemisorption is described through the net rate of adsorption (reaction with active sites) and desorption. Equilibrium adsorption isotherms are usually used to relate the gas phase concentrations to the solid surface concentrations. 

From the foregoing discussion, it is clear that the catalyst pellet is not only the heart of the catalytic reactor but also the hardest part of the system to model accurately. The difficulties associated with the modelling of the single pellet (especially the porous pellet) is due to the uncertainties associated with the intrinsic kinetics and the precise modelling of diffusion of mass and heat inside the pellet as well as the complex interaction between these two processes. The complex tortuous structure of porous catalyst pellet adds to the complexity. Different trials to estimate the tortuosity factor (which accounts for the complex tortuous structure of the pellet) theoretically have failed to give accurate results and this factor is usually estimated experimentally. 

By lowering the absolute reaction rate (e.g., low temperature) or by reducing the diffusion resistance (non-porous catalyst), the negative influence of mass transfer on the A/F window width can be counterbalanced. For the system studied here it has to be concluded that only a compromise between A/F window width in the lean range and absolute reaction rate can be attained. 

In any catalytic system not only chemical reactions per se but mass and heat transfer effects should be considered. For example, mass and heat transfer effects are present inside the porous catalyst particles as well as at the surrounding fluid films. In addition, heat transfer from and to the catalytic reactor gives an essential contribution to the energy balance. The core of modelling a two-phase catalytic reactor is the catalyst particle, namely simultaneous reaction and diffusion in the pores of the particle should be accounted for. These effects are completely analogous to reaction-diffusion effects in liquid films appearing in gas-liquid systems. Thus, the formulae presented in the next section are valid for both catalytic reactions and gas-liquid processes. 

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