Diffusion and reaction in pores. Effectiveness

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... 

A Rodrigues, R Quinta Ferreira, Effect of intraparticle convection, diffusion and reaction in a large-pore catalyst particle , AlChE Symp Ser, 1988, 84, 80-87... 

In terms of catalysis, important equilibrium processes include low-temperature gas adsorption (capillary condensation) and nonwetting fluid invasion, both of which are routinely used to characterize pore size distribution. Static diffusion in a Wicke-Kallenbach cell characterizes effective diffusivity. The simultaneous rate processes of diffusion and reaction determine catalyst effectiveness, which is the single most significant measure of practical catalytic reactor performance. 

In our discussion of surface reactions in Chapter 11 we assumed that each point in the interior of the entire catalyst surface was accessible to the same reactant concentration. However, where the reactants diffuse into the pores within the catalyst pellet, the concentration at the pore mouth will be higher than that inside the pore, and we see that the entire catalytic surface is not accessible to the same concentration. To account for variations in concentration throughout the pellet, we introduce a parameter known as the effectiveness factor. In this chapter we will develop models for diffusion and reaction in two-phase systems, which include catalyst pellets and CVD reactors. The types of reactors discussed in this chapter will include packed beds, bubbling fluidized beds, slurry reactors, and trickle beds. After studying this chapter you will be able to describe diffusion and reaction in two- and three-phase systems, determine when internal pore diffusion limits the overall rate of reaction, describe how to go about eliminating this limitation, and develop models for systems in which both diffusion and reaction play a role (e.g., CVD). 

Solution You should use diffusion coefficients to describe the simultaneous diffusion and reaction in the pores in the catalyst. You should not use mass transfer coefficients because you cannot easily include the effect of reaction (see Sections 16.1 and 17.1). 

Reaction rates of nonconservative chemicals in marine sediments can be estimated from porewater concentration profiles using a mathematical model similar to the onedimensional advection-diffusion model for the water column presented in Section 4.3.4. As with the water column, horizontal concentration gradients are assumed to be negligible as compared to the vertical gradients. In contrast to the water column, solute transport in the pore waters is controlled by molecular diffusion and advection, with the effects of turbulent mixing being negligible. 

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]. 

Information relating to the diffusion of metal-bearing compounds in catalytic materials at reaction conditions has been obtained indirectly through classic diffusion and reaction theory. Shah and Paraskos (1975) calculated effective diffusitivities of 7 x 10-8 and 3 x 10-8 cm2/sec for V and Ni compounds in reduced Kuwait crude at 760°F. These low values may be indicative of a small-pore HDS catalyst. In contrast, Sato et al. (1971) report that the effective diffusivity of vanadium compounds was one-tenth that of the nickel compounds on the basis of metal deposition profiles in aged catalysts. This large difference may be influenced by relative adsorption strengths not explicitly considered in their analysis. 

In this section we have presented and solved the BVPs associated with the diffusion and reaction that take place in the pores of a porous catalyst pellet. The results were expressed graphically in terms of the effectiveness factor rj versus the Thiele modulus d> for two cases One with negligible external mass and heat transfer resistances, i.e., when Sh and Nu —> oo, and another with finite Sh and Nu values. This problem is very important in the design of fixed-bed catalytic reactors. The sample results presented here have shown that for exothermal reactions multiple steady states may occur over a range of Thiele moduli d>. Efficient numerical techniques have been presented as MATLAB programs that solve singular two-point boundary value problems. 

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. 

A gas-solid reaction usually involves heat and mass transfer processes and chemical kinetics. One important factor which complicates the analysis of these processes is the variations in the pore structure of the solid during the reaction. Increase or decrease of porosity during the reaction and variations in pore sizes would effect the diffusion resistance and also change the active surface area. These facts indicate that the real mechanism of gas-solid noncatalytic reactions can be understood better by following the variations in pore structure during the reaction. 

Basically, reactant and product selectivities are mass transfer effects, where the diffusivities of the various species in practice frequently do not differ that extremely as indicated above. Instead, in most cases only a preferred diffusion of certain species is observed, a fact which often hinders a clear understanding of product shape selectivity. This is because the various products, during their way through the pore system, may be reacted when contacting the catalytically active surface of the wall. This combined effect of diffusion and reaction will be discussed in detail in the following, as it is of great importance for the product distribution in zeolite-catalyzed reactions. 

Wei [107] in 1982 was the first to come up with a continuous pseudohomogeneous model which allowed to simulate shape-selective effects observed during the alkylation of toluene using methanol to yield xylene isomers on a HZSM-5 catalyst. He treated diffusion and reaction of the xylene isomers inside the pores in a one-dimensional model. The isomer concentration at the pore mouth was set to zero, as a boundary condition. This allowed the model equations to be solved analytically, but it also limited the application of the results to small conversions. 

Therefore, the section shown in Figure 15, and the pore network in 3-D from which it arises, are both absolutely defined in a quantitative way. Inasmuch as the 3-D networks are felt to be a realistic representation of random pore spaces, it is feasible to compute directly several important macroscopic properties for the FCC powder particles. Amongst these properties are permeability and effective difflisivity, so that diffusion and reaction calculations relevant to gas-oil cracking in the FCC particles can be directly undertaken. Also important in this respect are calculations of deactivation due to coke laydown within the particles. It is also possible that the pore networks could be used to deduce strength and abrasion resistance of the particles. 

When gum formation proceeds, the minimum temperature in the catalyst bed decreases with time. This could be explained by a shift in the reaction mechanism so more endothermic reaction steps are prevailing. The decrease in the bed temperature speeds up the deactivation by gum formation. This aspect of gum formation is also seen on the temperature profiles in Figure 9. Calculations with a heterogenous reactor model have shown that the decreasing minimum catalyst bed temperature could also be explained by a change of the effectiveness factors for the reactions. The radial poisoning profiles in the catalyst pellets influence the complex interaction between pore diffusion and reaction rates and this results in a shift in the overall balance between endothermic and exothermic reactions. 

A great number of studies have been published to deal with relation of transport properties to structural characteristics. Pore network models [12,13,14] are engaged in determination of pore network connectivity that is known to have a crucial influence on the transport properties of a porous material. McGreavy and co-workers [15] developed model based on the equivalent pore network conceptualisation to account for diffusion and reaction processes in catalytic pore structures. Percolation models [16,17] are based on the use of percolation theory to analyse sorption hysteresis also the application of the effective medium approximation (EMA) [18,19,20] is widely used. 

A Bethe-tree is a particular case of more general networks considered in percolation theory. Sahimi and Tsotsis [1985] applied percolation theory and Monte Carlo simulation to deactivation in zeolites, approximated by a simple cubic lattice. Beyne and Froment [1990, 1993] applied percolation theory to reaction, diffusion and deactivation in the real ZSM-5 lattice. The finite rate of growth was described in terms of a polymerization mechanism. Pore blockage was reached in this small pore zeolite. It also affects the path followed by the diffusing molecules that becomes more tortuous, so that the effective diffusivity has to be expressed in terms of the blockage probability. 

The equations for simultaneous pore diffusion and reaction were solved independently by Thiele and by Zeldovitch [16,17]. They assumed a straight cylindrical pore with a first-order reaction on the surface, and they showed how pore length, diffusivity, and rate constant influenced the overall reaction rate. Their solution cannot be directly adapted to a catalyst pellet, since the number of pores decreases going toward the center and assuming an average pore length would introduce some error. The approach used here is that of Wheeler [18] and Weisz [19], who considered reactions in a porous sphere and related the diffusion flux to the effective diffusivity, Z). The basic equation is a material balance on a thin shell within the sphere. The difference between the steady-state flux of reactant into and out of the shell is the amount consumed by reaction. 

When diffusion and reaction occur simultaneously within a porous solid structure, concentration gradients of reactant and product species are established. If the various diffusional processes discussed in Section 12.2 are rapid compared with the chemical reaction rate, the entire accessible internal surface of the catalyst will be effective in promoting reaction because the reactant molecules will spread essentially uniformly throughout the pore structure, before they have time to react. Here only a small concentration gradient will exist between the exterior and interior of the particle, and there will be diffusive fluxes of reactant molecules in and product molecules out that suffice to balance the reaction rate within the particle. In... 

The effects of capillary condensation were included in the network model, by calculating the critical radius below which capillary condensation occurs based on the vapor composition in each pore using the multicomponent Kelvin Equation (23.2). Then the pore radius was compared with the calculated critical radius to determine whether the pore is liquid- or vapor-filled. As a significant fraction of pores become filled with capillary condensate, regions of vapor-filled pores may become locked off from the vapor at the network surface by condensate clusters. A Hoshen and Kopelman [30] algorithm is used to identify vapor-filled pores connected to the network surface, in which diffusion and reaction continue to take place after other parts of the network filled with liquid. It was assumed that, due to the low hydrogen solubility in the liquid, most of the reaction takes place in the gas-filled pores. The diffusion/reaction simulation is repeated, including only vapor-filled pores connected to the network surface by a pathway of other vapor-filled pores. 

The processes are transport of the gaseous reactant A in the vicinity of the liquid, transfer of this reactant through the gas-liquid interface into the liquid, transport of the dissolved gaseous reactant and of the liquid reactant B in the vicinity of the solid, transfer of both reactants at the external surface of the catalyst, through the liquid-solid interface, diffusion and reaction into the pores of the catalyst. The product undergoes similar mass transfer steps from the catalytic active sites to the liquid phase. In the case of a reaction with a heat effect, heat transfer steps associated to the proceeding mass transfer steps have to be considered. 

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