Thiele catalytic model

Ihe Thiele-Zeldowitsch model presented in die literature to describe the limitation of the catalytic activity by internal mass transport does not take into account die surface roughness. At temperatures higher than those where the reaction is starting to be trans-port-limited, the reaction front withdraws to the outermost layer of the catalyst body, viz., to a layer of a thickness of 10-100 mm. This layer is rough, and therefore very well accessible from die gas phase. The activity of the catalyst is consequently best described as controlled by external diffusion. Thus, at low gas velocities and high reaction rates, the catalytic activity is controlled by the miaoscopic surface roughness of the catalyst particles. 

The analysis of the literature data shows that zeolites modified with nobel metals are among perspective catalysts for this process. The main drawbacks related to these catalysts are rather low efficiency and selectivity. The low efficiency is connected with intracrystalline diffusion limitations in zeolitic porous system. Thus, the effectiveness factor for transformation of n-alkanes over mordenite calculated basing on Thiele model pointed that only 30% of zeolitic pore system are involved in the catalytic reaction [1], On the other hand, lower selectivity in the case of longer alkanes is due to their easier cracking in comparison to shorter alkanes. 

IS a modified Damkohler number = A nhsCno ts the dimensionless NH3 adsoiption constant, D, is the molecular diffusivity of species 1 is the effective intraporous diffusivity of species i evaluated according to the Wakao-Smith random pore model [411. Equation (4) is taken from Ref. 39. Equations (6)-(8) provide an approximate analytical solution of the intraporous diffusion-reaction equations under the assumption of large Thiele moduli (i.e., the concentration of the limiting reactant is zero at the centerline of the catalytic wall) the same equations are solved numencally in Ref. 36. 

Both studies show that at relatively low temperatures, i.e., during ignition of the catalyst, the rate-limiting step shifts from chemical kinetics to diffusion in the washcoat. This is clear from Fig. 7, computed using a one-dimensional model by Nakhjavan et al. [54]. Figure 7A shows the Thiele modulus and Fig. 7B an external diffusion limiting factor F versus dimensionless axial position in the reactor at various times on-stream for the catalytic combustion of propene in monolith reactors. The time is defined as the time after injection of the fuel in a preheated air flow. 

Sousa et al [5.76, 5.77] modeled a CMR utilizing a dense catalytic polymeric membrane for an equilibrium limited elementary gas phase reaction of the type ttaA +abB acC +adD. The model considers well-stirred retentate and permeate sides, isothermal operation, Fickian transport across the membrane with constant diffusivities, and a linear sorption equilibrium between the bulk and membrane phases. The conversion enhancement over the thermodynamic equilibrium value corresponding to equimolar feed conditions is studied for three different cases An > 0, An = 0, and An < 0, where An = (ac + ad) -(aa + ab). Souza et al [5.76, 5.77] conclude that the conversion can be significantly enhanced, when the diffusion coefficients of the products are higher than those of the reactants and/or the sorption coefficients are lower, the degree of enhancement affected strongly by An and the Thiele modulus. They report that performance of a dense polymeric membrane CMR depends on both the sorption and diffusion coefficients but in a different way, so the study of such a reactor should not be based on overall component permeabilities. 

Thiele, E.W., 1939. Relation between catalytic activity and size of particle. Ind. Eng. Chem. 31, 916-920. Vasquez, J.L., 2006. The Porous Medium Equations Mathematical Theory. Oxford University Press, Oxford. Yablonsky, G.S., Constales, D., Gleaves, J.T., 2002. Multi-scale problems in the quantitative characterization of complex catalytic materials. Syst. Anal. Model. Simul. 42, 1143-1166. 

The first reduction of the two-scale model that we consider is already included in the equations from Table 3.1 and refers to the reaction-diffusion problem inside catalytic particles. The treatment of this question based on the effectiveness factor concept (rf) is widely generalized in the literature, after the seminal works of Damkohler [77], Thiele [78], and Zeldovitch [79]. It may be defined for a reaction j with respect to the conditions prevailing at the pellet surface by Ref. [80] ... 

The methods used for modeling-supported PTC systems are all based on the standard equations developed for porous catalysts in heterogeneous catalysis (Chapter 6). These are expressed in terms of an overall effectiveness factor that accounts both for the mass transfer resistances outside the supported catalyst particles (film diffusion resistance, expressed as a Biot number) and within them (intraparticle diffusional resistance, expressed in terms of a Thiele modnlns). Then, for any given solid shape, the catalytic effectiveness factor can be derived as a function of the Thiele modulus A. Thus, for a spherical support solid, we have... 

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