Diffusion and catalytic reaction, isothermal

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

Most of the theory of diffusion and chemical reaction in gas-solid catalytic systems has been developed for these simple, unimolecular and irreversible reactions (SUIR). Of course this is understandable due to the obvious simplicity associated with this simple network both conceptually and practically. However, most industrial reactions are more complex than this SUIR, and this complexity varies considerably from single irreversible but bimolecular reactions to multiple reversible multimolecular reactions. For single reactions which are bimolecular but still irreversible, one of the added complexities associated with this case is the non-monotonic kinetics which lead to bifurcation (multiplicity) behaviour even under isothermal conditions. When the diffusivities of the different components are close to each other that added complexity may be the only one. However, when the diffusiv-ities of the different components are appreciably different, then extra complexities may arise. For reversible reactions added phenomena are introduced one of them is discussed in connection with the ammonia synthesis reaction in chapter 6. 

Reactants must diffuse through the network of pores of a catalyst particle to reach the internal area, and the products must diffuse back. The optimum porosity of a catalyst particle is deterrnined by tradeoffs making the pores smaller increases the surface area and thereby increases the activity of the catalyst, but this gain is offset by the increased resistance to transport in the smaller pores increasing the pore volume to create larger pores for faster transport is compensated by a loss of physical strength. A simple quantitative development (46—48) follows for a first-order, isothermal, irreversible catalytic reaction in a spherical, porous catalyst particle. 

Stoichacmetry and reaction equilibria. Homogeneous reactions kinetics. Mole balances batch, continuous-shn-ed tank and plug flow reactors. Collection and analysis of rate data. Catalytic reaction kinetics and isothermal catalytic radar desttpi. Diffusion effects. 

For the detailed study of reaction-transport interactions in the porous catalytic layer, the spatially 3D model computer-reconstructed washcoat section can be employed (Koci et al., 2006, 2007a). The structure of porous catalyst support is controlled in the course of washcoat preparation on two levels (i) the level of macropores, influenced by mixing of wet supporting material particles with different sizes followed by specific thermal treatment and (ii) the level of meso-/ micropores, determined by the internal nanostructure of the used materials (e.g. alumina, zeolites) and sizes of noble metal crystallites. Information about the porous structure (pore size distribution, typical sizes of particles, etc.) on the micro- and nanoscale levels can be obtained from scanning electron microscopy (SEM), transmission electron microscopy ( ), or other high-resolution imaging techniques in combination with mercury porosimetry and BET adsorption isotherm data. This information can be used in computer reconstruction of porous catalytic medium. In the reconstructed catalyst, transport (diffusion, permeation, heat conduction) and combined reaction-transport processes can be simulated on detailed level (Kosek et al., 2005). 

The conversion of cyclohexanes to aromatics is a highly endothermic reaction (AH 50 kcal./mole) and occurs very readily over platinum-alumina catalyst at temperatures above about 350°C. At temperatures in the range 450-500°C., common in catalytic reforming, it is extremely difficult to avoid diffusional limitations and to maintain isothermal conditions. The importance of pore diffusion effects in the dehydrogenation of cyclohexane to benzene at temperatures above about 372°C. has been shown by Barnett et al. (B2). However, at temperatures below 372°C. these investigators concluded that pore diffusion did not limit the rate when using in, catalyst pellets. 

We have used CO oxidation on Pt to illustrate the evolution of models applied to interpret critical effects in catalytic oxidation reactions. All the above models use concepts concerning the complex detailed mechanism. But, as has been shown previously, critical. effects in oxidation reactions were studied as early as the 1930s. For their interpretation primary attention is paid to the interaction of kinetic dependences with the heat-and-mass transfer law [146], It is likely that in these cases there is still more variety in dynamic behaviour than when we deal with purely kinetic factors. A theory for the non-isothermal continuous stirred tank reactor for first-order reactions was suggested in refs. 152-155. The dynamics of CO oxidation in non-isothermal, in particular adiabatic, reactors has been studied [77-80, 155]. A sufficiently complex dynamic behaviour is also observed in isothermal reactors for CO oxidation by taking into account the diffusion both in pores [71, 147-149] and on the surfaces of catalyst [201, 202]. The simplest model accounting for the combination of kinetic and transport processes is an isothermal continuously stirred tank reactor (CSTR). It was Matsuura and Kato [157] who first showed that if the kinetic curve has a maximum peak (this curve is also obtained for CO oxidation [158]), then the isothermal CSTR can have several steady states (see also ref. 203). Recently several authors [3, 76, 118, 156, 159, 160] have applied CSTR models corresponding to the detailed mechanism of catalytic reactions. 

If we analyze all the steps taking part in the process, the final result will be very complicated. The diffusion, adsorption, and desorption processes are fast enough in comparison with the chemical reaction. Therefore, the adsorption process is in equilibrium during the catalytic reaction, since it is a fast process, and as a result, we can use an adsorption isotherm, for example, the Langmuir isotherm to calculate the amount of reactant in the surface. 

The whole of the internal surface area of a porous catalyst will be available for the catalytic reaction if the rates of diffusion of reactant into the pores, and of product out of them, are fast compared with the rate of the surface reaction. In contrast, if the reactant diffuses slowly but reacts rapidly, conversion to product will occur near the pore entrances and the interior of the pores will play no role in the catalysis. Ion exchange resins are typical examples of catalysts for which such considerations are important (cf. Sect. 2.3). The detailed mathematics of this problem have been treated in several texts [49-51] and we shall now quote some of the main theoretical results derived for isothermal conditions. The parameters involved tend to be those employed by chemical engineers and differ somewhat from those used elsewhere in this chapter. In particular, the catalyst material (active + support) is present in the form of pellets of volume Vp and the catalytic rates vv are given per unit volume of pellet (mols m 3). The decrease in vv brought about by pore diffusion is then expressed by an effectiveness factor, rj, defined by... 

In many catalytic systems multiple reactions occur, so that selectivity becomes important. In Sec. 2-10 point and overall selectivities were evaluated for homogeneous well-mixed systems of parallel and consecutive reactions. In Sec. 10-5 we saw that external diffusion and heat-transfer resistances affect the selectivity. Here we shall examiineHEieHnfiuence of intrapellet res ahces on selectivity. Systems with first-order kinetics at isothermal conditions are analyzed analytically in Sec. 11-12 for parallel and consecutive reactions. Results for other kinetics, or for nonisothermal conditions, can be developed in a similar way but require numerical solution. ... 

A section of a fixed-bed catalytic reactor is shown in Fig. 13-4. Consider a small volume element of radius r, width Ar, and height Ar, through which reaction mixture flows isothermally. Suppose that radial and longitudinal diffusion can be expressed by Pick s law, with and Dj as effective diffusivities, based on the total (void and nonvoid) area perpendicular to the direction of diffusion. We want to write a mass balance for a reactant over the volume element. With radial and longitudinal diffusion and longitudinal convection taken into account, the input term is... 

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. 

DIFFUSION AND HETEROGENEOUS CHEMICAL REACTION IN ISOTHERMAL CATALYTIC PELLETS... 

What is the defining expression for the isothermal effectiveness factor in spherical catalysts Reactant A is consumed by three independent first-order irreversible chemical reactions on the interior catalytic surface. Your final expression should be based on mass transfer via diffusion and include the reactant concentration gradient at the external surface of the catalyst, where t) = 1. Define the intrapellet Damkohler number in your final answer. 

A quantitative strategy is discussed herein to design isothermal packed catalytic tubular reactors. The dimensionless mass transfer equation with unsteady-state convection, diffusion, and multiple chemical reactions represents the fundamental starting point to accomplish this task. Previous analysis of mass transfer rate processes indicates that the dimensionless molar density of component i in the mixture I must satisfy (i.e., see equation 10-11) ... 

A to products by considering mass transfer across the external surface of the catalyst. In the presence of multiple chemical reactions, where each iRy depends only on Ca, stoichiometry is not required. Furthermore, the thermal energy balance is not required when = 0 for each chemical reaction. In the presence of multiple chemical reactions where thermal energy effects must be considered becanse each AH j is not insignificant, methodologies beyond those discussed in this chapter must be employed to generate temperature and molar density profiles within catalytic pellets (see Aris, 1975, Chap. 5). In the absence of any complications associated with 0, one manipulates the steady-state mass transfer equation for reactant A with pseudo-homogeneous one-dimensional diffusion and multiple chemical reactions under isothermal conditions (see equation 27-14) ... 

Various parameters must be considered when selecting a reactor for multiphase reactions, such as the number of phases involved, the differences in the physical properties of the participating phases, the post-reaction separation, the inherent reaction nature (stoichiometry of reactants, intrinsic reaction rate, isothermal/ adiabatic conditions, etc.), the residence time required and the mass and heat transfer characteristics of the reactor For a given reaction system, the first four aspects are usually controlled to only a limited extent, if at aH, while the remainder serve as design variables to optimize reactor performance. High rates of heat and mass transfer improve effective rates and selectivities and the elimination of transport resistances, in particular for the rapid catalytic reactions, enables the reaction to achieve its chemical potential in the optimal temperature and concentration window. Transport processes can be ameliorated by greater heat exchange or interfadal surface areas and short diffusion paths. These are easily attained in microstructured reactors. 

Figure 1 shows the dependence effectiveness factor tj = roi/ri on Thiele modulus for the selected values of parameter p and for = 2, v = 2 (sphere), yt = 7, yi = 12, 5 = 1 (endothermic reactions) and xa(1) = 1. One can find that for Peffectiveness factor rj assumes in the certain ranges of Thiele modulus values much higher than unity. This means that in the cases discussed the internal diffusion, in contrast to the classical isothermal or endothermic catalytic reactions, may considerably increase the rate of the heterogeneous autocatalytic reactions. 

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