Pellets, mass transfer diffusion

This study was carried out to simulate the 3D temperature field in and around the large steam reforming catalyst particles at the wall of a reformer tube, under various conditions (Dixon et al., 2003). We wanted to use this study with spherical catalyst particles to find an approach to incorporate thermal effects into the pellets, within reasonable constraints of computational effort and realism. This was our first look at the problem of bringing together CFD and heterogeneously catalyzed reactions. To have included species transport in the particles would have required a 3D diffusion-reaction model for each particle to be included in the flow simulation. The computational burden of this approach would have been very large. For the purposes of this first study, therefore, species transport was not incorporated in the model, and diffusion and mass transfer limitations were not directly represented. 

The relative importance of the boundary-film and intra-pellet diffusion in mass transfer is measured by the Biot number. On the assumption that there is no accumulation of adsorbate at the external surface of a pellet, then ... 

Provided that the catalyst is active enough, there will be sufficient conversion of the pollutant gases through the pellet bed and the screen bed. The Sherwood number of CO is almost equal to the Nusselt number, and 2.6% of the inlet CO will not be converted in the monolith. The diffusion coefficient of benzene is somewhat smaller, and 10% of the inlet benzene is not converted in the monolith, no matter how active is the catalyst. This mass transfer limitation can be easily avoided by forcing the streams to change flow direction at the cost of some increased pressure drop. These calculations are comparable with the data in Fig. 22, taken from Carlson 112). 

A hydrocarbon is cracked using a silica-alumina catalyst in the form of spherical pellets of mean diameter 2.0 mm. When the reactant concentration is 0.011 kmol/m3, the reaction rate is 8.2 x 10"2 kmol/(m3 catalyst) s. If the reaction is of first-order and the effective diffusivity De is 7.5 x 10 s m2/s, calculate the value of the effectiveness factor r). It may be assumed that the effect of mass transfer resistance in the. fluid external Lo the particles may be neglected. 

In order to verify that the fixed bed and the micro-channel reactor are equivalent concerning chemical conversion, an irreversible first-order reaction A —) B with kinetic constant was considered. For simplicity, the reaction was assumed to occur at the channel surface or at the surface of the catalyst pellets, respectively. Diffusive mass transfer to the surface of the catalyst pellets was described by a correlation given by Villermaux [115]. 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

Bulk or forced flow of the Hagan-Poiseuille type does not in general contribute significantly to the mass transport process in porous catalysts. For fast reactions where there is a change in the number of moles on reaction, significant pressure differentials can arise between the interior and the exterior of the catalyst pellets. This phenomenon occurs because there is insufficient driving force for effective mass transfer by forced flow. Molecular diffusion occurs much more rapidly than forced flow in most porous catalysts. 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Surface Diffusion Rate Controlled At high gas velocities, the pellet surface byproduct concentration is maintained at an equilibrium value determined by the by-product concentration in the gas. In this condition, the mass transfer from the surface is balanced by the diffusion within the pellet to the surface. However... 

Micropore mass transfer resistance of zeoUte crystals is quantified in units of time by r /Dc, where is the crystal radius and Dc is the intracrystalline diffusivity. In addition to micropore resistance, zeolitic catalysts may offer another type of resistance to mass transfer, that is resistance related to transport through the surface barrier at the outer layer of the zeoHte crystal. Finally, there is at least one additional resistance due to mass transfer, this time in mesopores and macropores Rp/Dp. Here Rp is the radius of the catalyst pellet and Dp is the effective mesopore and macropore diffusivity in the catalyst pellet [18]. 

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... 

Figure 7-7 Reactant concentration profiles in directionx, which is perpendicular to the flow direction z expected for flow over porous catalyst pellets in a packed bed or sluny reactor. External mass transfer and pore diffusion produce the reactant concentration profiles shown.

In connection with multiphase diffusion another poorly understood topic should be mentioned—namely, the diffusion through porous media. This topic is of importance in connection with the drying of solids, the diffusion in catalyst pellets, and the recovery of petroleum. It is quite common to use Fick s laws to describe diffusion through porous media fJ14). However, the mass transfer is possibly taking place partly by gaseous diffusion and partially by liquid-phase diffusion along the surface of the capillary tubes if the pores are sufficiently small, Knudsen gas flow may prevail (W7, Bl). 

There are three distinct mass-transfer resistances (1) the external resistance of the fluid film surrounding the pellet, (2) the diffusional resistance of the macropores of the pellet, and (3) the diffusional resistance of the zeolite crystals. The external mass-transfer resistance may be estimated from well-established correlations (4, 5) and is generally negligible for molecular sieve adsorbers so that, under practical operating conditions, the rate of mass transfer is controlled by either macropore diffusion or zeolitic diffusion. In the present analysis we consider only systems in which one or other of these resistances is dominant. If both resistances are of comparable importance the analysis becomes more difficult. 

If zeolitic diffusion is sufficiently rapid so that the sorbate concentration through any particular crystal is essentially constant and in equilibrium with the macropore fluid just outside the crystal, the rate of mass transfer will be controlled by transport through the macropores of the pellet. Transport through the macropores may be assumed to occur by a diffusional process characterized by a constant pore diffusion coefficient Z)p. The relevant form of the diffusion equation, neglecting accumulation in the fluid phase within the macropores which is generally small in comparison with accumulation within the zeolite crystals, is... 

Using the computer programs discussed above, it is possible to extract from these breakthrough curves the effective local mass transfer coefficients as a function of CO2 concentration within the stable portion of the wave. These mass transfer coefficients are shown in Figure 15, along with the predicted values with and without the inclusion of the surface diffusion model. It is seen that without the surface diffusion model, very little change in the local mass transfer coefficient is predicted, whereas with surface diffusion effects included, a more than six-fold increase in diffusion rates is predicted over the concentrations measured and the predictions correspond very closely to those actually encountered in the breakthrough runs. Further, the experimentally derived results indicate that, for these runs, the assumption that micropore (intracrystalline) resistances are small relative to overall mass transfer resistance is justified, since the effective mass transfer coefficients for the two (1/8" and 1/4" pellets) runs scale approximately to the inverse of the square of the particle diameter, as would be expected when diffusive resistances in the particle macropores predominate. 

Mass transfer through the catalyst pellet occurs by diffusion and only ordinary molecular and Knudsen20 diffusion are considered. 

However, nonmonotonic kinetics alone will not produce multiplicity. Such kinetics have to be coupled with a diffusion process, either in the form of a mass-transfer resistance between the catalyst pellet surface and the bulk gas, or within the pores of the pellets. If the flow conditions and the catalyst pellets size are such that diffusional resistances between the bulk gas phase and the catalytic active centers are negligible and the... 

Comparisons of estimated diffusivity values on zeolites from sorption uptake measurements and those obtained from direct measurements by nuclear magnetic resonance field gradient techniques have indicated large discrepancies between the two for many systems [10]. In addition, the former method has often resulted in an adsorbate diffusivity directly proportional to the adsorbent crystal size [11]. This led some researchers to believe that the resistance to mass transfer may be confined in a skin at the surface of the adsorbent crystal or pellet (surface barrier) [10,11]. The isothermal surface barrier model, however, failed to describe experimental uptake data quantitatively [10,12]. 

Diffusional mass transfer processes can be essential in complex catalytic reactions. The role of diffusion inside a porous catalyst pellet, its effect on the observed reaction rate, activation energy, etc. (see, for example, ref. 123 and the fundamental work of Aris [124]) have been studied in detail, but so far several studies report only on models accounting for the diffusion of material on the catalyst surface and the surface-to-bulk material exchange. We will describe only some macroscopic models accounting for diffusion (without claiming a thorough analysis of every such model described in the available literature). 

FIGURE 4 Schematic diagram of a biporous adsorbent pellet showing the three resistances to mass transfer (external fluid film, macropore diffusion, and micropore diffusion). R9 pellet radius rc crystal radius. 

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