Fig. 9.4 Transport processes in RD with a solid non-porous catalytic packing condition on the liquid-solid mass-transfer process. In some cases one or more of these steps may be rate determining allowing some simplification in the modeling of the transport and reaction processes.
In many more cases, however, the cases we have a porous catalyst that is supported in some structure within the walls of the column. In these processes the list of transport and reaction processes is even longer than that above. As illustrated in Fig. 9.5, we must consider:
- transport of reactants from the bulk vapor to the bulk liquid,
- transport of reactants from the bulk liquid to the catalyst surface,
- diffusion with simultaneous reaction within the porous catalyst,
- transport of reaction products from the catalyst surface to the bulk liquid, and
- transport of the reaction products from the bulk liquid to the bulk vapor.
The first of these steps can be modeled using the Maxwell-Stefan equations as already discussed above. There are two approaches to modeling the remaining steps: a pseudo-homogeneous model or a heterogeneous model.
The first, and simplest, approach is to treat the reaction pseudo-homogeneously, whereby catalyst diffusion (including diffusion to the catalyst surface) and reaction are lumped into an overall reaction term. For heterogeneous reactions that are modeled in this way the liquid phase material balance is as given above and sj is given by the total amount of catalyst present on the stage under consideration. In this case, one only needs to specify catalyst mass and activity.
A more rigorous approach to modeling heterogeneous systems involving porous catalyst particles (Fig. 9.5) would require a complete description of mass transport to the catalyst and diffusion and reaction inside the catalyst particles.
Modeling multi-component mass transfer in porous media is complicated when the mean-free-path length of the molecules is of the order of magnitude of the pore diameter. The difficulties posed by this case may be circumvented by a method
Fig. 9.5 Transport processes in RD with a porous catalytic packing iihTl
Fig. 9.5 Transport processes in RD with a porous catalytic packing iihTl originally introduced by Maxwell in 1866 and developed further by Mason et al. . Maxwell suggested that the porous material itself be described as a supplementary 'dust' species, consisting of very large molecules that are kept motionless by some unspecified external force. The Chapman-Enskog kinetic theory is then applied to the new pseudo-gas mixture, in which the interaction between the dust and gas molecules simulates the interaction between the solid matrix and the gas species. In addition, one is no longer faced with the problem of flux and composition variations across a pore and problems related to catalyst geometry.
The dusty fluid model as developed by Krishna and Wesselingh  is a modification of the dusty gas model so as to be able to model liquid phase diffusion in porous media. For a non-ideal mixture we have
Dl is the effective Knudsen diffusion coefficient for species i in the porous catalyst.
The mass-transfer rates are obtained by multiplying the fluxes by the interfacial area of the catalyst particles. This is not as straightforward, as it looks, since, depending on the geometry of the catalyst, the cross-sectional area can change along the diffusion path. It is necessary to take catalyst geometry into account in the solution of these equations.
The dusty gas model is often used as the basis for the calculation of a catalyst effectiveness factor in chemical reactor analysis. The extension to non-ideal fluids noted above and its application in RD modeling has not been used as often, partly because of the somewhat greater uncertainty in the parameters that appear in the equations [1, 11].
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