Knudsen diffusivity parallel pores

Blue et al. have studied the dehydrogenation of butane at atmospheric pressure, using a chromia-alumina catalyst at 530°C. For a spherical catalyst size of dp = 0.32 cm the experimental data suggest a first-order rate constant of about 0.94 cm /(sec)(g catalyst). The pore radius is given as 110 A. Assuming Knudsen diffusivity at this low pressure and estimating the pore volume as 0.35 cm /g, predict an effectiveness factor for the catalyst. Use the parallel-pore model with a tortuosity factor of 2.0. 

The net diffusivity of component A within the pores of a catalytic pellet is obtained by adding mass transfer resistances for Knudsen diffusion and ordinary molecular diffusion, where convection reduces the resistance due to ordinary molecular diffusion but Knudsen flow occurs over length scales that are much too small for convective mass transfer to be important. This addition of resistances is constructed to simulate resistances in series, not parallel. Consider the trajectory of a gas molecule that collides with the walls of a channel or other gas molecules. In the pore-size regime where Knudsen and ordinary molecular diffusion are equally probable, these collisions occur sequentially, which suggests that gas molecules encounter each of these resistances in series. Hence, for binary mixtures. 

The parallel transport model considers the total flux as the contribution from the molecules travelling via surface diffusion and from the molecules travelling via Knudsen diffusion [27,36,49,50]. This model does not consider transition stages and is applicable to pores that remain roughly the same size throughout the entire membrane such as nanotube-based membranes. Gilron and Softer [27] presented the following expression. 

The parallel transport model assumes that surface diffusion and Knudsen diffusion are occurring simultaneously such that the total permeability is given by Equation (5.18). This model is explained in further detail earlier in this chapter and has been used by various groups [27,49,50]. Parallel transport is initially dominated by surface diffusion within the smaller pores where the surface concentration is high while the mode of Knudsen diffusion dominates within the larger pores. 

Figure 5.12 Model prediction of permeability as a function of temperature. Modes of transport are indicated for the following pore sizes activated diffusion (d = 6.8 A), surface diffusion (d = 10 A), Knudsen diffusion (d = 10 A), parallel transport (d = 10A), and resistance In series transport (d rraii = 6.8A, d/a,ge = loA, Xk = 0.8)...

Effective diffusion coefficients in catalyst particles are calculated as functions of bulk gas diffusion coefficients, pore volume distribution specified as particle porosity, 8p, as a function of pore radius and the so-called tortuosity factor, x, which describes the actual road a molecule must travel. The use of different effective diffusion models is discussed in the literature [199] [436] and performance of measurements in [221], Below is shown the basic parallel pore model, where the effective diffusion coefficient, De is calculated from the particle porosity, the tortuosity factor, and the diffusion coefficient in the bulk and the Knudsen diffusion coefficient, Dbuik and Dk [199] [389] [440] as ... 

In an intermediate range of pore sizes, both ordinary and Knudsen diffusion contribute to the transfer of the species in the media. In this range, the combined ordinary and Knudsen diffusion can be represented by assuming parallel resistances and expressed as... 

For the case of pellet, the presence of uneven pore size as well as the interconnection of pores and tortuous path of diffusion, the flux expression for the pellet case has to be derived from that for a cylindrical capillary (eq. 13.2-1) through some model about the structure of the pellet. One such model is the parallel path pore model and in this model the effective flux is calculated by summing the combined Knudsen and molecular binary diffusions over each increment of pore volume. The expression for the steady state effective flux is ... 

CH4 because of its lighter mass resulting in a higher molecular velocity and does not change with pore size. Parallel transport follows the same trend as surface diffusion in small pores and tends toward Knudsen behaviour as the pore sizes increase. Finally, the resistance in series transport model predicts a decrease in selectivity as the permeability of CH4 increases more rapidly than for CO2 with increasing pore size. 

The Transition Region For certain pore sizes, diffusion will not be purely molecular or purely Knudsen. Both types of diffusion will contribute to the overall flux. The theory for this transition region is complex. A workable approach is to assume that the two types of diffusion occur in parallel. This leads to... 

The pore size diameters where Knudsen flow dominates are below 0.05 pm, which is below the normal pore size seen in fuel cells. However, Knudsen flow can play a role within the catalyst layers of many fuel cells, in parallel with normal diffusion processes. 

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