Diffusion, bulk Knudsen

Despite the fact Chat there are no analogs of void fraction or pore size in the model, by varying the proportion of dust particles dispersed among the gas molecules it is possible to move from a situation where most momentum transfer occurs in collisions between pairs of gas molecules, Co one where the principal momentum transfer is between gas molecules and the dust. Thus one might hope to obtain at least a physically reasonable form for the flux relations, over the whole range from bulk diffusion to Knudsen streaming. 

It remains to check Chat our supposedly general equations (11.20) -(11.23) reduce to the previously obtained limiting forms at the extremes of bulk diffusion and Knudsen diffusion. The bulk diffusion limit is easily obtained by letting — and assuming that for all r, s,... 

As a result, collisions with the wall occur more frequently than with other molecules. This is referred to as the Knudsen mode of diffusion and is contrasted with ordinary or bulk diffusion, which occurs by intermolecular collisions. At intermediate pressures, both ordinaiy diffusion and Knudsen diffusion may be important [see Eqs. (5-239) and (5-240)]. 

By comparing the relative magnitude of the mean free path (z) and the pore diameter (27), it is possible to determine whether bulk diffusion or Knudsen diffusion may be regarded as negligible. Using the principles of the kinetic theory... 

Effective diffusivity. Resistance to diffusion in a catalyst pore is due to collisions with other molecules and with the walls of the pore. The corresponding diffusivities are called bulk diffusivity and Knudsen diffusivity DK. Many data and correlations of the former type exist the latter is calculable from the following formula (Satterfield, 1970, p. 42) ... 

In the device of Fig.6a (sensor with distributed cavity), depending on the dimensions of the pores of the diffusion barrier, the oxygen diffusion can be bulk diffusion or Knudsen diffusion(ll). In the former case, the sensor output (limiting current Ig) is given by Eq. (9). Knudsen diffusion occurs when the average pore diameter is much smaller than the mean free path of the gas molecules, in which case collisions between molecules and pore walls are the dominant events. In this case(121. Dq -k T /2,and... 

On the other hand, the more rigorous Maxwell-Stefan equations and the dusty gas model are seldom used in industrial reaction engineering applications. Nevertheless, the dusty gas model [64] represents a modern attempt to provide a more realistic description of the combined bulk and Knudsen diffusion mechanisms based on the multicomponent Maxwell-Stefan model formulation. Similar extensions of the Maxwell-Stefan model have also been suggested for the surface diffusion of adsorbed molecular pseudo-species, as well as the combined bulk, Knudsen and surface diffusion apparently with limited success [48] [49]. 

Basic Equations For many catalysts and reaction conditions (especially pressure) both bulk and Knudsen diffusion contribute to the mass-transport rate within the pore volume. For some years the proper combination of the two mechanisms was in doubt. About 1961 three independent investigations proposed identical equations for the rate of diffusion (in a binary gaseous mixture of A and B) in terms of the bulk diffusivity and Knudsen diffusivity If is the molal flux of A, it is convenient to represent the result as... 

A simpler approach is possible if either bulk diffusion or Knudsen diffusion is dominant at both ambient (reference) conditions and reaction conditions. For Knudsen flow, the effective diffusivity for any component can be calculated just from the ratio of temperatures and molecular weights, as indicated by Eq. (4.7). When He is the test gas. 

One or more of several different mechanisms may be responsible for the mass transfer process. These include ordinary bulk diffusion, Knudsen diffusion, surface diffusion, and bulk flow. For the majority of the catalysts and conditions used in industrial practice, the only significant mechanisms are bulk diffusion and Knudsen diffusion. The relative importance of these two processes depends on the relative values of the mean free path and the pore dimensions. 

The basic requirement in any study of internal diffusion is an understanding of the various modes of transport in a straight capillary bulk, Knudsen, configurational, and surface. Then this knowledge is extended to diffusion in the porous matrix of a pellet to formulate expressions for an effective diffusion coefficient. We confine our treatment in this book to listing in Table 7.3 the more important equations for direct use in estimation. Given in this table are the equations for bulk (for macropores with > 200 A), Knudsen (for micro-... 

Maxwell-Stefan Formulation for Bulk-Knudsen Diffusion in a Capillary... 

The following table summarises the formulas for the combined bulk-Knudsen diffusion. 

Table 8.6-1 Equation fonnulas for the case of bulk-Knudsen diffusion...

Stefan-Maxwell Approach for Bulk-Knudsen Diffusion in Complex System of Capillaries... 

We have presented the various constitutive flux equations for the general case of combined bulk-Knudsen and viscous flow in the last section. Now let us illustrate its application to the simple case of transient flow of a diffusing mixture in a capillary exposed to an infinite environment. 

Maxwell-Stefan formulation for bulk-Knudsen diffusion in capillary 470... 

Stefan-Maxwell approach for bulk-Knudsen diffusion in complex. . 487... 

The relative importance of intermolecular collisions and molecular-wall collisions can be characterized by Knudsen number Kn = T/mean free path and rfpore is the average pore diameter. When Kn bulk diffusion is dominant and when Kn 1 Knudsen diffusion is dominant. When Kn - both bulk diffusion and Knudsen diffusion have to be taken into account, which typically is the situation for SOFCs. 

Many reaction conditions, particularly the pressure, lead to the existence of diverse molecular concentrations in pore spaces where both bulk diffusion and Knudsen diffusion can prevail concurrently. In the so-called transition region, both mass dif-fusivities, (Dab) and Dk)a> are included in the formulation of a combined mass diffusivity, Dcomb- In the analysis of mass dif-fusivities, the key parameter to be considered is the size of the pore with respect to the mean free path of the molecules. 

The basic requirement in any study of internal diffusion is an understanding of the various modes of transport in a straight capillary bulk, Knudsen, configurational, and surface. This knowledge is then extended... 

Surface diffusion of adsorbed molecular species along the pore wall surface. This mechanism of transport becomes dominant for micro-pores and for strongly adsorbed species. Extensions of the Maxwell-Stefan bulk diffusion model have been proposed in order to provide a realistic description of the combined bulk, Knudsen and surface diffusion mechanisms, apparently with limited success [70, 71]. 

Burghardt and Aerts [12] proposed a method for evaluation of the pressure change in an isothermal porous pellet within which a single chemical reaction takes place, accompanied by mass transfer by Knudsen diffusion, bulk diffusion and viscous convective flow of the reacting mixture. The pressure change did also depend on the reaction and on the mixture composition on the pellet surface. It was concluded that the pressure changes in a catalyst pellet under conditions normally encountered in industry are most likely so small that they can be neglected in process simulations. 

Maxwellian diffusion (bulk molecular diffusion) in moderately large pores (macropores) or Knudsen diffusion in pores (micropores) which have a diameter smaller than the mean free path of the adsorbate molecules ... 

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