Knudsen diffusion structure

For the same reaction in a pellet of finely porous structure, where Knudsen diffusion controls, the appropriate dynamical equations sre (12.20) and (12.21) if we once more adopt approximations which are a consequence of Che large size of K. These again have a dimensionless form, which may be written... 

The internal structure of the catalyst particle is often of a complex labyrinth-like nature, with interconnected pores of a multiplicity of shapes and sizes, In some cases, the pore size may be less than the mean free path of the molecules, and both molecular and Knudsen diffusion may occur simultaneously. Furthermore, the average length of the diffusion path will be extended as a result of the tortuousity of the channels. In view of the difficulty of precisely defining the pore structure, the particle is assumed to be pseudo-homogeneous in composition, and the diffusion process is characterised by an effective diffusivity D, (equation 10.8). 

The ratio of the overall rate of reaction to that which would be achieved in the absence of a mass transfer resistance is referred to as the effectiveness factor rj. SCOTT and Dullion(29) describe an apparatus incorporating a diffusion cell in which the effective diffusivity De of a gas in a porous medium may be measured. This approach allows for the combined effects of molecular and Knudsen diffusion, and takes into account the effect of the complex structure of the porous solid, and the influence of tortuosity which affects the path length to be traversed by the molecules. 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

The effective diffusivity De is a characteristic of the particle that must be measured for greatest accuracy. However, in the absence of experimental data, De may be estimated in terms of molecular diffusivity, Dab (for diffusion of A in the binary system A + B), Knudsen diffusivity, DK, particle voidage, p, and a measure of the pore structure called the particle tortuosity, Tp. 

Mitrovic and Knezic (1979) also prepared ultrafiltration and reverse osmosis membranes by this technique. Their membranes were etched in 5% oxalic acid. The membranes had pores of the order of 100 nm, but only about 1.5 nm in the residual barrier layer (layer AB in Figure 2.15). The pores in the barrier layer were unstable in water and the permeability decreased during the experiments. Complete dehydration of alumina or phase transformation to a-alumina was necessary to stabilize the pore structure. The resulting membranes were found unsuitable for reverse osmosis but suitable for ultrafiltration after removing the barrier layer. Beside reverse osmosis and ultrafiltration measurements, some gas permeability data have also been reported on this type of membranes (Itaya et al. 1984). The water flux through a 50/im thick membrane is about 0.2mL/cm -h with a N2 flow about 6cmVcm -min-bar. The gas transport through the membrane was due to Knudsen diffusion mechanism, which is inversely proportional to the square root of molecular mass. 

With anodic oxidation very controlled and narrow pore size distributions can be obtained. These membranes mounted in a small module may be suitable for ultrafiltration, gas separation with Knudsen diffusion and in biological applications. At present one of the main disadvantages is that the layer has to be supported by a separate layer to produce the complete membrane/support structure. Thus, presently applications are limited to laboratory-scale separations since large surface area modules of such membranes are unavailable. 

The molecular diffusivity D must be replaced by an effective diffusivity De because of the complex internal structure of the catalyst particle which consists of a multiplicity of interconnected pores, and the molecules must take a tortuous path. The effective distance the molecules must travel is consequently increases. Furthermore, because the pores are very small, their dimensions may be less than the mean free path of the molecules and Knudsen diffusion effects may arise Equation 10.170 is solved in Volume 1 to give equation 10.199 for a catalyst particle in the form of a flat platelet... 

One of the most intriguing aspects of surface diffusion is the strong dependence of the diffusivity on sorbate concentration. The dependence of surface diffusivities on pressure, temperature and composition is much more complicated than those of the molecular and Knudsen diffusivities, because of all the complexities of porous medium geometry, surface structure, adsorption equilibrium, mobility of adsorbed molecules, etc. 

The prediction of the parameter values for mass transport through porous materials is too difficult, because we do not know how to take into account the complicated pore geometry as it is in reality. Thus, data for the effective diffusivity or effective molecular and Knudsen diffusivities and the permeability are still more accurately determined experimentally. Experiments of this kind also provide valuable information on the porous structure, such as the average pore size and pore size distribution. 

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

Resistance to mass transfer comes both from the membrane stmcture and the air present within the membrane pores. The resistance by the membrane structure (in the absence of air) can be described by Poiseuille flow. In presence of air within the membrane pores either Knudsen diffusion or molecular diffusion, or a combined Knudsen-molecular diffusion flow model can be used. 

Fig. 18. Determination of type of diffusion in a particle of unknown pore structure. Rate vs. remaining pressure differential obtained (A) in case of Knudsen diffusion (B) in case of ordinary diffusion.

Multicomponent diffusion in pores is described by the dusty-gas model (DGM) [38,44,46 8]. This model combines molecular diffusion, Knudsen diffusion, viscous flux, and surface diffusion. The DGM is suitable for any model of porous structure. It was developed by Mason et al. [42] and is based on the Maxwell-Stefan approach for dilute gases, itself an approximation of Boltzmann s equation. The diffusion model obtained is called the generalized Maxwell-Stefan model (GMS). Thermal diffusion, pressmn diffusion, and forced diffusion are all easily included in the GMS model. This model is based on the principle that in order to cause relative motion between individual species in a mixture, a driving force has to be exerted on each of the individual species. The driving force exerted on any particular species i is balanced by the friction this species experiences with all other species present in the mixture. Each of these friction contributions is considered to be proportional to the corresponding differences in the diffusion velocities. 

Adsorption of molecules proceeds by successive steps (1) penetration inside a particle (2) diffusion inside the particle (3) adsorption (4) desorption and (5) diffusion out of the particle. In general, the rates of adsorption and desorption in porous adsorbents are controlled by the rate of transport within the pore network rather than by the intrinsic kinetics of sorption at the surface of the adsorbent. Pore diffusion may take place through several different mechanisms that usually coexist. The rates of these mechanisms depend on the pore size, the pore tortuosity and constriction, the cormectivity of the pore network, the solute concentration, and other conditions. Four main, distinct mechanisms have been identified molecular diffusion, Knudsen diffusion, Poiseiulle flow, and surface diffusion. The effective pore diffusivity measured experimentally often includes contributions for more than one mechanism. It is often difficult to predict accurately the effective diffusivity since it depends so strongly on the details of the pore structure. 

The phosphorus deactivation curve is typical type C, and, according to the Wheeler model, this is associated with selective poisoning of pore mouths. Phosphorus distribution on the poisoned catalyst is near the gas-solid interface, i.e. at pore mouths, which confirms the Wheeler model of pore mouth poisoning for type C deactivation curves. Thus we may propose that in the fast oxidative reactions with which we are dealing, transport processes within pores will control the effectiveness of the catalyst. Active sites at the gas-solid interface will be controlled by relatively fast bulk diffusional processes, whereas active sites within pores of 20-100 A present in the washcoat aluminas on which the platinum is deposited will be controlled by the slower Knudsen diffusion process. Thus phosphorus poisoning of active sites at pore mouths will result in a serious loss in catalyst activity since reactant molecules must diffuse deeper into the pore structure by the slower Knudsen mass transport process to find progressively fewer active sites. 

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