Models considering pore diffusion

Models Considering Pore Diffusion. Pseudo steady state transport equation for SOj in a spherical porous solid reactant (CaO) can be written as follows ... 

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. 

Gu et al. (1990a and 1990b) proposed an even more reduced general rate model, which only considers pore diffusion inside the particles (pore diffusion model). Thus, Eq. 6.84 is replaced by Eq. 6.85 ... 

For Knudsen diffusion to take place, the lower limit for pore diameter has usually been set to 4>ore > 20 A. Gilron and Soffer have, however, discussed thoroughly how Knudsen diffusion may contribute to transport in even smaller pores, and from a model considering pore structure, shown that contributions to transport may both come from activated transport and Knudsen through one specific fiber. It may thus be difficult to know exactly when transport due to Knudsen diffusion is taking place. One way to approach this problem is to calculate the Knudsen number, Knudsen. for the system, which is 2/fi pore> where X is the mean free path. If Knudsen > 10, then the separation can be assumed to take place according to Knudsen diffusion. Therefore, if the preparation of the carbon membranes has been unsuccessful, one may get Knudsen diffusion. 

These models are classified into two groups. The models listed in the first group do not consider pore diffusion resistence. First group models are extensions of the unreact core model and they are limited to small particle sizes in the order of magnitude of a few microns. Pore diffusion resistence becomes important especially at high temperatures and at the initial stages... 

Models which do not consider pore diffusion Models which consider pore diffusion... 

Models of chemical reactions of trace pollutants in groundwater must be based on experimental analysis of the kinetics of possible pollutant interactions with earth materials, much the same as smog chamber studies considered atmospheric photochemistry. Fundamental research could determine the surface chemistry of soil components and processes such as adsorption and desorption, pore diffusion, and biodegradation of contaminants. Hydrodynamic pollutant transport models should be upgraded to take into account chemical reactions at surfaces. 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

For hydrophilic and ionic solutes, diffusion mainly takes place via a pore mechanism in the solvent-filled pores. In a simplistic view, the polymer chains in a highly swollen gel can be viewed as obstacles to solute transport. Applying this obstruction model to the diffusion of small ions in a water-swollen resin, Mackie and Meares [56] considered that the effect of the obstruction is to increase the diffusion path length by a tortuosity factor, 0. The diffusion coefficient in the gel, )3,i2, normalized by the diffusivity in free water, DX1, is related to 0 by... 

The lumped pore model (often referred to as the POR model) was derived from the general rate model by ignoring two details of this model [5]. The first assumption made is that the adsorption-desorption process is very fast. The second assumption is that diffusion in the stagnant mobile phase is also very fast. This latter assumption leads to the consequence that there is no radial concentration gradient within a particle. Instead of the actual radial concentration profile across the porous particle, the model considers simply its average value. 

The industrial rates obtained earlier from the pseudohomogeneous model actually include diffusional limits and are suitable for the specific reactor with the specific catalyst particle size for which the data was extracted. Such pseudohomogeneous models do not account explicitly for the catalyst packing of the reactor. On the other hand, heterogeneous models account for the catalyst explicitly by considering the diffusion of reactants and of products through the pores of the catalyst pellet. 

In eq 1 Dic is the effective diffusivity of species i in the reaction mixture which can be determined on the basis of various models of the diffusion process in porous solids. This aspect is discussed more fully in Section A.6.3. Difi is affected by the temperature and the pore structure of the catalyst, but it may also depend on the concentration of the reacting species (Stefan-Maxwell diffusion [9]). As Die is normally introduced on the basis of more or less empirical models, it may not be considered as a physical property, but rather as a model-dependent parameter. 

Sophisticated mathematical models based on the numerical simulation of the chromatographic process consider different kinetic and thermodynamic mechanisms [19], The theoretical approaches describe the biospecific adsorption of monovalent and multivalent adsorbates. They also account for the film mass transfer and pore diffusion contributions to the adsorption process and can be applied to analyze various complex experimental situations. Thus, ideally, the appropriate model will have to be selected to describe the actual chromatographic system. 

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. 

These assumptions give a simplified general rate model, which considers only film mass transfer and pore diffusion. The advantage of this simplified model is that it is possible to calculate the inverse Laplace transform of its solution in the Laplace domain, and obtain a time-domain solution provided that we may... 

The major difference between the various GRM models is due to the mechanism of intraparticle diffusion that they propose, namely pore diffusion, siuface diffusion or a combination of both, independent or competitive diffusion. The pore diffusion model assumes that the solute diffuses into the pore of the adsorbent mainly or only in the free mobile phase that impregnates the pores of the particles. The surface diffusion model considers that the intraparticle resistance that slows the mass transfer into and out of the pores proceeds mainly through surface diffusion. In the GRM, diffusion within the mobile phase filling the pores is usually assumed to control intraparticle diffusion (pore diffusion model or PDM). This kind of model often fits the experimental data quite well, so it can be used for the calculation of the effective diffusivity. If this model fails to fit the data satisfactorily, other transport formulations such as the Homogeneous Surface Diffusion Model (HSDM) [27] or a model that allows for simultaneous pore and siuface diffusion may be more successful [28,29]. However, how accurately any transport model can reflect the actual physical events that take place within the porous... 

Ramachandran and Smith obtained satisfactory agreement with experimental results on the reduction of nickel oxide with carbon monoxide (pore opening case) by considering the product layer diffusion coefficient as an adjustable parameter. Similarly, the model predicted pore closure and reaction die-off for the reaction of calcium oxide with sulfur dioxide, where the molar volume of calcium sulfate product is about three times that of the calcium oxide reactant. 

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