Diffusion Knudsens

Knox-Out2FM Knudsen cells Knudsen diffusion Knudsen s law Koavone [86115-11-9]... 

Continuous stirred tank reactor Dispersion coefficient Effective diffusivity Knudsen diffusivity Residence time distribution Normalized residence time distribution... 

This result was first observed experimentally by Graham and is called Graham s law of diffusion. Knudsen diffusion membranes have been used to separate gas isotopes that are difficult to separate by other methods, for example tritium from... 

In order to predict correctly the fluxes of multicomponent mixtures in porous membranes, simplified models based solely on Fields law should be used with care [28]. Often, combinations of several mechanisms control the fluxes, and more sophisticated models are required. A well-known example is the Dusty Gas Model which takes into account contributions of molecular diffusion, Knudsen diffusion, and permeation [29]. This model describes the coupled fluxes of N gaseous components, Ji, as a function of the pressure and total pressure gradients with the following equation ... 

The constitutive equations of transport in porous media comprise both physical properties of components and pairs of components and simplifying assumptions about the geometrical characteristics of the porous medium. Two advanced effective-scale (i.e., space-averaged) models are commonly applied for description of combined bulk diffusion, Knudsen diffusion and permeation transport of multicomponent gas mixtures—Mean Transport-Pore Model (MTPM)—and Dusty Gas Model (DGM) cf. Mason and Malinauskas (1983), Schneider and Gelbin (1984), and Krishna and Wesseling (1997). The molar flux intensity of the z th component A) is the sum of the diffusion Nc- and permeation N contributions,... 

The following diffusivities (Knudsen and binary) need to be determined from tabulated data, handbooks, correlations, theoretical equations, etc. ... 

In many studies the separation factor, which is indicative of the membrane s ability to separate two gases in a mixture, is predominantly governed by Knudsen diffusion. Knudsen diffusion is useful in gas separation mostly when two gases are significantly different in their molecular weights. In other cases, more effective uansport mechanisms are required. The pore size of the membrane needs to be smaller so that molecular sieving effects become operative. Some new membrane materials such as zeolites and other molecular sieve materials and membrane modifications by the sol-gel and chemical vapor deposition techniques are all in the horizon. Alternatively, it is desirable to tailor the gas-membrane interaction for promoting such transport mechanisms as surface diffusion or capillary condensation. 

Transfer mechanisms involved in SC CO2 permeation through such porous membranes can be convection (Poiseuille law), diffusion (Knudsen flow), and surface membrane interaction by adsorption, capillary condensation, etc. [11]. Mechanisms have been specifically investigated for nanofiltration and zeolite membranes. With a nanofilter presenting a pore diameter of about 1 nm, Sarrade [11] mentioned a Poiseuille flow associated with an irreversible CO2 adsorption on the micropore wall. Transfer... 

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. 

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. 

Bulk diffusivity, Knudsen diffusivities in micro- and macropores, molecular weights, macro- and micro porosities, and mean micro- and macropore radii... 

The solid particles in the bed always have a degree of porosity, regardless of whether the solid is a catalyst or not. The pores contained in each particle may have different characteristics (size, etc.). Thus, the diffusion of molecules on the solid surface is very important, which has various transport mechanisms, such as intraparticle diffusion, Knudsen diffusion, or surface diffusion. The latter, for example, depends on the surface characteristics, such as high or low surface area. Typically, one determines the effective diffusion encompassing both characteristics. 

The same set of transport mechanisms learnt in Chapter 7 is again considered in Chapter 8, but is dealt with in the framework of Maxwell-Stefan. This is the cornerstone in dealing with multicomponent diffusion in homogeneous media as well as heterogeneous media. We first address this framework to a homogeneous medium so that readers can grasp the concept of friction put forwards by Maxwell and Stefan in dealing with multicomponent systems. Next, we deal with diffusion of a multicomponent mixture in a capillary and a porous medium where continuum diffusion, Knudsen diffusion as well as viscous flow can all play an important role in the transport of molecules. 

As we have discussed in the introduction, there are basically four modes of transport of molecules inside a porous medium. They are free molecular diffusion (Knudsen), viscous flow, continuum diffusion and surface diffusion. 

We have considered separately the necessary flux equations for the cases of Knudsen diffusion and continuum diffusion. Knudsen diffusion usually dominates when the pore size and the pressure are small, and the continuum diffusion dominates when the pore size and pressure are large. In the intermediate case which is usually the case for most practical systems, we would expect that both mechanisms will control the mass transport in a capillary or a porous medium. In this section, we will consider this intermediate case and present the necessary flux equations. 

In the last sections, you have learnt about the basic analysis of bulk flow, bulk flow and Knudsen flow using the Stefan-Maxwell approach. Very often when we deal with diffusion and adsorption system, the total pressure changes with time as well as with distance within a particle due to either the nonequimolar diffusion or loss of mass from the gas phase as a result of adsorption onto the surface of the particle. When such situations happen, there will be an additional mechanism for mass transfer the viscous flow. This section will deal with the general case where bulk diffusion, Knudsen diffusion and viscous flow occur simultaneously within a porous medium (Jackson, 1977). 

We have presented above the three basic equations (8.8-1 to 8.8-3) for the case where bulk diffusion-Knudsen diffusion-viscous flows are simultaneously operating. What we will do in this section is to combine them to obtain a form which is useful for analysis and subsequent computation as we shall show in Chapters 9 and 10. 

Eqs. (8.8-12) or (8.8-13) are the basic constitutive flux equations for the case where molecular diffusion, Knudsen diffusion and viscous flow are all operating. They are in the form suitable to be used in the mass balance equation, which we shall show later in this chapter as well as in Chapters 9 and 10. 

We have presented the three different formulations for the bulk diffusion-Knudsen diffusion-viscous flow in Sections 8.8.1 to 8.8.3. Now we will show how these formulations can be used to derive useful equations for limiting cases which are often encountered in adsorption systems. 

Model II Pore Diffusion in the Particles is r.d.s. (Knudsen Diffusion, Shrinking Core Model] If pore diffusion (Knudsen diffusion at HV) in the silica particles limits the mass loss and we assume that an IL-free shell is formed (shrinking core model), the mass loss is given by... 

The model accounts for three different transport mechanisms, molecular diffusion, Knudsen diffusion, and viscous transport. The total diffusive flux in DGM results from molecular diffusion acting in series with Knudsen diffusion. The viscous porous media flow (Darcy flow) acts in parallel with diffusive flux. The DGM can be written as an implicit relationship among molar concentrations, fluxes, concentration gradients and pressure gradient as... 

The penneation through inorganic manbranes can be described in terms of transport across porous materials, where a penetrant diffuses through small pores according to the pores characteristics and its affinity with the pore walls. Thus the penetrant flux can be described with different mechanisms, as presented in Fig. 7.12 bulk diffusion, Knndsen diffusion, Knudsen diffusion conpled with surface diffusion, surface diffusion conpled with capillary condensation, and molecular sieving. 

Macropore diffusion has been widely studied in connection with its influence on the overall kinetics of heterogeneous catalytic reactions. Four distinct mechanisms of transport may be identified molecular diffusion, Knudsen diffusion, Poiseuille flow, and surface diffusion. The effective macropore diffusivity is thus a complex quantity which often includes contributions from more than one mechanism. Although the individual mechanisms are reasonably well understood, it is not always easy to make an accurate a priori prediction of the effective diffusivity since this is strongly dependent on the details of the pore structure. 

Open pores, which are connected with each other and also with the surface of the crystal. The form and distribution of the pores are most important for transport phenomena such as diffusion, Knudsen flow, and surface diffusion. Because of the technological importance of this field, it possesses an extensive literature, especially on procedures for determining porosity. We can only make mention of this literature here [12]. 

The process can be broadly classified as bulk diffusion, Knudsen diffusion, and surface diffusion. The molecular driving force of diffusion is the chemical potential difference created by a local population of a chemical species. Molecules tend to distribute uniformly across the space while migrating among like or unlike molecules. Bulk diffusion is the predominant mechanism when the pressures are high and pore sizes are large. On the other hand, at lower pressures, Knudsen diffusion prevails, when the mean free path of the molecules are larger than the pore size. When the molecules are adsorbed strongly on the pores or the pore sizes are too small, the mechanism of diffusion becomes surface diffusivity. 

Computational models Effective diffusivity D a See Wakao and Smith (1962) Bulk diffusivity, Knudsen diffusivities in... 

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