Transport diffusion pore wall

Unlike high-resolution NMR spectra of bulk solutions where NMR linewidths well below 1 Hz can be obtained routinely, NMR spectra of liquids permeating porous solids in most cases will not exhibit such a high spectral resolution. First of all, the interaction of liquid phase molecules with pore walls of the catalyst and rapid diffusion-driven intrapore transport will lead to a pronounced homogeneous broadening of the observed NMR lines. Smaller pore sizes and the presence of paramagnetic impurities in the solid material usually aggravate the situation and thus should be avoided. Another reason why NMR spectra of liquids in porous... 

Ordinary or bulk diffusion is primarily responsible for molecular transport when the mean free path of a molecule is small compared with the diameter of the pore. At 1 atm the mean free path of typical gaseous species is of the order of 10 5 cm or 103 A. In pores larger than 1CT4 cm the mean free path is much smaller than the pore dimension, and collisions with other gas phase molecules will occur much more often than collisions with the pore walls. Under these circumstances the effective diffusivity will be independent of the pore diameter and, within a given catalyst pore, ordinary bulk diffusion coefficients may be used in Fick s first law to evaluate the rate of mass transfer and the concentration profile in the pore. In industrial practice there are three general classes of reaction conditions for which the bulk value of the diffusion coefficient is appropriate. For all catalysts these include liquid phase reactions... 

In both situations the interaction of the medium inside the pore with the pore wall (1) is increased (2) or changed which affect the transport and separation properties (surface diffusion, multilayer adsorption) and/or help overcome equilibrium constraints in membrane reactors. Membrane modifications can be performed by depositing material in the internal pore structure from liquids (impregnation, adsorption) or gases. Several modification possibilities are schematically shown in Figure 2.3. Some results obtained by Burggraaf, Keizer and coworkers are summarized in Table 2.7. Composite structures on a scale of 1-5 nm were obtained. 

As the pore size decreases, molecules collide more often with the pore walls than with each other. This movement, intermediated by these molecule—pore-wall interactions, is known as Knudsen diffusion. Some models have begun to take this form of diffusion into account. In this type of diffusion, the diffusion coefficient is a direct function of the pore radius. In the models, Knudsen diffusion and Stefan—Maxwell diffusion are treated as mass-transport resistances in seriesand are combined to yield... 

A diameter of 20 A represents approximately the limiting pore size that can be measured by mercury intrusion. In pores smaller than this, transport becomes increasingly affected by molecule-pore wall interactions, and conventional theories based on molecular and Knudsen diffusion break down. The classification is somewhat arbitrary, however, since the point at which such effects become important also depends on the size of the diffusing molecule. Adsorption equilibrium in microporous adsorbents also depends to some extent on the pore size as well as on the nature of the surface, so control of the pore size distribution is important in the manufacture of an adsorbent for a particular separation. 

Good quality RO membranes can reject >95-99% of the NaCl from aqueous feed streams (Baker, Cussler, Eykamp et al., 1991 Scott, 1981). The morphologies of these membranes are typically asymmetric with a thin highly selective polymer layer on top of an open support structure. Two rather different approaches have been used to describe the transport processes in such membranes the solution-diffusion (Merten, 1966) and surface force capillary flow model (Matsuura and Sourirajan, 1981). In the solution-diffusion model, the solute moves within the essentially homogeneously solvent swollen polymer matrix. The solute has a mobility that is dependent upon the free volume of the solvent, solute, and polymer. In the capillary pore diffusion model, it is assumed that separation occurs due to surface and fluid transport phenomena within an actual nanopore. The pore surface is seen as promoting preferential sorption of the solvent and repulsion of the solutes. The model envisions a more or less pure solvent layer on the pore walls that is forced through the membrane capillary pores under pressure. 

More complicated and realistic models which allow the prediction of transport processes in porous media have been suggested, and have been validated in recent years. For example, it was realized that there might be significant contributions to the overall flux by components which are adsorbed at pore walls but possess a certain mobility [30]. To quantify such surface diffusion processes, a Generalized Stefan-Maxwell equation has been proposed [28] ... 

In macro- and mesoporous membrane layers the nature of the flow is determined by the relative magnitude of the mean free path X of the molecules and the pore size dp. When the mean free path of the gas molecules is much larger than the pore size, i.e. X dp, collisions of molecules with the pore walls are predominant and the mass transport takes place by the well-known selective Knudsen diffusion process. If the pore radius is much larger than the mean free path of the molecules and a pressure difference over the membrane exists the mass transport takes place by non-selective viscous flow. 

The driving force for the transport is provided by a concentration gradient as the reactant moves further towards the center of the pellet its concentration is decreased by reaction. The resistance to the transport mainly originates from collisions of the molecules, either with each other or with the pore walls. The latter dominate when the mean free path of the molecules is larger than the pore diameter. Usually both type of collisions are totally random, which amounts to saying that the transport mechanism is of the diffusion type. Hence the rate of transport, expressed as a molar flux in mol mp2 s-1, in the case of equimolar counterdiffusion can be written as ... 

Rieckmann and Keil (1997) introduced a model of a 3D network of interconnected cylindrical pores with predefined distribution of pore radii and connectivity and with a volume fraction of pores equal to the porosity. The pore size distribution can be estimated from experimental characteristics obtained, e.g., from nitrogen sorption or mercury porosimetry measurements. Local heterogeneities, e.g., spatial variation in the mean pore size, or the non-uniform distribution of catalytic active centers may be taken into account in pore-network models. In each individual pore of a cylindrical or general shape, the spatially ID reaction-transport model is formulated, and the continuity equations are formulated at the nodes (i.e., connections of cylindrical capillaries) of the pore space. The transport in each individual pore is governed by the Max-well-Stefan multicomponent diffusion and convection model. Any common type of reaction kinetics taking place at the pore wall can be implemented. 

The analysis of the effects of transport on catalysis has focused on a comparison of the availability of reacting species by diffusion to the rate of reaction on the catalytic sites. High-surface-area catalysts are usually porous. Comparison of transport to reaction rates has usually been based on Knudsen diffusion (by constricted collision with the pore walls) as the dominant mode of transport. DeBoer has noted that for small pores surface diffusion may dominate transport (192). Thiele modulus calculations may therefore not be valid if they are applied to systems where surface diffusion can be significant. This may mean that the direct participation of spillover species in catalysis becomes more important if the catalysts are more microporous. Generalized interpretations of catalyst effectiveness may need to be modified for systems where one of the reactants can spill over and diffuse across the catalyst surface. 

When the size of the pores is much smaller than the molecular mean free paths in the gas mixture, collisions of gas molecules with the pore wall are more frequent than inter-molecular collisions. This type of gas transport is also known as Knudsen diffusion. In this case each molecule acts independently of all the others and each component in a mixture behaves as though it were present alone. The movement of each molecule can be conveniently pictured as a random walk between the walls of pores. This leads to the following expression for the molar flux of species A ... 

Transport of the reactants by diffusion and convection out of the bulk gas stream, through a laminar boundary layer, to the outer surface of the catalyst particles, and further through the pore system to the inner surface (pore walls)... 

The preceding section assumed that the mass-transport mechanism in a fluid medium is dominated by molecule-molecule collisions. However, the mean free path of gases often exceeds the dimensions of small pores typical of solid catalysts. In this situation, called Knudsen diffusion, molecules collide more often with the pore walls than with other molecules. According to Equation (6.3.1), the Knudsen diffu-sivity of component A, D a, is proportional to r / , but is independent of both pressure and the presence of other species ... 

The transport of a sub-critical Lennard-Jones fluid in a cylindrical mesopore is investigated here, using a combination of equilibrium and non-equilibrium as well as dual control volume grand canonical molecular dynamics methods. It is shown that all three techniques yield the same value of the transport coefficient for diffusely reflecting pore walls, even in the presence of viscous transport. It is also demonstrated that the classical Knudsen mechanism is not manifested, and that a combination of viscous flow and momentum exchange at the pore wall governs the transport over a wide range of densities. 

In the molecular dynamics calculations the trajectories of methane molecules in the pore are followed using the equation of motion with appropriate temperature control. A diffuse reflection condition is applied at the pore wall. For the EMD simulations a collective transport coefficient obtained from autocorrelation of the fluctuating axial streaming velocity via a Green-Kubo relation [S]... 

It is clear from the above results that all three simulation techniques yield identical results for the transport coefficient in pores with diffusely reflectmg walls. Further, a combination of momentum transfer at the wall and viscous transport in the fluid suffices to explain the transport behavior of pure component fluids in mesopores. 

For gas transport in small pores (say, less than the 10 nm range) the sizes of which are no longer much larger than those of the gas molecules, the contribution of viscous flow can be neglected and other considerations need to be factored in the model. First, the gas molecules are considered to be hard sphere with a finite size and the gas diffusion process is assumed to proceed in the membrane pores by random walk. The membrane pores are assumed to consist of smooth-wall circular capillaries. In addition to gas molecules colliding with the membrane pore walls, adsorption on the pore wall and the associated surface flow or diffusion are considered. Adsorption also effectively reduces the pore size for diffusion. 

Surface diffusion, which represents activated transport of adsorbed species along the pore wall. 

If the mean free path of the water molecules is much greater than the pore size of the membrane (A >dp,Kn> 10), then the molecules will collide with pore walls and the transport mechanism follows Knudsen diffusion as shown in Figure 19.5. In this case, the membrane coefficient is calculated using Equation 19.26 [32]... 

Usually, the mean radius f of pores in the catalyst layer will be small compared to /free [101, 102]. This means that diffusing molecules will collide more frequently with pore walls than with each other. The prevailing mechanism of gas transport in... 

---