Types molecular diffusion

Pick s law States that the molecular diffusion of water vapor in a gas without appreciable displacement of the gas is analogous to the conduction of heat, and is governed by a similar type of law. 

Marchello and Toor (M2) proposed a mixing model for transfer near a boundary which assumes that localized mixing occurs rather than gross displacement of the fluid elements. This model can be said to be a modified penetration-type model. Kishinevsky (K6-K8) assumed a surface-renewal mechanism with eddy diffusion rather than molecular diffusion controlling the transfer at the interface. 

In laminar flow, a similar mixing process occurs when the liquid is sheared between two rotating cylinders. During each revolution, the thickness of the fluid element is reduced, and molecular diffusion takes over when the elements are sufficiently thin. This type of mixing is shown schematically in Figure 7.3 in which the tracer is pictured as being introduced perpendicular to the direction of motion. 

Dendrimer micelles of this type have been formulated as drug delivery vehicles. Dendrimers with a hydrophobic interior have been used to entrap a hydrophobic drug such as indomethacin. This is retained because of the hydrophilic periphery containing ethylene glycol functional groups, and is released slowly because of the collapsed configuration of the interior, through which molecular diffusion is obstructed. 

The second use of Equations (2.36) is to eliminate some of the composition variables from rate expressions. For example, 0i-A(a,b) can be converted to i A a) if Equation (2.36) can be applied to each and every point in the reactor. Reactors for which this is possible are said to preserve local stoichiometry. This does not apply to real reactors if there are internal mixing or separation processes, such as molecular diffusion, that distinguish between types of molecules. Neither does it apply to multiple reactions, although this restriction can be relaxed through use of the reaction coordinate method described in the next section. 

To study the effects of molecular diffusion and formation/destabilization of reaction fronts it is advised to rely on small flow-through chambers such as capillaries or cells of sheet-type cross-section [68]. These micro reactors simply provide the small-scale environment needed for such laboratory investigations. 

When (DEB), is much smaller than unity, the polymer relaxation is relatively rapid compared to diffusion. In this case, conformational changes take place instantaneously and equilibrium is attained after each diffusional jump. This is the type of diffusion encountered ordinarily and is called viscous diffusion. Therefore, the transport will obey classical theories of diffusion. When (DEB), is much larger than unity, the molecular relaxation is very slow compared to diffusion and there are no conformational changes of the medium within the diffusion time scale. In this case, Fick s law is generally valid, but no concentration dependence of the diffusion coefficient is expected. This is termed elastic diffusion. When (DEB), is in the neighborhood of unity, molecular rearrangment... 

Bulk or forced flow of the Hagan-Poiseuille type does not in general contribute significantly to the mass transport process in porous catalysts. For fast reactions where there is a change in the number of moles on reaction, significant pressure differentials can arise between the interior and the exterior of the catalyst pellets. This phenomenon occurs because there is insufficient driving force for effective mass transfer by forced flow. Molecular diffusion occurs much more rapidly than forced flow in most porous catalysts. 

For diffusion in the soil air-pores, a molecular diffusivity of 0.02 m2/h is reduced to an effective diffusivity using a Millington-Quirk type of relationship by a factor of about 20 to 10-3 m2/h. Combining this with a path length of 0.05 m gives an effective air-to-soil mass transfer coefficient kSA of 0.02 m/h, which is designated as U5. 

Let us consider the transport of one component i in a liquid solution. Any disequilibration in the solution is assumed to be due to macroscopic motion of the liquid (i.e. flow) and to gradients in the concentration c,. Temperature gradients are assumed to be negligible. The transport of the solute i is then governed by two different modes of transport, namely, molecular diffusion through the solvent medium, and drag by the moving liquid. The combination of these two types of transport processes is usually denoted as the convective diffusion of the solute in the liquid [25] or diffusion-advection mass transport [48,49], The relative contribution of advection to total transport is characterised by the nondimensional Peclet number [32,48,49], while the relative increase in transport over pure diffusion due to advection is Sh - 1, where Sh is the nondimensional Sherwood number [28,32,33,49,50]. 

There is apparently an inherent anomaly in the heat and mass transfer results in that, at low Reynolds numbers, the Nusselt and Sherwood numbers (Figures. 6.30 and 6.27) are very low, and substantially below the theoretical minimum value of 2 for transfer by thermal conduction or molecular diffusion to a spherical particle when the driving force is spread over an infinite distance (Volume 1, Chapter 9). The most probable explanation is that at low Reynolds numbers there is appreciable back-mixing of gas associated with the circulation of the solids. If this is represented as a diffusional type of process with a longitudinal diffusivity of DL, the basic equation for the heat transfer process is ... 

A further distinction has to be made between reactions taking place within a molecular crystal and reactions of a molecular crystal A with a molecular crystaT B (see Scheme 2 A and B can also be the same crystal) [7bj. Reactions of the former type, or intra-solid reactions, can be either under topochemical control, depending on the proximity of the reactants, or may imply extensive molecular motion within the crystal lattice. Reactions of the second type, or inter-solid reactions, can either take place on the crystallite surface or require molecular diffusion through the lattice. Inter-solid reactions are often accompanied by formation of eutectics. 

Bischoff and Levenspiel (B14) present some calculations using existing experimental data to check the above predictions about the radial coefficients. For turbulent flow in empty tubes, the data of Lynn et al. (L20) were numerically averaged across the tube, and fair agreement found with the data of Fig. 12. The same was done for the packed-bed data of Dorweiler and Fahien (D20) using velocity profile data of Schwartz and Smith (Sll), and then comparing with Fig. 11. Unfortunately, the scatter in the data precluded an accurate check of the predictions. In order to prove the relationships conclusively, more precise experimental work would be needed. Probably the best type of system for this would be one in laminar flow, since the radial and axial coefficients for the general dispersion model are definitely known each is the molecular diffusivity. 

Existence of a high degree of orientational freedom is the most characteristic feature of the plastic crystalline state. We can visualize three types of rotational motions in crystals free rotation, rotational diffusion and jump reorientation. Free rotation is possible when interactions are weak, and this situation would not be applicable to plastic crystals. In classical rotational diffusion (proposed by Debye to explain dielectric relaxation in liquids), orientational motion of molecules is expected to follow a diffusion equation described by an Einstein-type relation. This type of diffusion is not known to be applicable to plastic crystals. What would be more appropriate to consider in the case of plastic crystals is collision-interrupted molecular rotation. 

There are two major types of diffusion contributing to mass transport in the monolith washcoat (cf., e.g. Aris, 1975 Froment and Bischoff, 1979, 1990 Poling et al., 2001) volume (molecular) diffusion, Eq. (14), and Knudsen diffusion, Eq. (15), the latter one being dominant in small pores. 

In natural systems there are two types of transport phenomena (1) transport by random motion, and (2) transport by directed motion. Both types occur at a wide range of scales from molecular to global distances, from microseconds to geological times. Well-known examples of these types are molecular diffusion (random transport) and advection in water currents (directed transport). There are many other manifestations such as dispersion as a random process (see Chapters 24 and 25) or settling of suspended particles due to gravitation as a directed transport. For simplicity we will subdivide such transport processes into those we will call diffusive for ones caused by random motions and those called advective for ones resulting from directed motions. 

Here P(rfy,t) is the conditional probability of finding an interparticle vector in df about f at time t if it was in df0 about f0 at time zero, and P(f0) is the probability of finding the interparticle vector initially in df0. The interacting particles are of two types those within a given molecule, the inter-nuclear vectors of which are changed by molecular rotation, and those external to the molecule. The interparticle vectors between a nucleus inside a molecule and particles external to the molecule are changed mainly by diffusion. The sum in Equation 7 may be thus separated into inter-molecular (diffusion) and intramolecular (rotation) parts, and for the moment we ignore the latter. 

Up to this point we have been discussing diffusion in terms of molecular or free diffusion where the diffusion rate is determined by molecular collisions and the particle voids which are larger than the mean free path. In packed gas chromatographic columns the diffusion process follows other laws. Under these conditions we can encounter four types of diffusion. 

Free or Molecular Diffusion. The type used to explain the above model. Takes place between particles and in pores of diameter >0.1 um. Here we encounter largest values for Dg, that is, 10 1 - 1.0 cirm/sec. 

Solid diffusion takes place in pore diameters of about 0.001 pm(10A) (Dg values of order of 10-Jcm-/sec). Since most diffusion types (Volmer, Knudsen, and solid) are orders of magnitude smaller than molecular diffusion, they contribute little to the... 

Ordinary diffusion. This process results when there exists a region of high concentration and a region of low concentration. The migration is from the high to the lower concentration region in the axial direction of the column. This type of diffusion occurs on the molecular level resulting from the movement of molecules after collision. 

The diffusion of the sulfur molecules in the pores of the catalyst would be assumed to be mainly influenced by molecular diffusion. This type of diffusion generally occurs in the liquid phase. Since the same liquid was used in both the cases, the coefficient of diffusion would be assumed to be the same. 

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