Effective diffusion length

The limiting cases of greatest interest correspond to conditions in which the mean free path lengths are large and small, respectively, compared with the pore diameters. Recall from the discussion in Chapter 3 that the effective Knudsen diffusion coefficients are proportional to pore diameter and independent of pressure, while the effective bulk diffusion coefficients are independent of pore diameter and inversely proportional to pressure. 

Table 1.6 Characteristic quantities to be considered for micro-reactor dimensioning and layout. Steps 1, 2, and 3 correspond to the dimensioning of the channel diameter, channel length and channel walls, respectively. Symbols appearing in these expressions not previously defined are the effective axial diffusion coefficient D, the density thermal conductivity specific heat Cp and total cross-sectional area S, of the wall material, the total process gas mass flow m, and the reactant concentration Cg [114].

The geometry of the pore structure makes it impossible to determine accurately the effective length of the diffusion path. Interconnections within the pore structure, the tortuous character of individual pores, and variations in cross-sectional area along the pore length all contribute to the difficulty of the task. 

The likelihood that Fick s laws will be obeyed in a crystal containing dislocations is dependent upon the spacing between the defects. Provided that this spacing is much greater than the diffusion length (Dt)1/2, where D is the bulk diffusion coefficient, Fick s laws are obeyed, with an effective (measured) diffusion coefficient, Deff, given by... 

The reduction of the long-range diffusivity, Di by a factor of four with respect to bulk water can be attributed to the random morphology of the nanoporous network (i.e., effects of connectivity and tortuosity of nanopores). For comparison, the water self-diffusion coefficient in Nafion measured by PFG-NMR is = 0.58 x 10 cm s at T = 15. Notice that PFG-NMR probes mobilities over length scales > 0.1 /rm. Comparison of QENS and PFG-NMR studies thus reveals that the local mobility of water in Nafion is almost bulk-like within the confined domains at the nanometer scale and that the effective water diffusivity decreases due to the channeling of water molecules through the network of randomly interconnected and tortuous water-filled domains. ... 

While the diffusion coefficient D is a physical constant for a particular species, t denotes the migration time, which is proportional to the velocity of migration v [Eq. (2)] and the effective length of the capillary /. Hence, it follows that... 

G. I. Taylor s concept of the effective axial diffusion coefficient, which has proved so useful in combining variable axial advection with radial transfer into one parameter, works best when there is no exchange of a passive tracer with the pipe walls. An analogue of his method, which should be applicable when development lengths are large and there is exchange at the wall, has yet to be provided. It would be of great value. 

To implement these simulation approaches, the value of the liquid film mass transfer coefficient Kf is required, which for nonporous and porous HPLC particles, can be calculated from literature correlations derived for bath357,400,408 or column models.407,408 For the case with porous particles, the apparent pore liquid mass transfer coefficient Kp can be expressed as an effective pore diffusivity over an average effective diffusion path length, such that... 

The boundary conditions are dx/dz = Px at z =0 and dx/dz = 0 at z = 1. Here, P is the axial Peclet number, defined as the product of velocity and reactor length, divided by the effective axial diffusivity. 

Thus diffusion limitations decrease the yield twofold. These results may be generalized to include interface and intraparticle diffusion for bidisperse catalysts [5]. The effect of diffusion limitation on the concentration distribution over the reactor length can be calculated from Equations 8.24, 8.29 and 8.33. An example of such calculations is shown in Figure 8.4 for the case CM = 0, DtA = D and kjkx = 0.1. Although the rate of... 

The A term is related to multipath effects (eddy diffusion) that are independent of mobile phase flow rate. Analyte molecules can follow multiple pathways of differing lengths that spread the analyte molecules apart and cause peak broadening. Since the smaller the particles, the lower the difference among molecule walks along the column, the A term is linearly dependent on particle diameter, d, according to X, a structure factor ... 

The kinetic competition between charge transport and recombination in DSSCs has been analysed in terms of an effective carrier diffusion length given by... 

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