Renkin effect

Diffusivity in Pores and Fick s Laws Diffusion in Gas-Filled Pores Knudsen Effect Diffusion in Liquid-Filled Pores Renkin Effect Diffusion in the Unsaturated Zone of Soils... 

Diffusion in Liquid-Filled Pores Renkin Effect... 

In liquids, the mean free path is typically of the order of 10 10 m. Hence the Knudsen effect is not important (i.e., diffusing molecules collide with solvent molecules long before they typically arrive at a pore wall). However, diffusion is affected by a different mechanism, the viscous drag caused by the pore walls. This is known as the Renkin effect (Renkin, 1954). In essence, the ratio of pore diffusivity in the liquid-filled pore space and diffusivity in the free liquid, D(pore/D,free, is a function of the nondimensional parameter... 

This means that the Renkin effect sets in when dp becomes smaller than about 1000 dt and suppresses diffusion completely when dp approaches d,. Diffusing molecules have a typical size of 1O 8 to 10 7 cm. As an example, based on Renkin s original equation, for qin = 0.4 (that is, when the cross-section of the solute covers about (0.4)2 = 0.16 or 16% of the pore opening cross-section), the diffusivity is reduced to 12% of diffusivity in the free liquid. 

Note that 0 < 0g < 0 < 1. There are numerous other empirical expressions like Eq. 18-66 (see, e.g., Hillel, 1998). They all have in common that Diuz becomes virtually zero if 0g drops below 0.1. Remember that Diuz of Eq. 18-66 is the diffusivity which appears in Fick s second law (Eq. 18-60). In contrast, D,pore, which we introduced for the Knudsen and the Renkin effect, does not yet include the effect of tortuosity, but replaces D, on the right-hand side of Eq. 18-57. 

The permeability coefficients and molecular radii are known. The effective pore radius, R, is the only unknown and is readily calculated by successive approximation. Consequently, unknown parameters (i.e., porosity, tortuosity, path length, electrical factors) cancel, and the effective pore radius is calculated to be 12.0 1.9 A. Because the Renkin function [see Eq. (35)] is a rapidly decaying polynomial function of molecular radius, the estimation of R is more sensitive to small uncertainties in the calculated molecular radius values than it is to experimental variabilities in the permeability coefficients. The placement of the perme-ants within the molecular sieving function is shown in Figure 9 for the effective... 

Renkin EM. Effects of blood flow on diffusion kinetics in isolated perfused hindlegs of cats A double circulation hypothesis. Am J Physiol 1953 183 125-36. 

Constriction if the pore radius rp is not much larger than the radius of the diffusing solute molecule rm, the molecule frequently collides with the pore wall whereby its diffusion is impeded, the more so for rm/rv closer to unity. It is difficult to predict the magnitude of this effect. It can be roughly estimated by the semiempirical Renkin equation, which applies to diffusion of reasonably spherical molecules (or particles) in a straight cylindrical capillary ... 

When the solute dimension is no longer orders of magnitude smaller than the pore dimensions, the solute molecules experience an additional transport resistance due to the proximity of the pore wall. The effective diffusion coefficient is reduced further by a hindrance factor/ drag factor Gm i i,rp). If the solute molecules are assumed spherical of radius r, and diffusing through the centerline of the pore of radius, tp, Faxen s expression may be used to estimate Gor for (r,rp) < 0.5 (Lane, 1950 Renkin, 1954) ... 

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