Hydraulic-mean pore radius

Example 3.4.4 The utility of equation (3.4.86) for the determination of the solvent flux through a porous membrane will be briefly illustrated with an example worked out by Cheryan (1987). For an XMIOOA ultrafQtration (UF) membrane, the mean pore diameter (= 2 x hydraulic mean pore radius) = 17.5 nm the number of pores/cm of the top membrane surface area (skin) = 3 x 10 = membrane... 

These techniques involving the measurement of membrane permeability to a fluid (liquid or gas) lead to a mean pore radius (usually the effective hydraulic radius Th) whose quantitative value is often highly ambiguous. The flux of a fluid through a porous material is sensitive to all structural aspects of the material [129]. Thus, in spite of the simplicity of the method, the interpretation of flux data, even for the simplest case of steady state, is subject to uncertainties and depends on the models and approximations used. 

As already indicated, by applying the Kelvin equation (assuming hemispherical meniscus formation) and correcting for the adsorbed layer thickness, we are able to calculate the ranges of apparent pore width recorded in Table 12.5. The values of mean pore diameter, w, are obtained from the volume/surface ratio, i.e. by applying the principle of hydraulic radius (see Chapter 7) and assuming the pores to be non-intersecting cylindrical capillaries and that the BET area is confined to the pore walls. 

The mean pore hydraulic radius Fh for a porous solid is obtained through the relationship... 

The mean hydraulic pore radius RH (estimated by the MP method) varied between 3.2 and 7.0 A. This parameter was slightly affected by the surfactant/ precursor ratio (SAA/TEOS) but increased with the surfactant chain length X. 

S is the ratio of the surface area of the medium to its pore volume and stands for equivalent diameter of the pores. The hydraulic (mean) radius m is defined as the ratio of the average pore cross-sectional area to the average wet perimeter, in line with the concept of the equivalent loads (as explained in Section III). All the geometrical parameters from Eq. (19) can be estimated for particulars of the porous media. For example, in the case of aligned fibers, hydraulic radius and equivalent diameter can be expressed by ... 

The exact pore shape is usually unknown and cylindrical pores are generally assumed. Mikhail, Brunauer, and Bodor show in their paper that equation (9.17) is equally valid for parallel plate or cylindrical pores and that the mean hydraulic radius in Table 9.1, is the same as the separation between plates or the cylindrical radius. 

The mean hydraulic radius, n, of a group of meso-pores, is defined as... 

The ratio of specific pore volume to specific surface area, vja, has been used for many years as a simple means of characterizing the pore size. This volume-to-surface ratio, when applied to a group of pores, is known as the hydraulic radius, rh, and has an unambiguous physical significance provided that the pore geometry can be specified by a single parameter (Everett 1958). 

Structural models emerge from the notion of membrane as a heterogenous porous medium characterized by a radius distribution of water-filled pores. This structural concept of a water-filled network embedded in the polymer host has already formed the basis for the discussion of proton conductivity mechanisms in previous sections. Its foundations have been discussed in Sect. 8.2.2.1. Clearly, this concept promotes hydraulic permeation (D Arcy flow [80]) as a vital mechanism of water transport, in addition to diffusion. Since larger water contents result in an increased number of pores used for water transport and in larger mean radii of these pores, corresponding D Arcy coefficients are expected to exhibit strong dependencies on w. 

In order to estimate the pore size distributions in microporous materials several methods have been developed, which are all controversial. Brunauer has developed the MP method [52] using the de Boer t-curve. This pore shape modelless method gives a pore hydraulic radius r, which represents the ratio porous volume/surface (it should be realised that the BET specific surface area used in this method has no meaning for the case of micropores ). Other methods like the Dubinin-Radushkevich or Dubinin-Astakov equations (involving slitshaped pores) continue to attract extensive attention and discussion concerning their validity. This method is essentially empirical in nature and supposes a Gaussian pore size distribution. 

Assuming that the pore diameter is characterized by the mean half hydraulic radius of the pore system, further assuming complete wetting and spherical monosized particles, the following equation is obtained ... 

Assumption 4 is invoked to relate the pressure jump across the spherical interface with the actual bubble point pressure of the LAD screens. Therefore, the equivalent radius, flpore. of the vertical capillary tube that is isomorphic with the mean curvature, Hm, can be approximated using a hydraulic diameter of a circle inscribed within the complex triangular pore shape such that Equation (3.13) becomes ... 

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