Pore diameter, radius

Let us now assemble the complete set of dimensionless parameters for the problem. These are set out in Table 11.1, where the last column indicates the nature of their dependence on the external pressure p, the mean pore diameter and the pellet radius a. Symbols ft and 0... 

FIG. 4 Nomialized concentration distribution of a 0.1 molar 1 1 electrolyte in an uncharged cylindrical pore of radius five times the diameter of the ions. The dashed line, solid up-triangles, and solid down-triangles are the neutral solvent particles, cations, and anions, respectively, in an SPM model with 0.3 solvent packing fraction. The open symbols are for the cations and anions in the RPM model. 

Bubble Point Large areas of microfiltration membrane can be tested and verified by a bubble test. Pores of the membrane are filled with liquid, then a gas is forced against the face of the membrane. The Young-Laplace equation, AF = (4y cos Q)/d, relates the pressure required to force a bubble through a pore to its radius, and the interfacial surface tension between the penetrating gas and the liquid in the membrane pore, y is the surface tension (N/m), d is the pore diameter (m), and P is transmembrane pressure (Pa). 0 is the liquid-solid contact angle. For a fluid wetting the membrane perfectly, cos 0 = 1. 

Figure 28a shows the result of SAXS on sample BrlOOO. We used Guinier s formula (see eq. 6) for the small angle scattering intensity, I(k), from randomly located voids with radius of gyration, Rg. Although Guinier s equation assumes a random distribution of pores with a homogeneous pore size, it fits our experimental data well. The slope of the solid line in Fig. 28b gives R - 5.5 A and this value has been used for the calculated curve in Fig. 28a. This suggests a relatively narrow pore-size distribution with an equivalent spherical pore diameter of about 14A. Similar results were found for the other heated resin samples, except that the mean pore diameter changed from about 12 A for samples made at 700°C to about 15 A for samples made at 1100°C.

In the reactor we are interested in the position in the bed z or height of the bed L, in the pellet we are interested in the position x in the pellet with radius R, in the pore we are interested in distance x down the pore diameter (ipore, nd on the walls of the pore we are interested in reactions on the catalyst particle diameter r/pamcie-... 

Many heterogeneous reactions give rise to an increase or decrease in the total number of moles present in the porous solid due to the reaction stoichiometry. In such cases there will be a pressure difference between the interior and exterior of the particle and forced flow occurs. When the mean free path of the reacting molecules is large compared with the pore diameter, forced flow is indistinguishable from Knudsen flow and is not affected by pressure differentials. When, however, the mean free path is small compared with the pore diameter and a pressure difference exists across the pore, forced flow (Poiseuille flow see Volume 1, Chapter 3) resulting from this pressure difference will be superimposed on molecular flow. The diffusion coefficient Dp for forced flow depends on the square of the pore radius and on the total pressure difference AP ... 

Ksec = 0 when the molecule hydrodynamic radius is higher than the mean pore diameter. KSEC is 1 with small molecules, which can easily penetrate into the pores. The most important parameters influencing resolution are the pore volume, pore size distribution, and particle size. The separation domain is between the exclusion volume Va and the inclusion volume ( V0 + Vp). 

It is possible that the water-filled a-LTX channel, which is relatively wide ( 10A at its narrowest (Krasilnikov and Sabirov 1992 Orlova et al. 2000), can pass small molecules. Indeed, a-LTX channels inserted in the membranes of synaptosomes, NMJ nerve terminals, and receptor-transfected COS7 cells appear to pass fluorescein (Stokes-Einstein radius, Re = 4.5 A) and norepinephrine (Re < 4 A) (Davletov et al. 1998 Rahman et al. 1999 Volynski et al. 2000), shown in Figure 2 for comparison with 8-hydrated calcium ion (Rc = 4.2 A) and the toxin channel. Analysis of impermeant cations commonly used in channel studies reveals that a-LTX channels are poorly permeable (Hurlbut et al. 1994) to glucosamine H+(Re = 4.6 A) and not significantly permeable (Tse and Tse 1999) to N-methyl-D-glucamine (Re = 5.2 A), thus limiting the pore diameter by 10 A. 

Mercury porosimetry measurements for a partially sintered alumina preform showed a bimodal pore size distribution with neck diameter Dn = 0.15 pm [Manurung, 2001], As a comparison with the pore sizes and distribution of the preform measured by porosimetry, SEM micrographs (Fig. 5.1) were taken before and after infiltration. Based on SEM examination, the pores in the preform before infiltration ranged in size from r 0.1-0.5 pm. Assuming an average pore radius of 0.3 pm, this radius is approximately four times larger than the pore-neck radius (Dn = 0.15 pm, so pore radius = 0.075 pm) determined by mercury porosimetry. 

Specimen BET surface area - I l .rr Micro- Meso- Macro- porosity" porosity porosity" Average pore diameter, Dave Standard Pore radius of the deviation of maximum differential PSD, a pore volume, Surface fractal dimension, tfsurf,SP Low-end thickness, High-end thickness, ad. max... 

Pore diameter and pore radius are expressed in nm. Vp is the volume (ml/g) of liquid adsorbate, and can be calculated from the volume of adsorbed gas (VJ by means of the Gurvitsch rule22... 

Three mesoporous silica gels, with variable mean pore radius and specific surface area, have been studied. The substrates are named according to their approximated mean pore diameter. Measured values appeared to differ somewhat from the product names.30 The Kieselgels 40, 60 and 100 have a mean pore diameter of 4.2, 7.0 and 12.0 nm, respectively. Specific surface area increases with decreasing pore radius. Measured values, using the BET method, are given in table 9.3. 

Amorphous microporous silica membranes as discussed here, consist of a macroporous a-alumina support (pore diameter -100 nm) with a mesoporous y-alumina intermediate layer (Kelvin radius of 2.5 nm) and a microporous silica top layer (pore diameter -4 A) [1,2],... 

For the preparation of tubular silica membranes, commercially available mesoporous membranes [17] are used. These tubular supports have a total length of 25 cm and are enamelled at both ends, required for a gas-tight sealing with carbon seals to the reactor, so that an effective porous length of 20 cm remains. The tube consists of 4 layers. Layer 1, 2 and 3 consist of a-alumina with a thickness of 1.5 mm, 40 and 20 im and a pore diameter of 12, 0.9 and 0.2 im respectively. Layer 4 consists of y-alumina with a thickness of 3-4 im a Kelvin radius of 4 nm. A schematic drawing of the cross-section of a mesoporous support tube is provided in Figure 4. 

Notes-. hydrothermal treatment, 4dc is the calcination temperature, R is the average radius of TiCL particles, v is the anatase content in a sample (% of TiC>2 by mass), Dp is the average pores diameter, Vs is the total pores volume, Ssp is the specific surface area, and y is the quantum yield of the evolved hydrogen. 

The above values of pore diameter are the distances between the centres of the carbon atoms in the pore walls. From a practical standpoint (as pointed out in previous chapters) it has been customary to refer to the effective pore diameter (or width). In the case of a cylindrical buckytube, an approximate evaluation of the effective diameter is arrived at by allowing for the radius of the carbon atom (c. 0.17 nm). Thus, for the two model diameters considered by Maddox and Gubbins (i.e. 1.02 and 4.78 nm), the corresponding effective pore diameters are c. 0.7 and 4.4 nm. These are the values used in the following discussion. 

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. 

Determine the time to dry in 10% relative humidity air at 100°C a spherical green body of radius 10 cm with a void fi action of 0.4 filled with H2O. Assume that boundary layer mass or heat transfer is the rate limiting step. The temperature of the surface is the wet bulb temperature. This time is the maximum time for the constant rate period. Data mean particle size = 1.0 /am mean pore diameter = 0.1 /am. See Problem 14.1 for drying data on water and air. 

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