Conical pores

The chief cases that are the subject of the problems here are zero, first and second order in spheres, slabs and cylinders with sealed flat ends, problems P7.03.03 to P7.03.ll. A summary of calculations of effectiveness is in P7.03.02. The correlations are expressed graphically and either analytically or as empirical curve fits for convenience of use with calculator or computer. A few other cases are touched on L-H type rate equation, conical pores and changes in volume. Nonisothermal reactions are in another section. 

A first order catalytic reaction takes place in a conical pore of radius R and depth L. Find the concentration profile and the effectiveness. 

Fig. 9.19 Optical micrographs of (a) cylindrical pores showing a degradation effect at the pore tips, ascribed to a limitation of mass transport (15% HF, 3 V, 1200 s). (b) Conical pores do not show this degradation at the same depth (15% HF, 3 V, 720 s). After [Lei7],...

Capillary condensation can be illustrated by the model of a conical pore with a totally wetting surface (Fig. 2.12). Liquid will immediately condense in the tip of the pore. Condensation continues until the bending radius of the liquid has reached the value given by the Kelvin equation. The situation is analogous to that of a bubble and we can write... 

Figure 2.12 Capillary condensation into a conical pore with wetting and partially wetting surfaces.

Fig. 7M (a) The meniscus formed in conical pores of a gas-diffusion electrodes and (b) the current distribution near the three-phase boundary in one of the pores. 

Fig. 2.3. Schematic picture of pore types in a porous membrane, a Isolated pore b,f dead end pore c,d tortuous and/or rough pores (d) with constrictions (c) e conical pore.

Lord Kelvin realized that, instead of completely drying out, moisture is retained within porous materials such as plants and vegetables or biscuits at temperatures far above the dew point of the surrounding atmosphere, because of capillary forces. This process was later termed capillary condensation, which is the condensation of any vapor into capillaries or fine pores of solids, even at pressures below the equilibrium vapor pressure, Pv. Capillary condensation is said to occur when, in porous solids, multilayer adsorption from a vapor proceeds to the point at which pore spaces are filled with liquid separated from the gas phase by menisci. If a vapor or liquid wets a solid completely, that is the contact angle, 0= 0°, then this vapor will immediately condense in the tip of a conical pore, as seen in Figure 4.8 a. The formation of the liquid in the tip of the cone by condensation continues until the cone radius, r, reaches a critical value, rc, where the radius of curvature of the vapor bubble reaches the value given by the Kelvin equation (r = rc). Then, for a spherical vapor bubble, we can write... 

Figure 4.8 a. For completely wetting materials in a conical pore, the formation of the liquid in the tip of the cone by condensation continues until the cone radius, r, reaches a critical value, rc (r = rc). b. For partially wetting materials, where 6> 0°, the radius of curvature increases, so that (r > rc) and for a first approximation, r= rc cos 6. 

Two types of porous electrodes can be considered two- and three-phase systems, where the latter is the special case of a triphasic interface in fuel cells, where the gas, liquid, and solid coexist. In the former, the liquid reactant is dissolved in the electrolyte and transported to the active sites of the electrocatalyst. In each case, we can consider uniform, parallel, cylindrical, or conical pores that are topped at the bottom by the metal substrate and at the top by the electrolyte [19,20],... 

Current and Potential Distribution in a Conical Pore Electrode.394... 

GAS DIFFUSION ELECTRODE ASSEMBLY 16.3.1 Conical Pore Diffusion Electrode... 

If the pores have a mean radii less than the critical radius r of Equation 16.38, the electrolyte floods the pores on reversing the conditions, the pores get dry. One way to overcome this problem is the use of a quasi-conic pore system. 

FIGURE 16.5 (a) Quasi-conical pore for an electrocatalyst in a three-phase system gaseous reactant, liquid... 

If the C-face (0001) is anodized, conical pores with triangular cross-sections are observed, but the pores are not arranged into arrays of planes (images not shown). The conical pores intersect, forming an overall spongy network. Even though their typical size is very similar to that... 

Gracias s group utilized a dry etch technique for the fabrication of conical nanopores in Si substrates." They dispersed gold nanoparticles (GNPs) on the surface of a single Si crystal wafer and etched GNPs containing Si wafers with CF4/O2 plasma. They obtained conical pores with a small pore diameter as small as 20 nm with this method. The enhancement of the etch rate in the vicinity of Au NPs was explained by... 

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