Pore profiles

Homogeneous uniform pore profile through filter... 

Fig. 1.3 View of a pore profile illustrating the pore resistance (Rl... R3) and...

Figure 16-27 compares the various constant pattern solutions for R = 0.5. The curves are of a similar shape. The solution for reaction kinetics is perfectly symmetrical. The cui ves for the axial dispersion fluid-phase concentration profile and the linear driving force approximation are identical except that the latter occurs one transfer unit further down the bed. The cui ve for external mass transfer is exactly that for the linear driving force approximation turned upside down [i.e., rotated 180° about cf= nf = 0.5, N — Ti) = 0]. The hnear driving force approximation provides a good approximation for both pore diffusion and surface diffusion. 

Some studies of potential commercial significance have been made. For instance, deposition of catalyst some distance away from the pore mouth extends the catalyst s hfe when pore mouth deactivation occui s. Oxidation of CO in automobile exhausts is sensitive to the catalyst profile. For oxidation of propane the activity is eggshell > uniform > egg white. Nonuniform distributions have been found superior for hydrodemetaUation of petroleum and hydrodesulfuriza-tion with molybdenum and cobalt sulfides. Whether any commercial processes with programmed pore distribution of catalysts are actually in use is not mentioned in the recent extensive review of GavriUidis et al. (in Becker and Pereira, eds., Computer-Aided Design of Catalysts, Dekker, 1993, pp. 137-198), with the exception of monohthic automobile exhaust cleanup where the catalyst may be deposited some distance from the mouth of the pore and where perhaps a 25-percent longer life thereby may be attained. 

We apply the singlet theory for the density profile by using Eqs. (101) and (103) to describe the behavior of associating fluids close to a crystalline surface [120-122], First, we solve the multidensity OZ equation with the Percus-Yevick closure for the bulk partial correlation functions, and next calculate the total correlation function via Eq. (68) and the direct correlation function from Eq. (69). The bulk total direct correlation function is used next as an input to the singlet Percus-Yevick or singlet hypernetted chain equation, (6) or (7), to obtain the density profiles. The same approach can be used to study adsorption on crystalline surfaces as well as in pores with walls of crystalline symmetry. 

In Fig. 15 we show similar results, but for = 10. Part (a) displays some examples of the adsorption isotherms at three temperatures. The highest temperature, T = 1.27, is the critical temperature for this system. At any T > 0.7 the layering transition is not observed, always the condensation in the pore is via an instantaneous filling of the entire pore. Part (b) shows the density profiles at T = 1. The transition from gas to hquid occurs at p/, = 0.004 15. Before the capillary condensation point, only a thin film adjacent to a pore wall is formed. The capillary condensation is now competing with wetting. 

First we are looking for the adsorption of a fluid consisting of particles of species m, in a slit-like pore of width H. The pore walls are chosen normal to the z axis and the pore is centered at z = 0. Adsorption of the fluid m, i.e., the matrix, occurs at equihbrium with its bulk counterpart at the chemical potential The matrix fluid is then characterized by the density profile, p (z) and by the inhomogeneous pair correlation function A (l,2). The structure of that fluid is considered... 

We also have studied fluid distribution in the pore H = 6 (Fig. 12(b)) at Ppq = 4.8147 and at two values of Pp, namely at 3.1136 (p cr = 0.4) and at 7.0026 (pqOq = 0.7 Fig. 12(b)). In this pore, we observe layering of the adsorbed fluid at high values of the chemical potential Pp. The maxima of the density profile pi(z) occur at distances that correspond to the diameter of fluid particles. With an increase of the fluid chemical potential, pore filhng takes place primarily at pore walls, but second-order maxima on the density profile pi (z) are also observed. The theory reproduces the computer simulation results quite well. 

The principal effect of the presence of a smooth wall, compared to a free surface, is the occurrence of a maximum in the density near the interface due to packing effects. The height of the first maximum in the density profile and the existence of additional maxima depend on the strength of the surface-water interactions. The thermodynamic state of the liquid in a slit pore, which has usually not been controlled in the simulations, also plays a role. If the two surfaces are too close to each other, the liquid responds by producing pronounced density oscillations. 

Fig. 10 shows the radial particle densities, electrolyte solutions in nonpolar pores. Fig. 11 the corresponding data for electrolyte solutions in functionalized pores with immobile point charges on the cylinder surface. All ion density profiles in the nonpolar pores show a clear preference for the interior of the pore. The ions avoid the pore surface, a consequence of the tendency to form complete hydration shells. The ionic distribution is analogous to the one of electrolytes near planar nonpolar surfaces or near the liquid/gas interface (vide supra). 

Fig. 12 shows the atom density profiles as a function of the coordinate z along the pore axis. Oxygen atom positions correlate with positive and negative surface charges as a result of direct hydration and through hydrogen bonds, respectively. In a similar manner the maxima of the and CP density... 

FIG. 12 Normalized density profiles, along the pore axis. Top oxygen... 

Fig. 14 shows mobility profiles aeross two pores with a filling of 74 and 96 pereent. The figure displays the mean-square displaeement after 10 ps as a funetion of the p eoordinate of the water moleeule at initial time. The mobility is highest in the eenter of the pore and lowest near the pore surfaee. When the pore is almost eompletely filled (full line), the mobihty is more or less eonstant in the range 0 < / < 10 A and then deereases monotonieally. 

Dead-end Pores Dead-end volumes cause dispersion in unsteady flow (concentration profiles ar> ing) because, as a solute-rich front passes the pore, transport oceurs by molecular diffusion into the pore. After the front has passed, this solute will diffuse back out, thus dispersing. 

Elution profile of a large macromolecule (excluded from pore.s) (V —V )... 

It is obvious that these physical defects are dangerous in their own right but it is also possible for them to lead to subsequent corrosion problems, e.g. pitting corrosion at superficial non-metallic inclusions and crevice corrosion at pores or cracks. Other weld irregularities which may give rise to crevices include the joint angle, the presence of backing strips and spatter (Fig. 9.29). Butt welds are to be preferred since these produce a crevice-free profile and, furthermore, allow ready removal of corrosive fluxes. 

The small pore size and the uniform distribution result in capillary forces which should allow wicking heights and thus battery heights of up to 30 cm. Due to the cavities required for gas transfer and under the effect of gravity, the electrolyte forms a filling profile, i.e., fewer cavities remain at the bottom than at the top. Therefore with absorptive glass mats a rather flat battery... 

For a given set of conditions (lithology, climate, slope, etc.), there is presumably an optimum soil thickness that maximizes the rate of bedrock weathering (Fig. 9-3) (Carson and Kirkby, 1972 Stallard, 1985). For less than optimum soil thicknesses, there is insufficient pore volume in the soil to accept all the water supplied by precipitation and downhill flow. Excess water runs off and does not interact with the subsurface soil and bedrock. In contrast, water infiltrates and circulates slowly through thick soils (especially where forested If profile thicknesses greatly... 

This gives the concentration profile inside the pore, a l). The total rate of reaction within a pore can be found using the principle of equal rates. The reaction rate within a pore must equal the rate at which reactant molecules enter the pore. Molecules enter by diffusion. The flux of reactants molecules diffusing into a pore of diameter dp re equals the reaction rate. Thus,... 

Figure 2. Density profiles Illustrating effect of pore width on layering structure. Theory with 6 - oo LJ fluid.

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