Pores, capillary condensation

A similar model has been applied to the modeling of porous media with condensation in the pores. Capillary condensation in the pores of the catalyst in hydroprocessing reactors operated close to the dew point leads to a decrease of conversion at the particle center owing to the loss of surface area available for vapor-phase reaction, and to the liquid-filled pores that contribute less to the flux of reactants (Wood et al., 2002b). Significant changes in catalyst performance thus occur when reactions are accompanied by capillary condensation. A pore-network model incorporates reaction-diffusion processes and the pore filling by capillary condensation. The multicomponent bulk and Knudsen diffusion of vapors in each pore is represented by the Maxwell-Stefan model. 

The process of adsorption takes place when the concentration of the adsorptive is greater than the equiUbrium value vahd for the given temperature however, desorption requires a fluid concentration of the adsorptive which is smaller than the equilibrium concentratiom An adsorption isotherm favorable for adsorption is unfavorable for desorption and vice versa. Condensation of gases or vapors and solidification or crystalhzation will start when the relative supersaturation becomes > 1. In the case of adsorbents with capillary or very narrow pores, capillary condensation is observed for relative saturations adsorption isotherm vahd for adsorption and desorption can sometimes be ejqrlained, see Fig. 2.4-2. Sohd materials exposed to drying (see Chap. 10) often show such hysteresis behavior which can sometimes be explained by the ciu-vature of the liqttid sttrface in capillaries The radius of this surface is greater in the case of adsorption in comparison to the radius valid for a desorption process, see Fig. 2.4-2. 

At adsorption temperatures below the critical temperature of the component to be adsorbed, the adsorbent pores may fill up with liquid adsorpt. This phenomenon is known as capillary condensation and enhances the adsorption capacity of the adsorbent. Assuming cylindrical pores, capillary condensation can be quantitatively described with the aid of the Kelvin equation, the degree of pore filling being inversely proportional to the pore radius. 

Exanqiles of this pore blocking effect on the desorpdon branch of the isottom Imve been well docummited in the porosimetry literature. Several years ago Kraemer [11] pointed out that in an ink botde-type pore, capillary condensation on the adsorption branch would be dominated by the radius of... 

Liu et al. [27] has developed a 2D partial flooding model that considered pore size distribution of GDL to explain flooding. Two different kinds of pores, hydrophobic pores and hydrophilic pores, were dealt with. For hydrophilic pores, capillary condensation of liquid water will occur before saturation, and water will condense first in smaller pores and then in bigger ones. On the contrary, for hydrophobic pores, water condensation occurs in some extent of oversaturation and first in bigger pores. The fault of water flooding is embedded in this model, assuming that GDL is composed of hydrophilic pores and hydrophobic pores with different diameters. At every local position, they have the same property of pore diameter distribution. The GDL is partially flooded to different extent along the channel. 

As a general rule, adsorbates above their critical temperatures do not give multilayer type isotherms. In such a situation, a porous absorbent behaves like any other, unless the pores are of molecular size, and at this point the distinction between adsorption and absorption dims. Below the critical temperature, multilayer formation is possible and capillary condensation can occur. These two aspects of the behavior of porous solids are discussed briefly in this section. Some lUPAC (International Union of Pure and Applied Chemistry) recommendations for the characterization of porous solids are given in Ref. 178. 

Below the critical temperature of the adsorbate, adsorption is generally multilayer in type, and the presence of pores may have the effect not only of limiting the possible number of layers of adsorbate (see Eq. XVII-65) but also of introducing capillary condensation phenomena. A wide range of porous adsorbents is now involved and usually having a broad distribution of pore sizes and shapes, unlike the zeolites. The most general characteristic of such adsorption systems is that of hysteresis as illustrated in Fig. XVII-27 and, more gener-... 

Adsorbents such as some silica gels and types of carbons and zeolites have pores of the order of molecular dimensions, that is, from several up to 10-15 A in diameter. Adsorption in such pores is not readily treated as a capillary condensation phenomenon—in fact, there is typically no hysteresis loop. What happens physically is that as multilayer adsorption develops, the pore becomes filled by a meeting of the adsorbed films from opposing walls. Pores showing this type of adsorption behavior have come to be called micropores—a conventional definition is that micropore diameters are of width not exceeding 20 A (larger pores are called mesopores), see Ref. 221a. 

The basis of the classification is that each of the size ranges corresponds to characteristic adsorption effects as manifested in the isotherm. In micropores, the interaction potential is significantly higher than in wider pores owing to the proximity of the walls, and the amount adsorbed (at a given relative pressure) is correspondingly enhanced. In mesopores, capillary condensation, with its characteristic hysteresis loop, takes place. In the macropore range the pores are so wide that it is virtually impossible to map out the isotherm in detail because the relative pressures are so close to unity. 

If the adsorbent contains mesopores, capillary condensation will occur in each pore when the relative pressure reaches a value which is related to the radius of the pore by the Kelvin equation, and a Type IV isotherm will... 

The model proposed by Zsigmondy—which in broad terms is still accepted to-day—assumed that along the initial part of the isotherm (ABC of Fig. 3.1), adsorption is restricted to a thin layer on the walls, until at D (the inception of the hysteresis loop) capillary condensation commences in the finest pores. As the pressure is progressively increased, wider and wider pores are filled until at the saturation pressure the entire system is full of condensate. 

This widespread conformity to the Gurvitsch rule constitutes powerful support for the capillary condensation hypothesis in relation to Type IV isotherms. It is perhaps hardly necessary to stress that in order to test data for conformity to the rule it is essential that the stage which corresponds to the complete filling of the pores shall be clearly identifiable—as in the... 

As already indicated in Section 3.1, the study of mesoporous solids is closely bound up with the concept of capillary condensation and its quantitative expression in the Kelvin equation. This equation is, indeed, the basis of virtually all the various procedures for the calculation of pore size... 

From the Kelvin equation it follows that the vapour pressure p over a concave meniscus must be less than the saturation vapour pressure p°. Consequently capillary condensation of a vapour to a liquid should occur within a pore at some pressure p determined by the value of r for the pore, and less than the saturation vapour pressure—always provided that the meniscus is concave (i.e. angle of contact <90°). 

It must always be borne in mind that when capillary condensation takes place during the course of isotherm determination, the pore walls are already covered with an adsorbed him, having a thickness t determined by the value of the relative pressure (cf. Chapter 2). Thus capillary condensation occurs not directly in the pore itself but rather in the inner core (Fig. 3.7). Consequently the Kelvin equation leads in the first instance to values of the core size rather than the pore size. The conversion of an r value to a pore size involves recourse to a model of pore shape, and also a knowledge of the angle of contact 0 between the capillary condensate and the adsorbed film on the walls. The involvement of 0 may be appreciated by consideration... 

In calculations of pore size from the Type IV isotherm by use of the Kelvin equation, the region of the isotherm involved is the hysteresis loop, since it is here that capillary condensation is occurring. Consequently there are two values of relative pressure for a given uptake, and the question presents itself as to what is the significance of each of the two values of r which would result from insertion of the two different values of relative pressure into Equation (3.20). Any answer to this question calls for a discussion of the origin of hysteresis, and this must be based on actual models of pore shape, since a purely thermodynamic approach cannot account for two positions of apparent equilibrium. 

We consider first a cylinder closed at one end, B (Fig. 3.11(a)). Capillary condensation commences at that end to form a hemispherical meniscus r, and are equal to one another and therefore to r , which in turn is equal to r, the radius of the core (cf. Equation (3.7) and Fig. 3.7). Thus capillary condensation, to fill the whole pore, takes place at the relative pressure... 

Fig. 3.11 Capillary condensation in cylindrical pores, (a) Cylinder closed at one end, B. The meniscus is hemispherical during both capillary condensation and capillary evaporation, (h) and (c) Cylinder open at both ends. The meniscus is cylindrical during capillary condensation and hemispherical during capillary evaporation. Dotted lines denote the...

As Everett points out, however, the analogy of a pore as a narrownecked bottle is over-specialized, and in practice a series of interconnected pore spaces rather than discrete bottles is more likely. The progress of capillary condensation and evaporation in pores of this kind (cf. Fig. 3.13) has been discussed by de Boer, and more recently by Everett. ... 

Fig. 3.15 (a) A pore in the form of an interstice between close-packed and equal-sized spherical particles. The adsorbed him which precedes capillary condensation is indicated, (b) Adsorption isotherm (idealized). 

When the relative pressure falls to pj/p", the second group of pores loses its capillary condensate, but in addition the film on the walls of the first group of pores yields up some adsorbate, owing to the decrease in its thickness from t, to t. Similarly, when the relative pressure is further reduced to pj/p°, the decrement (nj-Wj) in the uptake will include contributions from the walls of both groups 1 and 2 (as the film thins down from tj to fj), in addition to the amount of capillary condensate lost from the cores of group 3. It is this composite nature of the amount given up at each step which complicates the calculation of the pore size distribution. 

An essential feature is the involvement of 6A, the additional area of multilayer exposed during the particular step as the group of pores loses its capillary condensate. 5A is calculated from the volume and radius of the group, using the geometry of the cylinder (column 15). The total area of multilayer which is thinned down during any step is obtained by summing the SA contributions in all the lines above the line of the step itself (column 16). 

Consider stage i in the desorption process where the thickness of the adsorbed film is and the pores of radius r, have just lost their capillary condensate. The volume of multilayer lining the pores of any radius r, where r > ri, will then be (since the pores are cylindrical) ... 

The volume i>r,) of the multilayer film on the walls of all pores which have already given up their capillary condensate is... 

At the point where capillary condensation commences in the finest mesopores, the walls of the whole mesopore system are already coated with an adsorbed film of area A, say. The quantity A comprises the area of the core walls and is less than the specific surface A (unless the pores happen to be parallel-sided slits). When capillary condensation takes place within a pore, the film-gas interface in that pore is destroyed, and when the pore system is completely filled with capillary condensate (e.g. at F in Fig. 3.1) the whole of the film-gas interface will have disappeared. It should therefore be possible to determine the area by suitable treatment of the adsorption data for the region of the isotherm where capillary condensation is occurring. 

Note that /4 = 0 when capillary condensation is complete.) Integration by measurement of the area under the curve of ln(p°/p) against n between the stated limits therefore gives the value of A, which is the area of the walls of the cores, not of the pores (cf. Fig. 3.28). 

A difficulty in using the method is that of identifying the ptoint F, where capillary condensation commences. This is usually taken as the lower closure point of the loop but as was pointed out in Section 3.5, capillary condensation can occur without hysteresis if the pores are of an appropriate shape—such as wedge-like—before the irreversible condensation responsible for the hysteresis loop sets in. The uncertainty arising from this cause is considerable, since the curve of ln(p°/p) is very steep in this region (cf. Fig. 3.28). 

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