Pore filling pressure correlations

It has been shown that much better agreement between the HK and the DFT pore filling pressure correlations is obtained if the same form of the gas-solid potential and the same potential parameter values are used in comparing the two models [111]. 

Fig. 20. (a) Model nitrogen adsorption isotherms at 77 K calculated using a modified Kelvin-BET method for carbon slit pores of physical width (reading from left to right) 10.0,11.4,14.3, 21.4 and 42.9 A [138]. (b) Comparison of pore filling pressure correlations for DFT (points) and the MK-BET method (line) for nitrogen adsorption in carbon slit pores at 77 K [139]. 

In this paper, a modified HK method is presented which accounts for spatial variations in the density profile of a fluid (argon) adsorbed within a carbon slit pore. We compare the pore width/filling pressure correlations predicted by the original HK method, the modified HK method, and methods based upon statistical thermodynamics (density functional theory and Monte Carlo molecular simulation). The inclusion of the density profile weighting in the HK adsorption energy calculation improves the agreement between the HK model and the predictions of the statistical thermodynamics methods. Although the modified Horvath-Kawazoe adsorption model lacks the quantitative accuracy of the statistical thermodynamics approaches, it is numerically convenient for ease of application, and it has a sounder molecular basis than analytic adsorption models derived from the Kelvin equation. 

Figure 5 Comparison of filling pressure correlations for argon adsorption in carbon slit pores at 77 K using the original HK method, modified HK method, and DFT.

Fig. 19. Comparison of pore fill ing correlations developed to interpret experimental measurements of the nitrogen condensation pressure at 77 K in MCM-41 samples that have different pore diameters [129]. The dotted and short-dashed lines denote the results for the original Kelvin equation, i.e.. / = 0 in Eq. (40), with a cylindrical and hemispherical meniscus assumed respectively for the adsorption and desorption cases. The long-dashed line shows the pore filling correlation for the MK model of Eq. (40), and the solid line shows the result when the pore width is adjusted by an additional factor of 0.3 nm.

Trens et al. [11], have correlated the intersection of the desorption branch with the adsorption branch at the low pressure (referred to as the reversible pore filling or rpf ) with thermodynamic properties. Specifically, it seems to follow the Clausius-Clapeyron equation and follows that relationship expected from corresponding states relationship. This indicates that the rpf is characteristic of a first-order gas-liquid transition. The enthalpy of this transition is somewhat higher than the hquid-gas transition in llie bulk, which should not be surprising since the interaction of the solid with the adsorbate should supply an extra energy. 

It is interesting to observe that a fair correlation can be found between the pore size evaluated by the Washbum-Laplace model and the pore size evaluated by the BJH model of nitrogen adsorption in the case of SBA-15 [12] and other materials with interconnected pores [13], In the case of gas adsorption, the surface defects are filled at a lower pressure and do not affect the pressure of capillary condensation [10]. However, the BJH model does not take into account the effects of curvature on condensation and systematically underevaluates the size of the mesopores [7, 14]. 

It is well established that the pore space of a mesoporous solid fills with condensed adsorbate at pressures somewhat below the prevailing saturated vapor pressure of the adsorptive. When combined with a eorrelating function that relates pore size with a critical condensation pressure, this knowledge can be used to characterize the mesopore size distribution of an adsorbent from its adsorption isotherm. The correlating function most commonly used is the Kelvin equation [1], Refinements make allowance for the reduction of the physical pore size by the thickness of the adsorbed film existing at the critical condensation pressure [1-2]. Still further refinements adjust the film thickness for the curvature of the pore wall [3]. 

Qe is the energy transferred per imit total area of the particle normal to the direction of heat transfer. The effective thermal conductivities of catalyst pellets are remarkably low because of the pore structure. The contribution of the thermal conductivity of the solid skeleton is little, since the extremely small heat transfer areas existing at solid-solid contact points offer substantial resistance to heat transfer. The gas phase filling the void spaces in the pores also participates in hindering heat conduction experimental results indicate that decreases as Gp increases. At low pressures, when the mean free path of molecules is greater than or equal to pore size, increases with total pressure since free-molecule conduction starts to dominate. There are no general correlations for predicting Ae from the physical properties of the solid and fluid phases involved. An approximate correlation based on the thermal conductivities of the individual phases and the porosity of the particle has been proposed ... 

To illustrate, suppose an increase in external gas pressure causes liquid to condense into a small channel. If the other end is connected to a bigger cavity the liquid would simply evaporate from that end of the channel into the bigger cavity until equilibrium was reached. Adsorption filling is determined by the size of the pore correlated to relative pressure whereas desorption is determined by the branching interconnectivity of the porous network [79]. 

If flux measurements are started when the relative vapor pressure of the eon-densable gas equals unity, all the pores in the membrane should be elosed (i.e., filled with condensed liquid), avoiding any diffusive flux of the noneondensable gas through the membrane. When the relative vapor pressure is slightly below unity the liquid contained in the largest pores starts to vaporize diereby opening these pores. The Kelvin equation allows correlation of die relative pressure with the size of the pores opened to flux (Kelvin radius, ). The measured flow of the noncondensable gas can be easily translated in terms of pore number onee die appropriate gas transport model is taken into account. By decreasing steadily die relative pressure until all the pores are opened, both the differential and integral pore size distributions can be obtained. 

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