Cylindrical pore model

4 Theoretical evaluation of hysteresis loops 3.4. 1 Cylindrical pore model 

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A procedure involving only the wall area and based on the cylindrical pore model was put forward by Pierce in 1953. Though simple in principle, it entails numerous arithmetical steps the nature of which will be gathered from Table 3.3 this table is an extract from a fuller work sheet based on the Pierce method as slightly recast by Orr and DallaValle, and applied to the desorption branch of the isotherm of a particular porous silica. 

The equations for effectiveness factors that we have developed in this subsection are strictly applicable only to reactions that are first-order in the fluid phase concentration of a reactant whose stoichiometric coefficient is unity. They further require that no change in the number of moles take place on reaction and that the pellet be isothermal. The following illustration indicates how this idealized cylindrical pore model is used to obtain catalyst effectiveness factors. 

The Effectiveness Factor for a Straight Cylindrical Pore Second- and Zero-Order Reactions. This section indicates the predictions of the straight cylindrical pore model for isothermal reactions that are zero- and second-... 

Plots of effectiveness factors versus corresponding Thiele moduli for zero-, first-, and second-order kinetics based on straight cylindrical pore model. For large hr, values of r are as follows ... 

As Figure 12.3 indicates, it is also possible to obtain an analytical solution in terms of the straight cylindrical pore model for the case of a zero-order reaction. Here the dimensionless... 

The measured value of k Sg is 0.716 cm3/(sec-g catalyst) and the ratio of this value to k ltTueSg should be equal to our assumed value for the effectiveness factor, if our assumption was correct. The actual ratio is 0.175, which is at variance with the assumed value. Hence we pick a new value of rj and repeat the procedure until agreement is obtained. This iterative approach produces an effectiveness factor of 0.238, which corresponds to a differs from the experimental value (0.17) and that calculated by the cylindrical pore model (0.61). In the above calculations, an experimental value of eff was not available and this circumstance is largely responsible for the discrepancy. If the combined diffusivity determined in Illustration 12.1 is converted to an effective diffusivity using equation 12.2.9, the value used above corresponds to a tortuosity factor of 2.6. If we had employed Q)c from Illustration 12.1 and a tortuosity factor of unity to calculate eff, we would have determined that rj = 0.65, which is consistent with the value obtained from the straight cylindrical pore model in Illustration 12.2. 

The analyses of simultaneous reaction and mass transfer in this geometry are similar mathematically to those of the straight cylindrical pore model considered previously, because both are essentally one-dimensional models. In the general case, the Thiele modulus for semiinfinite, flat-plate problems becomes... 

Figure 16. The pore-size distribution for sol—gel-derived birnessite Na(5Mn02 20 as processed into three pore-solid nanoarchitectures xerogel, ambigel, and aerogel. Distributions are derived from N2 physisorption measurements and calculated on the basis of a cylindrical pore model. (Reprinted with permission from ref 175. Copyright 2001 American Chemical Society.)...

The pore diameter on the abscissa is calculated by employing a particular pore model, usually to the intrusion branch. As a matter of convenience, a cylindrical pore model is traditionally applied. On the ordinate, steep changes in the cumulative diagram are reflected as peak maxima in the incremental curve. From several possible representations (incremental, differential, log differential), the log differential plot seems to be the most revealing, since the areas under the peaks are proportional to the pore volume [79]. Data that can easily derived from mercury intrusion are the pore size distribution, median or average pore size, pore volume, pore area, bulk and skeletal density, and porosity. 

Wheeler s treatment of the intraparticle diffusion problem invokes reaction in single pores and may be applied to relatively simple porous structures (such as a straight non-intersecting cylindrical pore model) with moderate success. An alternative approach is to assume that the porous structure is characterised by means of the effective diffusivity. (referred to in Sect. 2.1) which can be measured for a given gaseous component. In order to develop the principles relating to the effects of diffusion on reaction selectivity, selectivity in isothermal catalyst pellets will be discussed. 

An approximate quantitative treatment was first proposed by Glueckauf127) in terms of a cylindrical pore model which yields... 

Wheeler<26) considered the problem of chemical selectivity in porous catalysts. Although he employed a cylindrical pore model and restricted his conclusions to the effect of pore size on selectivity, the following discussion will be based on the simple geometrical model of the catalyst pellet introduced earlier (see Fig. 3.2 and Section 3.3.1). 

Because of the conceptual and mathematical simplicity of these models, they have been used recently to describe adsorption in zeolite and clay structures,53,54,55,56 based on either a cylindrical pore model or a packed sphere model. 

The results obtained with the cylindrical pore geometry (Table 9.2) are in reasonable agreement with the reported experimental data [97], However, for the H-Y zeolite, the cylindrical pore model did not provide a good result, since the pore system of the zeolite Y resembles a three-dimensional cylindrical system [115], The appropriate model for the zeolite Y is the spherical geometry pore [107] in this regard, the results reported in Table 9.3 shows that only the zeolite Y is properly described with the spherical geometry pore model [97],... 

Experimental (- eXHp2) and Calculated (- c) Values for the Cylindrical Pore Model for H-ZSM-5, H-MOR, H-Beta, H-USY, and H-MCM-41... 

For the five mixtures, the cumulative mesoporous volume, Feds, and mesoporous surface area, S edB, and are both linear decreasing functions of the micropore content y (Figure 2b). The cumulative specific surface area SedB is definitely a better estimator of the mesoporous surface than the specific surface S xt computed Ifom the t-plot. The lUPAC classification states that mesopores are pores whose width is larger that 2 nm. In the case of the cylindrical pore model retained for the pore size distribution, this is equivalent to radii larger than 1 nm. It should however be stressed that the calculation of the cumulative surface and volume of the mesopores must not be continued at lower pressures than the closing of the hysteresis loop (gray zones of Figures 3a and 3b). If a black box analysis tool is used and if the calculation is systematically continued down to 1 nm, severe overestimation of the mesopores surface and volume may occur. 

The total pore volume Vpi has been estimated from the amount of gas. Fads, adsorbed at a relative pressure PI Po=Q.99. Assuming a straight cylindrical pore model, V(d) has been determined according to the BJH (BARRET, JOYNER and HALENDA) method. The porosity is defined by the silica network density 2.2 g/cm and the specific pore volume 0.227 cm /g. It follows c 30%. 

Figure 2 Simple-cylindrical-pore model for a catalyst. (From Ref. 9.)...

The MIP results were calculated using the Washburn equation [3] (see below) assuming a cylindrical pore model graphs representing the intrusion volume versus pore diameter were plotted. 

Specific surface area values of SDDP bulk and size fractions measured by gas adsorption are approximately in the same range (3.3-3.7m /g). Specific surface areas as determined from MIP results assuming a cylindrical pore model are in keeping with those obtained by gas adsorption technique (2.9-3.7mVg). 

The pore size distribution is derived, assuming a cylindrical pore model, from the intrusion volume-pressure curve using the Washburn law dp = -Ay cos0) / P, where y is the surface tension of mercury (484 mN/m), 6 the solid/mercury contact angle (130°) and P the pressure exerted by the mercury. 

The adsorption-desorption isotherms of nitrogen at -196 °C obtained on all the catalysts under investigation were mainly of Type IV of Brunauer s classification [16], exhibiting hysteresis loops closed at P/Po ranging between 0.25 and 0.55. The adsorption data are summarized in Table 1, including BET-C constant, specific surface area(SBEj), total pore volume (Vp), estimated from the saturation values of the adsorption isotherms and average pore radius (r P), assuming cylindrical pore model for which superscript (cp) was used. 

This technique, developed by Eyraud [140] modified by Katz et al. [143] and recently by Cuperus et al. [141], is based on the controlled blocking of pores by capillary condensation of a vapour (e.g. CCli, methanol, ethanol, cyclohexane), present as a component of a gas mixture, and the simultaneous measurement of the gas flux through the remaining open pores of the membrane. The capillary condensation process is related to the relative vapour pressure by the Kelvin equation. Thus for a cylindrical pore model and during desorption we have... 

The solid line is the experimental curve, estimated by electron microscopy on the basis of a cylindrical pore model. The broken line is the theoretical curve calculated from the experimental S. and Pr ( 2 = 0.177 pm and Pr = 0.206) using Equation 32. For this membrane, we can calculate Ng directly from the experimental using the relation Nq = l/(irS. ) and X from the experimental Fr... 

Broens, Bargeman and Smolders( ) reported on the use of nitrogen sorption/desorption method for studying pore volume distributions in ultrafiltration membranes. The pore volume distributions were calculated for a cylindrical capillary model. More recent results from the same laboratory are published in this volume ( ). In our view, applicability of cylindrical pore models for asymmetric membranes should be verified, rather than assumed. This can be done, for example, by analysis of both branches of the sorption isotherm. For a reasonable model choice, the two pore volume distributions should be in substantial agreement. 

In these beds, pore size is determined by the number of nearest neighbors (coordination number) n, the sphere radius r, and the type of packing geometry. Two radii characterize the pore size one for the "throat" and one for the "cavity" of the pore (18). Isotherms have been calculated similar to those of Reference (2.), for polysulfone (density 1.370 g/cm ) spheres for values of n 4,6,8,10 (tetrahedral, primitive cubical, body-centered cubical, body-centered tetragonal geometries, respectively). Nitrogen vapor at -195.6°C was assumed and the adsorbed layer thickness was calculated with Halsey s equation (15) as in the cylindrical pore model. Calculated isotherms are plotted in Figure 5. 

For studies where the periodicity of the graphite surface plays a role in the determination of properties, (e.g., low-temperature determinations of the structure of layers adsorbed on graphite), the Fourier expanded molecule-surface potential of Steele is commonly used [4—6, 19]. For complex geometries such as heterogeneous surfaces (see Bojan et al., coal pores [20]) and fuUerenes [21] (Martinez-Alonso et al., Ar on C q), a fuU sum of the direct atom—atom potentials is needed. In the recent simulation studies of carbon nanotubes, some studies have used asummed atom-atom potential description (e.g., see the work of Stan et al. [22]) while others use a continuum cylindrical pore model [23, 24]. 

In the case of a three-phase electrocatalytic system, such as those of fuel cells, the cylindrical pore model is not applicable since it does not consider the problem of the partial pressure of the reactant in the gas phase. In this case, an equilibrium between the gaseous pressure inside the pore (which tends to force the electrolyte out of it) and the capillary forces of the electrolyte (which tend to flood the electrolyte away from the pore) must occur. This is known as the stable meniscus condition ... 

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