Cylinder pore model

The planar symmetric pore consists of two parallel walls with the distance H between them which infinitely range into the x- and j/-direction of the pore-fixed coordinate system. The 2-axis stands perpendicularly on the i-j/-plane as the normale of both walls. The cylinder pore model places its j/-axis as the rotational axis. The z-axis stands perpendicularly on the pore wall as in slit-like pores and runs through the middle of the pore. Hence the x- differs from the y-axis inside the cylinder pore in opposite to the slit-like pore. This fact turns out to be important even for the adsorption of fluids which consists of non-spherical particles. 

Figure 15.1 Simple cylinder pore model of size-exclusion chromatography. In reality, not all pores are equal in size.

The working out of these ideas will be illustrated by reference to a number of simple pore models the cylinder, the parallel-sided slit, the wedge-shape and the cavity between spheres in contact. 

The procedures described so far have all required a pore model to be assumed at the outset, usually the cylinder, adopted on the grounds of simplicity rather than correspondence with actuality. Brunauer, Mikhail and Bodor have attempted to eliminate the over-dejjendence on a model by basing their analysis on the hydraulic radius r rather than the Kelvin radius r . The hydraulic radius is defined as the ratio of the cross-sectional area of a tube to its perimeter, so that for a capillary of uniform cross-section r is equal to the ratio of the volume of an element of core to... 

A more detailed treatment has been given by Gurfein and his associates who chose as their pore model a cylinder with walls only one molecule thick. A few years later, Everett and Fowl extended the range of models to include not only a slit-shaped pore with walls one molecule thick, but also a cylinder tunnelled from an infinite slab of solid and a slit formed from parallel slabs of solid. 

All pore sizes are according to the slit shaped model Cylinder-shaped pore model (diameter)... 

Cylinder-shaped pore model Analase phase... 

Fig. 3-6. Cross-sections through various geometric pore models (ri) uniform capillary, (B) indented capillary (cylinders with different radii), (C) indented capillary (sphere with cylinders), (D) sinusoidal capillary, and (E) prefractal capillary.

Fig. 43. Fits of two-dimensional hollow-cylinder microenvironment model to pore-water NH/ profiles at FOAM (see Table V for model values). Solid lines represent measured profile, dashed represent model.

Fio. 44. Fits of two-dimensional hollow-cylinder microenvironment model to pore-water... 

The parallel-pore model provides an in-depth description of the void volume fraction and tortuosity factor Tor based on averages over the distribution in size and orientation, respectively, of catalytic pores that are modeled as straight cylinders. These catalyst-dependent strncture factors provide the final tools that are required to calculate the effective intrapellet diffusion coefficients for reactants and prodncts, as well as intrapellet Damkohler numbers. The following conditions are invoked ... 

This EVB model was employed to the systematic study of model pores or channels. Taking a simphstic view, the polymer was regarded as a rigid framework in which slab or cylinder pores of constant thickness or radius, respectively, are formed. Within this approach, proton transport in pores has been studied as a function of a variety of generic structural and dynamical features of the polymer and operational parameters of the working fuel cell (such as temperature and humidity). These studies revealed a number of factors determining the proton mobihty, such as the width of the channel, distance between the sidechains, and their flexibility. The main lessons of the simulations and the theoretical analysis were ... 

As mentioned before, the carbonation of CaO is a typical non-catalytic, gas-solid reaction. As snch, it has been extensively modelled using either random pore or grain models. The random pore model was developed and first applied by Bhatia and Perlmntter [57] to model the sulphation of lime and subsequently extended by Sun et al. [65]. The pores were assumed as an assembly of randomly oriented cylinders of uniform diameter, which initially overlapped (Fig. 6.17). The initial increase in the reaction rate was attributed to the growth of the surface area of the CaO-CaCOs interface, which is, however, overshadowed in later stages by the intersection of the growing surfaces, leading subsequently to a decrease in the reaction rate. 

In this umtk, we aim at j dning insists at a molecular level on the hysteretic behavior of adsorption/desorption isoftienns of several fluids rxinfined in a rather large collection of pore models tlut include sin e pores of varioiui gennetries (cylinders, ellipsoids, constricted pores and Vycor). 

As there is a relationship between the macromolecular coils size and the pore opening diameters, it is possible to calculate the dependence of accessible pore volume V on pore di une -ters d. Eor the model of cylinder pore shape we used the following equation... 

Both (i) and (ii) necessitate recourse to a model of pore shape. By far the commonest, chosen on grounds of simplicity, is the cylinder but the slit model is being increasingly used where the primary particles are plate-like, and the model where the pore is the cavity between touching spheres is beginning to receive attention. 

For each group of pores, the pore volume 6v is related to the core volume by means of a model, either the cylinder or the parallel-sided slit as the case may be. Allowance is made for the succession of film thicknesses corresponding to the progressive thinning of the multilayer in each pore, as desorption proceeds. Thus for group i, with radius rf when the film thickness is tj j > i) and the core volume is the pore volume 6vf will be given by... 

We have studied, by MD, pure water [22] and electrolyte solutions [23] in cylindrical model pores with pore diameters ranging from 0.8 to more than 4nm. In the nonpolar model pores the surface is a smooth cylinder, which interacts only weakly with water molecules and ions by a Lennard-Jones potential the polar pore surface contains additional point charges, which model the polar groups in functionalized polymer membranes. 

The most important mass transfer limitation is diffusion in the micropores of the catalyst. A simplified model of pore diffusion treats the pores as long, narrow cylinders of length The narrowness allows radial gradients to be neglected so that concentrations depend only on the distance I from the mouth of the pore. Equation (10.3) governs diffusion within the pore. The boundary condition at the mouth of the pore is... 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

Above we considered a porous catalyst particle, but we could similarly consider a single pore as shown in Fig. 5.36. This leads to rather similar results. The transport of reactant and product is now determined by diffusion in and out of the pores, since there is no net flow in this region. We consider the situation in which a reaction takes place on a particle inside a pore. The latter is modeled by a cylinder with diameter R and length L (Fig. 5.36). The gas concentration of the reactant is Cq at the entrance of the pore and the rate is given by... 

We have recently tested the Tx model described above by obtaining T, measurements in powder samples with known S/V. Samples used were constructed from fumed silica (CAB-O-SIL M-5 and TS-500, Cabot Corp.), and were either hydrophilic (M-5) or treated by the manufacturer to be hydrophobic (TS-500). Powder of each type was pressed into a polycarbonate cylinder, with a degree of compression controlling the pore space volume of each sample. These materials have a very high specific surface area (200 m2 g 1 for M-5, 212 m2 g-1 for TS-500), which is not expected to change significantly even at the maximum compaction pressure used. 

B) Surface and stick representation of the model, viewed down the fibril axis. (The top coil is shown as sticks, with the remainder showing van derWaals radii.) The glutamine side chains are proposed to form hydrogen-bonded stacks parallel to the fibril axis. The large diameter of the cylinder results in a pore down the center. Both panels were generated with Pymol (DeLano, 2002). 

Analytical expressions for the fluorescence decay in the case of RET between donor and acceptor molecules randomly distributed in various models of restricted geometries (spheres, cylinders, etc.) that mimic simple pores have been established by Klafter and Blumen (1985). The donor decay can be written in a form similar to that of Eq. (9.36) ... 

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