Pores modelling

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. 

In the pioneer work of Foster the correction due to film thinning had to be neglected, but with the coming of the BET and related methods for the evaluation of specific surface, it became possible to estimate the thickness of the adsorbed film on the walls. A number of procedures have been devised for the calculation of pore size distribution, in which the adsorption contribution is allowed for. All of them are necessarily somewhat tedious and require close attention to detail, and at some stage or another involve the assumption of a pore model. The model-less method of Brunauer and his colleagues represents an attempt to postpone the introduction of a model to a late stage in the calculations. 

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 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... 

In order to allow for the thinning of the multilayer, it is necessary to assume a pore model so as to be able to apply a correction to Uj, etc., in turn for re-insertion into Equation (3.52). Unfortunately, with the cylindrical model the correction becomes increasingly complicated as desorption proceeds, since the wall area of each group of cores changes progressively as the multilayer thins down. With the slit model, on the other hand, <5/l for a... 

To convert the core area into the pore area ( = specific surface, if the external area is negligible) necessitates the use of a conversion factor R which is a function not only of the pore model but also of both r and t (cf. p. 148). Thus, successive increments of the area under the curve have to be corrected, each with its appropriate value of R. For the commonly used cylindrical model,... 

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. 

Reverse osmosis models can be divided into three types irreversible thermodynamics models, such as Kedem-Katchalsky and Spiegler-Kedem models nonporous or homogeneous membrane models, such as the solution—diffusion (SD), solution—diffusion—imperfection, and extended solution—diffusion models and pore models, such as the finely porous, preferential sorption—capillary flow, and surface force—pore flow models. Charged RO membrane theories can be used to describe nanofiltration membranes, which are often negatively charged. Models such as Dorman exclusion and the... 

Hint. Use the pore model to estimate an isothermal effectiveness factor and obtain eff from that. Assume le =0.15 J/(m s K). 

In the case that the effective diffusion coefficient approach is used for the molar flux, it is given by N = —Da dci/dr), where Dei = (Sp/Tp)Dmi according to the random pore model. Standard boundary conditions are applied to solve the particle model Eq. (8.1). 

Models of regular geometrical pores with rectangular, spherical, cylindrical, and conical shapes have been developed for electrophoresis and gel chromatography media. Figure 7, from Ref 314, gives samples of these uniform structures. These uniform-pore models have been used more extensively in the analysis of gel filtration chromatography. 

Omstein [276] developed a model for a rigidly organized gel as a cubic lattice, where the lattice elements consist of the polyacrylamide chains and the intersections of the lattice elements represent the cross-links. Figure 7 shows the polymer chains arranged in a cubic lattice as in Omstein s model and several other uniform pore models for comparison. This model predicted r, the pore size, to be proportional to I/Vt, where T is the concentration of total monomer in the gel, and he found that for a 7.5% T gel the pore size was 5 nm. Although this may be more appropriate for regular media, such as zeolites, this model gives the same functional dependence on T as some other, more complex models. 

Squire [364] and Porath [300,301] developed geometrical pore models for gel chromatography media. Squire considered a gel with a set of conical, cylindrical, and rectangular crevices, and found the pore volume, assumed equal to the partition coefficient K y, to vary as... 

FIG. 7 Uniform-pore models (figure based on that of Ref. 276). (a) Cubic lattice. (From Ref. 276.) (b) Conical lattice. (From Refs. 300 and 301.) (c) Spherical lattice. (From Ref. 297.) (d) Cylindrical lattice. (From Ref. 3.) (e) Circular pores in rectangular sheets. (From Ref. 3.) (f) Rectangular pores. (From Ref. 364.)... 

Gel filtration chromatography has been extensively used to determine pore-size distributions of polymeric gels (in particle form). These models generally do not consider details of the shape of the pores, but rather they may consider a distribution of effective average pore sizes. Rodbard [326,327] reviews the various models for pore-size distributions. These include the uniform-pore models of Porath, Squire, and Ostrowski discussed earlier, the Gaussian pore distribution and its approximation developed by Ackers and Henn [3,155,156], the log-normal distribution, and the logistic distribution. 

Typical steady-state voltage-current characteristics in poreforming electrolytes are shown in Fig. 16. A number of authors have attempted to interpret these dependences.103 104 Ebihara et a/.105 used an equation based on a pore model and taking into account a rate-determining transport through the barrier part of... 

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... 

Computed from the random pore model of Wakao and Smith (22) and given here at 101.3 kPa and 723 K. 

Parallel plates, flow between, 15 720t Parallel plate viscometers, 21 735-736 Parallel-pore model, 25 306 Parallel pores, 25 301 Parallel synthesis, microwaves in, 16 549-552... 

The observation of pores in the anodic oxide with a density in the order of 1011 cnT2 [Agl] supports the so-called fluctuating pore model [Lel3]. This model assumes that randomly distributed pores in the oxides work as charge collecting centers, which lead to oscillations synchronized by the applied external electric field. It should be noted that the observed pore density corresponds well with the roughness at the oxide-electrolyte interface observed after the stress-induced transition of an anodic oxide, as shown in Fig. 5.5. 

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

GAMMA-ALUMINA WITHOUT PVA pore model slit shaped... 

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