Cylindrical single pore

Cylindrical Single Pore (Length L, Diameter dp) and nth-Order Reaction... 

Figure 2. Theoretically predicted size exclusion effects of different solute types (cylindrical pores of single pore size). (O) random coil (Rg) (%) hard sphere (R, = r VIJJ) CA rod (R, = L, VIT)...

Capacitance of a single pore is proportional to its surface area. Therefore, for a cylindrical NP it is proportional to 2m l, where r is the NP radius, and / is its length. Inner resistance of NP is proportional to / as well. The NP parameters depend on how the activated carbon powder was obtained but they are normally not affected by treating the powder when fabricating the electrode. 

Taking these effects into account, internal pore diffusion was modeled on the basis of a wax-filled cylindrical single catalyst pore by using experimental data. The modeling was accomplished by a three-dimensional finite element method as well as by a respective differential-algebraic system. Since the Fischer-Tropsch synthesis is a rather complex reaction, an evaluation of pore diffusion limitations... 

FIGURE 12.1 Representation of a single cylindrical catalyst pore and mass balance for... 

The modeling of the internal pore diffusion of a wax-filled cylindrical single catalyst pore was accomplished by the software Comsol Multiphysics (from Comsol AB, Stockholm, Sweden) as well as by Presto Kinetics (from CiT, Rastede, Germany). Both are numerical differential equation solvers and are based on a three-dimensional finite element method. Presto Kinetics displays the results in the form of diagrams. Comsol Multiphysics, instead, provides a three-dimensional solution of the problem. 

There are two main types of single-pore models. In the first, the approach of Giner and Hunter is taken in which there are straight, cylindrical gas pores of a defined radius. These pores extend the... 

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. 

The single cylindrical pore is of course not the geometry we are interested in for porouS catalysts, which may be spheres, cylinders, slabs, or flakes. Let us consider first a honeycomb catalyst of thickness It with equal-sized pores of diameter cfp, as shown in Figure 7-14. The centers of the pores may be either open or closed because by symmetry there is no net flux across the center of the slab. (If the end of the pore were catalytically active, the rate would of course be sHghtly different, but we will ignore this case.) Thus the porous slab is just a collection of many cylindrical pores so the solution is exactly the same as we have just worked out for a single pore. 

Construct a cylindrical cell of radius Rc centered on a single pore as illustrated in Fig. 16.13 and solve the diffusion problem within it using cylindrical coordinates and the same basic method employed to obtain... 

Analysis of behavior in single pores is certainly an excellent place to start an understanding of adsorption hysteresis. On the other hand, real porous materials eu e in most cases not simply described in terms of single pore behavior. At the very least a distribution of pores of different sizes should be contemplated. The first analysis of hysteresis loops using a theory of adsorption in single pores together with a pore size distribution was the independent domain theory of Everett and coworkers (Everett, 1967). The most sophisticated application of this kind of approach was made by Ball and Evans (1989) who used density functional theory for adsorption in a distribution of cylindrical pores and compared the hysteresis loops obtained with those for xenon adsorbed in Vycor glass. 

The random structure of the porous electrode, illustrated in Figure 13.11(a), leads to a distribution of pore diameters and lengths. Nevertheless, the porous electrode is usually represented by the simplified single-pore model shown in Figure 13.11(b) in which pores are assumed to have a cylindrical shape with a length i and a radius r. The impedance of the pore can be represented by the transmission... 

The impedance of a one-dimensional cylindrical pore structure with invariant interfacial impedance along the wall of the pores can be described in terms of the electrolyte resistance the impedance for a single pore, Z the impedance of the pore wall, Z ii the pore radius, r, and the pore length, 1, using the relationships ... 

An effective diffusivity can now be predicted by combining Eq. (1 l-l) for a single pore with this parallel-pore model. To convert D, which is based on the cross-sectional area of the pore, to a diffusivity based upon the total area perpendicular to the direction of diffusion, D should be multiplied by the porosity. In Eq. (11-1), x is the length of a single, straight cylindrical pore. To convert this length to the diffusion path in a porous pellet, X , from Eq. (11-22) should be substituted for x. With these modifications the diffusive flux in the porous pellet will be... 

The models mentioned so far are limited in their application as they represent only first order reaction kinetics with Fickian diffusion, therefore do not allow for multicomponent diffusion, surface diffusion or convection. Wood et al. [16] applied the algorithms developed by Rieckmann and Keil [12,44] to simulate diffusion using the dusty gas model, reaction with any general types of reaction rate expression such as Langmuir-Hinshelwood kinetics and simultaneous capillary condensation. The model describes the pore structure as a cubic network of cylindrical pores with a random distribution of pore radii. Transport in the single pores of the network was expressed according to the dusty gas model as... 

The selectivity may be converted into terms of molecular wei t of various solutes by using the repressions given in Table 1. The maximal selectivity of a support is thus 0.7 per decade in molecular weic. As Kwn tiy Yau et al., the pore sh (i.e., cylindrical or open-slab pores) has only a small influence on the selectivity of the support (107). For a single pore size support the -range 0.1 to 0.9 will oorreqxxid to a solute size ratio... 

The combined diffusivity, Dcomb> calculated for a single cylindrical pore is based on the cross-sectional area of the pore perpendicular to the direction of diffusion. A catalyst particle consists of an assembly of single pores. Therefore, the ultimate aim is to find the effective diffusivity of the porous catalyst particle, Dg, based on the total area exposed by the cross sections of all the pores in the particle, which constitutes the total mass transfer area normal to the direction of diffusion. 

The geometric model is used to convert Dcomb of the single pore to De of the entire particle comprising an array of cylindrical pores. 

If a catalyst pellet (of any shape) has well-structured pores that are of imiform diameter d and length L and the pores are uniformly distributed throughout the volume of the pellet, then the overall rate equation can be derived by accounting for the rate of diffusion and rate of reaction in one single pore within the catalyst pellet. Consider a cylindrical pore of diameter d and length L (Figure 4.24) in a catalyst pellet in contact with a gas stream containing reactant A at concentration Ag- AS is the concentration of A in the gas at the pore mouth on the outer surface of the catalyst pellet. 

We here restrict ourselves to the presentation of a fairly simple example for the AI (7.81) originally proposed by Jaroniec and Choma [7.21, 7.62]. Assuming the sorbent material to include only micropores of simple cylindrical shape of different diameters which do not interconnect, adsorption on a single pore can be described by the DR-isotherm (7.79) with N = 2. Assuming also the miaopores to be statistically distributed according to a T-distribution function [7.63] of degree (n), one gets from (7.81) the isotherm... 

Figure 4.5.17 Representation of (a) a single cylindrical catalyst pore and (b) mass balance for an elementary slice of the pore. Adapted from Levenspiel (1999).

While several simplifying assumptions needed to be made so as to derive an analytical model, the model captures all relevant physical processes. Specifically, it employed thermodynamic equilibrium conditions for temperature, pressure, and chemical potential to derive the equation of state for water sorption by a single cylindrical PEM pore. This equation of state yields the pore radius or a volumetric pore swelling parameter as a function of environmental conditions. Constitutive relations for elastic modulus, dielectric constant, and wall charge density must be specified for the considered microscopic domain. In order to treat ensemble effects in equilibrium water sorption, dispersion in the aforementioned materials properties is accounted for. 

The model domain is a unit cell that contains a single pore, as shown schematically in Figure 2.19. The dimensionless microscopic swelling variable of the unit cell model is given by rj = Vp/vo, where Vp is the volume of the cylindrical pore and vq is the volume of the unit cell under dry conditions. The polymer volume fraction per unit cell is given by [Pg.104]

Length of cylindrical pore in PEM (cm). Chapter 2 Length of cylindrical reference pore in PEM (cm). Chapter 2 Single cell thickness (cm)... 

Ultrafiltration membrane (Whatman, Anotop 10), syringe (SGE, 10 mL), holder (Millipore, 13 mm) were assembled as shown in Fig. 7.2. The thicknesses (pore diameters) of the top and bottom layers of ultrafiltration membrane are 59 xm (200 nm) and 1 p,m (20 or 100 nm), respectively. The two layers contain a nearly equal number of cylindrical pores namely, each smaller pore is under a large 200-nm one, which prevents possible interference of the flow fields generated by different small pores at their entrances, i.e., each smaller pore is isolated, so that our study nearly resembles a single pore experiment even many pores are actually used. In each solution, we added an appropriate amount of short linear polystyrene chains with a size smaller than the small pore. They can pass through the small pore by diffusion even without any flow so they served as an internal... 

The single pore model is equivalent to the model for a slab-like particle. In fact, the conservation equation for the liquid phase (Eq. 1.5) can be written in a general form applicable to slab-like, cylindrical, and spherical particles (Lee 1984). The catalyst distribution at the end of the pore-filling period (Z), which is the approximate distribution that results after fast drying, is given by ... 

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