Single cylindrical pore diffusion

Mass Balance of a Single Cylindrical Pore and Diffusive... 

For a single cylindrical pore of length L and a reactant A diffusing into the pore, where a first-order reaction takes place at the pore surface, the power law rate expression... 

After passing through the boundary layer, the molecules of adsorbate diffuse into the complex structure of the adsorbent pellet, which is composed of an intricate network of fine capillaries or interstitial vacancies in a solid lattice. The problem of diffusion through a porous solid has attracted a great deal of interest over the years and there is a fairly good understanding of the mechanisms involved, at least for gas phase diffusion. Here, diffusion within a single cylindrical pore is considered and, then, the pore is related to the pellet as a whole. 

SECTION 11-1 GASEOUS DIFFUSION IN SINGLE CYLINDRICAL PORES... 

In the absence of experimental data it is necessary to estimate from the physical properties of the catalyst. In this case the first step is to evaluate the diffusivity for a single cylindrical pore, that is, to evaluate D from Eq. (11-4). Then a geometric model of the pore system is used to convert D to for the porous pellet. A model is necessary because of the complexity of the geometry of the void spaces. The optimum model is a realistic representation of the geometry of the voids, with tractable mathematics, that can be described in terms of easily measurable physical properties of the catalyst pellet. As noted in Chap. 8, these properties are the surface area and pore volume per gram, the density of the solid phase, and the distribution of void volume according to pore size. 

The earliest studies of diffusion and reaction in catalysts were by Thiele, Damkoehler, and Zeldowitsch." Thiele considered the problem from the standpoint of a single cylindrical pore (see Prob. 11-9). Since the catalytic area per unit length of diffusion path does not change in a straight cylindrical pore whose walls are catalytic, the results are of the form of those for a flat plate [Eqs. (11-55) and (11-56)]. 

Let us first consider diffusion in an idealized single cylindrical pore. Pick s law for a binary system with equimolar counter diffusing occurring is ... 

The equations for diffusion in a single cylindrical pore are established. 

The mechanisms of mass transfer in the pore volume have to be considered for the development of diffusion equations for the single cylindrical pore [9, 16, 29]. Pore diffusion may occur by one or more of three mechanisms ... 

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 combined diffusivity, Dcomh is calculated for the single cylindrical pore. 

A single cylindrical pore of length L and radius of r (=d) located in a microscopic section of the catalyst particle is generally used for modeling the diffusion-reaction process (Figure 2.3). The steady-state component mass balance for a control volume extending over the cross section of the pore includes diffusion of reactant into and out of the control volume as well as reaction on the inner wall surface. The simple case taken as an example is that of an isothermal, irreversihle first-order reaction ... 

Diffusion and Reaction in a Single Cylindrical Pore within the Catalyst Pellet... 

Single Cylindrical Pore In contrast to the interplay of chemical reaction with interfacial mass transfer, a reaction at the walls of the pores of a solid catalyst and the internal mass transfer by pore diffusion are not consecutive processes. For a single cylindrical pore of length L and a reactant A diffusing into the pore, where a first-order reaction takes place at the pore surface, we obtain ... 

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

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... 

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. 

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 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... 

Figure 5.10 Potential energy difference (W) for a single oxygen molecule at the entrance of a carbon cylindrical pore of diameter d. The pore regions where the diffusion mechanisms (activated diffusion, surface, and Knudsen flow) dominate are separated by the critical pore sizes dm, (where W = 0) and d,f (where W = 0.04eV), indicated by...

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. 

The capillary wall is composed of a single layer of endothehal cells about 1 /tm thick. Lipid soluble substances (e.g., O2) can diffuse across the entire wall surface, whereas water soluble substances are restricted to small aqueous pathways equivalent to cylindrical pores 8 to 9 nm in diameter (e.g., glucose in most capillaries in capillaries with tight junctions and few fenestrations (brain, testes), glucose is predominantly transported). Total pore area is about 0.1% of the surface area of a capillary. The permeability of the capillary wall to a particular substance depends upon the relative size of the substance and the pore ( restricted diffusion). The efficiency of diffusive exchange can be increased by increasing the number of perfused capillaries (e.g., heart and muscle tissue from rest to exercise), since this increases the surface area available for exchange and decreases the distances across which molecules must diffuse. 

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... 

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