Diffusion single pore

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... 

Natural rocks seldom have a single pore size but rather a distribution of pore sizes. If all pores are in the fast-diffusion limit, have the same surface relaxivity and have no diffirsional coupling, then the pores will relax in parallel with a distribution of relaxation times that corresponds to the distribution of the pore sizes. The magnetization will decay as a sum of the exponentials as described by Eq. (3.6.4). 

It should be noted that the decomposition shown in Eq. 3.7.2 is not necessarily a subdivision of separate sets of spins, as all spins in general are subject to both relaxation and diffusion. Rather, it is a classification of different components of the overall decay according to their time constant. In particular cases, the spectrum of amplitudes an represents the populations of a set of pore types, each encoded with a modulation determined by its internal gradient. However, in the case of stronger encoding, the initial magnetization distribution within a single pore type may contain multiple modes (j)n. In this case the interpretation could become more complex [49]. 

The rate of production of V within a single pore must be equal to the net rate of diffusion out of the pore. 

The earliest models of fuel-cell catalyst layers are microscopic, single-pore models, because these models are amenable to analytic solutions. The original models were done for phosphoric-acid fuel cells. In these systems, the catalyst layer contains Teflon-coated pores for gas diffusion, with the rest of the electrode being flooded with the liquid electrolyte. The single-pore models, like all microscopic models, require a somewhat detailed microstructure of the layers. Hence, effective values for such parameters as diffusivity and conductivity are not used, since they involve averaging over the microstructure. 

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. 

This problem may look at first to be the same geometry as for diffusion in a single pore, but this situation is quite drSerent hr the single pore we had reaction controlled by diffusion down the pore, while in the tube wall reactor we have convection of reactants down the tube. 

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... 

Rajagopalan and Luss (1979) developed a theoretical model to predict the influence of pore properties on the demetallation activity and on the deactivation behavior. In this model the change in restricted diffusion with decreasing pore size was included. Catalysts with slab and spherical geometry composed of nonintersecting pores with uniform radius but variable pore lengths were assumed. The conservation equation for diffusion and first-order reaction in a single pore of radius rp is... 

The model formulated by Ahn and Smith (1984) considered partial surface poisoning for HDS and pore mouth plugging for HDM reactions. The conservation equations with first-order reactions for metal-bearing and sulfur-bearing species were based on spherical pellet geometry rather than on single pores. Hence, a restricted effective diffusivity was employed... 

When people consider confinement effects, they consider mainly an increase in the encounter probability inside a single pore and therefore, expect an acceleration of the reaction. Such in-pore acceleration has been quantified by Tachiya and co-workers for diffusion-limited reactions through the so-called confinement factor [see Eq. (11.58) in Ref. 40]. From this treatment, confinement effects are expected to disappear when the reaction radius is less than one tenth of the confinement radius. Considering the reaction radii of radiolytic species, no acceleration by confinement should be expected for pore diameter larger than 10 nm. For smaller pore size, acceleration of the recombination reactions within spurs would be critical in the determination of radiolytic yields in the nanosecond time range. However, the existence of such an acceleration of radiolytic reactions has not been suggested in the nanosecond pulse radiolysis of zeolites and has still to be assessed using picosecond pulse radiolysis. 

Simple single pores as uniform diameter cylinders date back some 50 years [8]. As indicated in Fig. 2, a simple diffusion reaction balance describes surface catalysis in such pores. This analysis is the basis of almost all textbook treatments [9]. The approximation of typically complex labyrinthal pore spaces by such a simplified model is certain to introduce inadequacies into process simulation. 

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... 

In principle, Eq. (8) can be used to compute Cl in a single pore if the pore radius, a, is known. Implicit in Eq. (8) is the assumption that the tip does not interfere with the molecular diffusion from the pore. This is a reasonable approximation when the tip radius is very small compared with the pore radius, i.e., rt a. However, in many experiments, rt is comparable to or even larger than a, and the tip can not be treated as a noninterfering probe when it is placed directly at the pore opening. Thus, Eq. (8) is useful only as an approximate relationship between the tip current and ft. Furthermore, the radius of the pore, a, may not be known a priori in real samples. A quantitative method of analysis that circumvents both of these problems is presented in Sec. II.D. Regardless of tip interference effects, the key result here is that the magnitude of the SECM tip current is proportional to the molecular flux in the pore. 

The effective diffusion coefficient accounts for the rate of diffusion of the protein through the complex, porous polymer matrix. Incorporation of an effective diffusion coefficient is necessary because protein molecules do not diffuse through the pure polymer phase, but must find a path out of the slab by diffusing through a tortuous, water-filled network of pores. Z>eff is assumed to be independent of position in the slab Equation 9-14 is justified in this case by local averaging over a volume that is large compared to a single pore. Characteristic desorption or protein release curves are represented by Equation 9-18, with substitution of Dgn- for Z),.p ... 

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