Parallel-pore model

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... 

Q. To proceed further at this point one has to specify a pore model for the catalyst, and a model for the active site distribution. Froment and co-workers have examined a variety of cases such as single pore models (single-ended pores and pores open on both sides) with both deterministic and stochastic active site distributions, the bundle of parallel pores model and various tree-like models of the porous structure, which were earlier used by Pismen (40) to describe transport and reaction in porous systems. Such treelike models contain interconnected pores but lack any closed loops and are usually called Bethe networks or lattices. They are completely characterized by their coordination number Z, which is the number of pores connected to the same site of the network. 

Fresh Zeolite. The diffusivities within usual porous catalyst (pore radius a few nm) can be estimated by the parallel pore model (18) or the random pore model (19). However, configurational diffusion occurs within the pores of zeolites (pore diameter < 1 nm) and there are only a few reports on the measurement or estimation methods of the diffusivities of zeolites, especially at higher temperature range (20,21). Here we will review the results of ZSM-5, which first explains the diffusivity of fresh ZSM-5, then the results of coke loaded ZSM-5. 

A single pore is evidently an oversimplified model of a catalyst particle A relatively stralghtforvard extension, accounting up to a certain extent for the pore size distribution, is the parallel pore model. This is still unsatisfactory with complex, multilayered catalysts and/or when blockage by metal or coke deposition occurs For such cases the location of the blockage matters and pore branching and types of interconnection between pores become of importance. 

The Parallel-pore Model Wheeler proposed a model, based on the first three of these properties, to represent the monodisperse pore-size distribution in a catalyst pellet. From p and Vg the porosity e is obtained from Eq. (8-16). Then a mean pore radius d is evaluated by writing equations for the total pore volume and total pore surface in a pellet. The result, developed as Eq. (8-26), is... 

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... 

Blue et al. have studied the dehydrogenation of butane at atmospheric pressure, using a chromia-alumina catalyst at 530°C. For a spherical catalyst size of dp = 0.32 cm the experimental data suggest a first-order rate constant of about 0.94 cm /(sec)(g catalyst). The pore radius is given as 110 A. Assuming Knudsen diffusivity at this low pressure and estimating the pore volume as 0.35 cm /g, predict an effectiveness factor for the catalyst. Use the parallel-pore model with a tortuosity factor of 2.0. 

With the aid of the parallel-pore model, it is possible to calculate the product of Papp and Sm, which allows one to express heterogeneous kinetic rate laws in pseudo-volumetric form. The parallel-pore model is discussed in greater depth later in this chapter. For the present discussion, it is only necessary to visualize straight cylindrical pores of length L and radius (raverage)- If E represents the total number of pores, then the void volume is... 

ESTIMATING TORTUOSITY FACTORS AND INTRAPELLET POROSITY BASED ON THE DISTRIBUTION IN ORIENTATION AND SIZE OF CATALYTIC PORES VIA THE PARALLEL-PORE MODEL... 

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 ... 

TABLE 21-3 Normalized Orientational Part of the Distribution Function in the Parallel-Pore Model for Several Cases and the Corresponding Tortuosity Factors... 

Finally, the parallel pore model was used to account for the actual distribution of molecular size and pore size. Eq. 3.S.d-9 for communicating pores, with an average value for tortuosity, was utilized ... 

E. H. Cwirko and R. G. Carbonell, Interpretation of Transport-CoefTicients in Nafion Using a Parallel Pore Model, Journal of Membrane Science, 67,227 (1992). Y. M. Volfkovich, V. S. Bagotzky, V. E. Sosenkin, and I. A. Blinov, The Standard Contact Porosimetry, Colloids and Surfaces a-Physicochemical and Engineering Aspects, 187, 349 (2001). 

In [82] different model variants of pore-space evolution (random network, serial and parallel pore models) were compared to each other. A morphology of equally swelling parallel pores gives the most favorable a(w) relations with steepest increase of proton conductivity at small water contents. Results obtained for such a morphology are in good agreement with conductivity data of Dow membranes, which possess shorter pendant sidechains than Nafion. 

The two major pore models that have been used extensively over the years for practical purposes are the parallel-pore model proposed by Wheeler in 1955 [5, 9] and the random-pore model proposed by Wakao and Smith in 1962 [34]. Among the more recent advanced models are the parallel cross-linked pore model [35] and pore-network models [36, 37]. 

The original semiempirical parallel-pore model represents a monodisperse pore-size distribution and makes use of the measurable physical properties, Sg, Vg, ps, and Gp. The complex particle with porosity Gp is replaced by an array of straight and parallel cylindrical pores of radius a, much like a honeycomb structure. The mean pore radius a is simply calculated by assuming that the sum of the inner surface areas of all the n pores in an array nlnaL) is equal to the total surface area Sg and the sum of all the pore volumes nna L) is equal to the experimental pore volume V [5] ... 

Gm) and micropores (Gp), as well as separate mean pore radii for macropores (am) and micropores (a ). In the microregion, df, is calculated in the same manner as in the parallel-pore model using Sg and Vg, while in the macroregion related to the void spaces between primary particles, Am is obtained from pore-volume distribution. When Gm = 0, a monodisperse model similar to the parallel-pore model is obtained [9, 17, 34]. 

The only other variable required for predicting D is the length of the diffusion path, which is the thickness of the particle multiplied also by the adjustable tortuosity factor, S, that accounts for distorted diffusion pathways and also for varying pore cross sections in interconnections and constrictions the value of 5 varies between V2 and 10 but is typically 3 or 4 in most industrial catalysts. The simplest geometric model which is still commonly used in practical applications for estimating is the parallel-pore model ... 

Effective diffusion coefficients in catalyst particles are calculated as functions of bulk gas diffusion coefficients, pore volume distribution specified as particle porosity, 8p, as a function of pore radius and the so-called tortuosity factor, x, which describes the actual road a molecule must travel. The use of different effective diffusion models is discussed in the literature [199] [436] and performance of measurements in [221], Below is shown the basic parallel pore model, where the effective diffusion coefficient, De is calculated from the particle porosity, the tortuosity factor, and the diffusion coefficient in the bulk and the Knudsen diffusion coefficient, Dbuik and Dk [199] [389] [440] as ... 

Cwirko, E. H. and CarboneU, R. G. 1992a. Interpretation of transport coefficients in Nafion using a parallel pore model. 67(2-3), 227-247. 

The same equation can be deriv on the basis of the parallel pore model which pictures transport as occurring through a number of parallel capillaries of the same size 14 The detailed derivation for this is describ elsewhere (Teramoto, M. et al., Ind, Eng, Chem. Res. in press.). 

Smith [134] discusses the simple parallel-pore and random-pore models, which are extensively adopted in chemical reactor engineering describing the effective diffusiv-ities of porous material. The parallel-pore model presents a mono-disperse pore-size... 

Broad Pore-Size Distribution Several models have been developed to estimate the effective diffusivity in catalysts with broad pore-size distributions." One of the simplest is the parallel pore model of Johnson and Stewart. In their approach, the value ofDp ir) is calculated by averaging over the whole range of pore sizes ... 

The previously described models for the structural parameters Bq, Kq, and T, while straightforward, are oversimplified. More complex equations have been developed to describe diffusion in solids with broad pore size distribution. The model of Johnson and Stewart [37] (henceforth called the parallel-pore model) assumes that each pore makes its contribution to the total diffusive flux independently of the others. For isobaric conditions,... 

Satterfield and Cadle [38] determined the tortuosity factors for 17 commercially manufactured, pelleted catalysts and catalyst supports using the parallel-pore model. Except for two materials that had been calcined at very high temperatures, all tortuosity factors fell between 3 and 7. For about half the catalysts, the tortuosity factor was about 4, regardless of macroporosity or composition. 

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