Tortuosity parallel-pore model

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. 

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 ... 

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 ... 

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. 

Desig- nation Nominal Size Surface Area (m /g) Total Void Fraction Dt X 10s (cm2/sec) Average Tortuosity Factor t, Parallel-Path Pore Model r = 2K./5. (A) r Based on Average Pore Radius... 

The literature data on the tortuosity factor r show a large spread, with values ranging from 1.5 to 11. Model predictions lead to values of 1/e s (8), of 2 (parallel-path pore model)(9), of 3 (parallel-cross-linked pore model)(IQ), or 4 as recently calculated by Beeckman and Froment (11) for a random pore model. Therefore, it was decided to determine r experimentally through the measurement of the effective diffusivity by means of a dynamic gas chromatographic technique using a column of 163.5 cm length,... 

Early efforts to model catalyst deactivation either utilized simplified models of the catalyst s porous structure, such as a bundle of nonintersecting parallel pores, or pseudo-homogeneous descriptions in terms of effective diffusivities and tortuosity... 

More general models for the porous structure have also been developed by Johnson and Stewart [60] and by Feng and Stewart [43], called the parallel cross-linked pore model. Here, Eqs. 3.5.b-4 to 6 or Eq. 3.5.b-7 are considered to apply to a single pore of radius r in the solid, and the diffusivities interpreted as fte actual values rather than effective diffusivities corrected for porosity and tortuosity. A pore size and orientation distribution function /(r, Q), similar to Eq. 3.4-2, is defined. Then /(r, Q)dri is the fraction open area of pores with radius r and a direction that forms an angle Q with the pdlet axis. The total porosity is then... 

Influence on Electrolyte Conductivity In porous separators the ionic current passes through the liquid electrolyte present in the separator pores. Therefore, the electrolyte s resistance in the pores has to be calculated for known values of porosity of the separator and of conductivity, o, of the free liquid electrolyte. Such a calculation is highly complex in the general case. Consider the very simple model where a separator of thickness d has cylindrical pores of radius r which are parallel and completely electrolyte-filled (Fig. 18.2). Let / be the pore length and N the number of pores (all calculations refer to the unit surface area of the separator). The ratio p = Ud (where P = cos a > 1) characterizes the tilt of the pores and is called the tortuosity factor of the pores. The total pore volume is given by NnrH, the porosity by... 

Capillary theory uses the simplest model, whereby pores within a solid material are represented as parallel capillaries of equal diameters in the porous solid. The analogy is between the tortuous pore-system of the solid and the cylindrical pores of the capillaries. The equation for k is then derived from the Hagen-Poiseuille equation for streamline flow through straight circular capillaries taking account of the tortuosity of material s pores. The tortuosity is defined as the ratio of the actual length of the flow channel to the length of the porous medium. 

Hereby the pores are modeled by a set of parallel cylindrical pores with diameter d and a tortuosity t. is the so-called Adzumi factor taking values between 0.81 and 1. A/ is the molar mass of the gas molecules and T and Rq are the temperature and the gas constant. P is the specific permeability, i.e., the permeability normalized to the geometrical dimensions of... 

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