Fractionation models tortuosity

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 ... 

Here, Da is diffusion coefficient in the amorphous phase alone, oc is the volume fraction of crystalline polymer, and t is a scalar quantity that denotes the tortuosity of diffusional path of the solute. The value of Da may be estimated by the Peppas-Reinhart model if the amorphous regions of the polymer are highly swollen. This substitution yields... 

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... 

In Equation (9.6), x is the direction of flux, nt [mol m-3 s 1 ] is the total molar density, X [1] is the mole fraction, Nd [mol m-2 s 1] is the mole flux due to molecular diffusion, D k [m2 s 1] is the effective Knudsen diffusion coefficient, D [m2 s 1] is the effective bimolecular diffusion coefficient (D = Aye/r), e is the porosity of the electrode, r is the tortuosity of the electrode, and J is the total number of gas species. Here, a subscript denotes the index value to a specific specie. The first term on the right of Equation (9.6) accounts for Knudsen diffusion, and the following term accounts for multicomponent bulk molecular diffusion. Further, to account for the porous media, along with induced convection, the Dusty Gas Model is required (Mason and Malinauskas, 1983 Warren, 1969). This model modifies Equation (9.6) as ... 

It is evident from the concepts presented that no tortuosity factor is involved in this model. The actual path length is equal to the distance coordinate in the direction of diffusion. To apply Eq. (11-25) requires void fractions and mean pore radii for both macro and micro regions. The mean pore radii can be evaluated for the micro region by applying Eq. (11-21) to this region. However, must be obtained from the pore-volume distribution, as described in Sec. 8-7. The mean pore radii are necessary in order to calculate 2>k)m om Eq. (11-27). 

Do these models of diffusion in complicated microenvironments help us predict rates of drug movement in tissues Unfortunately, insufficient experimental measurements are available to test predictions rigorously, but the models compare favorably to the data that are available. Nicholson et al. [90] have measured the effective diffusion coefficient for a variety of compounds in the brain tortuosity can be predicted from these measured values (Figure 4.22). Since these measurements were made in the same tissue, the porosity or extracellular volume fraction should be equal (the extracellular volume fraction of the brain is 20%). All of the measured tortuosities are in the range predicted by the various models for media with this porosity ( 2 to 30, cf. Figure 4.18). 

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... 

Ca, is the fluid reactant concentration in the pore, Rp the pore radius. D,p in this model may be a harmonic mean of the bulk and Knudsen diflusion coefficient with real geometries it would be a true effective difTusivity including the tortuosity factor and an internal void fraction. D p is an effective diffiisivity for the mass transfer inside the solid and is a correction factor accounting for the restricted availability of reactant surface in the region where the partially reacted zones interfere. For Jt(y) < LJ2 (shown in Fig. 4.5-2) or j>2 < J f e factor ( = 1 for L/ > R y) > L/2 or >i < y < yj the factor = 1 — (40/x) where tgB = (2/L) Ji (y) - (L/2) for y < yi the factor C 0, where R(y) is the radial position of the reaction front. It is clear from Eq. 4.S-1 that no radial concentration gradient of A is considered within the pore. 

The diffusion data was fitted by a two-component diffusion model and an alternate classification of the two components was discussed by Perkins and Batchelor (2012) in combination with traditional definitions of water types within such systems. At intermediate moistme contents, the diffusion coefficients of both components of water were observed to decrease with deCTeasing moisture content because SBW contribution increases. The relative mass fractions of water in each component were near equal and the ratio between the two diffusion coefficients remained near constant, implying that there was interaction between the two water components under these moisture conditions. Through plane and in-plane studies show an increase in the interaction effects and tortuosity of the diffusion motion for water through the sheet compared to that within the plane of the sheet. The fiber ultrastructme had no measured effect on the diffusion behavior (Perkins and Batchelor 2012). 

In an effective properties model, the porous microstructures of the SOFC electrodes are treated as continua and microstructural properties such as porosity, tortuosity, grain size, and composition are used to calculate the effective transport and reaction parameters for the model. The microstmctural properties are determined by a number of methods, including fabrication data such as composition and mass fractions of the solid species, characteristic features extracted from micrographs such as particle sizes, pore size, and porosity, experimental measurements, and smaller meso- and nanoscale modeling. Effective transport and reaction parameters are calculated from the measured properties of the porous electrodes and used in the governing equations of the ceU-level model. For example, the effective diffusion coefficients of the porous electrodes are typically calculated from the diffusion coefficient of Eq. (26.4), and the porosity ( gas) and tortuosity I of the electrode ... 

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