Calculated tortuosities

Table V. Calculated tortuosities for lOO-um particles with 27-nm pore dia Surface T n...

In MD studies, a value of 2 is frequently assumed for the tortuosity factor to predict the MD fluxes (Schofield et al., 1987). The calculated tortuosity factor by Khayet et al. (2004a, 2004b) was 1.59, 2.14, and 2.12 for the membranes TF200, GVHP, and HVHP, respectively. 

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... 

The equations used to calculate permeability coefficients depend on the design of the in vitro assay to measure the transport of molecules across membrane barriers. It is important to take into account factors such as pH conditions (e.g., pH gradients), buffer capacity, acceptor sink conditions (physical or chemical), any precipitate of the solute in the donor well, presence of cosolvent in the donor compartment, geometry of the compartments, stirring speeds, filter thickness, porosity, pore size, and tortuosity. 

The permeability coefficients and molecular radii are known. The effective pore radius, R, is the only unknown and is readily calculated by successive approximation. Consequently, unknown parameters (i.e., porosity, tortuosity, path length, electrical factors) cancel, and the effective pore radius is calculated to be 12.0 1.9 A. Because the Renkin function [see Eq. (35)] is a rapidly decaying polynomial function of molecular radius, the estimation of R is more sensitive to small uncertainties in the calculated molecular radius values than it is to experimental variabilities in the permeability coefficients. The placement of the perme-ants within the molecular sieving function is shown in Figure 9 for the effective... 

TJ, tight junction LS, lateral space. b Tortuosity is the tortuous length of the lateral space divided by the height of the cell. All physical dimensions are measured by electron microscopy using transverse sections of cell monolayers. c Calculated as (cell height — TJ length) X tortuosity. 

The measured value of k Sg is 0.716 cm3/(sec-g catalyst) and the ratio of this value to k ltTueSg should be equal to our assumed value for the effectiveness factor, if our assumption was correct. The actual ratio is 0.175, which is at variance with the assumed value. Hence we pick a new value of rj and repeat the procedure until agreement is obtained. This iterative approach produces an effectiveness factor of 0.238, which corresponds to a differs from the experimental value (0.17) and that calculated by the cylindrical pore model (0.61). In the above calculations, an experimental value of eff was not available and this circumstance is largely responsible for the discrepancy. If the combined diffusivity determined in Illustration 12.1 is converted to an effective diffusivity using equation 12.2.9, the value used above corresponds to a tortuosity factor of 2.6. If we had employed Q)c from Illustration 12.1 and a tortuosity factor of unity to calculate eff, we would have determined that rj = 0.65, which is consistent with the value obtained from the straight cylindrical pore model in Illustration 12.2. 

In the special case of an ideal single catalyst pore, we have to take into account that diffusion is quicker than in a porous particle, where the tortuous nature of the pores has to be considered. Hence, the tortuosity r has to be regarded. Furthermore, the mass-related surface area AmBEX is used to calculate the surface-related rate constant based on the experimentally determined mass-related rate constant. Finally, the gas phase concentrations of the kinetic approach (Equation 12.14) were replaced by the liquid phase concentrations via the Henry coefficient. This yields the following differential equation ... 

The leak rate through a porous seal volume can be computed as flux times area and converting from moles to volume with the ideal gas law. The seal void fraction and tortuosity have the same effect as in the hydrodynamic leakage calculations. 

Reid, Sherwood and Prausnitz [11] provide a wide variety of models for calculation of molecular diffusion. Dr is the Knudsen diffusion coefficient. It has been given in several articles as 9700r(T/MW). Once we have both diffusion coefficients we can obtain an expression for the macro-pore diffusion coefficient 1/D = 1/Dk -i-1/Dm- We next obtain the pore diffusivity by inclusion of the tortuosity Dp = D/t, and finally the local molar flux J in the macro-pores is described by the famiUar relationship J = —e D dcjdz. Thus flux in the macro-pores of the adsorbent product is related to the term CpD/r. This last quantity may be thought of as the effective macro-pore diffusivity. The resistance to mass transfer that develops due to macropore diffusion has a length dependence of R]. 

Tortuosity is a long-range property of a porous medium, which qualitatively describes the average pore conductivity of the solid. It is usual to define x by electrical conductivity measurements. With knowledge of the specific resistance of the electrolyte and from a measurement of the sample membrane resistance, thickness, area, and porosity, the membrane tortuosity can be calculated from eq 3. 

While a good equivalent-circuit representation of the transport processes in a fuel cell can lead to an increased understanding, it is not as good as taking a 1-D sandwich model and taking it into the frequency domain. These models typically analyze the cathode side of the fuel cell. °2.3i3 3i4 pj g j ost comprehensive is probably that of Springer et al. °2 The use of impedance models allows for the calculation of parameters, like gas-phase tortuosity, which cannot be determined easily by other means, and can also allow for the separation of diffusion and migra-... 

These calculations were done using the typical thickness of a washcoated SCT substrate ( 7 micron) and the effective diffusivity based on the diffusion coefficient (4.148 x 10 m s ) calculated for a real gas matrix and taking representative values of the tortuosity (3.0), porosity (0.4) and constriction (0.8) factors. 

If data are available on the catalyst pore- structure, a geometrical model can be applied to calculate the effective diffusivity and the tortuosity factor. Wakao and Smith [36] applied a successful model to calculate the effective diffusivity using the concept of the random pore model. According to this, they established that ... 

The pure compound rate constants were measured with 20-28 mesh catalyst particles and reflect intrinsic rates (—i.e., rates free from diffusion effects). Estimated pore diffusion thresholds are shown for 1/8-inch and 1/16-inch catalyst sizes. These curves show the approximate reaction rate constants above which pore diffusion effects may be observed for these two catalyst sizes. These thresholds were calculated using pore diffusion theory for first-order reactions (18). Effective diffusivities were estimated using the Wilke-Chang correlation (19) and applying a tortuosity of 4.0. The pure compound data were obtained by G. E. Langlois and co-workers in our laboratories. Product yields and suggested reaction mechanisms for hydrocracking many of these compounds have been published elsewhere (20-25). 

It is necessary to point out a difficulty with regard to the integration of the flux equations in a real membrane. If, for example, membranes with a pore structure are concerned, the final result which one calculates for a complicated network of capillaries which run in all directions and which are interconnected is different from what is calculated for the model which only contains pores which run perpendicularly to the membrane surface, but when proceeding from the local parameters (e.g. Oik s or diffusion coefficients) to the integral ones, an extra parameter occurs in the resulting expressions, which accounts for the nature of the pore structure (tortuosity factor). 

In the porous medium, diffusion is affected by the porosity and tortuosity of the medium itself therefore Knudsen diffusion is computed as well as the ordinary diffusion. Eventually, an effective diffusion coefficient is calculated that depends on the ordinary and Knudsen diffusion coefficients and on the ratio between porosity and tortuosity of the medium (Equation (3.58)). 

It follows from a comparison of Eqs. 1.51 and 1.52 that in the calculation of the actual EOF mobility, the effect of tortuosity is taken into account so that peo, packed can be used to calculate the zeta potential of the packing. In the literature of electroki-netic phenomena in porous media [44], constriction factor is employed to describe the effect arising from the variability of cross-section along a given flow path. However, in packed CEC columns this factor is estimated to be in the range 0.93-1.0 and so may be neglected in the following treatment. 

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,... 

The overall process can be affected by pore diffusion and external mass transfer. Molecular diffusion coefficients DPB may be calculated by Aspen Plus. Effective pore diffusion may be estimated by the relation DP = Dpb( j,/tp) = 0.1 DPE, in which ep is the particle porosity and rp the tortuosity. Furthermore, the Thiele modulus and internal effectiveness can be calculated as ... 

It should be pointed out that the characteristics of polymer structure (e.g., porosity, tortuosity, steric hindrance, mesh size, etc.) should be determined in order to calculate the diffusion coefficient of a specific molecule in a particular polymer. For cross-linked polymers, additional polymer properties should be characterized. Even though there are methods to determine these properties, a simple mathematical relationship between the diffusion coefficient of a solute and its molecular weight has been used due to the complexity of the experiment ... 

In this work extraction models are developed for three different particle geometries using the shrinking core concept. Model calculations will be compared and fitted to extraction data of toluene and 1,2 dichlorobenzene (DCB) from shallow packed beds in order to obtain values of the efective diffusivity (De) and the tortuosity factor. 

A simple way to calculate catalyst performance is to use the effective diffusivity. It can be estimated by Equation 3.26 for gases or Equation 3.27 for liquids. The porosity-tortuosity ratio ejyp for these calculations can be found with the correlation s between sjyp and the porosity ep itself. Probst and Wohlfahrt [28] took all available data and checked the proposed correlations. They observed that, for different types of porous systems, different correlations have to be used. They distinguished four systems on the basis of their preparation (Table 3.4). 

What pressure gradient is necessary to promote a given flow through a cell wall Because the interfibrillar spaces, or interstices, in a cell wall have diameters near 10 nm (100 A Fig. 1-13), we will let r be 5 nm for purposes of calculation. (Complications due to the tortuosity of the aqueous channels through the interstices will be omitted here. We will also assume that the pores occupy the entire cell wall.) For Jv equal to 1 mm s-1, Equation 9.11b... 

Since theoretical calculation of effectiveness is based on a hardly realistic model of a system of equal-sized cylindrical pores and a shaky assumption for the tortuosity factor, in some industrially important cases the effectiveness has been measured directly. For ammonia synthesis by Dyson and Simon (Ind. Eng. Chem. Fundam., 7, 605 [1968]) and for SO2 oxidation by Kadlec et al. Coll. Czech. Chem. Commun., 33, 2388, 2526 [1968]). 

Ifp li increases linearly with time then p2i can be calculated by p2i = Po — Pih in Eq. (3.206). Here, bottom and top compartments have been considered to have the same volume. It is not difScult to observe that the parameters requiring identification are rv / and r i / where r is the mean pore radius and i / the tortuosity. In this case, in accordance with relation (3.189), the function for the minimization will be written as follows ... 

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