Tortuosity coefficient

It is common in many practical battery designs to immobilize a liquid electrolyte phase within a porous solid insulator. The electrolyte conductivity and ohmic loss in such a system are determined by the number of pores, their size, shape and tortuosity. The tortuosity coefficient, /3, is defined as the ratio of the mean distance covered by an ion traversing a porous matrix, to the direct distance of one side of the matrix to the other. The relative reduction in the conductivity of an electrolyte solution caused by confining it in a porous solid is called the conductivity attenuation, 0. For a matrix of uniform cylindrical pores it is given by... 

The characteristics of the experimental aquifer were independently determined from appropriate flowthrough column experiments or obtained directly from the literature. The dry bulk density of the sand ph= 1.61 kg/1, and the aquifer porosity 0=0.415 were evaluated by gravimetric procedures. The dimensionless retardation factor, R= 1.31, of the aqueous-phase TCE was determined from a column flowthrough experiment. The tortuosity coefficient for the aquifer sand was considered to be x =1.43 [75]. The molecular diffusion coefficient for the aqueous-phase TCE is D=0.0303 cm2/h [76]. The pool radius is r=3.8 cm. Bromide ion in the form of the moderately soluble potassium bromide salt was the tracer of choice [77 ] for the tracer experiment conducted in order to determine the longitudinal and transverse aquifer dispersivities a =0.259 cm and a-,— 0.019 cm, respectively. The experimental pool contained approximately 12 ml of certified ACS grade (Fisher Scientific) TCE with solubility of Cs=1100 mg/1 [78]. 

When volume V is occupied by a porous solid, eq 11 is generally made to include a tortuosity coefficient [), the pore fraction e (fraction of the support grain volume occupied by the pore space) and an interaction coefficient K between the precursor and the support (K = 1 if there is no interaction) ... 

Apparently the parameters of stochastic models are quite different from those of classic (deterministic) models where the permeability, the porosity, the pore radius, the tortuosity coefficient, the specific surface, and the coefficient of the effective diffusion of species represent the most used parameters for porous media characterization. Here, we will present the correspondence between the stochastic and deterministic parameters of a specified process, which has been modelled with a stochastic and deterministic model in some specific situations. 

Fluid saturation, which defi nes oil-water contact, was determined using wireline log interpretation and RFT data. Shaliness was obtained through density/neutron log data and the salinity of the formation water was analysed from RFT sampled water. Additional parameters used in the calculation, such as saturation and cementation exponents (n and m, respectively), and tortuosity coefficient (a), were measured in the laboratory. 

The tortuosity coefficient [33] is the ratio of open porosity to permeable... 

The term yfl comes from the fact that we assume the pore axis has an average angle of 45° against the surface (tortuosity coefficient), so ... 

Ruthven (gen. refs.) summarizes methods for the measurement of effective pore diffusivities that can be used to obtain tortuosity factors by comparison with the estimated pore diffusion coefficient of the adsorbate. Molecular diffusivities can be estimated with the methods in Sec. 6. 

Diffusivity and tortuosity affect resistance to diffusion caused by collision with other molecules (bulk diffusion) or by collision with the walls of the pore (Knudsen diffusion). Actual diffusivity in common porous catalysts is intermediate between the two types. Measurements and correlations of diffusivities of both types are Known. Diffusion is expressed per unit cross section and unit thickness of the pellet. Diffusion rate through the pellet then depends on the porosity d and a tortuosity faclor 1 that accounts for increased resistance of crooked and varied-diameter pores. Effective diffusion coefficient is D ff = Empirical porosities range from 0.3 to 0.7, tortuosities from 2 to 7. In the absence of other information, Satterfield Heterogeneous Catalysis in Practice, McGraw-HiU, 1991) recommends taking d = 0.5 and T = 4. In this area, clearly, precision is not a feature. 

The coefficient s increases witfi increasing tortuosity and decreasing porosity. It is independent of pore radius wfien the values of these parameters are constant the decrease in radius r that occurs while the total porosity is kept unchanged will be compensated for by an increase in the number of pores, N. 

For all the essential nutrient ions, the diffusion coefficient, Du is essentially the same with a value of around 10 cm s whereas the water flux at the root surface is typically of the order 10 cm s for soils at around field capacity. The tortuosity factor typically scales with the volumetric moisture content over quite a wide range of moisture content, i.e., / 0. As the soil becomes drier, the water flux will decline much faster than the tortuosity factor due to the typi-... 

In order to verify the conditions of this averaging process, one has to relate the displacements during the encoding time - the interval A between two gradient pulses, set to typically 250 ms in these experiments - with the characteristic sizes of the system. Even in the bulk state with a diffusion coefficient D0, the root mean square (rms) displacement of n-heptane or, indeed, any liquid does not exceed several 10 5 m (given that = 2D0 A). This is much smaller than the smallest pellet diameter of 1.5 mm, so that intraparticle diffusion determines the measured diffusion coefficient (see Chapter 3.1). This intrapartide diffusion is hindered by the obstades of the pore structure and is thus reduced relative to D0 the ratio between the measured and the bulk diffusion coeffident is called the tortuosity x. More predsely, the tortuosity r is defined as the ratio of the mean-squared displacements in the bulk and inside the pore space over identical times ... 

Fig. 3.3.4 Variation of the tortuosity x inside the catalyst pellets during coking and regeneration, obtained by measuring the self-diffusion coefficient of n-heptane at room temperature.

Since it was proposed in the early 1980s [6, 7], spin-relaxation has been extensively used to determine the surface-to-volume ratio of porous materials [8-10]. Pore structure has been probed by the effect on the diffusion coefficient [11, 12] and the diffusion propagator [13,14], Self-diffusion coefficient measurements as a function of diffusion time provide surface-to-volume ratio information for the early times, and tortuosity for the long times. Recent techniques of two-dimensional NMR of relaxation and diffusion [15-21] have proven particularly interesting for several applications. The development of portable NMR sensors (e.g., NMR logging devices [22] and NMR-MOUSE [23]) and novel concepts for ex situ NMR [24, 25] demonstrate the potential to extend the NMR technology to a broad application of field material testing. 

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... 

For hydrophilic and ionic solutes, diffusion mainly takes place via a pore mechanism in the solvent-filled pores. In a simplistic view, the polymer chains in a highly swollen gel can be viewed as obstacles to solute transport. Applying this obstruction model to the diffusion of small ions in a water-swollen resin, Mackie and Meares [56] considered that the effect of the obstruction is to increase the diffusion path length by a tortuosity factor, 0. The diffusion coefficient in the gel, )3,i2, normalized by the diffusivity in free water, DX1, is related to 0 by... 

From the magnitudes of the diffusion coefficients, it is evident that under the conditions cited the majority of the mass transport will occur by Knudsen diffusion. Equation 12.2.9 and the tabulated values of the porosity and tortuosity may be used to determine the effective diffusivity. 

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 ... 

Substrate transport through the film may be formally assimilated to membrane diffusion with a diffusion coefficient defined as12 Ds = Dch( 1 — 9)/pjort. In this equation, the effect of film structure on the transport process in taken into account in two ways. The factor 1—0 stands for the fact that in a plane parallel to the electrode surface and to the coating-solution interface, a fraction 9 of the surface area in made unavailable for linear diffusion (diffusion coefficient Dcj,) by the presence of the film. The tortuosity factor,, defined as the ratio between the average length of the channel and the film thickness, accounts for the fact that the substrate... 

The reduction of the long-range diffusivity, Di by a factor of four with respect to bulk water can be attributed to the random morphology of the nanoporous network (i.e., effects of connectivity and tortuosity of nanopores). For comparison, the water self-diffusion coefficient in Nafion measured by PFG-NMR is = 0.58 x 10 cm s at T = 15. Notice that PFG-NMR probes mobilities over length scales > 0.1 /rm. Comparison of QENS and PFG-NMR studies thus reveals that the local mobility of water in Nafion is almost bulk-like within the confined domains at the nanometer scale and that the effective water diffusivity decreases due to the channeling of water molecules through the network of randomly interconnected and tortuous water-filled domains. ... 

In the following sections I discuss the components of the diffusion coefficient so defined in turn. Note all the components of D are altered by flooding the soil. As well as increasing the cross-sectional area for diffusion, represented by flooding affects the geometry and tortuosity of the soil pore network, represented by /l and fs, and solute sorption on the soil solid, represented by dCt/dC. 

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]. 

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