Path tortuosity

Groundwater contamination The mixing of a contaminant with a non-contaminant phase. The mixing is due to the distribution of flow paths, tortuosity of flow paths, and molecular diffusion. 

PET film. The overall improvement in barrier properties of PET upon biaxial deformation, followed by heat-setting, is the combined result of higher deformation-induced crystallization and orientation. The role of crystallinity may be more significant than that of orientation, because it reduces both the diffusion rate (increasing the permeant s path tortuosity ) as well as its solubility, while orientation reduces only the diffusion rate. 

For diffusion in a porous material, the effective diffusion coefficient is assumed to depend on two factors, pore shape and diffusion path tortuosity [103] ... 

Figure 9.60 reveals an increase in crack-path tortuosity due to crack deflection by hard and coarser ZrB2 particles. A closer observation of Fig. 9.60b also reveals the... 

Physically, the tortuosity is defined as the ratio of the actual distance traversed by the species between two points to the shortest distance between those two points. By combining Eqs 9.7 and 9.9, the tortuosity can be evaluated. With the Bruggeman factor of 3.5, the pore path tortuosity representative of the resistance to oxygen diffusion is calculated around 4 and the proton path tortuosity indicative of the ion transport resistance through the electrolyte phase around 18. The high proton path tortuosity could be attributed to the significantly low volume fraction (e g. <20%) of the electrolyte phase typically considered in the standard PEFC catalyst layers. 

The permeability of small penetrant molecules through an organic matrix is determined by the solubility and diffusivity of the small molecule in the matrix as well as by the mean-square displacement (total path length traveled) divided by the sample thickness. In principle, the addition of a filler in the polymer matrix is expected to affect the solubility and diffusivity of a penetrant molecule, especially in the vicinity of the filler (i.e in the filler-polymer interfacial region and at least one polymer Rg away from the filler surface). Also, it is expected that fillers will affect the path tortuosity (mean-square displacement of penetrant versus film thickness) directly, when penetrants are forced to travel around impermeable fillers, and indirectly, when fillers induce polymer chain aUgnment or alignment and modification of polymer crystallites. ... 

Theoretical approaches on the barrier properties of nanocomposites beat fillers as impermeable nonoverlapping particles and assume no permeability changes in the polymer matrix. Effectively, this means that the permeability of the composite will be smaller than the permeability of the matrix (unfilled polymer) by a factor equal to path tortuosity in the composite (simply assuming that the penetrant path cannot cross any filler particles). This path tortuosity was calculated by Nielsen for completely aligned filler particles (aU fillers have then-larger surface parallel to the film surfaces, but there is no order in the filler center of mass), and its contribution to the composite permeability was derived to be... 

FIGURE 2.9 Comparison of theoretical models quantifying the effect of path tortuosity on the permeability of a composite Nielsen model [eq. (2.8)], Friedrickson-Bicraano [eq. (2.10)], modified Nielsen [eq. (2.9)], and Cussla--Aris [eq. (2.11)]. 

FIGURE 2.10 Theoretical predictions based on path tortuosity [eq. (2.9)], as a function of (a) filler aspect ratio a = 1 to 1000 (b) filler aspect ratio and alignment (5 = 1 perfect smectic alignment—dashed lines S = 0 random orientation —solid lines) (c) filler aspect ratio for a constant volume fraction (pv = 5%. (d) Comparison of the same theoretical predictions (parameters as indicated) with experimental values for water vapor permeabilities in various polymer-montmorillonite nanocomposites. (From Refs. 39-41.)... 

In reality, the length of the flow path in the hypothetical capillaries in the bed is larger than L due to the path tortuosity. Experimental measurements indicate that the following equation (the Blake-Kozeny equation) is instead more accurate as long as fi < 0.5 and dpp Votp (1 - c)) <10 ... 

Dpi is smaller than the diffusivity in a straight cylindrical pore as a result of the random orientation of the pores, which gives a longer diffusion path, and the variation in the pore diameter. Both effects are commonly accounted for by a tortuosity factor Tp such that Dp = DjlXp. In principle, predictions of the tortuosity factor can be made if the... 

Tortuosity is defined as the relative average length of a flow path (i.e., the average length of the flow paths to the length of the medium). It is a macroscopic measure of both the sinuosity of the flow path and the variation in pore size along the flow... 

Tortuosity t is basically a correction factor applied to the Kozeny equation to account for the fact that in a real medium the pores are not straight (i.e., the length of the most probable flow path is longer than the overall length of the porous medium) ... 

The size distribution of the particles (with particles of a uniform size, there is increased voidage between the particles and a lower tortuosity, that is deviation from the linear path for the fluid flowing between the particles, making for improved filterability), and... 

Tye [38] explained that separator tortuosity is a key property determining the transient response of a separator (and batteries are used in a non steady-state mode) steady-state electrical measurements do not reflect the influence of tortuosity. He recommended that the distribution of tortuosity in separators be considered some pores may have less tortuous paths than others. He showed mathematically that separators with identical average tortuosities and porosities can be distinguished by their unsteady-state behavior if they have different distributions of tortuosity. 

The ratio of the overall rate of reaction to that which would be achieved in the absence of a mass transfer resistance is referred to as the effectiveness factor rj. SCOTT and Dullion(29) describe an apparatus incorporating a diffusion cell in which the effective diffusivity De of a gas in a porous medium may be measured. This approach allows for the combined effects of molecular and Knudsen diffusion, and takes into account the effect of the complex structure of the porous solid, and the influence of tortuosity which affects the path length to be traversed by the molecules. 

The >eff across the porous medium for this example is linearly related to the porosity of the path, which is in turn simply the ratio of the open cross-sectional area to the total cross-sectional area. There are no constriction or tortuosity effects in this example i.e., t = 1 and... 

One must be very careful in reviewing the older, and some more recent, literature in consideration of the tortuosity and constriction factors some work has attempted to separate these two factors however, more modem developments show that they cannot be strictly decoupled. This aspect will be particularly important when reviewing the barrier and tortuous-path theories of electrophoresis, as discussed later. 

In a more extensive development of the tortnons-path and barrier theories, Boyack and Giddings [45] considered the transport of solnte in a simple geometrical system similar to that used in the diffnsion analysis of Michaels [241] bnt with added tortuosity effects. The effective mobility in this system was found to be... 

Boyack and Giddings [45] considered the tortuosity effects to be separable from the constriction effects. In their derivation they assumed that the electric field in the constricted channel decreased proportionally to the decrease in cross-sectional area and changes in path length. The field was assumed not to penetrate the barrier. The effect of constriction can be written in terms of porosity and L as... 

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... 

The tortuosity factor appears as a squared term because it decreases the concentration gradient and increases the diffusive path length. Using a cubic lattice model and inquiring how many steps a diffusing molecule needs to take to get around an obstacle, 0 was derived to be... 

---