Porous media tortuosity factor

When macroscopic diffusion models are compared to experimental data, high values of tortuosity (corresponding to low values of Detr) are obtained (Tables I and II). For a random porous medium, tortuosity values due to windiness ofthe diffusional path should be between 1 and 3 (Pismen, 1974 Bhatia, 1986). Because the overall tortuosities predicted by the macroscopic models are much larger, other physical properties ofthe matrix must influence the rate of protein diffusion within the matrix. Microgeomet-ric models and percolation theory have been used to study the factors that might control protein diffusion within these polymer matrices. 

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 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 in situ concentration gradient along the real diffusion path is reduced by tortuosity t. Thus the in situ flux is reduced by the same factor. This effect is incorporated in the porous medium diffusivity Dipm. If the pores are not too narrow, we get ... 

It is noted that straight capillary tubes may not portray the complex structure of the porous medium. Thus, in practice, the factor 32 in Eq. (5.321) is commonly replaced by an empirical parameter known as tortuosity. The tortuosity accounts for the tortuous paths of the porous medium. 

In rocks and soils, the only appreciable diffusion of gases occurs in the voids or pores, which may be occupied by air, water or a mixture of both. Migration over any ap preciable distance is possible only if the soil pores are continuous with each other. Collisions of the gaseous molecules with liquids or solids impede their progress, so that diffusion in a porous medium is slower than in a free space. The important factors are the shape, size, tortuosity... 

Tortuosity factor or tortuosity x was first introduced by Carman in 1937 [12,13] by reference to a direction that corresponds to a given macroscopic flow. It was defined as the square of the ratio of the "effective average path length" in the porous medium to the shortest distance L measured along the direction of macroscopic flow... 

Tortuosity factor The distance a particle must travel to pass through a porous medium divided by the overall length of the medium. 

A similar process will occur in a static fluid in a porous medium. The diffusion process in this case will be hindered by the presence of the solid phase. The cross sectional area across which diffusion can take place is reduced by a factor that is equal to the volumetric water content, 6. The tortuosity of the pores increases the microscopic distance across which... 

The tortuosity, r, of a porous medium is defined as the ratio of the distance between two fixed points and the tortuous passage followed by a fluid element of a single fluid saturated in the porous medium when traversing the two points. It may be viewed as a line porosity because, by definition, tortuosity is a one-dimensional property of the porous medium. It can be related to the formation factor, or formation conductivity factor, Ft by... 

In this equation, d is the tortuosity of the porous medium (between 0 and 1 0.5 would be typical for a consolidated sandstone), Z>nuid is the diffusion coefficient in pure fluid and i is a retardation factor, defined as... 

To analyze the flow through a porous medium, we can, as before, model the medium as a collection of parallel cylindrical microcapillaries. As noted in Section 4.7, the actual sinuous nature of the capillaries may be accounted for by the introduction of an empirical tortuosity factor. The results for electroosmotic flow through a capillary are then readily carried over to the porous medium by using Darcy s law (Eq. 4.7.7) and, for example, the Kozeny-Carman permeability (Eq. 4.7.16). 

The tortuosity describes the ratio of the (average) incremental distance that an ion/molecule must travel to cover the direct distance in the direction of diffusion (Berner, 1980) and is thus the factor, which describes the increase in travel distance in the porous medium. Since the definitions of tortuosity in literature indicate small however substantial differences (an excellent overview of different procedures, equations and definitions is given by Boudreau (1997)), the definition used here is described briefly. The diffusion coefficient is corrected by the squared tortuosity. The square of the tortuosity is called tortuosity factor (Carman, 1937). This factor represents the... 

Tortuosity is a measiue of the extent to which the path traversed by fluid elements deviates from a straight-line in the direction of overall flow and may be defined as the ratio of the average length of the flow paths to the distance travelled in the direction of flow. Though the tortuosity depends on voidage and approaches imity as the voidage approaches unity, it is also affected by particle size, shape and orientation in relation to the direction of flow. For instance, for plate like particles, the tortuosity is greater when they are oriented normal to the flow than when they are packed parallel to flow. However, the tortuosity factor is not an intrinsic characteristics of a porous medium and must be related to whatever one-dimensional flow model is used to characterise the flow. 

The tortuosity factor q is the square of the tortuosity. Thus the Knudsen flux equation for a porous medium obtained from the parallel capillaries model is given by ... 

Here 8 is the porosity of the porous medium, and q is the tortuosity factor. The inclusion of the porosity and the tortuosity factor was proved in Section 7.4 for the case of Knudsen flow. 

In the last expression, tortuosity t > 1 was added as a generalized factor to allow for a correction factor if the capillaries are not straight. Then, Th should be used instead of h [19]. Obviously, the other approach exploiting the permeability tensor offers better opportunity to account for the internal geometry of the porous medium because many theoretical predictions are... 

In addition to the reduced cross-sectional area available for diffusion and pore space tortuosity, Grathwohl (1998) describes a factor that accounts for the constrictiveness of pores relative to solute molecular size the same idea applies to diffusing particles. The porous medium may contain small pores which are not accessible due to size exclusion... 

Due to the visualization of a porous medium as an ensemble of large dust molecules in the Dusty G ls Model pore structure properties such as porosity, tortuosity, and pore size distribution are not directly included. All information on pore structure characteristics is contained in the permeability constants Co, Ci, and Ca. Heteroporosity as originating from a wide pore size distribution is not accounted for specifically. On the other hand the Dusty Gas Model has the etdvantage to allow a separation of the influence of pore structure characteristics on the different transport mechanisms. The influence of the adsorbent material pore structure on gas phase mass transport is incorporated through the parameters Co, Ci, and C2 resp. They are determined by flux experiments for the specific adsorbent material (refs. 4, 6). The values for the different trstructural parameters such as representative pore diameter dp, porosity p, and tortuosity factor Tp by the expressions ... 

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