Tortuosity of the porous medium

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... 

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) ... 

Kq depends not only on the structure of the porous medium (i.e., porosity, tortuosity, and grain size distribution), but also on the viscosity of the liquid that flows through the pores, and finally on the acceleration of gravity, g. In contrast, the permeability, ,... 

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. 

This relation, which is analogous to PoiseuiUe s relation, gave rise to various models taking into account the irregularity of the porous medium (tortuosity, noncircular sections, etc.). Carman-Kozeny s model is a simple and usually precise model which leads to the following expression of D ... 

Areal Porosity and Tortuosity. Areal porosity or areosity, Ap, is defined as the effective areal ratio of the open pore cross-section to the bulk space. A more strict definition of areosity was introduced by Ruth and Suman (56). However, the areosity as defined by them is not a property of the porous medium only but a property of both the porous medium and the transport strength of the fluid such as the flow strength and electric current strength. The areal porosity is undoubtedly a very useful quantity for a bundled or ensemble passage model because it represents the ratio of the total passage cross-sectional area to the total cross-sectional area of the porous medium at a given planar section. 

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... 

Single-phase fluid flow in porous media is a well-studied case in the literature. It is important not only for its application, but the characterization of the porous medium itself is also dependent on the study of a single-phase flow. The parameters normally needed are porosity, areal porosity, tortuosity, and permeability. For flow of a constant viscosity Newtonian fluid in a rigid isotropic porous medium, the volume averaged equations can be reduced to the following the continuity equation,... 

A capillary tube model can be used to estimate the permeability of the medium before fines deposition or release has occurred. The Car-man-Kozeny equation uses the diameter of the substrate particles, dg, and the tortuosity of the medium, r, to evaluate the effective permeability of the porous medium. 

The void area fraction in (21-76) is based on the fractional area in a plane at constant x that is available for diffusion into catalysts with rectangular symmetry. A rather sophisticated treatment of the effect of g 6) on tortuosity is described by Dullien (1992, pp. 311-312). The tortuosity of a porous medium is a fundamental property of the streamlines or lines of flux within the individual capillaries. Tortuosity measures the deviation of the fluid from the macroscopic flow direction at every point in a porous medium. If all pores have the same constant cross-sectional area, then tortuosity is a symmetric second-rank tensor. For isotropic porous media, the trace of the tortuosity tensor is the important quantity that appears in the expression for the effective intrapellet diffusion coefficient. Consequently, Tor 3 represents this average value (i.e., trace of the tortuosity tensor) for isotropically oriented cylindrical pores with constant cross-sectional area. Hence,... 

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. 

Capillary theory uses the simplest model, whereby pores within a solid material are represented as parallel capillaries of equal diameters in the porous solid. The analogy is between the tortuous pore-system of the solid and the cylindrical pores of the capillaries. The equation for k is then derived from the Hagen-Poiseuille equation for streamline flow through straight circular capillaries taking account of the tortuosity of material s pores. The tortuosity is defined as the ratio of the actual length of the flow channel to the length of the porous medium. 

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... 

Darcy s law involves coefficient k, which is the intrinsic permeability of the porous medium. Its dimension is square meters. This quantity depends on the dimensions of the porous medium (grain or pore size) and on the geometrical characteristics (porosity, tortuosity) defined in the previous section. 

A porous medium (Figure 14.7(b)) of thickness L is thus represented by a network of circnlar capillary tubes of diameter D. The length of each tube is denoted by /. It is greater than the thickness L of the porous medium, due to the tortuosity of the charmels running through the porous medium. We, therefore, write t = UL. 

The tortuosity t is introduced in [14.26] in order to express the Darcy law in terms of the thickness of the porous medium rather than the length of the tubes. We recover in [14.26] the first form of Darcy s law (equation [14.1]). The intrinsic permeabihty can then be expressed as a parameter depending on the geometrical characteristics of the capillary tube network ... 

The geometrical parameters of the porous medium are the grain size d and the number of grains per unit volume p. The geometrical parameters of the network of capillary tubes are the diameter D of the tubes and the number of tubes passing through a unit surface in the Oxy plane. Tortuosity i is a common parameter of the porous medium and the capillary-tube network model. The Kozeny-Carman formula expresses, based on [14.27], the dependence of k on d, t, and e, establishing a link between and D, on the one hand, and between p and d, on the other. These relations are obtained under the hypothesis that the porous medium and the network of capillary tubes have the same porosity e and the same specific area Os... 

As such, it is a property of the porous medium, which in this case is the fiber reinforcement and is anisotropic. Its value depends on the porosity of the reinforcement, the dimensions of the capillary passages in various directions in the reinforcement and the tortuosity of these passages [4]. Darcy s law for a linear and slow steady state flow through a... 

In this work solid-gas chromatography is used to measure dynamic diffusion coefficients of argon in various porous solids. Mercury porosimetry is used to study the internal macroporosity and macro-morphology of these solids. Finally, an attempt is made to elucidate a relationship between the tortuosity measured from the transport experiment and the internal structure of the porous medium as characterized by porosimetry. 

Diffusion within the largest cavities of a porous medium is assumed to be similar to ordinary or bulk diffusion except that it is hindered by the pore walls (see Eq. 5-236). The tortuosity T that expresses this hindrance has been estimated from geometric arguments. Unfortunately, measured values are often an order of magnitude greater than those estimates. Thus, the effective diffusivity D f (and hence t) is normally determined by comparing a diffusion model to experimental measurements. The normal range of tortuosities for sihca gel, alumina, and other porous solids is 2 < T < 6, but for activated carbon, 5 < T < 65. 

A microscopic description characterizes the structure of the pores. The objective of a pore-structure analysis is to provide a description that relates to the macroscopic or bulk flow properties. The major bulk properties that need to be correlated with pore description or characterization are the four basic parameters porosity, permeability, tortuosity and connectivity. In studying different samples of the same medium, it becomes apparent that the number of pore sizes, shapes, orientations and interconnections are enormous. Due to this complexity, pore-structure description is most often a statistical distribution of apparent pore sizes. This distribution is apparent because to convert measurements to pore sizes one must resort to models that provide average or model pore sizes. A common approach to defining a characteristic pore size distribution is to model the porous medium as a bundle of straight cylindrical or rectangular capillaries (refer to Figure 2). The diameters of the model capillaries are defined on the basis of a convenient distribution function. 

Mixing Due to Obstructions The tortuosity of the flow channels in a porous medium means that fluid elements starting a given distance from each oilier and proceeding at the same velocity will not reniain tlie same distance apart. 

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... 

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. 

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)). 

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