Average tortuosity

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. 

Desig- nation Nominal Size Surface Area (m /g) Total Void Fraction Dt X 10s (cm2/sec) Average Tortuosity Factor t, Parallel-Path Pore Model r = 2K./5. (A) r Based on Average Pore Radius... 

Fig. 3. Microporous membranes are characterized by tortuosity, T, porosity, S, and their average pore diameter, d. (a) Cross-sections of porous membranes containing cylindrical pores, (b) Surface views of porous membranes of equal S, but differing pore size.

Gravity, or centrifugation rarely provide enough hydrostatic pressure to force Hquid into nonwetting pores. If the Hquid wets the soHd the clump density increases as gas is displaced from the interior. It is best if submersion does not occur until the Hquid has completely displaced gas from the pores (Fig. 2). For wetting Hquids and pores with average diameter, D-pQ- y and tortuosity,the length, to which Hquid is pulled into a bed of powder by... 

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. 

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... 

Coimectivity is a term that describes the arrangement and number of pore coimections. For monosize pores, coimectivity is the average number of pores per junction. The term represents a macroscopic measure of the number of pores at a junction. Connectivity correlates with permeability, but caimot be used alone to predict permeability except in certain limiting cases. Difficulties in conceptual simplifications result from replacing the real porous medium with macroscopic parameters that are averages and that relate to some idealized model of the medium. Tortuosity and connectivity are different features of the pore structure and are useful to interpret macroscopic flow properties, such as permeability, capillary pressure and dispersion. 

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 ... 

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... 

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 addition to the packed bed acting as an ultrafilter, the porous frits used at both ends of the column may act as very effective filtering devices. Thus a 2-vim porosity frit would have an average pore radius of 1 lun. Because of the tortuosity and relatively wide pore-size distribution present in frits, it would be safe to assume that it contains much smaller crevices which can entrap macromolecules. 

Because the firn is ventilated by atmospheric air while the bubbles are forming over a period of time and ice depths, the air eventually trapped in the bubbles is a time-integrated sample that is younger than the snow deposit itself. For example, in one recent study (Smith et al., 1997), the air bubbles were, on average, 220-700 years younger than the ice in which they were embedded, but the difference can be as much as several thousand years (e.g., see Rommelaere et al., 1997). These exchange processes with the atmosphere, gas diffusion, and the porosity and tortuosity of the ice pores have to be taken into account in relating the depth of the core to the age of the trapped air. 

The membrane tortuosity (r) reflects the length of the average pore compared to the membrane thickness. Simple cylindrical pores at right angles to the membrane surface have a tortuosity of one, that is, the average length of the pore is the... 

Here, E n = 0 on Sp (Neumann type boundary condition), where n is the unit outward normal from the pore region, and T> is compact. E can be interpreted as the microscopic electric field induced in the pore space when a unit macroscopic field e is applied, assuming insulating solid phase and uniform conductivity in the pore fluid. Its pore volume average is directly related to the tortuosity ax ... 

The average velocity in the column for the above case can be easily determined using the velocity profile given in Eq. 1.32 and accounting for the tortuosity in the bed... 

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