Geometric tortuosity

Within a porous body the flow of a fluid is resisted by viscous and geometric (tortuosity) effects. A porous media friction term is therefore added to the right hand side of the momentum equation. The physical meaning of different terms in the equation is explained in sect 3.4.6. 

Since 3D images are given by voxels, that is, a 3D image can be seen as a 3D array, another possibility to define geometric tortuosity is based on the voxel representation of the pore phase see, for example, [55, 57, 61]. 

A shghtly different approach to determine geometric tortuosities is described by Jorgensen [57], which is also based on a voxel representation of the microstructure. 

Table 24.1 Mean values and standard deviations (SDs) of geometric tortuosity, pore size, and coordination number, and mean relative length of MST.

Data Porosity Geometric tortuosity Pore size Coordination number Relative length of MST... 

Figure 24.19 Histograms of geometric tortuosities with varying binder modelings (a) original data (b) complete cell filling (c) partial cell filling with rg = 30 (d) partial cell filling with rg =18 (e) partial cell filling with rg = 6. Reprinted with permission from [5]. Copyright (2008) Journal of the Electrochemical Society.

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. 

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... 

The tortuosity for pore-filling liquids is ideally a purely geometric factor but can, in principle, depend on the fluid-surface interaction and the molecular size if very small pores are present such as in zeolites (see Chapter 3.1). To obtain a measure for a realistic situation, we have used n-heptane as a typical liquid and have computed x... 

In addition to the criticisms from Anderman, a further challenge to the application of SPEs comes from their interfacial contact with the electrode materials, which presents a far more severe problem to the ion transport than the bulk ion conduction does. In liquid electrolytes, the electrodes are well wetted and soaked, so that the electrode/electrolyte interface is well extended into the porosity structure of the electrode hence, the ion path is little affected by the tortuosity of the electrode materials. However, the solid nature of the polymer would make it impossible to fill these voids with SPEs that would have been accessible to the liquid electrolytes, even if the polymer film is cast on the electrode surface from a solution. Hence, the actual area of the interface could be close to the geometric area of the electrode, that is, only a fraction of the actual surface area. The high interfacial impedance frequently encountered in the electrochemical characterization of SPEs should originate at least partially from this reduced surface contact between electrode and electrolyte. Since the porous structure is present in both electrodes in a lithium ion cell, the effect of interfacial impedances associated with SPEs would become more pronounced as compared with the case of lithium cells in which only the cathode material is porous. 

If data are available on the catalyst pore- structure, a geometrical model can be applied to calculate the effective diffusivity and the tortuosity factor. Wakao and Smith [36] applied a successful model to calculate the effective diffusivity using the concept of the random pore model. According to this, they established that ... 

Both Knudsen and molecular diffusion can be described adequately for homogeneous media. However, a porous mass of solid usually contains pores of non-uniform cross-section which pursue a very tortuous path through the particle and which may intersect with many other pores. Thus the flux predicted by an equation for normal bulk diffusion (or for Knudsen diffusion) should be multiplied by a geometric factor which takes into account the tortuosity and the fact that the flow will be impeded by that fraction of the total pellet volume which is solid. It is therefore expedient to define an effective diffusivity De in such a way that the flux of material may be thought of as flowing through an equivalent homogeneous medium. We may then write ... 

The tortuosity is also included in the geometric factor to account for the tortuous nature of the pores. It is the ratio of the path length which must be traversed by molecules in diffusing between two points within a pellet to the direct linear separation between those points. Theoretical predictions of r rely on somewhat inadequate models of the porous structure, but experimental values may be obtained from measurements of De, D and e. 

Here, e = Ju jiuj = (1 — i)6/2 is the complex viscous skin depth parameter, and 4>, aoo, and C are purely geometrical parameters, respectively the porosity, tortuosity, and C = 2/A, where A is a pore size parameter characterizing transport properties of the porous material [6],... 

The results from this analysis can now be used to construct geometrically accurate models of the diffusive transport in porous polymers. Previous models of diffusion in these polymers have used an empirically determined tortuosity factor as a lumped parameter to account for the retardation of release by all mechanisms (7-8). 

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. 5-249). The tortuosity x that expresses this hindrance has been estimated from geometric arguments. Unfortunately,... 

Fitted values of Dp and F are given in Table I, together with values of the tortuosity (r) determined from Equation 1. The tortuosity is reasonably constant, as it should be for a geometric factor, and has a value typical of beds of spheres. Therefore, it can be concluded safely that the transport mechanism in the pores is ordinary bulk gas diffusion, and in particular, there is no evidence of surface diffusion. 

In practice it is simpler to treat the tortuosity x as an empirical factor and to determine it experimentally (see below). The same holds for other geometrical parameters like 0 and (3, discussed in connection with Eqs. (9.6 and (9.2). In porous pellets of packed particles a correlation of the type e/x = constant is frequently found [3]. The validity of this expression is not shown however for low values of the porosity (e < 0.30) and very small pore sizes. Experimental tortuosity values generally fall in the region 2 < x < 5, but in special cases much larger values have been reported. Leenaars et al. [17] reported values of x = 6-7 for membranes consisting of a packing of plate-shaped (boehmite, gamma alumina) particles. 

---