Tortuosity model

CPSM-Tortuosity. This model consists of an empirical correlation (Appendix I, eq. I-l) that is based on CPSM-Nitrogen predictions of intrinsic pore size distribution and nominal pore length (i.e. NJ data. The CPSM-Tortuosity Model [13] enables realistic predictions of pore structure Tortuosity Factors in satisfactory agreement with relevant literature data, [18]. Normally, high tortuosity factors correspond to low pore structure connectivities and usually wide hysteresis loops. 

More detailed aspects of transport in heterogeneous media have been given in the excellent reviews of the subject by Barrer (55) and Petropolus (51) The models described by these authors and others include the effects of size, shape, and anisotropy of the crystalline phase on the tortuosity. Models of highly ordered anisotropic media have been demonstrated to have tortuosities in the range of 30, which reflects the rather dramatic role that orientation can have on the barrier properties of semicrystalline polymers. 

The correction on the tortuosity term in Kozeny-Carman equation was initiated by Foscolo et al. (89) however, they repeated the error of Kozeny s work, which was pointed out later by Epstein (65). The tortuosity model of Foscolo et al. (89) was not in agreement with experiments, although the same line of investigation was continued by Puncochar and Drahos (66). 

However, the data clearly suggest that the effective diffusion coefficient depends on molecular weight this effect is not predicted by these models of porous structure (i.e., the tortuosity models in Figure 4.18 do not depend on molecular size). The tortuosity for a small water-soluble molecule, the tracer ion TMA, is 2 (Figure 4.22). Large molecules have tortuosity values greater than 2 and the tortuosity increases with molecular size. For larger molecules, this tortuosity —which is estimated from the effective diffusion coefficient— must reflect a decrease in diffusion rate due to actual tortuosity in the extra-... 

Alternatively, the capillary network model constitutes a significant improvement over the aforementioned mentioned tortuosity model, since it can provide realistic modeling, especially for systems involving membranes partially blocked by condensed vapors. In this model the degree of connectivity of the pores, z, is replacing the less tangible tortuosity factor t. The estimation of the z can be based on gas and condensed vapor relative permeability measurements, presented in this section. 

With impermeable fillers like GO nanosheets, the tortuosity model yields the closest predictions of permeability of the nanocomposites. Taking into account the random orientation of the GO nanosheets in the polymer matrix, the value of the order parameter S (Eqn (8.13)) was evaluated to be 1, 0.5, and —0.5 for three different orientation angles namely, 0°, 45°, and 90°, respectively. The permeability coefficient for polymer nanocomposites for each value of order parameter S was calculated using Eqn (8.14), and the results are depicted in Table 8.22. 

The value of permeability obtained by the series model is zero, indicating that a perfectly aligned planar arrangement of GO nanosheets in direction obstructing the fluid path will practically make the thin nanocomposite film impermeable. The values obtained for the tortuosity model for the range of orientation angles from 0° to 90° lie in the permeability boundary values predicted by the series and the parallel models. 

Series model Parallel model Geometric-mean model Tortuosity model ... 

The classical (Nielsen) tortuosity model, a model for the torturous zigzag diffusion path of a permeant in an exfoliated polymer-clay nanocomposite when used as a gas barrier [6],... 

O2 gas permeability of neat PLA and various PLACNs as function of OMSFM content (wt.%) at 20 C and 90% relative humidity. The filled circles represent the experimental data and the line based on Nielsen tortuosity model (Equation (3.1)) by considering L/D equal to 275. Reproduced from Sinha Ray, Yamada, Okamoto, Ogami and Ueda by permission of American Chemical Society, USA. 

Despite the large success of the tortuosity model in the polymer nanocomposites literature, no one has adopted this second model. It is probably due to the numerous numerical parameters that are not obtainable experimentally. TTie original idea, however, is so interesting that very recently proposed a new model... 

Nielsen s tortuosity model [4] is perhaps the most cited model for the estimation of barrier property of flake-filled membranes. It predicts that Fq/F/ increases linearly with (a. However, as indicated by our nmnerical analysis as well as the experimental data on some systems, this simple model is likely to work well only at low (ct ) levels. The other model proposed by Cussler et al. [5] predicts that when (a ) is large, Fo/Ff increases more rapidly, at a leading rate of The experimental data on a number of systems show that the latter model predicts the permeability behavior reasonably well at high (ct )... 

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