## 4 Results and Discussion
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Expression** & **Range of Application** & **References** \\ \hline \(\tau=2-\varepsilon\) & \begin{tabular}{l} Porous structures with a hyperbola of \\ revolution as a pore model \\ \end{tabular} & \begin{tabular}{l} [PERSON] (1892) [46] \\ [PERSON] (1958) [48] \\ \end{tabular} \\ \hline \(\tau=\frac{1}{\sqrt{\varepsilon}}\) & \begin{tabular}{l} Composite heterogeneous porous media \\ \end{tabular} & \begin{tabular}{l} [PERSON] (1935) [47] \\ \end{tabular} \\ \hline \(\tau=\frac{1}{\sqrt{\varepsilon}}\) & \begin{tabular}{l} Partially saturated homogeneous isotropic \\ monodisperse sphere packing \\ \end{tabular} & \begin{tabular}{l} [PERSON] (1959) [49] \\ [PERSON] and [PERSON] (1974) [51] \\ \end{tabular} \\ \hline \(\tau=1-\frac{ln(\epsilon)}{2}\) & \begin{tabular}{l} Overlapping spheres \\ \end{tabular} & \begin{tabular}{l} [PERSON] et al. (1987) [53] \\ [PERSON] (1981) [54] \\ [PERSON] and [PERSON] (1974) [51] \\ [PERSON] (1963) [52] \\ \end{tabular} \\ \hline \(\tau=1-ln(\epsilon)\) & \begin{tabular}{l} Random arrays of freely overlapping cylinders \\ \end{tabular} & \begin{tabular}{l} [PERSON] and [PERSON] (1983) [50] \\ \end{tabular} \\ \hline \(\tau=\frac{1}{\varepsilon}\) & \begin{tabular}{l} Catalytic fractal porous media \\ \end{tabular} & \begin{tabular}{l} [PERSON] et al. (1991) [45] \\ \end{tabular} \\ \hline \(\tau=\left(\frac{2-\varepsilon}{\varepsilon}\right)^{2}\) & \
\begin{tabular}{l} Cation-exchange resin membrane \\ \end{tabular} &
\begin{tabular}{l} [PERSON] and [PERSON] (1955) [44] \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Empirical expressions of tortuosity vs. porosity. However, the so-defined _Effective Porosity_ has the drawback that it is not comprised between zero and unity. Therefore, analogously to what done by [PERSON] (2011) [24] and [PERSON] et al. (2012) [25] for non-overlapping particles, we can define here a _Normalized Effective Porosity_\(\varepsilon_{Eff}^{*}\) ranging within [0,1] by subtracting the limit volume from both total and void volume (Equation (11)):  
\[\varepsilon_{Eff}^{*}\equiv\frac{\varepsilon-\varepsilon_{Lim}}{1-\varepsilon_ {Lim}}=\frac{V_{V}-V_{Lim}}{V_{T}-V_{Lim}} \tag{11}\]  
It can be noticed that, for \(\varepsilon_{Min}\) equal to zero, the nominal porosity \(\varepsilon\) and the normalized effective porosity \(\varepsilon_{Eff}^{*}\) are coincident. The functionality of tortuosity \(\tau\) with \(\varepsilon_{Eff}^{*}\) is shown in Figure 13 for all the structures considered. The normalized data representation allows recognition of similarities and differences among structures. In particular, we notice that all structures show the same trend for a \(\varepsilon_{Eff}^{*}\) values higher than around 0.55, whereas the behaviour of the FCC, BCC, and Tetragonal structures is very similar in a wider range (higher than around 0.2). Therefore, the morphological difference is more relevant in the range of lower \(\varepsilon_{Eff}^{*}\), in which tortuosity is very sensitive to the particular shape of the void channels. From tortuosity, we can finally plot the trend of connectivity factor \(\varphi\) (Figure 14). It is also noticeable that the equation developed by [PERSON] and [PERSON] represents an upper bound for connectivity of all structures, this being expected from Figure 11. However, it is interesting to find out the morphological reasons for that. Specifically, the virtual structure considered by [PERSON] and [PERSON] is peculiar, as it is ideally composed of non-overlapping spherical particles with an infinitely-disperse size, which means that porosity can progressively decrease by adding smaller and smaller particles that progressively fill the voids. In such an ideal morphology, the structure shows voids being completely interconnected even closely to the zero-porosity condition and, thus, connectivity should be the maximum possible, representing an upper bound in a {\(\varphi\) vs. \(\varepsilon\)} plot for all structures.