## 1 Introduction
Moreover, it is pointed out that the diffusional tortuosity considered in the present paper coincides with the tortuosity used to evaluate the effective diffusivity (Equation (2)) [14; 15; 16; 17; 18]:  
\[D_{i,Eff}=D_{i}\frac{\varepsilon}{\tau} \tag{2}\]  
where \(\varepsilon,D_{i}\) and \(D_{i,Eff}\) are porosity, free diffusivity, and effective diffusivity of the \(i\)th species, respectively. Some scholars prefer writing Equation (2) by distinguishing a more proper tortuosity factor \(\overline{\tau}\) as follows (Equation (3)):  
\[D_{i,Eff}=D_{i}\frac{\varepsilon}{\overline{\tau}^{2}} \tag{3}\]However, such a distinction is not important for our purpose, as both parameters depend on geometry only. Therefore, we will use Equation (2) as the reference effective diffusivity definition, following the same approach as that of the pioneers of the volume average method [19; 20; 21; 22]. The choice to define a connectivity factor as the inverse of tortuosity has been driven by several reasons. First, tortuosity can be viewed as a measure of the difficulty for a molecule to go from one point of a structure to another one within the available void space. Hence, the more the structure is tortuous, the less the connection between two voids is, whilst, on the contrary, a lower tortuosity implies a \"more favourable\" connection. In fact, as we will show in the following sections, tortuosity tends to infinity when a structure reaches a specific limit value of porosity (non-necessarily zero), below which all the internal voids are completely isolated from each other, as briefly sketched in Figure 1b. This situation can occur, for example, in real catalytic devices subjected to a high degree of sintering, resulting in a loss of specific area [23]. Second, as [PERSON] et al. (1987) clearly state, the diffusional tortuosity is a parameter depending on geometry only [19], and, consequently, the same holds also for the here-defined connectivity factor (Equation (1)), whose value is comprised within the range [0, 1] because of the tortuosity ranges within [1, \(\infty\)]. Based on the previous considerations, we consider it straightforward to write the effective diffusivity in terms of connectivity factor \(\varphi\) (Equation (4)):  
\[D_{i,Eff}=D_{i}\varepsilon\cdot\varphi \tag{4}\]  
This is in line with the physical concept that the effective diffusivity increases with increasing connectivity, as one may expect from the dynamics of the mass transfer phenomenon. This allows us to further state that connectivity, and, dually, tortuosity, is not much related to the value of porosity, but rather to the porosity distribution within a structure. Figure 1: Sketch of two different structure morphologies: (**a**) Interconnected voids (with a certain value of tortuosity) and (**b**) Completely disconnected voids (zero connectivity and infinite tortuosity). We mention this fact because in a number of studies several scholars try to relate tortuosity and porosity to each other, as if this were generally possible [24]. However, this can be done only for the structures with a single geometrical degree of freedom, like regular structures of mono-sized spherical particles [25], structures with porosity tending to unity ([PERSON], 1881) [26], and infinitely-disperse random structures of particles ([PERSON] and [PERSON], 1973) [27]. For other types of structures, the two parameters are completely independent of each other and, thus, should be evaluated independently. After defining the connectivity factor, the next step for its quantitative measure consists in using a systematic way to evaluate tortuosity. In the past, tortuosity was generally considered a factor whose value was to be estimated by data regression.