## 3 Results
### Use of HSDM Formulation for Interpreting Dynamic Adsorption
#### 3.4.2 Dimensionless Analysis of Diffusional Regimes  
In the precedent paragraph, we employed the homogeneous surface diffusion model (HSDM) for simulating Na+/M+ (or 2 Na+/M++) exchanges in the fixed-bed column and predicting the adsorption dynamics of nuclides. In the following section, we interpret dimensionless numbers relative to column operations (Biot, \(B_{i}\); Stanton, \(S_{i}\); solute distribution constant, \(D_{g}\)) with the aim of gaining information regarding the shapes of breakthrough curves and breakthrough times, as well as the types of diffusional regimes involved [105; 108; 109; 110]. In this work, the dimensionless numbers \(B_{i}\), \(S_{t}\), and \(D_{g}\) (as defined in Section 2.9) were determined from experimental adsorption results for different elements in alkali-activated brick, and their values are listed in Table 4. Globally, characteristic dimensionless numbers ranged from 1091 to 5237 for \(B_{i}\), from 162 to 256 for \(D_{g}\), and from 296 to 778 for \(S_{t}\). Ratio \(Ed=S_{t}/B_{i}\) was also considered in the dimensionless analysis, knowing that its implication on the adsorption process was directly related to the individual values of the \(B_{i}\) and \(S_{t}\) numbers. The very large values of \(B_{i}\) revealed that film mass transfer occurred very faster than internal diffusion, and hence, internal surface diffusion dominated the adsorption process. The high Biot number values also permitted us to better understand the influence of \(D_{s}\) on the dynamics of breakthrough curves. Indeed, since the obtained \(B_{i}\) values were high and \(D_{s}\) and \(B_{i}\) were inversely proportional, these should lead to low \(D_{s}\) values, as evidenced experimentally (\(D_{s}\) values ranging from 2.00 \(\times\) 10\({}^{-12}\) to 6.66 \(\times\) 10\({}^{-12}\) m\({}^{2}\cdot\)s\({}^{-1}\)). Dimensionless analysis further confirmed the occurrence of slow diffusional regimes, which were characterized by smoothed wavefronts. Moreover, accord  
Figure 10: Experimental and simulated breakthrough curves relative to the dynamic adsorption of cations (Cs+, Rb+, Co2+ and Sr2+) on to brick composite.  
ing to [PERSON] and his co-workers [108], longer computational times are expected to be observed for large values of \(D_{g}\) in an elevated adsorbent medium. In the studied adsorption systems, as high time-scales were given by \(D_{g}\), weak internal diffusions took place and significantly controlled the adsorption process. This finding was supported by the weak values of \(Ed\), which also confirmed the involvement of slow adsorption diffusion.  
The effective diffusion coefficients of the ions (Cs\({}^{+}\), Rb\({}^{+}\), Co\({}^{2+}\), and Sr\({}^{2+}\)) adsorbed onto the brick-based composite were determined afterwards compared with published diffusion data relative to the mobility of these cations in industrial and natural minerals, like cements, clays, and clay rocks/soils (see Table 5).  
Overall, one can conclude that the \(D_{\mathit{eff}}\) values calculated here were within the range suggested for materials with close mineralogy, like cements alone, those associated with zeolites Na-A/Na-P [111; 112; 113; 114; 115], or even slag-blended geopolymers [115] (see Table 5).