## 3 Results and Discussion
### Adsorption and Desorption Curves of DAC Systems  
The mathematical model provides the breakthrough curves for the adsorption of CO\({}_{2}\), for the considered chemisorbents and physisorbents. As already mentioned, these plots show the ratio between CO\({}_{2}\) concentration at the bed outlet and the CO\({}_{2}\) feed concentration, as a function of time. Figure 3 shows these trends for MOFs. Overall, in each Figure, three different sections can be identified as a function of time. At first, the concentration is zero because the step function of CO\({}_{2}\) has not passed the end of the air contactor. After a defined time (the breakthrough time, where the outlet CO\({}_{2}\) concentration rises to 5% of the inlet CO\({}_{2}\) concentration), the concentration increases because CO\({}_{2}\) reaches the end of the adsorber, and the sorbent continues to capture CO\({}_{2}\) until it is in equilibrium with the feed. An adsorption front moves along the bed with time, defining a mass transfer zone. Finally, the sorbent does not capture any more CO\({}_{2}\), because the entire bed is in equilibrium with the CO\({}_{2}\) in the feed. No additional mass transfer occurs between gas and solid phases. This is defined here as the equilibration time (outlet concentration is 95% of the inlet CO\({}_{2}\) concentration) [98]. The difference between breakthrough and equilibration times defines the length of the mass transfer zone. Generally, this value depends on the value of the mass transfer rate: a higher mass transfer rate creates a shorter mass transfer zone [98]. The higher mass transfer rates imply lower mass transfer resistance, and a higher rate of adsorption [99]. Our results show that the shortest mass transfer zone of 146.2 s is present for MOF-177 with the highest mass transfer rate of 1.49 \(\times\) 10\({}^{-1}\) s\({}^{-1}\). The greatest mass transfer zone of 703.5 s is present for MOF-5 due  
Figure 2: Diagram of the experimental setup used for the validation of the mathematical model [74]. to the lowest mass transfer rate, of 1.9 \(\times\) 10\({}^{-2}\) s\({}^{-1}\). On the other hand, for MIL-101 a mass transfer zone of 331.2 s is obtained, for which the value of k\({}_{\rm l}\) is of 3.3 \(\times\) 10\({}^{-2}\) s\({}^{-1}\). These results suggest that the dynamic capacity of the adsorbent bed with MOF-177 is higher than other sorbents. Reducing the dynamic mass transfer zone increases the dynamic capacity of the bed [100]. Higher values of k\({}_{\rm l}\), as for MOF-177, allow a steeper curve compared to others, suggesting that the bed is being utilized more efficiently [101]. Figure 3 also suggests that the bed with MIL-101 sorbent has a very low residence time (the time that the air has spent inside the adsorber), as also reported in the work of [PERSON] et al. [55] for a MIL-101-based sorbent. Breakthrough curves for amine-based sorbents are shown in Figure 4.  
The same three sections are present. However, a greater mass transfer zone is present for these chemisorbents compared the previous sorbents, due to a lower mass rate constant of 4 \(\times\) 10\({}^{-4}\) s\({}^{-1}\) for the silica gel-based sorbent, and of 3.8 \(\times\) 10\({}^{-3}\) s\({}^{-1}\) for the cellulose-based sorbent. In fact, the mass transfer zone for the SI-AEATPMS is around 14,380 s, while for APDES-NFC-FD it is 1915 s. A higher mass transfer resistance is present in chemisorbents. A steeper breakthrough curve in these last sorbents is due to a higher bed porosity, the effect of which is studied in the work of [PERSON] et al. [97]. However, even though chemisorbents are characterized by lower adsorption time constants, the breakthrough curve is shifted to later times, suggesting a stronger affinity between the adsorbent and CO\({}_{2}\)[102]. Since a chemical reaction occurs, a greater adsorption capacity is present, as shown in Figures 5 and 6, reporting, respectively, the trend of loading for physisorbents and chemisorbents as a function of time and length of adsorbent bed.