## 2 Mathematical Modelling  
### Sorbents and Adsorbent Bed  
Here, the MOFs (MIL-101, MOF-5 and MOF-177) (physisorbents) are compared against two amine-functionalized chemisorbents which have been suggested by Climeworks to be of interest. In particular, the APDES-NFC-FD sorbent is used by this company in its plant located in Switzerland. These amine-based sorbents are: SI-AEATPMS ([N-(2-aminoethyl)-3-aminopropyl]trimethoxysilane (AEATPMS) grafted on silica gel) and APDES-NFC-FD (3-aminopropylthethyldiethoxysilane (APDES) onto nanofibrilated cellulose (NFC)), both belonging to class 2. MOFs are considered because of their high porosity, ordered and well-characterized porous structures, and adjustable chemical functionality which makes them potentially very attractive for CO\({}_{2}\) adsorption [66]. These properties are purported to allow them to capture and release CO\({}_{2}\) with fast kinetics and high reversibility over many cycles. Little data are present in the literature about CO\({}_{2}\) capture using these kinds of sorbents. In contrast, amine-functionalized sorbents are selected for study because their strong bonds with CO\({}_{2}\) allow significant uptake at low CO\({}_{2}\) partial pressure [4]. The selected sorbents are produced through synthesis routes described in the literature, as follows. The synthesis of MOF-5 starts with a reaction between Zn(NO\({}_{3}\))\({}_{2}\cdot\)6H\({}_{2}\)O and 1,4 benzene dicarboxylic acid in N,N\"-dimethylformamide solvent at 373 K [67]. After cooling, the mother liquor was decanted from the crystalline material, and the white solid was washed with dimethylformamide [68]. Properties such as the BET specific surface area, average par ticle size, and micropore volume have been evaluated and reported by [PERSON] et al. [69], and are respectively 3570 m\({}^{2}\)/g, 2 um, and 1.25 cm\({}^{3}\)/g. MOF-177 can be obtained through a room temperature technique, mixing benzenetribenzoic acid and zinc acetate dehydrate with diethylformamide. The obtained compound was filtered, washed with diethylformamide, then chloroform was added to remove impurities. After that, the MOF was thermally activated under vacuum at 393 K [70]. Pore textural properties were provided by [PERSON] and [PERSON] [71]: BET specific surface area, pore diameter, and pore volume are, respectively, 1218 m\({}^{2}\)/g, 14.4 A, and 0.748 cm\({}^{3}\)/g. For the synthesis of chromium terephthalate metal-organic framework (MIL101) a mixture of benzene-1,4-dicarboxylic acid, chromium(III) nitrate nonahydrate, hydrogen fluoride, and water was heated at 493 K, forming a green crystalline powder. After a natural cooling, the obtained material was filtered to separate the MIL-101 powder [72]. This MOF is characterized by a high BET surface area (>3000 m\({}^{2}\)/g), a large pore diameter (29-34 A), and a huge pore volume (\(\approx\)2 cm\({}^{3}\)/g) [73]. The SI-AEATPMS sorbent has been synthetized by drying silica gel beads inside a natural convection oven at 383 K, before loading with AEATPMS, in a two-neck flask equipped with a reflux condenser. The mixture was heated at 423 K in a silicone oil bath for 3 h. After cooling to 298 K, the solid product was washed in diethyl ether, and dried in ambient air for 24 h [74]. BET surface area, pore volume, and average pore size were 216 m\({}^{2}\)/g, 0.65 cm\({}^{3}\)/g, and 9.6 nm. The other amine-functionalized sorbent, APDES-NFC-FD, has been obtained by adding APDES to NFC hydrogel to obtain a suspension of 10% _w/w_ of APDES, and incubated for 2 h. This compound was then added dropwise into liquid N\({}_{2}\) to produce a freeze-dried frozen granulate. The obtained material was then subjected to thermal treatment at 393 K under N\({}_{2}\) flow in a natural convection oven. The final grains had a BET surface area of 12.2 m\({}^{2}\)/g, [75]. Data regarding particle and density diameter, and bed porosity for the considered MOFs are reported in Table 4, while the same parameters for amine-based sorbents are shown in Table 5.