## 2 Mathematical Modelling
### Economic Analysis  
Total costs, as the sum of annualized capital (CAPEX) and operating (OPEX) costs, were evaluated according to the procedure suggested by [PERSON] and [PERSON] for a fluid-solid process type with a detailed factorial estimate [92]. Day zero costs are calculated in terms of fixed capital investment (direct and indirect costs, contractor's fee and contingency) and working capital, while annualized capital costs are evaluated considering a lifetime for the process of 20 years, with an interest rate of 10%. Installation, instrumentation and control, piping, electrical, buildings, yards improvements, service facilities, and land are considered among direct costs and as a percentage of total equipment costs. Engineering and supervision, construction expenses are for indirect costs and as a percentage of total equipment costs.  
The equipment costs of an adsorbent bed and fan are calculated from Matches [93], considering carbon steel as the material. The used cost calculator provides an order of magnitude that is useful for a comparison of different processes as carried out in this work and inputs for the cost evaluation are reported in Table S2 of Supporting Information.  
These costs are based on 2014, then are adjusted for inflation using the following correlation adopted from the Chemical Engineering Plant Cost Index (CEPCI) (see Equation (24)) [3]:  
\[\frac{Cost_{2021}}{Cost_{2014}}=\frac{CEPCI_{2021}}{CEPCI_{2014}} \tag{24}\]  
where the values of \(CEPCI_{2014}\) and \(CEPCI_{2021}\) are, respectively, 576.1 and 655.9 [94]. A location factor is assumed for CAPEX, assuming that the plant is located in Switzerland. The updated location factor from 1993 [83] to 2021 is 0.58 [84]. OPEX are evaluated as the sum of direct production costs (raw materials and utilities and others as a percentage of capital costs such as maintenance and repairs, operating supplies, patent and royalty), fixed charges (depreciation, local taxes, insurance as a percentage of capital costs), administrative costs, distribution and selling costs, and research and development costs all evaluated as a percentage of the overall production cost.  
For the calculation of OPEX, the process is assumed to be fully automated (which indicates that the costs are a base level), while the costs of sorbents, cooling water, waste heat, and electricity are shown in Table 9. Waste heat and electricity are provided by a municipal waste incinerator (MWI), as for the Clineworks plant. The price of MOFs is set to 1.95 USD/kg (the best-case scenario cost for a synthetic sorbent, assuming an efficient scalability of production) while the price of cooling water is of 0.08 USD/m\({}^{3}\)[95]. Electricity has a cost of 0.1 USD/kWh [96], while waste heat costs 0.024 USD/kWh [96]. SI-AEATPMS sorbent has a cost of 1.3 USD/kg, while the other, amine-based sorbent has a price of 1.16 USD/kg (these costs are weighted on raw material mass fractions and raw material costs). It is assumed that all adsorbents have a lifetime of 5 years. Currently sorbents used in industry (such as zeolites, activated carbons, and silicas and aluminas) show lifetimes of 7-10 years, while for MOFs a lifetime of 5 years has been defined [95]. For this reason, we are adopting a conservative estimation on the existing knowledge, although the cycle numbers per day can be different among different sorbents.