7 Direct air capture (DAC) is an auspicious technology in pursuing negative CO2 emissions. A 8 promising process is temperature vacuum swing adsorption (TVSA) employing amine functionalised 9 adsorbents such as Lewatit® VP OC 1065, which is selected as a benchmark sorbent in this study. To 10 further improve process design, and critically lower costs, detailed modelling of DAC cycles is 11 imperative. However, the multi-component adsorption on these materials, particularly the cooperative 12

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... orption of CO2 and H2O, is crudely understood, and yet to be described in mathematical terms, 13 prohibiting sound modelling efforts. Here, we commit in-depth understanding of the effect of humidity 14 on CO2 adsorption and demonstrate how this impacts modelling of DAC cycles. We present two novel 15 mechanistic co-adsorption isotherm models to describe water's effect on CO2 adsorption and find a good 16 fit to original experimental co-adsorption data. We also show the considerable improvement in 17 predictions of these models when compared to an empirical co-adsorption isotherm model from 18 literature. A detailed TVSA DAC cycle process model is then used elucidating how different co-19 adsorption models affect the predicted process performance. It is found that the two novel isotherm 20 models generate similar results and Pareto fronts, whilst the minimum work equivalent calculated using 21 the more conservative of the two models is found to be 2.49 MJ kg -1 for the case study considered. 22 These mathematical descriptions laid out will lead to more accurate modelling and optimisation of 23 cyclic DAC adsorption processes, prompting a greater understanding of the material-process 24 combinations ideal for DAC and how costs can be driven down in the years to come. Importantly, they 25 allowed us to independently benchmark a Climeworks type DAC process, providing key DAC 26 performance data to the public domain. 27 2