Analysis of direct capture of CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {CO}}_{2}$$\end{document} from ambient air via steam-assisted temperature–vacuum swing adsorption

In this work, direct air capture (DAC) via adsorption is studied through the design and analysis of two temperature–vacuum swing adsorption (TVSA) cycles. In the first part, a novel way of describing the adsorption of CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {CO}}_{2}$$\end{document} in presence of water vapor is proposed for co-adsorption kinetic and thermodynamic data gathered from the literature. Secondly, two TVSA cycle designs are proposed: one with a desorption step via external heating, and one with a steam purge. A schematic method for the determination of the cycle step times is proposed and a parametric study on the operating conditions is performed via cycle simulations using a detailed, first principles model. Finally, the two cycles are compared in terms of CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {CO}}_{2}$$\end{document} production and energy consumption. The parametric study on the desorption time shows that there is a desorption time yielding the highest CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {CO}}_{2}$$\end{document} production at low energy consumptions. Low evacuation pressures are necessary to reach high CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {CO}}_{2}$$\end{document} production, but higher evacuation pressures show to be always favorable in terms of specific electrical energy requirements. A steam purge requires an additional thermal energy cost, but it not only allows decreasing the specific electrical energy consumptions, it also enhances CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {CO}}_{2}$$\end{document} desorption kinetics and allows reaching higher CO2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hbox {CO}}_{2}$$\end{document} productions at milder evacuation pressures. The results of this work present the possibility to directly relate the availability of power and heat to the design of the cycle.


Introduction
A major cause of climate change is the increasing concentration of greenhouse gases, one of which being CO 2 , which concentration has rapidly increased from 280 ppm in the preindustrial era to 407 in 2018 (IEA 2019). As a consequence, the global temperature has increased, posing a threat to the environment, and in the effort of limiting this increase to 1.5-2 • C, a transition to net-zero CO 2 emissions by 2050 is necessary (IPCC 2018). CO 2 emissions mitigation technologies have been widely studied and developed to tackle these issues; many of which focus on CO 2 capture technologies at the emission source. Assuming all of the emitted CO 2 coming from point sources, such as power plants and industry, can be removed with these technologies or avoided altogether by using renewables as energy source, there is still the need to account for unavoidable residual emissions coming from sectors such as agriculture and aviation (IPCC 2018). These account for about one fourth of the global greenhouse gas emissions, and cannot be taken care of using conventional CO 2 capture systems. Direct air capture (DAC), a removal technology in which CO 2 is captured directly from ambient air, has the potential to deal with these residual emissions, and has the additional advantage of being able to address both present and past emissions. DAC technologies can contribute to netnegative CO 2 emissions, when the captured CO 2 is coupled to carbon storage systems (Fuss et al. 2018), or to net-zero CO 2 emissions, when it is used for the production of carbonrich synthetic fuels that are powered by renewable energy sources (Davis et al. 2018;Sutter et al. 2018).
In DAC processes, the goal is to separate CO 2 from the other components in the air, which are mainly N 2 , O 2 and water vapor (Socolow et al. 2011). Due to the low concentration of CO 2 in the air, chemical sorbents with high affinity to CO 2 are good options for the separation. The diluted concentration presents an additional challenge in the process design and operation due to the fact that high volumes of air need to be processed to extract a significant amount of CO 2 ; in a process with 100% recovery, approximately 1400 m 3 of air at normal conditions need to be processed in order to capture 1 kg of CO 2 , thereby making it more of a CO 2 extraction technology than an air purification technology. On the other hand, advantages might include that with respect to point-source CO 2 capture systems, such as post-combustion capture from flue gases, there is no requirement that the capture unit be located at the source of CO 2 emission. This does not mean, however, that a DAC unit can be placed anywhere, since both its functioning and cost will highly depend on the availability and type of energy sources, on the availability of CO 2 storage and/or transportation and on the atmospheric conditions of the chosen location.
Today, most DAC applications capture CO 2 either via absorption in aqueous solutions or via adsorption on amine-functionalized solid sorbents. In both processes, the CO 2 in the air is uptaken by the sorbent, leaving the CO 2 -reduced air components to exit the column in gaseous form (Baciocchi et al. 2006;Keith et al. 2018). In absorption processes, CO 2 reacts with the aqueous sorbent, for instance KOH, forming H 2 O and K 2 CO 3 . In a second step, calcium hydroxide is added to K 2 CO 3 , forming a solid calcium carbonate with the CO 2 in solution and regenerating the aqueous solvent. To then release CO 2 from the carbonate via calcination, temperatures of around 900 • C and a pure-oxygen atmosphere are required Zeman 2007). In amine-functionalized adsorption processes, CO 2 binds to the amines forming a carbamate.
The removal of CO 2 and sorbent regeneration differ greatly with respect to absorption processes, as CO 2 is typically removed by vacuum and heating at regeneration temperatures under 110 • C (Darunte et al. 2016;Gebald et al. 2011;. Although absorption is used as the benchmark technology for various CO 2 -gas separations, the high-quality heat required for the solvent regeneration makes studying adsorptionbased DAC processes, where the low-grade heat could be eventually provided by waste heat, very promising and interesting. A great part of the research on adsorption-based DAC is focused on novel materials that have high CO 2 capacities in DAC-relevant conditions. Gebald et al. (2011) and developed amine-based solid silica particles and experimentally tested the material's stability on adsorption-desorption cycles. Darunte et al. (2016) showed how using metal organic frameworks as solid supports for amines could have an advantage over the silica-based sorbents due to their high specific surface area. Whilst the research on the potential of novel materials continues, there is a growing need for a study of the direct air capture process as a whole in order to acquire a better understanding of DAC process performance in terms of capture, regeneration, selectivity, pressure drop and energy requirements. Detailed reviews and technoeconomic assessments of DAC processes can be found in the literature (Fasihi et al. 2019;Sanz-Pérez et al. 2016), where existing DAC processes are compared in terms of type of process, materials used, CO 2 productivities, energy consumptions and costs. What is lacking is a detailed understanding of the impact that the process design and operation has on the performance. To address this caveat, this work sets out to simulate a cyclic DAC adsorption process, based on a detailed representation of the adsorption phenomena, to inform optimal DAC process design. A detailed, first principles, in-house model is used. For the process simulations, the equilibrium and kinetic data on the binary adsorption of CO 2 and H 2 O on the amine-functionalized solid sorbent developed by Gebald et al. (2014) are used. As using steam as a purge has proven to be feasible on amine-supported sorbents (Bos et al. 2019;Li et al. 2010), and given its potential to enhance CO 2 desorption during regeneration, cycle designs with and without steam purge are presented and compared in terms of their CO 2 production and energetic requirements. With the aim of understanding how to minimize the energy consumption, a parametric study on the desorption time, evacuation pressure and on the steam purge velocity allows the determination of the crucial elements that influence the electrical and thermal energy requirements of a direct air capture process.

Modeling
In this work, we use a standard one dimensional model for gas adsorption in fixed beds (Casas et al. 2012). The model is described in detail in Sect. 2.1. The constitutive equations are chosen to describe an amine-appended nano fibrillated cellulose material that has been specifically studied and developed for DAC applications (Gebald et al. 2014;Wurzbacher et al. 2012Wurzbacher et al. , 2016. Using the data from the three experimental works cited, we have determined the adsorption isotherms and transport parameters, as described in Sect. 2.2. The adsorbent used in the three papers mentioned is always the same, but the amine content, and therefore, the adsorbent's maximum uptake capacity differs. As a consequence we have been able to determine model parameters that are plausible, but not necessarily accurate in describing any of the three specific adsorbents. This choice is consistent with the scope of this work, where the focus is on the effect of operating conditions on DAC performance, hence plausibility of the model parameters is more important than accuracy.

Column model
The model here proposed is a detailed first principles model of a transient, one-dimensional cylindrical column of length L. The model consists of energy and material balances involving the gas phase, the solid phase and the column wall. The balances form a set of partial differential equations (PDE) that are solved transiently until a cyclic steady-state is reached. The following assumptions are made: -One-dimensional model in the axial direction, therefore there are no concentration, temperature or velocity radial gradients; -The solid and gas phases are in thermal equilibrium; -The mass transfer rate is described using a linear driving force model (LDF); -Axial dispersion and axial conductivity are assumed to be negligible under the conditions of the simulations and in the scope of the analysis presented in this work; -The pressure drop is described using the Ergun equation; -The heat capacities, the viscosity, the isosteric heat of adsorption and the heat and mass transfer coefficients are constant.
The component and overall mass balances are therefore the following: (1) where z and t are the independent variables of space and time; c and q are the total fluid and adsorbed phase concentration, y i is the molar fraction of each component i; u is the superficial gas velocity; t is the total void fraction; b is the adsorbent bed density, and n is the total number of components, three in our case. The mass transfer for each component is calculated using a linear driving force: where k i is the lumped mass transfer coefficient and q * i is the adsorbed phase concentration of each component at equilibrium, calculated based on the isotherm models described in Sect. 2.2.1. Two energy balances describing the heat transfer between the fluid phase, solid phase, column wall and outside (heating/cooling jacket) are needed: where T, T w and T wf are the column, column wall and working fluid temperature; p is the total pressure in the column; C g , C s , C ads and C w are the fluid, solid, adsorbed phase and wall heat capacities; H j is the heat of adsorption for component j; h L and h w are the heat transfer coefficients from the column to the column wall and from the column wall to the environment respectively; R in and R out are the inner and outer wall radius. The model parameters are reported in Table 1. The low values of the mass transfer coefficients and the large heat capacity of the adsorbent lead to conditions where neglecting axial dispersion and axial conductivity (see fourth bullet at the beginning of this section) is a reasonable assumption.
The energy and material balances are solved in Fortran using the finite volume method, where the PDEs are discretized along the space coordinate into equally-spaced cells. The values at the cell boundaries are calculated using the Total Variation Diminishing (TVD) scheme with a Van Leer flux limiter to avoid oscillations (LeVeque 2002). The PDEs are solved in time using the LSODES solver on Fortran. The boundary conditions for each step of the process are Table 3 and explained in Sect. 2.3. For more information on the model used, a detailed description of it can be found in previous works (Casas et al. 2012).

System characterization
The main criterion for selecting the adsorbent used for the process simulations in this work is the availability of equilibrium and kinetic data on the binary adsorption of CO 2 and H 2 O in DAC-relevant conditions. For this study, the adsorption equilibrium and kinetic data of CO 2 and H 2 O on APDES-NFC, that has been experimentally obtained by Gebald et al. and Wurzbacher et al. (Gebald et al. 2014;Wurzbacher et al. 2012Wurzbacher et al. , 2016, is used. APDES-NFC is an amine-functionalized nanofibrillated cellulose that binds to CO 2 , leading to the formation of a carbamate when in a dry gas and to ammonium bicarbonate when in the presence of water vapor.

Adsorption isotherms
The single and binary equilibrium data of H 2 O and CO 2 on APDES-NFC have been extracted from Figs. 3 and S2 (Gebald et al. 2014) and the isotherm fitting has been done in this work. Figure 1a shows the water equilibrium data both in the case of a single-component equilibrium experiment ( p CO 2 = 0 kPa, Solid heat capacity C s (J K −1 kg −1 ) 2070 Heat transfer solid markers) and in the case of co-adsorption of CO 2 and H 2 O ( p CO 2 = 0.045 kPa, open markers) on APDES-NFC (Gebald et al. 2014). In their work, Gebald et al. postulate that H 2 O physisorbs to a silanol group, a cellulose hydroxyl group or an unreacted amine group when CO 2 is absent, or via zwitterion deprotonation when CO 2 is present and already adsorbed on the solid. The equilibrium data exhibits a type II isotherm, which is typical for adsorption on cellulosebased materials (Thommes et al. 2015); as commonly used to describe H 2 O adsorption on such materials (Bratasz et al. 2011), the Guggenhein-Anderson de Boer (GAB) was used to model the water uptake q H 2 O , as a function of the relative humidity, x: where c m , c G and K ads are the three GAB parameters. The GAB parameters were estimated using a nonlinear leastsquares function and are represented in Table 2. The open markers, showing data from equilibrium experiments of a binary mixture in which CO 2 is present, fall right on the GAB isotherm, which suggests that the water uptake is barely influenced by the presence of CO 2 . This is confirmed by other results reported in the literature (Veneman et al. 2014). Therefore for the process simulations, H 2 O equilibrium loading is modeled using the single-component GAB isotherm for both single and binary mixtures. Figure 1b shows the CO 2 uptake on APDES-NFC under dry ( p H 2 O = 0 kPa, solid markers) and wet ( p H 2 O = 2.55 kPa, open markers) conditions at temperatures ranging between 296 and 343 K. The dry gas containing CO 2 shows a favorable isotherm that can be accurately described by the Toth isotherm, where the loading is a function of both CO 2 partial pressure and temperature as given by:  where n s , b and t are the temperature-dependent Toth parameters. Here, the reference temperature was set to 296 K and n s0 , b 0 and t 0 were estimated by fitting the corresponding data points. Data at 296, 323 and 343 K were then used to estimate , and the isosteric heat of adsorption, H 0 , which are reported in Table 2. Although all parameters were re-estimated in this work, H 0 was found to be in the same range as that in Gebald et al.'s work. Contrary to what was observed for water adsorption in the presence of CO 2 , the CO 2 uptake is indeed influenced by the presence of water vapor in the gas. Results show that the CO 2 uptake is enhanced when water is present; for a given temperature, the shape of the isotherm becomes slightly steeper (= higher affinity) and the uptake at higher partial pressures increases (Serna-Guerrero et al. 2008). To describe the co-operative adsorption of CO 2 in humid conditions, Wurzbacher et al. used an empirical approach, by introducing an enhancement factor that takes into account the presence of water (Wurzbacher et al. 2016). Although the relation holds for some data points, it fails to describe the behavior of others. Similar to a recent work (Hefti and Mazzotti 2018), a new isotherm model, accounting for the dependence on the water loading, is proposed in this work to describe CO 2 uptake in binary conditions. Using the Toth isotherm as a basis, the new isotherm model proposed in the following equation accounts for the water uptake dependence in the maximum uptake term, n s , and in the affinity coefficient, b: The water dependence is introduced in such a way that when water vapor is present in the gas, both the isotherm affinity and the maximum uptake increase, whereas when absent, the model reduces to the single-component Toth isotherm of Eq. 7. The available equilibrium data shown in Fig. 1 was used to estimate the parameters and , so as the modified Toth isotherm was determined, as shown by the dashed curves. The isotherm parameters are presented in Table 2.

Heat and mass transfer
The heat and mass transfer coefficients were estimated by fitting the model equations to dynamic experiments of CO 2 and H 2 O coadsorption on APDES-NFC (Wurzbacher et al. , 2016. The mass transfer coefficients were estimated by a linear square fitting of the one dimensional mass balances and the linear driving force assumption of Eq. 3 to the experimental breakthrough curves. The mass transfer coefficient for CO 2 has been estimated to be 2 × 10 −4 s −1 , which is in line with the values found in a recent work on the same material and those on an amine-functionalized MOF under DAC conditions Ng et al. 2018); that of water has been found to be 2 × 10 −3 s −1 . A calibration of the maximum CO 2 and H 2 O uptake was necessary in order to compensate for the difference in amine content between the paper reporting the breakthrough experiments ) and that used for the determination of the adsorption isotherms (Gebald et al. 2014). Experimental breakthrough curves of CO 2 and H 2 O respectively at ca. 400 ppm and a relative humidity of 40% are shown in Fig. 2a together with the simulation results obtained using the estimated parameters . The comparison between measurements and simulations shows a good fitting.
Similarly, the energy balances of Eqs. 4-5 were used to estimate the heat transfer coefficients h L and h W from experimental data of the desorption of an inert gas, namely the wall and gas temperature plotted for both experiments and simulations in Fig. 2b (Wurzbacher et al. 2016). Using the heat transfer coefficients reported in Table 1, the agreement between the two is satisfactory.

Cycle design
In this work, a steam-assisted temperature-vacuum swing adsorption (S-TVSA) cycle is presented, as shown in Fig. 3. The cycle is made up of four steps; (1) adsorption, (2) blowdown, (3) heating and (4) desorption. In the adsorption step, a fan is used to blow air at atmospheric temperature T L and pressure p H through the packed column until the bed reaches almost complete saturation of CO 2 in the bed. Atmospheric air is modeled as a ternary mixture of CO 2 , H 2 O and air at a constant composition of 400 ppm of CO 2 , with a relative humidity of 50% and air composition of 98.8%. Air is made up of mainly N 2 and O 2 , which have low affinity to the material. For this reason, air is modeled as an inert compound made entirely up of N 2 . At the end of the adsorption step, CO 2 and H 2 O are both in gas and solid phase. After the adsorption step, the blowdown step takes place, where the air inlet is closed and the column is brought down to the evacuation pressure, p L , with a vacuum pump at the column outlet. In this short step, air is partially removed from the gas phase in the column and released back into the atmosphere. In the following heating step, heat is transferred to the column through the external heating jacket at the desorption temperature T H . Like in the blowdown step, the stream exiting the column during heating is mainly composed of air and is therefore released back into the atmosphere. The desorption step is modeled with two variants; in one case, it is modeled as a heating step, where the two driving forces for desorption, heat and low pressure, are provided by the external jacket and by the vacuum pump (cycle B). The second alternative is to use a steam purge, that not only gives an additional temperature swing through direct contact with the adsorbent material, but also adds to the pressure swing-driven desorption through purge by displacement (cycle A). Superheated steam enters the column at the desorption temperature, T H , and evacuation pressure p L . During CO 2 and H 2 O desorption, to reduce the electrical energy requirements and to avoid problems in the pump due to the formation of liquid water among compression, a water trap is added before the compressor, where H 2 O is condensed out of the hot stream.
The boundary conditions for each step are reported in Table 3. During the blowdown step, the pressure at the outlet is assumed to decrease with an exponential decay p(t) that takes into account small flow resistances in the experimental setup (Casas et al. 2013). During the other steps, it is assumed to be kept constant using a back pressure regulator at either ambient pressure, i.e. p(t) = p H , or under vacuum, i.e. p(t) = p L .

Performance indicators
To be able to accurately describe the performance of direct air capture cycles, some performance indicators must be defined and consequently evaluated. The purity of the product stream, , depends on the mass content of the recovered CO 2 with respect to the air components that need to be separated: where m P is the mass of each component in the product steam. Water is not included in this calculation because it is condensed out of the CO 2 -H 2 O stream. The CO 2 production rate is defined as a function of the mass of recovered CO 2 per cycle time, t cycle : In this work the energy requirements are divided into thermal energy consumption, Q, and the mechanical energy consumption, W. The thermal energy consumption is defined as the sum of energy required for the external heating and for the purge steam.
where T amb is the ambient temperature, m H 2 O is the total mass flow of steam being produced, C H 2 O and C v are the heat capacities of liquid water and water vapor, H v is the enthalpy of vaporization, T H is the desorption temperature at which the steam enters the column and T v = f (p L ) is the water boiling point at the desorption pressure. The heat requirements for the external heating bed include the heat transfer from the jacket to the wall and from the wall to the bed; those for the steam production include heating liquid water from ambient temperature to boiling point, that for water vaporization, and that to heat the saturated steam from boiling point to the desorption temperature. The mechanical energy consumption is the sum of the energy needed for the fan and for the vacuum pump, that is described as an isothermal compression: where F is the volumetric flow rate entering the column during a step, p in and p out are the inlet and outlet pressures of the gases entering and exiting the column, ̇n is the molar flow rate being processed in the compressor, R is the ideal gas constant, p H and p L are the high (ambient) and low (vacuum) pressures of the compressor and are the fan and vacuum pump efficiencies, reported in Table 4.

Results and discussions
The column design is a very important step of the direct air capture process design. Processing such high volumes of air means that the pressure drop in the adsorption step could significantly contribute to the electrical energy consumptions. To lower pressure drops, shorter columns (of around 0.5-2 cm (Climeworks 2019; Gebald et al. 2012;Yu and Brilman 2017) are preferred to longer columns, typically used in post-combustion capture. For the sake of adsorbing significant amounts of CO 2 , the short bed length often leads to pancake-like designs. In this work, an adsorbent bed of 1 cm in length and 8 cm in diameter is used. The length of the bed allows for pressure drops of under 1 mbar for all simulations. The diameter was specifically chosen to match the cross-sectional area of heat transfer experiments setup (Wurzbacher et al. 2016), so that heat transfer is modeled as precisely as possible, given the lumped heat transfer coefficients in the model. The determination of the optimal operating conditions is done via parametric studies on the design variables, which can range from the stream flow rates and compositions, working fluid temperatures, evacuation pressure, step times and so on. To reduce computational cost, the degree of freedom of the design variables can be reduced by establishing rigorous criteria to determine the step times of steps 1-3, respectively t 1 , t 2 and t 3 . In this section, these criteria will be introduced and applied to the cycle (Fig. 4), in order to perform effectively the sensitivity analyses on the desorption time t 4 , the evacuation pressure p L and the steam flow rate F s (Figs. 5, 6, 7). The operating conditions used for all simulations in this work are summarized in Table 5.

Step time determination
For all simulations shown in this work, the times of steps 1-3 are set using criteria that are based on physical arguments related to the purpose of each step of the cycle. In the adsorption step, the goal is to maximize the bed's capacity to adsorb CO 2 hence to fully saturate the column with CO 2 . This is done because, contrarily to post-combustion capture (PCC), the CO 2 recovery, defined as the ratio between the CO 2 captured in the product stream over the amount of CO 2 fed with ambient air during one cycle, is not a limiting factor. When the column is saturated, the CO 2 in the air flows through the column and back into the atmosphere, therefore not contributing to net-positive emissions like in the case of PCC. However, the CO 2 mass transfer rate is slow, which entails that adsorption times are very long, as shown in Fig. 2a. Therefore, the adsorption time has been defined as the time at which the last gridpoint of the bed reaches 90% saturation starting from a fully regenerated column, therefore ensuring CO 2 saturation even when the bed is initially not entirely regenerated. The adsorption time is therefore maintained constant for all simulations. Once the CO 2 is captured, blowdown occurs with the goal of bringing the column down to evacuation pressure p L . For such a column, the time it takes to do so is very short and is almost independent of the chosen evacuation pressure, therefore it is also maintained constant for all simulations. In the heating step, the goal is to prepare the column for CO 2 desorption, and this differs based on the cycle design and evacuation pressure. In cycle B, where there is no steam purge in the desorption step, the only condition that must be satisfied is the removal of air ( y Air < 20% ) from the column so as to always ensure CO 2 purity in the product stream during desorption ( > 0.996 for all simulations). The time it takes to satisfy this condition is a function of the chosen evacuation pressure. In cycle A, to avoid condensation of the water vapor in the steam purge, the column must also reach a minimum temperature T M ; we have chosen 5 • C above the boiling point, which also is a function of p L . Therefore, the minimum time it takes to fulfill both conditions corresponds to the heating step time t 3 , which, as expected, increases with the evacuation pressure. Figure 4 shows profiles that are obtained in a cyclic steady-state with fixed operating conditions on a cycle operating at a 50 mbar evacuation pressure and a steam flow rate of 25 cm 3 /s: in pink, the adsorption step; in dark gray, the blowdown step; in light gray, the heating step; in blue, the desorption step. Figure 4a shows the adsorbed CO 2 profile at the exit of the column, showing how 90% of its adsorption capacity in that point is reached, which is 2.02 mmol/g. It also shows that very little CO 2 is lost during blowdown and heating due to their short step times and temperatures. At these conditions, full regeneration is obtained at desorption times that are approximately two times longer than the adsorption time. The pressure profile shows that the transition from ambient pressure, p H , to the evacuation pressure p L , occurs almost instantly. During blowdown, only part of the air in the column is removed, as seen in Fig. 4c. The remaining air is removed in the heating step, where the minimum temperature to avoid steam condensation, calculated as a function of p L using Antoine's equation, is also reached. Figure 4f shows the relative humidity of the gas exiting the column. The decrease in relative humidity in the blowdown step is due to the little desorption of H 2 O in this step, which consequently decreases by very little the temperature. The relative humidity profile in the cycle shows that, although steam only contains water vapor, it enters the column at such high temperature and low pressure, that the relative humidity decreases during desorption, meaning that water is actually desorbing from the adsorbent and not adsorbing, as seen in Fig. 4e.  Figure 5 compares the CO 2 production rate to the specific electrical and thermal energy requirements for different process designs. Each curve represents a different evacuation pressure in the range 30 mbar to 200 mbar, and each point on the curve represents a different desorption time, t 4 , ranging from 2000 to 50,000 s, where 30,000 s is long enough for full regeneration to occur. The arrow on each curve shows the direction of increasing t 4 . The curves exhibit similar trends, showing that there is a desorption time yielding highest CO 2 production rate corresponding to an energy consumption close to the lowest values. Lower evacuation pressures yield higher driving forces for CO 2 desorption, which explains the difference in peak in CO 2 production between the curves. In Fig. 5a, the direction of t 4 suggests that operating at low desorption times is not convenient in terms of specific electrical energy consumptions because similar production rates are achieved at lower specific electrical energy consumptions when desorption is long. However, working at desorption times longer than the peak in production is not convenient either, because in all cases the specific electrical energy consumption reaches a minimum, under which it is not possible to operate. At long t 4 , the components in the gas and solid phase reach equilibrium and the outlet stream exits the column at constant composition and continuously decreasing flow rate. The CO 2 in the product stream therefore almost ceases to increase, as does the electrical energy consumption of the vacuum pump due to the continuously decreasing molar flow rate of the product stream. On the other hand, the thermal energy consumption does not cease to increase, as seen in Fig. 5b, since the heat requirements for steam production are constant throughout desorption. Therefore, although the denominator of the specific thermal energy reaches a constant value, the numerator continues to increase, which explains why the direction of t 4 in Fig. 5b is inverted with respect to Fig. 5a. This result suggests that, thermal energetically speaking, lower desorption times are favorable.

Effect of evacuation pressure
The overall trend of the electrical energy curves show a slight trade-off between high CO 2 production rates and low energy consumptions. A similar trade-off is not observed in the thermal energy curves. Instead, the peak in CO 2 production increases with decreasing specific thermal energy, suggesting that it would be always best to operate at low evacuation pressures.

Effect of steam flow rate
The effect of the steam flow rate has been studied for p L = 50 mbar at desorption times between 2000 and 50,000 s. For constant p L , the cyclic capacity increases with the steam flow rate, yielding higher amounts of CO 2 extracted from equally-saturated adsorbent beds. For each curve in Fig. 6a, the electrical energy contribution of the first three steps of the cycle is equal due to the identical operating conditions. As increasing F s in the desorption step enhances CO 2 desorption, the molar flow rate that the compressor must process is higher. Therefore, the total electrical energy increases with F s , but at high t 4 , the specific electrical energy decreases because it is compensated by the higher production of CO 2 (in the denominator). This is why no trade-off is observed, like in the case of Fig. 5a, and why the minimum in specific electrical energy is fundamentally constant for all F s . Figure 6b shows a CO 2 -thermal energy trade-off when varying the amount of steam purge. The increase in thermal energy consumption with F s is a direct consequence of the need for higher steam production. The gray curve, corresponding to the profile of cycle B with no steam, and the purple, which corresponds to low steam flow rates, exhibit an inversion in the direction of t 4 with respect to the higher steam flow rates. In the lower range, the direction is similar to that of the electrical energy, where higher desorption times yield lower specific energy consumptions for the same CO 2 production. Cycle B even reaches a minimum, similar to Fig. 6a. When steam is absent, in fact, the thermal energy entirely goes into externally heating the adsorbent bed through the wall. When the entire bed has reached desorption temperature, assuming that there are no heat losses, no thermal energy needs to be additionally provided and the total thermal energy consumption also ceases to increase. When steam is added, its production rate needs to be accounted for throughout desorption. For higher steam flow rates, the specific thermal energy consumption for high t 4 exceeds that of low desorption times, showing that low desorption times are favorable. Overall, the peak in production rate in cycle B with no steam yields a decrease in production of 9% and in specific thermal energy consumptions of 70% with respect to the peak at flow rates of F s = 25 cm 3 /s. Figure 7 shows productivity-evacuation pressure isolines for different steam flow rates. The circle, square and triangle markers correspond to CO 2 production rates of 1.9 × 10 −5 , 2.2 × 10 −5 and 2.45 × 10 −5 kg/h, respectively. The solid, dashed and dotted lines correspond to evacuation pressures of 30, 50, and 100 mbar, respectively. The colors of the markers represent the steam flow rate, as in Fig. 6.   Fig. 7 Electrical vs. thermal energy consumption for CO 2 production rate isolines of 1.9 (circles), 2.2 (squares) and 2.45 × 10 −5 kg/h (triangles) and evacuation pressures of p L : 30 (solid), 50 (dashed) 100 mbar (dotted). The color scheme corresponds to the steam flow rate, increasing from left to right: F s : 0 (gray), 5 (purple), 10 (yellow), 17.5 (orange), 25 cm 3 /s (blue) (Color figure online) This figure is a good illustration of the importance of the cycle design. By looking at the different curves at equal evacuation pressure, one can immediately notice that for low CO 2 production rates, such as 1.9 × 10 −5 kg/h in the figure, it is favorable to operate at high evacuation pressures so as to reduce electrical energy consumptions. At constant levels of vacuum, the different steam purge flow rates result in an electrical-thermal energy trade-off. Higher steam flow rates indeed decrease the specific electrical energy consumption. However, this decrease is minimal with respect to the significant increase in the associated thermal energy consumption, suggesting that a steam purge might not be convenient. However, it is also clear that not all CO 2 production rates can be achieved at high p L , which is clearly indicated by the absence of the 2.45 × 10 −5 kg/h productivity isoline at 100 mbar. Such a rate can be produced at lower evacuation pressures, as demonstrated by the triangle curves at 50 and 100 mbar. However, even at such low pressures, it is not possible to reach such production rate without steam (as indicated by the absence of the gray triangular markers). To increase CO 2 production, one must either decrease the evacuation pressure or increase the amount of steam purge. Steam could therefore become a valuable option, when low vacuums are not industrially feasible and when there is low-quality waste heat available. This is an important result because it shows how the availability and type of energy source highly influences how the optimal cycle is designed.

Conclusions
In this work, direct air capture via adsorption is studied through the design and analysis of two temperature-vacuum swing cycles. A novel way of describing the CO 2 −H 2 O co-adsorption, a rigorous way of designing a cycle, and  Column press. p H (bar) 1 Jacket temp.
T H ( • C) 95 Step time t 3 = t 3 (p L ) (s) 704 [477,704,969,1120,1732 Step time  a parametric study on various operating conditions are presented. It is well known, and our simulations confirm it, that direct air capture comes at the cost of a substantial energy penalty. Part of it (the larger fraction) is thermal energy for column regeneration, and part is electrical energy for air blowing and column evacuation. The trade-off between the two forms of energy is quantified by our model and well illustrated by the maps of Fig. 7. Results show how it is energetically favorable to operate at high evacuation pressures and low steam purge rates (or no steam at all) to obtain low CO 2 production rates. High CO 2 production can instead be achieved at deep vacuum levels or alternatively by increasing the steam purge; the choice will depend on the availability of power and heat.
In fact, where and how direct air capture is deployed will depend strongly on the absolute and relative availability of affordable and low-carbon power and heat, besides that of a permanent (storage) or temporary (utilization) CO 2 sink. The design of the direct air capture plant and of its cycle and operation will have to account for this. Therefore, the availability of a sound model and process simulator as presented here will be indispensable to quantify the trade-offs and realize an optimal design and operation, in terms of productivity and energy intensity.