Ambient wind conditions impact on energy requirements of an offshore direct air capture plant

This study proposes an off-grid direct air (carbon) capture (DAC) plant installed on the deck of an offshore floating wind turbine. The main objective is to understand detailed flow characteristics and CO2 dispersion around air contactors when placed in close proximity to one another. A solid sorbent DAC design is implemented using a commercially deployed air contactor configuration and sorbent. The sorbent is assumed to undergo a temperature vacuum swing adsorption cycle. Computational fluid dynamics (CFD) is used to determine the local conditions entering each unit based on varying wind speed and angle. Two-dimensional (2D) simulations were used to determine the pressure drop through a detailed air contactor design considering the sorbents APDES-NFC, Tri-PE-MCM, MIL-101(Cr)-PEI-800, and Lewatit VP OC 106. Only APDES-NFC was explored for further analysis in the three dimensional (3D) CFD dispersion model. 3D simulations were used to model flow patterns and CO2 dispersion using passive scalars. A worst case scenario is analyzed for all DAC units in adsorption mode with fans running simultaneously. 2D simulations show an under utilization of contactor length, and quantify pressure loss curves for four common sorbents. One commercially deployed sorbent is considered for further analysis; a pressure drop of 390.62 Pa is experienced for a flow velocity of 0.73 m s−1 through a 1.5m×1.5m×1.5m contactor. Using 3D simulations, fan energy demands are computed based on flow velocities and applied pressure gradients. There is found to be a decrease in overall fan power demand as wind speed increases. High wind speeds can passively drive the adsorption process with fans shut off at certain wind directions. This occurs at an average contactor inlet velocity of 17.5 m s−1, correlating to a hub height (150 m) wind speed of 24 m s−1. Thermal energy demands are computed based on inlet CO2 concentrations entering downstream units. Thermodynamic work for desorption based on an assumed second law efficiency is compared to thermal energy for desorption computed using a simplified isotherm method, taking into account the impacts of humidity on CO2 adsorption. Going from 414.72 ppm to 300 ppm CO2 inlet concentration requires an additional 63.9 kWh (t- CO2)−1 using the 2nd law thermodynamic efficiency method and 25.9 kWh (t- CO2)−1 using the isotherm method. Contactor arrangement, wind angles, and wind speeds have a significant impact on flow patterns experienced, and resulting CO2 dispersion. High wind speeds assist in CO2 dispersion, resulting in higher inlet concentrations to downstream DAC units and decreased thermal energy requirement.


Introduction
Meeting the climate targets set out in the Paris Agreement will require an aggressive global effort to decarbonize, both in terms of carbon emission reductions, as well as direct or indirect carbon removal from the atmosphere or oceans respectively.It is well understood among the scientific community that CDR is now a necessary component to meeting climate goals [1].DAC is a carbon negative technology used for directly capturing carbon dioxide (CO 2 ) from atmospheric air and has a number of distinct advantages over alternative approaches.A significant advantage of DAC is that it is location independent.CO 2 is well mixed within the atmosphere so its concentration variations are small when comparing different locations.Changes in performance are expected as a result of varying ambient conditions, such as temperature and humidity.Choosing optimal locations for DAC is dependent on proximity to resources including energy, water, and sufficient CO 2 storage.So although locationally independent, it cannot be placed just anywhere.
Captured CO 2 can also be utilized for a variety of purposes, known as carbon capture and utilization.This includes purposes such as drop in synthetic fuels when combined with hydrogen, as well as alternative fuels such as methanol [2][3][4].Additionally CO 2 can be used in chemical feed stocks/products, carbonated beverages, concrete production and many others [5][6][7][8][9][10].This provides a pathway to generate an additional revenue stream.The goal of this study is to provide a negative emissions solution, therefore the focus was on carbon sequestration, and utilization pathways were not considered in the scope.
As the world's population continues to increase, competition for land and resources become stronger.This, along with utilization of vast offshore wind resources and technologies, drives the motivation to explore locations where direct land use competition is not an issue, such as far from shore.Along with having plentiful basalt for sequestration, many offshore sites are also in close proximity to renewable energy resources.This provides an opportunity to use otherwise difficult to capture and transport renewable energy to drive a negative emissions process.Building DAC offshore allows for a site to be chosen that is in close proximity to renewable energy resources, as well as a storage location, eliminating the need to transport CO 2 from shore.
An opportunity for energy savings exists in using ambient wind conditions to decrease the fan power required to overcome the large pressure drops encountered by DAC devices.The proposed design is placed on-board floating offshore wind turbines, which by nature are placed in regions with steady high wind speeds.Current studies estimating plant wide energy consumption assume zero ambient wind [11][12][13][14], computing a fan power required to overcome a certain reactor pressure drop to meet a designed volumetric flow rate.Fan power has been found to account for approximately 3%-57% of the total plant energy requirement [11,12,15].
Typical wind speeds for floating offshore wind show annual averages of 6-11.3 m s −1 at 100 m hub height [16].During times of low power availability, one could envision operational schemes for DAC where fans are left turned off, allowing air to passively pass through the contactors.Although sufficient volumetric flow may not be achieved to reach the desired capture rate, cycle times for adsorption could be modified to allow a longer period for air to contact sorbent.All power available could then be directed to the desorption process of as many units as possible, allowing for the plant to continue running at a lower output.
This study is motivated by the opportunity to provide a CDR solution that harnesses currently unused but abundant renewable energy far offshore and does not contribute to land competition, while choosing sites adjacent to plentiful and secure CO 2 storage.Its important to note that this comes with additional complexities of offshore operation.This means that the energy used in the process does not directly remove clean electricity capacity that could be used elsewhere to decrease the carbon intensity of the grid.Of course, conducting any project requires financial investment, so competition exists regardless.DAC is an energy intensive process, and should be driven by renewable power to maximize the net climate benefits.
Two main DAC technologies exist on a commercial scale, solid sorbent (SS), and liquid solvent/aqueous approaches.A detailed summary of the technology was conducted by McQueen et al [17].The two methods follow a similar overall process, but differ in design and implementation.In either case, CO 2 is absorbed/adsorbed onto a chemical capture material, either liquid or solid, by using fans to pass large volumes of air over the capture material.Once the substance has reached a saturated state, heat and pressure are used to drive the CO 2 off into a concentrated output stream and the capture material is regenerated for further use.
LS DAC is designed to be run as a continuous process, whereas SS DAC runs as a batch-wise process, using multiple units in parallel to enable continuous operation.LS DAC historically has relied on scaling up unit size, taking advantage of economies of scale based on unit size to drive cost reductions.SS DAC uses small modular units and scales out to drive cost reductions using economies of scale based on number of units produced.The temperature requirements of the processes also differ.LS DAC requires heat around 900 • C [15,18], which is typically met using combustion of natural gas.Temperatures at this level can be achieved using electrical heating as well, but is far more challenging.In contrast, SS DAC requires heat on the order of 100 • C [19,20].Heat transfer is commonly done using direct electrical heating, jacket heating by passing heat transfer fluid through internal tubes surrounding sorbents, or passing steam through the contactor.
The pressure losses within the contactor overcome by the use of fans contribute significantly to the overall electricity consumption.As stated previously, fans can account for between 3% and 57% of the total plant energy requirement.Carbon engineering (LS DAC) reported 61 kWh (t-CO 2 ) −1 [15] electricity consumption that is 16.7% of the total electrical consumption, and 3.3% of the total energy consumption (electrical plus thermal energy).In the case of SS DAC, Bos et al [11] reports fan electrical consumption of 31.9% of the total energy consumption.Sinha et al [12] reports a fan energy consumption of 44.4% to 57.0% of total plant energy use.
One of the major implications of operating SS DAC units in an offshore environment is the relatively high humidity.As humidity increases, water adsorption in the contactor also increases.Co-adsorption of water severely impacts the energy requirements of the system.Using a conservative estimate for CO 2 to H 2 O desorption ratio of 1:4, the heat energy requirement for desorption per mol of CO 2 is more than tripled ( [21].As humidity increases, there is also an enhancing effect on CO 2 adsorption as shown by Wurzbacher et al [22].Few studies have quantified this effect because co-adsorption measurements are complex.Experimental data is available for only select sorbents, and as a result in modeling of DAC, humidity enhancing effects on CO 2 adsorption are often neglected.
Different approaches have been used to capture this effect when modeling DAC energy requirements.Wurzbacher et al [23] used an empirical enhancing factor dependent on the relative humidity and CO 2 partial pressures.Stampi-Bombelli et al [24] built on this by embedding the H 2 O isotherm within the CO 2 isotherm, but this requires extensive experimental data on the sorbent of interest.Elfving et al [25] explored the implications of ambient temperature and humidity on CO 2 adsorption capacity and found that cold humid air had the highest adsorption capacity.Sanz-Pérez et al discusses the implications of temperature and humidity on both LS DAC and SS DAC [26].
Plant wide energy estimates for SS DAC have been completed, but to the best knowledge of the author, none have taken into account the local atmospheric conditions such as varying CO 2 concentration downstream of other contactors and incoming wind speed.The goal of the present study is to build an understanding of the effects of varying wind speeds and angles on total energy consumption of a plant.Additionally, the present study aims to produce pressure loss curves for a commercially deployed air contactor design using CFD which is not available in literature.Due to DAC technology's relatively nascent nature, most detailed information remains proprietary.The pressure drop through SS DAC units is currently only available at a lab scale; very little detail is given about operation at commercial scale [19].As such, a 'generic DAC contactor geometry' is applied and tuned to industry stated values using various sorbents.Energy estimates are sensitive to the chosen sorbent and configuration in which they are laid out.This makes producing accurate energy estimates at scale difficult, especially for academic studies.The three main contributions from this paper include analysis of: (1) two-dimensional (2D) pressure loss curves using different adsorbents of a commercially deployed contactor design and scale, (2) three-dimensional (3D) CO 2 dispersion model of a DAC plant installed onboard a floating offshore wind turbine, (3) the change in energy consumption due to inlet CO 2 concentration and local wind speed, and (4) energy modeling and thermal energy requirements for sorbent at varying CO 2 inlet concentrations.
The paper is structured as follows.Section 2 discusses the methodology and design setup for the analysis.Section 3 presents the results alongside discussion from the CFD and energy modeling, and section 4 presents the summary of the major findings.

Methods
A steady state RANS model supplemented with a Eulerian advection-diffusion dispersion model of passive scalars is used to model the dispersion of CO 2 .Passive scalars are not actively involved in the flow physics of the simulation; rather they are tracked and analyzed during post processing after solutions to other flow variables are solved.This assumes the species present is in low concentration, and is considered a fluid.Passive scalars are commonly seen in dispersion studies, however, the majority of studies examine a passive scalar source in pollution dispersion applications rather than a passive scalar sink (removal) as is used in the current study.
CFD is used to determine: (1) the pressure loss curve for an air contactor, (2) the local CO 2 dispersion around air contactors, and (3) pressure gradient applied by each fan/momentum source and volumetric flow rate.CFD outputs are fed into a spreadsheet model for energy consumption.

Initial plant design
Currently, the largest DAC installation is the Iceland Orca plant at 4000 t-CO 2 y −1 [19] from the SS DAC company, Climeworks.It uses eight shipping container sized units, each capturing about 500 t-CO 2 y −1 using a mixture of electrical and geothermal heat.The combined thermal and electrical energy consumption is estimated to be about 2000 kWh (t-CO 2 ) −1 [27], equal to a constant load of 127 kW at 500 t-CO 2 y −1 if run with a 90% capacity factor to allow for maintenance and shutdowns.Thermal energy is about 75% of the total energy, 1500 kWh (t-CO 2 ) −1 , and is assumed to be met using electrical means.
The floating offshore platform for the 15 MW IEA reference turbine [28] was used for this analysis.Based on an average annual capacity factor of 45% [29], the average power output is 6750 kW.An approximation using a capacity factor, as opposed to a second-by-second dynamic power analysis, removes site specific attributes and wind profiles, and allows modeling takeaways to be applied more broadly to various locations.To assume continuous operation at this power output, sufficient energy storage is assumed to be installed on board.Efficiency losses due to the energy conversion are not accounted for in the present analysis; rather a conservative estimate of the number of contactors is used.Round trip efficiencies have a large range depending on the energy storage system, with estimates between 20% and 98% [30].One attractive option for onboard logistics is energy storage using hydrogen, by electrolysis of seawater.As DAC is a thermally intensive process, hydrogen could be combusted to meet the thermal demand, and potentially produce a greater round trip efficiency.
At 6750 kW average power output, and each DAC consuming an average of 127 kW, 54 DAC units could be installed on board, giving an annual capture capacity of 27 000 t-CO 2 y −1 .It is hypothesized that due to the tight spacing of air contactors, local concentration will be decreased at the entrance to downstream contactors, thus they would have a larger energy requirement.Assuming this consequence is large, a conservative estimate of 30 DAC units (three contactor banks, each with five DAC units stacked vertically into two rows) are placed symmetrically in a triangular arrangement around the base of the turbine.This gives a target capture capacity of about 15 000 t-CO 2 y −1 .This configuration can be seen in figure 1.In future analysis, if the consequences of CO 2 mixing are small, an additional three banks of contactors could be added in the same triangular configuration inside of the outer contactor banks.Each additional bank could contain eight units (two rows of four).The dimensions of the platform are adapted from the reference turbine [31].The floater design does not contain a deck, however the DAC units are assumed to be mounted on catwalks extending between the three outside floats.
The DAC units are mounted to promote horizontal airflow as this allows ambient wind to assist in overcoming the pressure loss of the air contactor, thus reducing fan power required.This configuration can be seen deployed commercially in the Orca plant discussed earlier.Another possible configuration is to mount the fans vertically, drawing air in the top, and expelling CO 2 depleted air out the sides or visa versa [18,32].A central vertical fan may be shared between multiple DAC units, which would decrease the capital cost of each unit.However, it does not take advantage of local wind conditions, as upstream contactors may be aligned with the wind direction, but the downstream/opposite facing contactors will be forced to blow upwind.For the purpose of this study, the goal is to examine the extent that ambient wind can assist DAC, and reduce energy consumption, so the horizontal fan configuration was chosen.
The rotor at the top of the turbine was not included in current analysis.The main goal of the present study is to understand how fan work can be reduced by incoming horizontal wind flow, and to understand the order of magnitude of local CO 2 depletion in the region of DAC units.It can be seen in figure 15 from Uchida et al [33], that the impacts to the ABL profile are minimal in the region immediately below the rotor.The majority of the impacts to velocity profile are seen downstream of the rotor within the stream tube.For the sake of this analysis, the impacts from the rotor acting overhead are deemed to be negligible.This is meant to serve as a worst case analysis.In reality the rotor may induce increased mixing, as it could transport un-depleted air from above down to the region of the DAC units, but this is beyond the scope of the current analysis and would require a further, more detailed investigation.

CFD part 1-2D modeling
The goal of this section is present the 2D modeling scheme used to obtain a pressure loss curve for a commercial scale and design SS DAC unit using a variety of sorbent materials.Gebald et al [34] and Sauerbeck et al [35].present a patent for a commercial design of an SS DAC air contactor.This design was re-produced for this analysis, with a cross sectional view shown in figure 2(a); employing horizontal sheets of SS particles arranged in an accordion manner.Simplification to a 2D analysis is completed by taking a 2D cross section through the center of the contactor, and applying symmetry in the vertical and transverse directions as shown in figure 2(b).Each sorbent sheet was modeled using a porous media model within the commercial software Simscale.The fixed coefficient porous media model [36] was used in the 2D analysis.The area averaged pressure was interpreted at the inlet and outlet to obtain to pressure drop through the contactor by applying numerous inlet velocities to produce a pressure loss curve.This result is then applied in part 2, section 2.3 to simplify the mesh complexity and computational effort when introduced into a larger 3D domain.
The fixed coefficient porous media model is applied to each sorbent sheet.This model adds a momentum source term to the governing RANS equations being solved in the back end of the commercial software.Users specify coefficients for all three directions, and the software internally solves a pressure gradient dependent on the flow velocity.The pressure gradient ∇p is: where ρ ref is the density of the fluid, v s is the superficial fluid velocity, and A and B are coefficients input based separately for each direction x, y, and z.In Simscale, the coefficients are specified as α, and β, however, to avoid confusion with other symbols used in this analysis, the variables are discussed as constants A and B instead.In this case, isotropic flow is expected so the coefficients are equal in all three directions.In order to find appropriate values for these coefficients, the Ergun equation ( 2) is applied.The Ergun equation expresses a friction factor through packed beds with spherical particles based on a modified Reynolds number.The pressure drop through a packed bed of spherical particles is shown in equation ( 2) Here, L is the length/thickness of the bed in the direction of flow, D p the spherical particle diameter of the packing, ρ is density of the fluid, µ the dynamic viscosity of the fluid, v s the superficial velocity, and ϕ the void fraction/porosity of the bed.The parameters used for each sorbent are summarized in table S1.
By matching terms in equations ( 1) and ( 2), the coefficients A and B are solved.The resulting equations are shown in equations ( 3) and ( 4) The goals of incrementally increasing model complexity with 2D simulations was to first verify the coefficient porous media model worked as intended, apply it to a lab scale configuration with known pressure drop, and then scale the model up to a commercial size unit.This was done with the three incremental sub models: First, to verify the validity of the coefficients, the outputs of the CFD model are compared with the results of the Ergun equation.This is done with a simple single vertical sheet of porous material.Particles of sorbent APDES-NFC are assumed to be packed into a 1 cm thick sheet, and air is passed through perpendicular through the thickness.The pressure drop across the sheet is analyzed at the domain inlet and outlet.Once verified, this methodology is applied to other materials with known spherical particle diameter, and density and applied to other geometrical configurations.This result is shown in the supplemental information in table S3.
Second, the fixed coefficient model is applied to a lab scale, air contactor configuration using a stack of horizontal sheets of APDES-NFC sorbent as shown in figure 2. The pressure loss obtained from the CFD analysis was compared with the pressure loss curve presented in the patent [34].
Third, this geometric configuration, figure 2, was then scaled up to an appropriate sizing for commercial operation, with dimensions suggested by Sabatino et al [13].Sorbent sheets with the same thickness and spacing were arranged in the same manner as described above, and a pressure drop curve was obtained for four sorbents of interest by applying the same representative inlet velocities to each one, matching the flow described in the patent.Four common sorbents discussed in the paper by Sabatino et al [13] were modeled to find the pressure loss curves.The sorbents include APDES-NFC, Tri-PE-MCM, MIL-101(Cr)-PEI-800, and Lewatit VP OC 106.The sorbent physical properties are summarized in table S1 and the calculated coefficients A and B for each sorbent are shown in table S2 in the supplemental information.Additionally, the dimensions of simulated scenarios are shown in figures S1-S3.
Model closure was obtained using the steady-state RANS equations, with the k-ω-SST turbulence model using commercial CFD software Simscale, run on the OpenFOAM framework.The k-ω-SST model is one of the most frequently used in industry, combining two common turbulent models; the k-ω for near wall, low Reynolds number flows; and switching to the k-ϵ model in the free stream.Outputs were evaluated once residual values reached below 1 × 10 −4 .

2D-mesh generation
Simscale only supports 3D simulations, however, a 2D simulation can be performed by creating a thin mesh with a small number of cell depth in one direction and applying a periodic boundary condition (BC) at both faces.To obtain this, a mesh was created using the hex dominant parametric setting.A base mesh with perfect cubes for each cell was generated with a cell size of 1 mm in all three directions.Five cells depth into the page produced the best results.For the single vertical sheet, a mesh was generated with 172.5k cells.For the 0.5 m contactor, a mesh with 212.6k cells, and 442.4k cells for the 1.5 m contactor simulations.

2D-BCs
Six BCs were applied to the internal flow domain, shown in figure 2(b).A uniform velocity inlet is prescribed at the leftmost face, in the positive x-direction.Symmetry was applied in the positive-y direction on the top and bottom faces.A periodic BC was applied on the sides into and out of the page (+−z).This results in a repeating slice of the sorbent sheets.This assumption ignores the no slip condition at the contactor edges, as this is deemed insignificant in the overall pressure drop.The gauge pressure was set to zero at the outlet face.

2D-flow regime
To determine the expected level of turbulence, the Reynolds number was computed in key areas.The largest Reynolds number is encountered at the inlet of the sheets where the flow is constricted, as shown in figure 2(b).In this region, a hydraulic diameter of a square duct is used, with a value of 2 cm.The density of air is assumed to be 1.204 kg m −3 , and dynamic viscosity of 1.825 × 10 −5 for air 20 • C. The average velocity in the constriction is 2.9 m s −1 .This gives a Reynolds number of 3826.A critical Reynolds number of 2300 is generally accepted for internal flow in circular ducts [37], so the flow is expected to be in the transition region.As such, an appropriate turbulence model must be applied in the CFD analysis.

CFD part 2-3D modeling
The k-ϵ turbulence model is used for all 3D modeling discussed in the following sections.The k-ϵ model is commonly used for wind studies around buildings and dispersion flow problems using large external domains concerned with bulk flow characteristics [38][39][40][41][42].A diffusion transport model utilizing passive scalars was supplemented using Eulerian advection-diffusion equations.Outputs were evaluated once residual values fell below 1 × 10 −4 .The tools used in the 3D analysis are the pressure loss curve porous media model [36], momentum sources [43], and passive scalars [44].
The geometry shown in figure 1 is used in this section, applied to the 3D domain as shown in figure 3(a).Each contactor bank consists of a sorbent region, represented with a porous media, and fan region, represented with a momentum source, as shown in figure 3(b).The sorbent region is assumed to have a depth of 1.5 m as per Sabatino et al [45], and the fan region is assumed a depth of 1 m.The pressure drop curve obtained using the methods from part 1 is applied to the porous zone, utilizing the pressure loss curve model in Simscale.Momentum sources calculate the force required (i.e.pressure gradient), and apply it to meet an average flow velocity through the applied geometry.
A standard shipping container has dimensions 12.19 m × 2.6 m × 2.44 m (L × H × W), which was used to roughly determine the size of the modeled units.Commercially, a shipping container sized DAC unit may be comprised of multiple adjacent units.However, in the case of this analysis, units were modeled by one fan region, and one sorbent region, neglecting any gaps between adjacent units to simplify meshing.The focus of the current study was analyzing gross flow characteristics at a length scale of meters downstream.As a result, each contactor bank (two vertical rows of five DAC units) includes one fan region, and one sorbent region.A thin (0.3 m) container shell contains both the sorbent and fan region to direct the flow through the two regions.Three contactor banks are placed symmetrically in a triangular arrangement around the turbine base as shown in figure 3(a).
Passive scalars were used to model the dispersion of CO 2 depleted air exiting each DAC unit.Passive scalars are assumed to be transported in the flow, and do not actively change the flow physics of the simulation.Passive scalar transport can be described by the transport equation, also known as the convection-diffusion equation: where c is the scalar field, j is the total (convective + diffusive) flux of c through the boundary, S is the source or sink term inside the domain.The flux, j, is divided into two terms, the diffusive, and convective terms respectively in the full transport equation: where ∂c ∂t represents the time variation of the scalar quantity, D is the diffusivity (m 2 s −1 ), c is the transported scalar quantity, u is the velocity of the means which transports this quantity (m s −1 ), and S S and S R are the pure source term and the reaction source term respectively.S R is often neglected in engineering applications.
In Simscale, the value of c is tracked by the term T 1 , but to avoid confusion with temperature, c is used throughout this analysis.The variable c can have any unit assigned to it, as long as it is kept consistent in all aspects of model setup, and post processing.In this analysis, c is considered as a concentration, with units of (mg-CO 2 m −3 ), which can be converted to more understandable units of concentration such as ppm.An atmospheric concentration of 414.72 ppm CO 2 [46] corresponding to 758.55 mg m −3 .A sample calculation for this unit conversion is shown in the supplemental information.This value is assigned as a fixed value at the velocity inlet along the inlet and top of the domain, at the bottom BC as well as set as an initial condition in the entire domain.In simulation set-up, values of the diffusion coefficient (D) are defined, as well as the turbulent Schmidt number (Sc t ), representing the ratio between turbulent transport of momentum and the turbulent transport of mass.At 20 • C, D for CO 2 in air is 0.16 cm 2 s −1 [47], and Sc t is 0.7, based on the flow characteristic and is left at the default value.
Volumetric passive scalar sources (sinks) are assigned to the porous zone of each contactor.Passive scalar sources/sinks require a flux to be defined.In this case, a mass per volume flux (J v ) is defined in units of (mg (m 3 • s) −1 ).The flux is back calculated based on a target outlet concentration of the contactor.Based on the adsorption breakthrough curve shown by Gebald et al [34], and Bajamundi et al [48], CO 2 concentrations at the outlet of the contactor range from 10 to 20 ppm at the beginning of an adsorption cycle, and reach levels close to ambient toward the end of the cycle.An average value was interpreted from the figures of 150 ppm.In practice, adjacent DAC units would be in different phases of adsorption and desorption, however, a worst case scenario of all units in adsorption is used in this analysis.With all units in desorption mode, an outlet concentration is set homogeneously across the entire sorbent region, and is assigned a value representative of the beginning of an adsorption cycle.Similar values are reported for aqueous DAC systems, so this result could be applied to other systems as well.Keith et al [15] reports an air contactor outlet concentration of about 105 ppm CO 2 for aqueous based DAC process.
In order to calculate the concentration flux, one must understand the number of particles entering the contactor, and specify the amount that must be removed in order to achieve a desired outlet concentration.The amount of CO 2 particles (mg s −1 ) entering the contactor ( ṁenter ) is: where σ contactor is the cross sectional area of the contactor inlet, u contactor the velocity, and c the concentration entering.
The quantity of particles (mg s −1 ) removed by the volumetric passive scalar source ( ṁremoved ) is solved by re-arranging equation ( 8) The volumetric mass flux value (j v ), in units (mg (m 3 • s) −1 ) is solved using equation ( 9) Based on a contactor velocity (u contactor ) of 0.73 m s −1 , and an inlet area (σ contactor ) of 316.94 m 2 , a flux value of −188.58 mg (m 3 • s) −1 is applied to each volumetric source.A constant flux value was specified for all cases based on a design velocity, however, in reality, the flux would vary depending on actual flow velocity through contactors.Results were post processed in Paraview, and presented in units of ppm as it is more readily understandable as it relates to atmospheric concentrations.
Three nominal wind speeds are analyzed at a hub height of 150 m: cut-in (3 m s −1 ), rated (10.59 m s −1 ), and cut-out (25 m s −1 ).The turbine is assumed to have fixed mooring direction, with 0 • in the positive y-direction.This is expected to align with the predominant wind direction, however, as the wind direction varies, the turbine rotor will yaw about the tower to align with the wind direction, while the platform below remains fixed in place.Bidirectional fans are assumed to be used, so that fan direction will change to best align with the flow to minimize pushing air upstream.Wind angle is varied from 0 • to 180 • at 30 • increments.The fan direction with respect to the flow can be seen in figure 4 above.
It is hypothesized that at high wind speeds, upstream contactor banks may experience strong enough wind speeds to enable them to run passively, without turning fans on to increase velocity through the sorbent region.To test this hypothesis, a single bank of contactors, containing only a sorbent region is placed perpendicular to the incoming flow, and run at varying wind speeds from 3 to 25 m s −1 defined at hub height (150 m).The actual velocity experienced by the contactors is measured at contactor center height, 17.9 m.Velocity at the outlet of the sorbent region is measured to investigate whether sufficient volumetric air flow is met.An average outlet velocity of 0.73 m s −1 or greater is required to meet the volumetric flow rate to enable turning of the fans.As a result, the full scale simulation is modified accordingly, turning off fans where possible.

3D domain size
The domain size of external aerodynamic simulations can have a large impact on model accuracy, but also has a large impact on the number of cells in the mesh, and as a result computational effort.Abu-Zidan et al [49] aimed to optimize the domain size by comparing the results of many external simulations.They concluded that the previous recommendations by Franke et al [50] seemed overly conservative in most cases.Franke et al [50] recommends 5H upstream, 15H downstream, 5H in a lateral direction, and 6H a vertical direction.In this case, H refers to the tallest building height.Abu-Zidan et al [49] recommends 3H upstream, 3H downstream, 3H laterally, and 4H vertically.To remain conservative, the recommended domain size from Franke was used in this study, based on the height of the contactor units from the bottom of the domain.Future analysis could explore making the domain smaller, in line with Abu-zidan.
The domain dimensions were determined using a height (H) of 20 m from ocean surface to the top of the contactors.A distance of 15H was applied to one side as well as downstream, to account for the change in wind direction.Two domains/meshes were created, one for 0 • -90 • , and one for 91 • -180 • .For simulations from 0 • to 90 • , the domain has dimensions of 410 m × 410 m × 135 m in the x-y-z directions respectively.For simulations from 91 • -180 • , the domain has a size of 520 m × 506 m × 135 m in the x-y-z direction respectively.The domain dimensions are highlighted in figures 3(a) and S4 for 0 • -90 • , and 91 • -180 • respectively.The turbine tower base at water level is placed at the origin.The turbine tower extends far beyond the domain height, protruding through the top boundary.This may cause interference with the BCs, however, modeling based on this height will make the domain excessively large.The area of interest is at the bottom of the domain, where the CO 2 mixing predominantly occurs, and does not interact with the flow near the top of the domain.The turbine tower may cause the residuals to not decrease as low as if the domain was extended to fully encompass it.

3D mesh
A 3D mesh was generated using Simscale's automatic algorithm.This uses a mix of tetrahedral cells where necessary around key geometry, and hexahedral cells within the bulk flow.The global fineness was increased to seven out of ten.For simulations of the single contactor with porous region only, a mesh with 1.2M cells was generated.For the complete simulation, wind angles 0 • -90 • used a mesh with 8.3 million cells, and 8.2 million cells for angles 91 • -180 • .Mesh refinements were added in key areas to improve the resolution around flow features.Region refinements were implemented in cell zones and boundary layer (BL) inflation was implemented on all non-slip wall surfaces, discussed further in the following paragraphs.
Cell zones are used to group 3D regions of cells created in the CAD model.This allows properties to be applied to specified regions such.In this case, cell zones are used to define the porous media region, and the momentum source region.SimScale recommends having a minimum of five cells through the thickness of a cell zone, thus a region refinement was used to define a maximum edge length of 0.2 m inside the porous region and momentum source cell zones.
To achieve a horizontally homogeneous ABL profile with a sand grain roughness applied at the bottom of the domain, Blocken et al [39] suggests four basic requirements: (1) a sufficiently high mesh resolution in the vertical direction at the bottom face; (2) a homogeneous ABL upstream and downstream of the objects of interest; (3) the vertical distance from the center of the wall adjacent cell should be less than k s , and (4) k s and z 0 are related by equation ( 10) In this case, z 0 = 6.1 • 10 −3 m, equal to k s = 0.183 m.As a result the first layer cell height adjacent to the bottom plane needs to be greater than 0.366 m.This is obtained by adjusting the first layer height for the inflate BL refinement.

3D BCs
A variety of BCs are commonly used in literature.Typically six BCs are applied to the six sides of the rectangular domain: inlet, outlet, bottom, top, and two sides.Domain boundaries can be seen in figures 3(a) and (b).
A logarithmic ABL velocity profile is applied at the inlet.This resolves the expected wind velocity profile near earths surface, and is assumed to be fully developed over a sufficient upstream distance with the same ground roughness that is applied within the domain.This ensures that an internal BL profile does not develop within the domain.A wind angle of 0 • is in the negative y-direction.Wind angle increases in a clockwise rotation.This is shown in figures 3(a) and (b).The ABL profile follows the form of equations ( 11)-( 14) as per Blocken et al [39].An ABL profile in the domain was assured by running a simulation with an empty domain, shown in figure S12 of the supplemental information where u * is the wall function friction velocity (m s −1 ), u is the velocity (m s −1 ), u ref is the reference free stream velocity (m s −1 ) at reference height z ref (z ref = 150 m), K is the Von Karman constant equal to 0.41, z 0 is the aerodynamic roughness length (z 0 = 6.1 × 10 −3 m), z is the height at which the velocity is calculated, k is the turbulent kinetic energy (m 2 s −2 ), ϵ is the rate of dissipation of turbulent kinetic energy (m 2 s −2 ), and c µ is the turbulent viscosity constant equal to 0.09.The variables u, k, and ϵ are entered directly in CFD as a custom BC, applying a fixed value then inputting the above equations, while leaving pressure as zero gradient.
The bottom of the domain is defined by the no slip wall function with a sand grain roughness height (k s ).The relationship between z 0 and k s is given by equation ( 10) as per Blocken et al [39].Golbazi and Archer [51] recommend a median surface roughness height of 6.1 × 10 −3 , representative of a rough sea state.This corresponds to k s = 0.183 m.First layer height of the mesh along the bottom boundary is imperative to achieving a horizontally homogeneous BL as discussed in the previous section.
The outlet of the domain commonly has different BCs applied.In some studies [40,41], the normal gradients of all variables are set to zero, whereas in others [38,49] the static pressure, or in other words the gauge pressure is set to 0.Here the outlet pressure is set to the free stream value, and all other parameters set to zero-gradient.Setting zero gauge pressure across the outlet of a large external domain caused issues as it induces a small artificial pressure gradient.Applying the software solved free stream pressure at the outlet face effectively gets rid of any artificial pressure gradient caused by applying a strict zero gauge pressure.
To simulate different wind directions, unit vectors were applied to the velocity inlet BCs.Instead of using periodic side BCs, identical velocity inlet BC's were applied to two adjacent faces, and the free stream outlet BC applied on the opposite adjacent faces.Effectively this allows flow in and out through all four vertical faces of the domain.

Energy model
The minimum thermodynamic work is calculated to obtain a baseline of minimum plant wide energy demand based on inlet CO 2 concentration.Computing the real work based on a second law efficiency includes both electrical and thermal energy requirements, and is used as a baseline comparison.However, the work computed in this manner is only a function of the inlet inlet CO 2 concentration, and does not account for ambient wind conditions.The ambient conditions assumed are summarized in table S4 in the supplemental information.

Minimum thermodynamic work and second law efficiency
The minimum thermodynamic work (J (mol-CO 2 ) −1 ), wrev , for separating gas species can be computed as per Struchtrup [52]: where R is the universal gas constant, T is the ambient temperature, α is the separation efficiency, and X is the initial CO 2 mole fraction.Remaining air has CO 2 content αX.
In 2021, the global average concentration of CO 2 in the atmosphere was 414.72 ppm [46].In the results section to follow, a hypothetical worst case of inlet CO 2 concentration scenario are analyzed,as well as two middle scenarios.In the worst case, an upstream contactors outlet is directly connected to a downstream contactors inlet giving an inlet concentration of roughly 100 ppm.In the middle scenarios, 200 ppm and 300 ppm CO 2 , downstream contactors are placed some distance from upstream units and the downstream units experiences some decreased level of concentration between 100 and 414.72 ppm.These outlet concentrations are described previously in section 2.3.
An estimation of the actual work( w) can be computed using the second law thermodynamic efficiency (η II ), described by |w| = | wrev| ηII .The second law efficiency is computed through detailed analysis of specific plants.Long-Innes and Struchtrup [53] compute a second law efficiency of 7.8% using aqueous based DAC.SS systems are believed to fall within this range as per the National Academy of Sciences report on negative emissions [54], with middle-range scenarios between 7.6% and 11.4%.In this analysis, η II = 7.6% is assumed as a conservative estimate.

Electrical energy
The electrical energy consumption of the plant is composed of: the fan/blower to push air through the contactor over the adsorbent material, the vacuum pump to evacuate oxygen from the air contactor before heat is applied to avoid sorbent degradation, and the CO 2 compression to bring output CO 2 a desired pressure.The energy for the vacuum pump and CO 2 compression is based on design choices and does not vary greatly as a result of ambient conditions.This is assumed to be constant based on the designed setup, thus is not analyzed further in this study.
The fan power is a function of the pressure drop, and the volumetric flow rate through the contactor.As a result, the fan power is dependent on the ambient conditions entering the contactor.Conducting unit analysis along with laws of force, momentum, and power, an expression can be derived for the work from each fan.The force exerted on the flow represented by equation ( 16), and then the power required by equation ( 17) where S x ( kg m 2 •s 2 ) represents a direct momentum source term in the RANS momentum equations in the direction of the flow, V (m 3 ) is the volume of the momentum source, and u x (m s −1 )) is the normal velocity in the direction of flow.In this case, the average value of u x is obtained by integrating the normal flow velocity about the surface area the normal flow through the inlet area to each fan.
The momentum source used in the CFD model solves for the momentum source strength to meet a prescribed velocity.At each iteration, the pressure gradient applied to the momentum source volume is outputted as a result.The pressure difference, ∆p, experienced through the contactor, can be solved by ∆p = ∇pd, where ∇p is the pressure gradient, and d is the distance in the direction of flow.This can be used to compute the power consumed by the fan.Equation ( 17) is simplified to be used with the output from the CFD results as equation (18).From the CFD result, the area integral applied to the cross section perpendicular to the flow direction of the normal velocity across the fan inlet plane is taken to obtain the volumetric flow rate.A fan efficiency of 60% is assumed as per Sabatino et al [13] where η fan is the fan efficiency, ∆p is the pressure difference (Pa), d is the distance in the flow direction (m), and Vair is the volumetric flow rate (m 3 s −1 ).

Thermal energy-isotherm method
Thermal energy is required to strip the CO 2 molecules from the capture material.Typically this is met by passing steam through the module.The total equilibrium thermal energy (Q eq,th ) for desorption is comprised of the sensible (Q sensible ) and reaction heat (Q reaction ).It is described in equation (19).The sensible and reaction heat is further broken down into the components for CO 2 , H 2 O and the sorbent.The sorbent chosen for analysis in this study is the APDES-NFC sorbent, which has seen significant experimental work, and includes isotherms for CO 2 and H 2 O, in both dry and humid conditions The reaction heat for CO 2 is shown in equation (20), and for H 2 O in equation (21).The sorbents reaction heat is zero where ∆H iso is the isosteric heat of adsorption, ∆q is the equilibrium adsorption capacity, and M is the molar mass of each species.
The isosteric heat of adsorption can be solved using the Van't Hoff equation, which is a function of the temperature, and gas partial pressures during adsorption and desorption.This relationship is discussed in equation (42).
The equilibrium adsorption capacity, ∆q CO2 , and ∆q H2O describe the difference in the quantity of species adsorbed during the adsorption phase, q ads , and the quantity of species re-adsorbed during the desorption phase, q des : ∆q = q ads − q des .( Water adsorption, and re-adsorption during desorption phase, is described by the Guggenhein-Anderson de Boer isotherm model, also with parameters fit using experimental data obtained from literature summarized in table S5 in the supplemental information.Adsorption of CO 2 , and re-adsorption of CO 2 during the desorption phase, is described by the modified Toth isotherm equation, with parameters fit using experimental data from literature summarized in table S6 in the supplemental information.
The concentrations/partial pressures during desorption can be computed assuming all CO 2 and H 2 O adsorbed is desorbed in the subsequent phase.The partial pressures can be computed using Dalton's law of partial pressures.This allows the computation of the desorption capacities, q H2O,des and q CO2,des , and further the equilibrium loading capacities, ∆q H2O and ∆q CO2 .
The sensible heat for CO 2 is shown in equation ( 23), for H 2 O in equation ( 24), and for the sorbent in equation ( 25) Here c p is the molar specific heat capacity.The values for the specific heat capacity of CO 2 , c p,CO2 are interpolated from values in the NIST webbook [55].The c p,H2O is comprised of liquid and vapor phases, and values are interpolated from Struchtrup [52] for the vapor phase, and the engineering toolbox for the liquid phase [56].The c p,sorbent for APDES-NFC used the value from Wurzbacher et al [23].

Results and discussion
The results obtained from the CFD model are fed into the energy model.CFD results are shown in section 3.1, and the energy modeling results are shown in section 3.2.Further analysis, including the full scale model to explore the impacts of CO 2 mixing, and fan work were completed based on the sorbent APDES-NFC, as this sorbent has received significant attention in literature, as well as likely been commercially deployed.

2D CFD pressure drop simulations
This section produced a pressure loss curve for a contactor with realistic dimensions for commercial scale.In order to do this, the porous media was first verified by comparing the CFD results from a 2D vertical sheet, and the calculated pressure loss from the Ergun equation.The results matched very closely, summarized in table 1.
Next, the fixed coefficient model was applied to the configuration in the patent by Gebald et al [34] discussed earlier, to verify that the modeled geometry and sorbent properties, produced similar pressure drops to the pressure loss curve given in the patent.The modeled results closely matched the pressure loss curve, indicating the model accurately described the configuration used in the patent.The results are summarized in table 1.At low velocity, the largest error was found, with a 51% difference, however, the velocity of interest showed a decent correlation with an error of 6.1%.The CFD model uses periodic BCs on the side walls, and symmetry on the top and bottom and thus ignores the pressure loss associated with the friction from the wall at the sides and top of the contactor.This is likely a source of deviation between the results, and thus some large errors associated with it.This is more significant at lower velocities, where friction losses make up a larger portion of the overall pressure loss through the system.This verification provided good confidence that the model was accurately describing a realistic air contactor, and was then scaled up to a length of 1.5 m.The pressure loss data for APDES-NFC, Tri-PE-MCM, MIL-101(Cr)-PEI-800, and Lewatit VP OC 106 is shown in table 2. Only the sorbent APDES-NFC was considered for the following 3D simulations.
The flow through the sorbent sheets follows a typical jet pattern as expected.Air enters the domain on the left, is accelerated as it enters the flow channels/constrictions, diffuses through the porous material, and accelerates again through the outlet.This behavior can be observed in figure 5(a) for the 0.5 m contactor length simulation.A non-symmetric flow pattern is encountered at the outlet, which could be due to many reasons.The turbulence encountered in the wake will have oscillations of a certain frequency, which may cause the jet to favor one side or another.
It can be seen in figure 5(b) that the most of the flow diffuses through the porous zone only at the ends of the channel.This means that majority of the sorbent is not having air convectively passed over it, and thus is relies purely on diffusion to adsorb CO 2 .This can be understood from looking at the transport equation from earlier, equation (6).The two terms of interest are the diffusive flux, ∇ • (D∇c), and the convective flux, ∇ • (uc).At 20 • C, D for CO 2 in air is 0.16 cm 2 s −1 [47].The concentration gradient (∇c) from the flow channel to the adjacent particles is also expected to be relatively small.This results in a small change of overall concentration as a result of diffusive transport.With zero velocity through the sheet, the convective term is zero.The present 2D modeling does not account for CO 2 adsorption, and this could be explored in future work to gain a better understanding of the sorbent usage efficiency.
This suggests that alternate sorbent sheet configurations should be considered to maximum convective air flow past all particles in the sheet.If only the ends are coming into contact with the air, they will lose their adsorption capacity much quicker than the rest of the sheet, and will require replacement earlier.The whole sheet would likely be replaced at once, resulting in sorbent usage less than its full capacity.Sorbents are known to contribute the largest cost to DAC installations as they typically need replacement every 0.5-2 years.The uniform pressure buildup through the contactor length can be seen in figure 5(c).

3D CFD-wind driven velocity increase
The velocity throughput results of the single contactor bank with a porous region only placed perpendicular to incoming wind flow can be seen in figure 6.
It is clear from figure 6 that the desired output velocity of 0.73 m s −1 is not reached until an input velocity of roughly 17.48 m s −1 at contactor center height (24 m s −1 at hub height).With a cut-out wind speed of 25 m s −1 , majority of the operating zone of the turbine will require additional fan work.Above 24 m s −1 , the unit should still pass the required volume of air through the contactor to meet the desired capture rate, thus fans could be turned off, and the energy required to power the fans could be saved.In reality, this finding highlights the need for further air contactor refinement to minimize pressure drop, especially at low velocities.Doing so would allow the contactor to act passively in more of the wind turbines operational range.Air contactors placed at ground level do not reach the passive flow range as shown in figure S6 in the supplemental information.
Based on this conclusion, certain fans were turned off based on the velocity inlet angle when run at the cut-out wind speed of 25 m s −1 .The upstream fans are assumed to be turned off, whereas the downstream fans are left on as the upstream fans disrupt the local wind speed entering downstream units below the point  at which they meet the required flow velocity.This is shown in figure S5 within the supplemental information.

3D CFD-concentration entering contactor
The local mixing of CO 2 at cut-in wind speed (3 m s −1 ) can be seen on a cutting plane from above through the center of the contactor bank in figure 7. The cutting planes can be seen in figure S7 in the supplemental information.A wind angle of 0 • is shown in figure 7(a), 90 • in figure 7(b), and 180 • in figure 7(c).
Average CO 2 concentration entering downstream contactors can reach levels as low as about 280 ppm.This occurs in the case with a 3 m s −1 at 30 • .As wind speed increases, the concentration entering  downstream units also increases.At higher wind speeds, more CO 2 is brought into the domain, mixing with the depleted region at the contactor outlet.A top view plane of 0 • , 90 • , and 180 • wind angles at cut-in, rated, and cut-out wind speeds are shown in figures S9, S10 and S11 respectively in the supplemental information.This clearly shows the relationship of increasing CO 2 dispersion with increasing wind speed.Increased CO 2 dispersion means higher concentrations entering downstream units, thus decreased thermal energy demand.
A side view of CO 2 concentration leaving one contactor, and entering a downstream contactor is shown in figure 8. Majority of the CO 2 depleted air leaving the upstream contactor does not have CO 2 diffuse vertically into it from the surrounding air.Note that the upstream contactor is cut on an angle, whereas the downstream contactor is cut perpendicular to the flow direction.

Energy modeling results
From the CFD model, the pressure drop curve, discussed in section 3.1.1,and the wind driven velocity through the contactor, discussed in section 3.1.2,effect the energy consumed by the fan.The CO 2 dispersion, discussed in section 3.1.3,effects the thermal energy requirement.

Second law efficiency work
With η II = 7.6%, the real work required for 100 ppm, 200 ppm, 300 ppm, and 414.72 ppm is shown in figure 9.
This change in energy is non-negligible, especially as DAC is typically considered to be deployed at very large scales.Commercial DAC plants have been proposed on the order of 1 Mt-CO 2 y −1 .Under a worst case scenario with CO 2 concentration of 100 ppm entering downstream units, an additional 287.9 × 10 6 kWh (t-CO 2 ) −1 is required, an increase of 18% from 400 ppm.This is not practical as one would not expect to install units in this manner.However, some level of CO 2 depletion is expected to enter enter downstream units-assume 300 ppm for arguments sake.This results in an additional 65.6 × 10 6 kWh (t-CO 2 ) −1 (4% increase).Studies suggest DAC deployment in the multiple Gt-CO 2 y −1 level by about mid century, potentially reaching 10-20 Gt-CO 2 y −1 by the centuries end [54].At this level, the additional energy requirement can certainly not be ignored, especially for a technology whose biggest criticisms lies in its high energy demands compared to other decarbonization approaches.

Electrical energy-fans
The total fan power for all wind speeds and directions is shown in figure 10.A hypothetical fan power for zero velocity is shown, where the average contactor velocity is 0.73 m s −1 through an area of 316.9 m 2 , with a pressure drop of 391 Pa as per table 2. This results in a fan power of 455 kW.It is clear from the figure that there is a decrease in overall fan power at all three wind speeds.There is a slight increase in total power required from 3 m s −1 to 10.59 m s −1 , but this is likely due to negative pressures at the contactor outlet due to the contactor wake region.As a general trend, there is a drop in power requirement with increasing wind speed.At 25 m s −1 , upstream fans can be turned off as per figure 6, allowing the contactor to passively contact with air.Downstream contactors are kept on as they are within the wake region of the upstream contactors, thus will not be at the required inlet velocity.
Further details for contactor banks 1-3 can be seen in figures 11(a)-(c).At high wind speeds, wind angles perpendicular to the upstream contactors allow the fans to be turned off, and the minimum overall power is consumed.This becomes a control problem where based on the wind direction and speed, fans can be ramped up and down appropriately.

Thermal energy-isotherm method
The thermal energy breakdown computed using the isotherm data is presented in figure 12.
It can be seen that the sorbent sensible heat comprises majority of the thermal requirement of the plant (41%-43%).This aligns with values stated in literature using SSs [11,13].The mass of sorbent required for capture, as well as its specific heat capacity have a large impact on the overall thermal energy requirement.Minimizing the mass of sorbent required as well as specific heat capacity are areas of major focus as the next generation of SSs are developed.Materials with high affinity for CO 2 allow for less sorbent to be used as a result.Recovery of sensible heat is discussed by Bos et al [11].Recovery of sensible heat in fixed bed operation is difficult, and can be done by recovering heat from a hot bed at the end of a desorption cycle, and a cold bed at the beginning of a desorption cycle.Gebald et al [58] present an invention to recover sensible heat using a glycol loop and external thermal energy storage in a stratified tank.In their particular case, thermal energy demand decreases from 3334 kWh (t-CO 2 ) −1 to 2245 kWh (t-CO 2 ) −1 by recovering sensible heat.
The real thermodynamic work computed in section 3.2.1 is based on total plant energy which includes both electrical and thermal energy.Electrical energy does not vary greatly with a change in concentration as it is comprised of fan energy, and compression energy which is more a related to mass flow.As such, a comparison can be drawn between the two methods.Computed using the real work method, there was a  change of energy from 414.72 ppm to 300 ppm of 63.9 kWh (t-CO 2 ) −1 .Using the isotherm method, there is a change in thermal energy of 25.9 kWh (t-CO 2 ) −1 .Although the thermal energy requirement is higher when computed using the isotherm method than with the second law efficiency approach, the values are still well within the bounds presented in The National Academy of Sciences report on negative emissions [54].It is outside of the middle range, but within the full range of η II = 2%-24%.The higher energy values are likely a result of not accounting for thermal heat recovery and varying isotherm parameters by using different sorbents.

Conclusions
The results show that there is a clear impact on overall DAC plant energy consumption based on ambient wind conditions.Increased wind speeds decrease fan power requirement, as well as induce increased CO 2 mixing near air contactors.As a result of increased CO 2 mixing, plant wide thermal energy requirement is decreased.Wind approach angle also effects the fan power requirement and local CO 2 dispersion.A decreased fan energy requirement is observed.When the wind angle is perpendicular to upstream contactors.This is particularly dominant at high wind speeds, where upstream contactors can be shut off entirely at wind speeds above 24 m s −1 at hub height (150 m) correlating to an average contactor bank inlet velocity of 17.5 m s −1 .
Using CFD, pressure loss curves for a commercially deployed air contactor design are generated for four sorbents of interest, which can be used to improve future energy analysis of DAC.A pressure loss of 391 Pa was found for a flow velocity of 0.73 m s −1 through sheets of APDES-NFC sorbent arranged in an accordion manner with a length of 1.5 m.Breakdown of thermal energy requirement for sorbent APDES-NFC at varying CO2 inlet concentrations using isotherm method.Relative humidity of 66% assumed [57].Operating conditions reported in table S4 within the supplemental information.Further iteration of contactor layouts could be completed in future work to better understand impacts from alternate configurations.Iterating on the current methodology to include alternate arrangements, as well as simulating various phasing of adsorption/desorption cycles would give a broader insight in future studies.Ultimately, the phasing and operation of the plant could be optimized based on the power available at different wind speeds.
DAC is an energy intensive process, so all efforts should be made to optimize energy use, and plants should be carefully designed to allow for maximum CO 2 dispersion between air contactors.

Figure 1 .
Figure 1.DAC air contactor configuration mounted on the platform of a 15 MW floating offshore wind turbine.Sorbent zone shown in blue, and the fan zone shown in green.Contactor cross section can be seen in the appendix.

Figure 2 .
Figure 2. (a) Cross section of a stack of sorbent sheets used in a solid sorbent air contactor.(b) 2D cross section of a simplified geometry representing a DAC air contactor.

Figure 4 .
Figure 4. Fan directions with respect to incoming wind direction.

Figure 5 .
Figure 5. 2D CFD result of contactor with length of 0.5 m, sorbent sheet thickness of 1 cm, and air flow channels of 1 cm.Shown is the (a) velocity magnitude (m s −1 ), (b) velocity in the vertical direction (m s −1 ), and (c) the uniform pressure buildup through the reactor (Pa).

Figure 6 .
Figure 6.Porous zone output velocity as a function of wind speed with a contactor bank mounted 15 m above ground/water.Wind speed is measured using an upstream probe point at a height of 17.9 m, the center of the contactor bank.Target throughput velocity is 0.73 m s −1 , shown with the horizontal yellow line.

Figure 7 .
Figure 7. CFD result showing dispersion of CO2 in domain at cut in wind speed (3 m s −1 ) with inflow angle of wind angle (a) 0 • , (b) 90 • , and (c) 180 • .Wind direction shown with yellow arrow.Shown projected on the top plane (x-y plane) through the center of the air contactor bank, at height z = 17.9 m.Cut plane A-A shown in figure 7(a), and further cut planes are shown in figures S7 and S8 in the supplemental information.

Figure 8 .
Figure 8. CO2 mixing shown on a side view orientation.Cutting plane placed 15 m in the x-direction, parallel with the y-z plane.Cutting plane shown as A-A in figure 7(a) Wind enters at 3 m s −1 with an angle of 0 • (negative y-direction).

Figure 10 .
Figure 10.Total fan power (kW) for all three contactor banks at zero (theoretical), cut-in, rated, and cut-out wind speeds.

Figure 12 .
Figure12.Breakdown of thermal energy requirement for sorbent APDES-NFC at varying CO2 inlet concentrations using isotherm method.Relative humidity of 66% assumed[57].Operating conditions reported in table S4 within the supplemental information.

Table 2 .
Pressure loss data obtained by 2D modeling a 1.5 m long contactor, 1 cm thick sorbent sheets, and 1 cm flow channels.