The cost of capturing carbon dioxide (CO₂) from ambient air must be significantly reduced for Direct Air Capture (DAC) to be a viable climate change solution. This study examines ion-exchange resins with quaternary ammonium cations for moisture-driven DAC, potentially more economical than temperature swing methods. The Moisture-Temperature-Vacuum Swing Adsorption (MTVSA) process model considers temperature, relative humidity, and wind speed to evaluate their effects on sorbent CO₂ loading, cycle time, and net water loss. MTVSA performs well at lower humidity levels. Meteorological data from National Oceanic and Atmospheric Administration (NOAA) inform site selection for the MTVSA model. The sorbent captures CO₂ from ambient air when dry and releases CO₂ upon wetting. Two isotherm models, a modified Langmuir isotherm for CO₂ loading and the Flory-Huggins model for water loading, are used to estimate CO₂ capture and water loss at varying temperatures, relative humidity, and wind speed. The CO₂ capture model predicts higher efficiency during low relative humidity periods along with increased net water consumption. The Flory-Huggins model helps assess net water consumption, which varies with ambient conditions. Combining CO2 productivity with net water loss enables future economic analysis to determine carbon capture costs.
Research Article
Variability Analysis of Moisture Swing Sorbents in Passive Direct Air Capture Systems
https://doi.org/10.21203/rs.3.rs-7782108/v1
This work is licensed under a CC BY 4.0 License
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Direct Air Capture
Isotherm Model
Moisture Swing
Flory-Huggins
Langmuir
Reaction Kinetics
The Intergovernmental Panel on Climate Change (IPCC) has underscored the escalating challenge posed by greenhouse gas (GHG) emissions, primarily carbon dioxide (CO2), stemming from anthropogenic activities. So far emissions have resulted in a global temperature rise of 1.2°C above pre-industrial levels, with an anomaly reaching temperatures at 1.66oC above in 2023, and a projected increase of up to 4°C by 2100 if CO2 emissions are not curbed1–3.
Amidst this climate crisis, direct air capture (DAC) emerges as a pivotal carbon dioxide removal (CDR) technology. DAC offers a pathway to extract CO2 directly from ambient air, producing a concentrated CO2 stream suitable for storage 4,5.
In contrast to DAC technologies that rely on mechanical systems such as fans or blowers to facilitate air contact with a CO2 sorbent material, passive DAC relies on wind for air contact with minimal or no external energy input for capture 6. Relative to fan-driven contactors, the dominant energy input in passive systems shifts from air handling to regeneration; for amine-functionalized sorbents, regeneration can proceed at ≤ 100°C, enabling the use of low-grade heat, and steam-assisted regeneration can further reduce electricity requirements 7–9. The lower energy requirements coupled with modular units allow passive DAC units to be sited near CO2 storage/utilization sites or where low-grade heat is abundant, positioning it as a promising approach for cost-effective and energy-efficient CO₂ capture 4,5.
Passive DAC can be regenerated by temperature swing sorption or moisture swing sorption. Specifically, temperature swing sorption methods require high energy consumption during CO2 release due to elevated temperatures and reduced pressure conditions 10,11. However, the novel concept of moisture swing sorbents, as first proposed by Lackner 12, offers a promising avenue for reducing energy consumption by capitalizing on the free energy of water evaporation—an energy resource more economical than the heat required for temperature swing regeneration 13,14. Scaling DAC technologies economically hinges on innovative device design, advanced sorbents with improved sorption capacity, lower regeneration energy requirements, and a deeper process understanding facilitated by modeling 10,15,16.
This study focuses on moisture swing sorbents and constructs a mathematical model that estimates CO2 productivity as a function of ambient weather parameters. This is a vital step towards the realization of a commercial-scale passive DAC technology, capable of capturing, separating, and processing ambient CO2 into pipeline-ready form. Notably, this work represents a novel meteorology-driven screening model for a moisture swing passive DAC that maps ambient weather parameters to CO2 productivity and water-loss rate at a representative DAC site. The study's core objective is to unravel the dynamic behavior of a DAC system under varying ambient conditions by formulating a mathematical model to discern CO2 capacity fluctuations within a moisture-swing passive DAC system.
2.1 Process and Objective
In Passive DAC, the ambient weather conditions, primarily temperature (T), relative humidity (RH), and wind speed (\(\:{\varvec{v}}_{\varvec{w}\varvec{i}\varvec{n}\varvec{d}}\)), play a key role in determining the carbon capture performance of the sorbent. This study involves analysis of weather variability on the carbon capture process. The acquired weather data from the National Oceanic and Atmospheric Administration (NOAA) underwent preprocessing, including handling missing values, converting data types (e.g., datetime, float), and interpolating gaps, ensuring suitability for integration into a mathematical model 17. The moisture swing model developed serves as a robust tool for calculating both the CO2 productivity and water loss from the sorbent during a single sorbent cycle.
The moisture swing sorbent on which our modeling is based is Excellion I-200, a mixed-matrix quaternary ammonium ion exchange resin fabricated by SnowPure LLC, California 18. To approximately represent this sorbent's behavior, two fundamental isotherm models are utilized: a modified Langmuir isotherm for equilibrium loading with CO2 and the Flory-Huggins equation for equilibrium loading with water. It is essential to recognize that these isotherm models serve as a preliminary decoupled proxy, enabling initial performance modeling in the absence of a rigorous co-adsorption model required for precise prediction of the ion exchange resin's adsorption behavior. The primary objective of this study is to develop an initial modeling framework to understand the performance variability of moisture swing DAC sorbents influenced by ambient weather parameters. This model is designed with sufficient flexibility to allow future integration of a more accurate co-adsorption model that better represents the actual behavior of the sorbent.
Temperature effects are critical to the modeling approach. By assuming a specific temperature dependence of the equilibrium constant \(\:{\varvec{K}}^{\varvec{{\prime\:}}}\), one can incorporate a temperature dependence into the Langmuir model. By contrast, the Flory-Huggins equation, expressed in terms of relative humidity, already incorporates temperature effects through the temperature dependence of the water vapor-liquid equilibrium. These two models, operating synergistically, allow us to comprehensively characterize the sorbent's behavior under varying conditions.
Our analysis extends beyond static modeling and encompasses the dynamic sorbent cycle, consisting of capture and regeneration phases. Throughout this cycle, CO₂ loading and unloading rates are governed by a reduced-order linear-driving-force (LDF) rate expression that drives q toward the stage equilibrium (\(\:{\varvec{q}}^{\varvec{*}}\)). In the capture stage, the lumped mass-transfer rate constant \(\:{\varvec{k}}_{\varvec{r}\varvec{a}\varvec{t}\varvec{e}}\) is parameterized directly by ambient wind speed and temperature (with relative humidity entering primarily through \(\:{\varvec{q}}^{\varvec{*}}\)), enabling us to relate meteorology to time-per-cycle and hence productivity. For water, the Flory-Huggins equilibrium model is used to estimate the equilibrium swing between regeneration and capture conditions, from which net water loss per cycle is computed given the stage durations.
Detailed adsorption models for DAC often (i) use activity-based co-adsorption thermodynamics that explicitly couple CO₂ and H₂O at the surface 7,13,19, or (ii) employ fixed-bed partial differential equation (PDE) models with co-adsorption isotherms and multi-parameter kinetic closures 8,9,19. These approaches are powerful for device-scale design but require extensive sorbent- and device-specific parameters (site densities, heats, activity coefficients, fitted co-adsorption parameters, axial dispersion/heat transfer, wall heat transfer, pressure drop, etc.) and typically hinge on local CO₂/H₂O concentrations at the surface. In contrast, our intent here is complementary: a parsimonious, meteorology-driven screening model that maps long-run temperature, relative humidity, and ambient wind speed to CO₂ productivity and water-loss rate across sites using NOAA data. The chosen decoupled proxy (modified Langmuir for CO₂; Flory-Huggins for H₂O) minimizes parameter burden while retaining a path to replace the proxies with co-adsorption forms when detailed inputs become available. This enables geospatial assessment and site triage without committing to device-specific PDE detail.
Cycle scope: Fig. 1 summarizes the Moisture–Temperature–Vacuum Swing Adsorption (MTVSA) cycle. The cycle comprises: (i) capture (ambient contact), (ii) isolate, (iii) pre-evacuate, (iv) steam regeneration, (v) condense & product take-off, and (vi) re-expose.
In this paper we explicitly model only the (i) capture stage and (iv) steam-regeneration stage, because these are the stages whose kinetics and equilibria depend on ambient weather conditions (T, RH, \(\:{\varvec{v}}_{\varvec{w}\varvec{i}\varvec{n}\varvec{d}}\)). The remaining steps—(ii) isolate, (iii) pre-evacuate, (v) condense & product take-off, and (vi) re-expose—are treated as weather-agnostic logistics. We fix their setpoints/conditions (e.g., \(\:{\varvec{t}}_{\varvec{r}\varvec{e}\varvec{g}}\) = 1200s,\(\:\:{\varvec{T}}_{\varvec{r}\varvec{e}\varvec{g}}={50}^{\varvec{o}}\varvec{C}\), \(\:\varvec{R}{\varvec{H}}_{\varvec{r}\varvec{e}\varvec{g}}=95\varvec{\%}\), and \(\:{\varvec{p}}_{\varvec{c}{\varvec{o}}_{2}\:\varvec{r}\varvec{e}\varvec{g}}=600\:\varvec{P}\varvec{a}\), as specified) and do not co-optimize them. The variability analysis presented here isolates how ambient T, RH, and \(\:{\varvec{v}}_{\varvec{w}\varvec{i}\varvec{n}\varvec{d}}\) control CO2 productivity and water-loss rate; fixed thermal energy demands for vacuum generation, evacuation, condensation, and product recovery are outside the present scope. During regeneration we supply humidified steam to 95% RH at 50°C (exogenous low-pressure steam or recirculated vapor). The CO₂/H₂O mixture is subsequently cooled so that water is condensed and returned, and CO₂ is withdrawn; the energy for these logistics is outside our scope. We do not assume autothermal steam generation by sorption; rather, humidification/steam is provided externally (or by recycle). ‘Free energy of sorption’ is used in the thermodynamic sense to explain why moisture promotes desorption, not to imply zero external heat.
The notional device is a planar, open-cell passive contactor exposed to free-stream wind and operated in batch mode, alternating ambient capture with steam-humid regeneration. During capture, CO₂ uptake follows a reduced-order LDF kinetics closure with a lumped rate constant that depends on T, RH, and \(\:{\varvec{v}}_{\varvec{w}\varvec{i}\varvec{n}\varvec{d}}\); we select the capture duration via an approach-to-equilibrium optimizer (ψ) to maximize per-cycle CO2 productivity (see Supplemental Information, S3). When the optimized capture state is reached, the module is isolated, pre-evacuated, and transferred to a regeneration chamber where moisture, mild heat, and low CO2 partial pressure (50°C, 95% RH, \(\:{p}_{c{o}_{2}\:reg}=600\:Pa\), \(\:{t}_{reg}\) = 1200s) drive desorption of CO₂ before the module is re-exposed to ambient.
2.2 CO2 Isotherm Model
For moisture-swing DAC sorbents, Wang et al. (2013) proposed a linear relationship between the Gibbs free energy (\(\:\varDelta\:{\varvec{G}}_{\varvec{o}}\)) of adsorption of CO2 and the relative humidity in fraction (\(\:{\varvec{h}}_{\varvec{r}}\)) 20:
$$\:\varvec{\Delta\:}{\varvec{G}}_{\varvec{o}}=\varvec{a}+\varvec{b}\left(1+\varvec{\beta\:}{\varvec{h}}_{\varvec{r}}\right)\varvec{*}{\varvec{h}}_{\varvec{r}}+\varvec{c}\varvec{*}(\varvec{T}-{\varvec{T}}_{\varvec{o}})$$
1
Here, the parameters \(\:\varvec{a}\), \(\:\varvec{b}\), and\(\:\:\varvec{c}\) are obtained by fitting the model to experimental data, with values of -32.2 kJ/mol, 15.45 kJ/mol, and 0.02564 kJ/mol-K, respectively 20. \(\:{\varvec{T}}_{\varvec{o}}\) represent the average temperature maintained during the experiments in the study by Wang et al. (2013) as 288.15 K, \(\:\varvec{\beta\:}\)=0 at \(\:{\varvec{T}}_{\varvec{o}}=288.15\:\varvec{K},\) and T denotes the ambient temperature in Kelvin. Following Wang et al. (2013), β is defined such that at the reference temperature \(\:{\varvec{T}}_{\varvec{o}}\) the humidity-coupling correction vanishes (β = 0), and departures from \(\:{\varvec{T}}_{\varvec{o}}\) are captured by the \(\:\varvec{c}(\varvec{T}-{\varvec{T}}_{\varvec{o}})\) term. Given the small magnitude of the fitted coefficient c, the model predicts only a weak dependence of CO₂ adsorption on temperature.
The apparent equilibrium constant (\(\:\varvec{K}\varvec{{\prime\:}}\)), which provides the sorbent affinity for CO2 at fixed water vapor pressure, can be derived from \(\:\varDelta\:{\varvec{G}}_{\varvec{o}}\) as:
$$\:{\varvec{K}}^{\varvec{{\prime\:}}}=\mathbf{e}\mathbf{x}\mathbf{p}(-\frac{\varvec{\Delta\:}{\varvec{G}}_{\varvec{o}}}{\varvec{R}\varvec{T}})$$
2
Illustrative calculation: at \(\:T=298\:K\) and \(\:{h}_{r}=0.3\), Eq. (1) gives \(\:{\Delta\:}{G}^{o}\approx\:-27.3\:kJ\:mo{l}^{-1}\); Eq. (2) then yields \(\:{K}^{{\prime\:}}\approx\:6*{10}^{4}\:P{a}^{-1}\).
Here, \(\:\varvec{R}\) corresponds to the universal gas constant, which is 8.314*10− 3 kJ/mol-K.
To determine the CO2 sorbent saturation (\(\:\varvec{\theta\:}\)), the modified Langmuir isotherm model is applied,
$$\:\varvec{\theta\:}=\frac{{\varvec{K}}^{\varvec{{\prime\:}}}{\varvec{p}}_{\varvec{C}{\varvec{O}}_{2}}}{1+{\varvec{K}}^{\varvec{{\prime\:}}}{\varvec{p}}_{\varvec{C}{\varvec{O}}_{2}}}$$
3
In the above equation, \(\:{\varvec{p}}_{\varvec{C}{\varvec{O}}_{2}}\:\)represents the relative partial pressure of CO2 against a reference pressure of 1 atm (101,325 Pa).
2.3 Sorbent Loading
The equilibrium loading that can be approached during ambient conditions during the capture cycle (\(\:{\varvec{q}}_{\varvec{e}\varvec{q}\:\varvec{c}\varvec{a}\varvec{p}}\)), in mol CO2/kg sorbent, is calculated as the product of the sorbent equilibrium saturation (\(\:\varvec{\theta\:}\)) and the maximum stoichiometric sorbent loading (\(\:{\varvec{q}}_{\varvec{m}\varvec{a}\varvec{x}}\)), which is estimated to be 1.18 mol CO2/kg sorbent as derived from the equivalent capacity provided on the Manufacturing Specification Sheet for Excellion I-200 resins21. The equilibrium loading approached at the end of regeneration (\(\:{\varvec{q}}_{\varvec{e}\varvec{q}\:\varvec{r}\varvec{e}\varvec{g}}\)), in mol CO2/kg sorbent, is given by the product of the regeneration equilibrium fraction (\(\:{\varvec{\theta\:}}_{\varvec{r}\varvec{e}\varvec{g}}\)) and \(\:{\varvec{q}}_{\varvec{m}\varvec{a}\varvec{x}}\):
$$\:{\varvec{q}}_{\varvec{e}\varvec{q}\:\varvec{c}\varvec{a}\varvec{p}}=\varvec{\theta\:}\varvec{*}{\varvec{q}}_{\varvec{m}\varvec{a}\varvec{x}}\:$$
4
$$\:{\varvec{q}}_{\varvec{e}\varvec{q}\:\varvec{r}\varvec{e}\varvec{g}}={\varvec{\theta\:}}_{\varvec{r}\varvec{e}\varvec{g}}\varvec{*}{\varvec{q}}_{\varvec{m}\varvec{a}\varvec{x}}$$
5
We refer to amount of CO2 that is unloaded from the sorbent as it swings from an equilibrium with ambient condition to an equilibrium with temperature and moisture conditions in the regenerator as the net equilibrium CO2 loading (\(\:{\varvec{q}}_{\varvec{e}\varvec{q}\:\varvec{n}\varvec{e}\varvec{t}}\)), which is given in mol CO2/kg sorbent.
$$\:{\varvec{q}}_{\varvec{e}\varvec{q}\:\varvec{n}\varvec{e}\varvec{t}}={\varvec{q}}_{\varvec{e}\varvec{q}\:\varvec{c}\varvec{a}\varvec{p}}-{\varvec{q}}_{\varvec{e}\varvec{q}\:\varvec{r}\varvec{e}\varvec{g}}$$
6
2.4 Flory-Huggins Theory
Flory-Huggins model can serve as a theoretical approximation to model the water adsorption in Excellion I-200 sorbent in lieu of experimental data available from the literature 22. The Flory-Huggins equation, represented as Eq. 7, mathematically describes the relationship between the natural logarithm of the water activity (\(\:\varvec{a}\)) (water activity = relative humidity / 100%) and the volume fraction of water sorbed in the sorbent or polymer (ϕ2):
$$\:\mathbf{ln}\varvec{a}=\mathbf{ln}{\varvec{\varphi\:}}_{2}+\left(1-{\varvec{\varphi\:}}_{2}\right)+\varvec{\chi\:}{\left(1-{\varvec{\varphi\:}}_{2}\right)}^{2}$$
7
The Flory-Huggins interaction parameter (\(\:\varvec{\chi\:}\)) is a dimensionless quantity that characterizes the interaction between the polymer and the solvent and is influenced by temperature and pressure. \(\:\varvec{\chi\:}\) can be obtained for Excellion I-200 sorbent based on the water-to-carbonate ratio against water activity experimental data available in the literature 23 (see Supporting Information, S1). Once χ is known for the sorbent, the water loading for the sorbent can be obtained from ϕ2 at a particular water activity (a) (see Supporting Information, S2).
2.5 Reaction kinetics
We model the CO2 moisture-swing uptake with a first-order linear-driving-force (LDF) rate expression, with a mass transfer rate constant \(\:{\varvec{k}}_{\varvec{r}\varvec{a}\varvec{t}\varvec{e}}\), 9,24
$$\:\frac{\varvec{d}\varvec{q}}{\varvec{d}\varvec{t}}={\varvec{k}}_{\varvec{r}\varvec{a}\varvec{t}\varvec{e}}\left({\varvec{q}}^{\varvec{*}}-\varvec{q}\right)$$
8
which results in an exponential decay of the difference between the actual loading state and the equilibrium state. The sorption is first-order with respect to the solid-phase driving force \(\:\left({\varvec{q}}^{\varvec{*}}-\varvec{q}\right)\). For a finite capture time \(\:{\varvec{t}}_{\varvec{c}\varvec{a}\varvec{p}}\), the loading will approach but not reach the equilibrium level. A similar argument can be made for the discharge of CO2 from the sorbent during regeneration. Because regeneration is conducted at 50°C, lower CO2 partial pressure of 600 Pa, and 95% RH with a fixed 1,200 s hold, we assume near-equilibrium unloading under these setpoints. This is an optimistic upper bound consistent with moisture-assisted desorption behavior reported under humid regeneration; device-specific kinetics could reduce the approach to equilibrium and proportionally lower throughput.
The LDF rate equation is akin to the reduced-order treatment used in cycle modeling (e.g., Elfving et al.)9. In this study, \(\:{\varvec{q}}^{\varvec{*}}\) is the equilibrium loading (capture: ambient \(\:\varvec{T},{\varvec{h}}_{\varvec{r}},\:{\varvec{p}}_{\varvec{C}{\varvec{O}}_{2}};\:\)regeneration: set \(\:\varvec{T},{\varvec{h}}_{\varvec{r}},\:{\varvec{p}}_{\varvec{C}{\varvec{O}}_{2}\:\varvec{r}\varvec{e}\varvec{g}}\)).
With this assumption, it is possible to introduce the ratio \(\:\varvec{\psi\:}\) as
$$\:\varvec{\psi\:}=\frac{{\varvec{q}}_{\varvec{a}\varvec{c}\varvec{t}\:\varvec{n}\varvec{e}\varvec{t}}}{{\varvec{q}}_{\varvec{e}\varvec{q}\:\varvec{n}\varvec{e}\varvec{t}}}=1-\mathbf{e}\mathbf{x}\mathbf{p}(-{\varvec{k}}_{\varvec{r}\varvec{a}\varvec{t}\varvec{e}}{\varvec{t}}_{\varvec{c}\varvec{a}\varvec{p}})$$
9
In optimizing capture time, the dimensionless ratio, denoted by \(\:\varvec{\psi\:}\), plays a crucial role in balancing speed and collection rate. This ratio can be dynamically optimized to maximize the actual net CO2 loading (\(\:{\varvec{q}}_{\varvec{a}\varvec{c}\varvec{t}\:\varvec{n}\varvec{e}\varvec{t}}\)) under varying weather conditions. A design optimization model was developed to dynamically optimize \(\:\varvec{\psi\:}\) based on instantaneous weather conditions (see Supporting Information, S3).
By re-arranging Eq. 9, one can express the capture time, in terms of \(\:\varvec{\psi\:}\), as
$$\:{\varvec{t}}_{\varvec{c}\varvec{a}\varvec{p}}=-\frac{\mathbf{log}\left(1-\varvec{\psi\:}\right)}{{\varvec{k}}_{\varvec{r}\varvec{a}\varvec{t}\varvec{e}}}$$
10
Unlike fan-driven PSA/TSA/TVSA cycles with controlled superficial velocity, our passive contactor uses ambient wind as the air mover, so we regress a lumped \(\:{\varvec{k}}_{\varvec{r}\varvec{a}\varvec{t}\varvec{e}}\)against ambient wind velocity and temperature, Eq. (11)–(12). Thus, we map meteorological drivers (wind and temperature; relative humidity acts primarily through \(\:{\varvec{q}}^{\varvec{*}}\)) to \(\:{\varvec{k}}_{\varvec{r}\varvec{a}\varvec{t}\varvec{e}}\) for a passive device, while retaining an LDF structure compatible with standard CO2 adsorption from humid air. Because the lumped \(\:{\varvec{k}}_{\varvec{r}\varvec{a}\varvec{t}\varvec{e}}\) is influenced by wind speed and temperature, we adopt a linear parameterization for their influence, via \(\:{\varvec{v}}_{\varvec{w}\varvec{i}\varvec{n}\varvec{d}}\) and \(\:\varvec{f}\left(\varvec{T}\right)\), based on wind-tunnel experimental data (see Supporting Information, S4).
$$\:{\varvec{k}}_{\varvec{r}\varvec{a}\varvec{t}\varvec{e}}={\varvec{c}}_{\varvec{r}\varvec{a}\varvec{t}\varvec{e}}\varvec{*}\varvec{f}\left(\varvec{T}\right)\varvec{*}{\varvec{v}}_{\varvec{w}\varvec{i}\varvec{n}\varvec{d}}$$
11
Here, \(\:{\varvec{c}}_{\varvec{r}\varvec{a}\varvec{t}\varvec{e}}\) is an empirical coefficient derived from limited data obtained in wind-tunnel experiments conducted at ASU’s Center for Negative Carbon Emissions, yielding a value of 1.5 * 10− 4 m− 1 (see Supporting Information, S4). The wind velocity (converted to m s− 1) data are obtained from NOAA weather records. Additionally, an Arrhenius-type term, denoted as \(\:\varvec{f}\left(\varvec{T}\right)\), is incorporated into the equation to account for the impact of temperature on the rate constant. The factor is assumed to be:
$$\:\varvec{f}\left(\varvec{T}\right)=\mathbf{e}\mathbf{x}\mathbf{p}\left[\varvec{b}\varvec{{\prime\:}}\varvec{*}\left(\frac{1}{{\varvec{T}}_{\varvec{a}\varvec{m}\varvec{b}}}-\frac{1}{{\varvec{T}}_{25}}\right)\right]$$
12
In this equation, \(\:{\varvec{T}}_{\varvec{a}\varvec{m}\varvec{b}}\) represents the ambient temperature, \(\:{\varvec{T}}_{25}\) refers to the temperature at which the wind tunnel experiment was conducted (25°C), and \(\:{\varvec{b}}^{\varvec{{\prime\:}}}=-9600{\varvec{K}}^{-1}\) represents the fit parameter based on experimental data for the sorbent (see Supporting Information, S4).
2.6 CO2 productivity
The Langmuir isotherm model (discussed earlier) provides an estimate of the equilibrium CO2 loading in terms of moles of CO2 per kilogram of sorbent. Using Eq. 13, the cycle-average mass of CO2 captured (\(\:{\varvec{m}}_{\varvec{C}{\varvec{O}}_{2}})\) in kilograms per kilogram of sorbent can be calculated.
$$\:{\varvec{m}}_{\varvec{C}{\varvec{O}}_{2}}={\varvec{q}}_{\varvec{e}\varvec{q}\:\varvec{n}\varvec{e}\varvec{t}}\varvec{*}\varvec{\psi\:}\varvec{*}{\varvec{M}\varvec{W}}_{\varvec{C}{\varvec{O}}_{2}}$$
13
Here, \(\:{\varvec{q}}_{\varvec{e}\varvec{q}\:\varvec{n}\varvec{e}\varvec{t}}\) represents the equilibrium net CO2 loading obtained from the Langmuir isotherm model; ψ is the dimensionless ratio that determines the net actual loading of the sorbent relative to its equilibrium loading; \(\:{\varvec{M}\varvec{W}}_{\varvec{C}{\varvec{O}}_{2}}\) represents the molecular weight of CO2, which is 0.044 kg/mol.
Additionally, the CO2 reaction kinetics enables us to estimate the capture time (\(\:{\varvec{t}}_{\varvec{c}\varvec{a}\varvec{p}}\)), in seconds. The sum of the fixed regeneration time (\(\:{\varvec{t}}_{\varvec{r}\varvec{e}\varvec{g}}\)) of 1200 seconds and the variable capture time constitutes the duration of each cycle (converted to hours). In this work, “productivity” is defined as the cycle-average mass of CO₂ captured per unit sorbent and per unit time (kg-CO₂ kg⁻¹ hr⁻¹), equivalent to the rate in Eq. (14).
$$\:{\varvec{r}}_{{\varvec{C}\varvec{O}}_{2}}=\frac{{\varvec{m}}_{\varvec{C}{\varvec{O}}_{2}}}{\left({\varvec{t}}_{\varvec{c}\varvec{a}\varvec{p}}+{\varvec{t}}_{\varvec{r}\varvec{e}\varvec{g}}\right)}$$
14
2.7 Net Water Loss
We describe H2O uptake/desorption at the stage endpoints with the Flory-Huggins equilibrium model and evaluate per-cycle H2O exchange over the same capture window that governs CO2. Because CO2 is rate-limiting in our passive cycle, the device remains in ambient contact until the CO2 loading attains its design approach-to-equilibrium fraction (\(\:\varvec{\psi\:}\)); accordingly, we scale the H2O endpoint swing by the same \(\:\varvec{\psi\:}\) as a time-window surrogate, without introducing a separate \(\:{\varvec{k}}_{{\varvec{H}}_{2}\varvec{O}}\) or implying identical intrinsic kinetics 9,24. Since H2O typically equilibrates faster, this treatment is conservative for per-cycle water-loss accounting.
Under this closure, the net water mass exchanged per kilogram of sorbent in a cycle is:
$$\:{\varvec{m}}_{{\varvec{H}}_{2}\varvec{O}}={\varvec{q}}_{{\varvec{H}}_{2}\varvec{O}\:\varvec{n}\varvec{e}\varvec{t}}\varvec{*}\varvec{\psi\:}\varvec{*}{\varvec{M}\varvec{W}}_{{\varvec{H}}_{2}\varvec{O}}$$
15
Here, \(\:{\varvec{q}}_{{\varvec{H}}_{2}\varvec{O}\:\varvec{n}\varvec{e}\varvec{t}}\) represents the net water loading obtained from Flory-Huggins isotherm model and Eq. 16, considering a specific relative humidity. The molecular weight of H2O (\(\:{\varvec{M}\varvec{W}}_{{\varvec{H}}_{2}\varvec{O}}\)) is 0.018 kg/mol.
After regeneration at high water activity, exposure to drier ambient air drives net desorption of H₂O until the sorbent equilibrates with the ambient relative humidity. Conversely, during the regeneration phase at higher relative humidity, the sorbent gains water. The regeneration condition aims for a high relative humidity. Because we model water uptake in sorbent with Flory-Huggins equation, which becomes singular at 100% relative humidity, a nominal relative humidity of 95% is chosen as a typical condition. Following pre-evacuation, the regeneration chamber primarily contains H2O and CO2, with water added for regeneration and CO2 desorbed from the sorbent. The net water loading (\(\:{\varvec{q}}_{{\varvec{H}}_{2}\varvec{O}\:\varvec{n}\varvec{e}\varvec{t}}\)) and the net water-loss rate (\(\:{\varvec{l}}_{{\varvec{H}}_{2}\varvec{O}}\)) within each cycle are calculated using Equations 16 and 17, respectively:
$$\:{\varvec{q}}_{{\varvec{H}}_{2}\varvec{O}\:\varvec{n}\varvec{e}\varvec{t}}={\varvec{q}}_{{\varvec{H}}_{2}\varvec{O}\:\varvec{r}\varvec{e}\varvec{g}}-{\varvec{q}}_{{\varvec{H}}_{2}\varvec{O}\:\varvec{c}\varvec{a}\varvec{p}}$$
16
$$\:{\varvec{l}}_{{\varvec{H}}_{2}\varvec{O}}=\frac{{\varvec{m}}_{{\varvec{H}}_{2}\varvec{O}}}{\left({\varvec{t}}_{\varvec{c}\varvec{a}\varvec{p}}+{\varvec{t}}_{\varvec{r}\varvec{e}\varvec{g}}\right)}$$
17
2.8 Representative Meteorological Site
Since moisture-swing sorbents are strongly affected by ambient conditions, it is important to estimate performance for a specific location. Here, a representative site, St. Johns, Arizona, is selected to perform a thorough analysis of site-specific variability. This analysis focuses on assessing the site's potential for CO2 productivity and its associated water loss for each cycle in response to varying ambient weather conditions. Hourly weather data spanning 17 years from 2006 to 2022 is taken from St. Johns weather station (ID: 93027) in Arizona. The data is sourced from the NOAA and its resources, including the Local Climatological Data (LCD) and Climate Data Online (CDO) tools provided by the National Climatic Data Center (NCDC) 25,26. To ensure data integrity, preprocessing steps are performed, including the removal of missing values, converting data types (e.g., datetime, float), and interpolating gaps, facilitating their integration into the mathematical model. Temperature and relative humidity data for a representative site were plotted to assess their variability (see Supporting Information, S5). Discrete raw wind speed data obtained from NOAA weather records were converted into continuous values using Monte Carlo randomization (see Supporting Information, S6). These processed meteorological datasets are then ready for utilization in the mathematical model, enabling the analysis of the impact of weather conditions on the moisture-swing process.
3.1 CO2 Isotherm Model
The sorbent's CO2 saturation, which depends on the relative humidity (\(\:{\text{h}}_{\text{r}}\)), can be determined using Eq. 3. During the moisture-swing cycle, the sorbent undergoes a transition from the bicarbonate (HCO3−) form, representing a fully loaded sorbent with CO2 (maximum θ value of 1), to the carbonate (CO32−) form, indicating a sorbent that has been unloaded during regeneration with steam (minimum θ value of 0) 27.
Figure 2 illustrates the relationship between sorbent saturation and water activity. It reveals that lower relative humidity levels lead to higher CO2 loading of the sorbent. The wide range of observed sorbent saturation is attributed to the variability in weather conditions over the past 17 years. The diverse dataset encompasses different combinations of temperature and relative humidity. Since sorbent saturation is primarily influenced by relative humidity and only weakly affected by ambient temperature, see Eqs. 1 and 3, a clear trend emerges where saturation decreases with increasing relative humidity. However, the weak dependence on temperature contributes to the wide range of variation at higher relative humidity.
These results underscore the substantial influence of relative humidity on both the capture and release of CO2 by the sorbent. These findings are consistent with the anticipated behavior of moisture-swing sorbents, where they capture CO2 in dry conditions and release it when exposed to moisture 20,27.
3.2 CO2 Regeneration
The moisture-induced cycle undergoes sorbent regeneration by utilizing the free energy released during water adsorption, reducing external heat demand 20. In this process, the loaded sorbent is regenerated using steam at a temperature of 50°C and a saturated vapor pressure of 12.33 kPa, enhancing moisture release of CO2 by a moderate temperature elevation.
During the regeneration process, it is assumed that an endpoint of 95% relative humidity is established, and the system is pre-evacuated to retain a mixture comprising primarily CO2 and H2O. Consequently, the aggregate system pressure represents the sum of the partial pressure of water vapor and CO2 within the regeneration chamber assuming ideal gas behavior per Dalton’s Law. At 95% relative humidity, the water vapor pressure is 11.72 kPa and we target an endpoint CO2 regeneration pressure (\(\:{\text{p}}_{\text{C}{\text{O}}_{2}\:\text{r}\text{e}\text{g}}\)) of 600 Pa.
Figure 3 depicts the sorbent saturation as a function of CO2 pressure at different regeneration temperatures and a constant relative humidity of 95%. At a CO2 regeneration pressure of 600 Pa and temperature of 50oC, as per the model, the sorbent will retain a minimum saturation of \(\:\theta\:\)=0.74. Under these conditions, this is the deepest CO2 unloading that can be achieved.
3.3 CO2 Productivity
The equilibrium net CO2 loading can be determined by utilizing sorbent saturation and regeneration, as outlined in Equations 4, 5, and 6. Similarly, sorbent kinetics, which encompass capture time determined by wind speed and sorbent properties, are captured in Equations 9, 10, and 11. Bringing together sorbent loading and kinetics enables the calculation of the CO2 productivity, as described in Equations 13 and 14.
The capture rate, as shown in Fig. 4, is found to vary significantly depending on the prevailing relative humidity conditions which is expected as the process is moisture-driven and relative humidity has a significant impact on overall CO2 productivity.
During the summer months, characterized by lower relative humidity, a higher CO2 capture rate is observed. This can be attributed to the favorable conditions for CO2 adsorption onto the sorbent surface when the relative humidity is low. As a result, the sorbent exhibits enhanced CO2 capture efficiency during these months.
In contrast, a distinct period of zero capture rate is observed for two winter months. During these times, ambient relative humidity conditions are such that the sorbent in equilibrium with ambient air would be at loading state below the fully unloaded state of the sorbent at the end of the regeneration step.
By understanding the relationship between relative humidity and CO2 capture rate, it becomes possible to strategically plan the operation of the capture system. Adjustments can be made to the cycling frequency or the duration of capture and regeneration phases to maximize the overall CO2 capture efficiency, considering the varying relative humidity conditions throughout the year. This also helps to geospatially locate the ideal site of carbon capture across the United States.
3.4 Net Water Loss
During the sorbent regeneration process, a relative humidity of 95% is maintained, resulting in significant water adsorption. However, when exposed to ambient conditions characterized by lower relative humidity, the sorbent loses moisture to the air, leading to a net loss of water to the environment. This is an essential characteristic of the moisture-swing cycle and required to avoid further energy addition. Figure 5 illustrates the trend of net water loss, with higher losses observed during the summer months due to the lower relative humidity. This highlights the challenge posed by increased water demand during the summer months, as the sorbent tends to dry out more rapidly in the atmosphere while producing higher amount of CO2.
The findings indicate that the water-to-carbonate ratio, as influenced by relative humidity, plays a crucial role in the sorbent's performance. Understanding these dynamics is essential for optimizing the CO2 capture process and addressing the challenges posed by varying environmental condition.
In summary, this study develops a model for a moisture swing sorbent.
The water adsorption isotherm is modeled by Flory-Huggins, which is typical for ion-exchange resins. The Flory-Huggins interaction parameter (χ) usually depends on the polymer properties, temperature and pressure for water adsorption, and the CO2 loading in the sorbent. From the analysis in this study, χ for Excellion I-200 resins is calculated using the experimental data, and a weak dependence is found against relative humidity. The dependence of χ on temperature and CO2 loading is not resolved here due to limited experimental data; therefore, we take an average value of \(\:{\chi\:}\) as 1.08 (reflecting its weak variation with water activity) to obtain a simplified relation for water loading as a function of relative humidity (see Supporting Information, S1-S2).
A modified Langmuir adsorption isotherm is used to determine the CO2 loading as a function of relative humidity. The free energy of binding can be approximated as a linear function of relative humidity under ambient conditions based on the model used by Wang et al., 2013 20.
The reaction kinetics are closed with a linear-driving-force (LDF) expression, and the approach-to-equilibrium rate constant is parameterized as \(\:{\text{k}}_{\text{r}\text{a}\text{t}\text{e}}={\text{c}}_{\text{r}\text{a}\text{t}\text{e}}\text{*}\text{f}\left(\text{T}\right)\text{*}{\text{v}}_{\text{w}\text{i}\text{n}\text{d}}\). A linear dependence on wind speed is adopted over the calibrated range based on wind-tunnel experiments; \(\:\text{f}\left(\text{T}\right)\) provides an Arrhenius-type temperature correction (see Supporting Information, S4). We use station wind speed as a measured environmental driver and do not impose an explicit RH dependence in \(\:{\text{k}}_{\text{r}\text{a}\text{t}\text{e}}\) to avoid double counting humidity effects already captured in the equilibrium \(\:{\text{q}}^{\text{*}}\). The mapping from free-stream wind to local mass transfer is embedded in the empirical coefficient \(\:{\text{c}}_{\text{r}\text{a}\text{t}\text{e}}\).
The entire formulation is brought together to determine the sorbent behavior at a particular location as a function of the varying ambient weather conditions (temperature, relative humidity, and wind speed). Because loading approaches equilibrium exponentially, we pose a design optimization model in terms of the attained fraction \(\:{\psi\:}\) (see Supporting Information, S3) to set the capture time; this yields cycle time, CO₂ productivity, and net water loss. Regeneration is assumed near-equilibrium at 50°C/95% RH.
This study aimed to elucidate the dynamic behavior of a passive direct air capture (DAC) system operating under varying ambient conditions. By developing a mathematical model, we quantified CO₂ capture capacity fluctuations using a parsimonious, meteorology-driven closure (modified Langmuir for CO₂; Flory-Huggins for H₂O plus LDF kinetics). The model incorporated preliminary approximations (linear wind-speed parameterization of \(\:\text{k}\_\text{r}\text{a}\text{t}\text{e}\); temperature effects via \(\:\text{f}\left(\text{T}\right)\); proxy, decoupled isotherms). Future work should refine these assumptions by introducing non-linear velocity scaling and co-adsorption thermodynamics/kinetics as data become available, providing a drop-in replacement for the proxies while keeping the same cycle algebra.
Further research should also expand applicability by integrating broader meteorological datasets and performing geospatial analyses across diverse climatic zones in the U.S. to identify optimal locations for deploying passive DAC systems based on site-specific temperature, humidity, and wind conditions. By formulating a model that estimates carbon capture productivity and water-loss rate directly from weather parameters, this work establishes a foundation for screening and scaling moisture-swing sorbents in real-world passive DAC applications.
Author contributions
Mohammad Abu Talha: Conceptualization, Investigation, Methodology, Coding, Visualization, Writing (original draft)
John Cirucci: Conceptualization and Supervision
Klaus S. Lackner: Validation and Writing (review & editing)
Conflicts of interest
Klaus S. Lackner is a co-inventor on IP owned by Arizona State University (ASU) related to direct air capture. ASU has licensed part of this IP to Carbon Collect and holds an ownership stake in the company. As an ASU employee, Lackner serves as a technical advisor to Carbon Collect and holds shares in the company, which also supports DAC research at ASU.
Data availability
The data and model scripts that support the findings of this study are openly available on GitHub at https://github.com/mdtalhabu/Passive-DAC-Model. The repository contains input data, code, and example instructions necessary to replicate the core analyses presented in this article.
Acknowledgements
The authors are grateful for the support from the Center for Negative Carbon Emissions (CNCE) at Arizona State University.
The information, data, or work presented herein was funded in part by the U.S. Department of Energy (DE-FE0032097). The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.
- Mitigation Pathways Compatible with Long-Term Goals. In: Climate Change 2022 - Mitigation of Climate Change; Intergovernmental Panel On Climate Change (Ipcc), pp. 295–408. Ed.; Cambridge University Press (2023). https://doi.org/10.1017/9781009157926.005
- Ipcc: Global Warming of 1.5°C: IPCC Special Report on Impacts of Global Warming of 1.5°C above Pre-Industrial Levels in Context of Strengthening Response to Climate Change, Sustainable Development, and Efforts to Eradicate Poverty, 1st edn. Cambridge University Press (2022). https://doi.org/10.1017/9781009157940
- Igini, M.: Global Temperatures Breach 1.5C Warming Threshold for Full Year. Earth Org. https://earth.org/critical-1-5c-global-warming-threshold-breached-over-12-month-period-for-first-time/ (accessed 2024-03-03).
- Azarabadi, H., Lackner, K.S.: Postcombustion Capture or Direct Air Capture in Decarbonizing US Natural Gas Power? Environ. Sci. Technol. 54(8), 5102–5111 (2020). https://doi.org/10.1021/acs.est.0c00161
- McQueen, N., Gomes, K.V., McCormick, C., Blumanthal, K., Pisciotta, M., Wilcox, J.A., Review of Direct Air Capture (DAC): Scaling up Commercial Technologies and Innovating for the Future. Prog Energy. 3(3), 032001 (2021). https://doi.org/10.1088/2516-1083/abf1ce
- Zhu, X., Xie, W., Wu, J., Miao, Y., Xiang, C., Chen, C., Ge, B., Gan, Z., Yang, F., Zhang, M., O’Hare, D., Li, J., Ge, T., Wang, R.: Recent Advances in Direct Air Capture by Adsorption. Chem. Soc. Rev. 51(15), 6574–6651 (2022). https://doi.org/10.1039/D1CS00970B
- Grossmann, Q., Saenz-Cavazos, P.A., Ferru, N., Williams, D.R., Mazzotti, M.: Measuring and Modeling Water and Carbon Dioxide Adsorption on Amine Functionalized Alumina under Direct Air Capture Conditions. Ind. Eng. Chem. Res. 64(13), 7165–7175 (2025). https://doi.org/10.1021/acs.iecr.4c04581
- Analysis of direct capture of CO2 from ambient air via steam-assisted temperature–vacuum swing adsorption | Adsorption. https://link.springer.com/article/10.1007/s10450-020-00249-w (accessed 2025-09-28)
- Elfving, J., Sainio, T.: Kinetic Approach to Modelling CO2 Adsorption from Humid Air Using Amine-Functionalized Resin: Equilibrium Isotherms and Column Dynamics. Chem. Eng. Sci. 246, 116885 (2021). https://doi.org/10.1016/j.ces.2021.116885
- Elfving, J., Bajamundi, C., Kauppinen, J., Sainio, T.: Modelling of Equilibrium Working Capacity of PSA, TSA and TVSA Processes for CO2 Adsorption under Direct Air Capture Conditions. J. CO2 Util. 22, 270–277 (2017). https://doi.org/10.1016/j.jcou.2017.10.010
- Jiang, L., Liu, W., Wang, R.Q., Gonzalez-Diaz, A., Rojas-Michaga, M.F., Michailos, S., Pourkashanian, M., Zhang, X.J., Font-Palma, C.: Sorption Direct Air Capture with CO2 Utilization. Prog Energy Combust. Sci. 95, 101069 (2023). https://doi.org/10.1016/j.pecs.2022.101069
- Lackner, K.S.: Capture of Carbon Dioxide from Ambient Air. Eur. Phys. J. Spec. Top. 176(1), 93–106 (2009). https://doi.org/10.1140/epjst/e2009-01150-3
- Kaneko, Y., Lackner, K.S.: Isotherm Model for Moisture-Controlled CO 2 Sorption. Phys. Chem. Chem. Phys. 24(24), 14763–14771 (2022). https://doi.org/10.1039/D2CP01131J
- Wade, J.L., Lopez Marques, H., Wang, W., Flory, J., Freeman, B.: Moisture-Driven CO2 Pump for Direct Air Capture. J. Membr. Sci. 685, 121954 (2023). https://doi.org/10.1016/j.memsci.2023.121954
- Azarabadi, H., Lackner, K.S.A., Sorbent-Focused: Techno-Economic Analysis of Direct Air Capture. Appl. Energy. 250, 959–975 (2019). https://doi.org/10.1016/j.apenergy.2019.04.012
- Shi, X., Lin, Y., Chen, X.: Development of Sorbent Materials for Direct Air Capture of CO2. MRS Bull. 47(4), 405–415 (2022). https://doi.org/10.1557/s43577-022-00320-7
- mdtalhabu/Passive: -DAC-Model: Mathematical model for Moisture swing passive DAC process. https://github.com/mdtalhabu/Passive-DAC-Model (accessed 2025-03-02)
- Bernatowicz, J.M., Snow, M.J., O’Hare, R.J.: Methods and Apparatus for the Formation of Heterogeneous Ion-Exchange Membranes. US6716888B2, April 6, https://patents.google.com/patent/US6716888B2/en (2004). accessed 2023-11-05).
- The impact of binary water–CO 2 isotherm models on the optimal performance of sorbent-based direct air capture processes - Energy & Environmental Science (RSC Publishing) DOI:10.1039/D1EE01272J. https://pubs.rsc.org/en/content/articlehtml/2021/ee/d1ee01272j (accessed 2025-09-28)
- Wang, T., Lackner, K.S., Wright, A.B.: Moisture-Swing Sorption for Carbon Dioxide Capture from Ambient Air: A Thermodynamic Analysis. Phys. Chem. Chem. Phys. 15(2), 504–514 (2013). https://doi.org/10.1039/C2CP43124F
- Iglesia, A.C., Clemente: S. MEMBRANE SPECIFICATIONS
- Lin, H., Freeman, B.D.: Gas and Vapor Solubility in Cross-Linked Poly(Ethylene Glycol Diacrylate). Macromolecules. 38(20), 8394–8407 (2005). https://doi.org/10.1021/ma051218e
- Shi, X., Xiao, H., Chen, X., Lackner, K.S.: The Effect of Moisture on the Hydrolysis of Basic Salts. Chem. – Eur. J. 22(51), 18326–18330 (2016). https://doi.org/10.1002/chem.201603701
- Sircar, S., Hufton, J.R.: Why Does the Linear Driving Force Model for Adsorption Kinetics Work? Adsorption 6 (2), 137–147. (2000). https://doi.org/10.1023/A:1008965317983
- Index of /data/local-climatological-data/access. https://www.ncei.noaa.gov/data/local-climatological-data/access/ (accessed 2024-03-03)
- Local Climatological Data Station Details: ST JOHNS INDUSTRIAL AIR PARK, AZ US, WBAN:93027 | Climate Data Online (CDO) | National Climatic Data Center (NCDC). https://www.ncdc.noaa.gov/cdo-web/datasets/LCD/stations/WBAN:93027/detail (accessed 2024-03-03)
- Wang, T., Lackner, K.S., Wright, A.: Moisture Swing Sorbent for Carbon Dioxide Capture from Ambient Air. Environ. Sci. Technol. 45(15), 6670–6675 (2011). https://doi.org/10.1021/es201180v
Competing interest reported. Klaus S. Lackner is a co-inventor on IP owned by Arizona State University (ASU) related to direct air capture. ASU has licensed part of this IP to Carbon Collect and holds an ownership stake in the company. As an ASU employee, Lackner serves as a technical advisor to Carbon Collect and holds shares in the company, which also supports DAC research at ASU.