Cycle-resolved dynamic modeling of atmospheric CO 2 separation systems under time-varying ambient and electricity conditions

1 Introduction

Adsorption-based separation technologies are increasingly considered for removing dilute CO₂ from atmospheric air streams. In these systems, solid sorbents selectively capture CO₂ during adsorption and release it during thermal or pressure-driven regeneration steps. Because the feed gas is atmospheric air, system performance is inherently influenced by environmental conditions such as temperature and humidity, which affect both adsorption equilibria and process thermodynamics (Young et al., 2021; Sendi et al., 2022).

Recent studies have highlighted that ambient temperature and humidity can significantly influence adsorption system performance, including CO₂ loading, energy demand, and cycle productivity (Young et al., 2021; Sendi et al., 2022; Wurzbacher et al., 2016; Schellevis et al., 2024; Wenzel et al., 2025; Jajjawi et al., 2026; Rezo et al., 2026). More recent weather-aware studies have also linked time-varying ambient conditions to DAC design, operation, sorbent selection, and net-removal performance (Wiegner et al., 2022; Postweiler et al., 2024; Jamdade et al., 2024; Shorey and Abdulla, 2025; Glaser et al., 2025). As a result, several analyses have incorporated regional or hourly ambient conditions to assess atmospheric CO₂ separation systems across different geographic locations (Postweiler et al., 2024; Jamdade et al., 2024; Glaser et al., 2025; Wang et al., 2024; McQueen et al., 2021; Fasihi et al., 2019). However, many existing approaches still rely on steady-state, cycle-averaged, or simplified process representations. Relatively few frameworks retain transient adsorption–desorption dynamics, temperature–humidity coupling, and cycle-to-cycle sorbent-state evolution in a form sufficiently fast for large-scale data generation, surrogate training, and cyclic optimization. As highlighted in the recent BCG–XPRIZE report (BCG & XPRIZE, 2025) and previous studies (Wiegner et al., 2022; Postweiler et al., 2024; Young et al., 2023a; Young et al., 2023b; Galanti et al., 2026), such operational control and process optimization are important levers for reducing the cost of scaling adsorption-based atmospheric CO₂ separation systems.

Detailed cyclic adsorption models can capture these coupled physical processes by resolving mass transfer, heat transfer, and adsorption equilibria within each cycle (Young et al., 2021; Sendi et al., 2022). While these models provide valuable insight into adsorption physics, they are computationally intensive and therefore difficult to apply to large-scale simulations that require thousands of cycles or long-term ambient condition-resolved analysis. This computational limitation creates a practical barrier for generating large datasets required for system design exploration, uncertainty analysis, or data-driven optimization (Esche et al., 2022; Ünal et al., 2022).

Reduced-order and surrogate modeling provide a pathway to preserve dominant process behavior while reducing the cost of repeated dynamic simulations (Esche et al., 2022; Chung et al., 2022). However, integrated DAC workflows that connect moisture-coupled cycle physics, cycle-to-cycle state propagation, large-scale data generation, and control-oriented optimization remain limited. This study addresses this gap through a physics-informed ROM that acts as an intermediate layer between cyclic adsorption modeling and data-driven optimization.

In this study, we: (i) develop a cycle-resolved dynamic ROM that retains CO₂-H₂O co-adsorption, adsorption–desorption kinetics, incomplete regeneration, and cycle-to-cycle sorbent-state memory; (ii) use the ROM to generate approximately 120,000 physically screened cycle transitions across environmental, operating, geometric, and initial-state conditions; (iii) train a neural-network surrogate to reproduce the resulting state transitions and performance outputs; and (iv) embed the surrogate within a model predictive control framework to demonstrate cyclic optimization under time-varying ambient and electricity conditions. The framework is intended for comparative screening, physics-structured dataset generation, and proof-of-concept optimization within the modeled domain, rather than equipment sizing or plant-specific performance prediction.

2 Methodology

2.1 System definition and modeling framework

The reduced-order model (ROM) describes a TVSA cycle with six sequential stages: vacuum, heating, desorption, cooling, pressurization, and adsorption. This stage sequence follows the general operation used in adsorption-based atmospheric CO₂ separation systems and is consistent with recent process and system studies (Schellevis et al., 2024; Wenzel et al., 2025; Jajjawi et al., 2026; Bouaboula et al., 2024).

The model links time-varying environmental inputs with controllable operating variables that govern cycle severity, throughput, and regeneration completeness. Environmental inputs include ambient temperature and relative humidity. Operating variables include regeneration temperature, vacuum pressure, step durations, and inlet superficial velocity. The framework is intended for ambient-condition-aware annual simulation and large parametric studies. For this purpose, a zero-dimensional ROM is used to retain the main adsorption, desorption, and energy-balance behavior while keeping the computation fast enough for large data generation and repeated simulations (Schellevis et al., 2024; Wenzel et al., 2025; Jajjawi et al., 2026; Rezo et al., 2026). The key assumptions of the reduced-order model are summarized in Table 1. These assumptions are grouped into physical, reduced-order, and empirical categories to clarify the scope and limitations of the model.

TABLE 1. Summary of key physical assumptions, reduced-order closures, and empirical corrections in the dynamic ROM framework.

The ROM is intended as a control-oriented state-transition model for large parametric screening, cycle-resolved dataset generation, surrogate training, and optimization. It retains the dominant thermodynamic, kinetic, moisture-coupled, and state-memory behavior but does not resolve axial gradients or equipment-level dynamics. Literature data are therefore used to assess performance magnitude and environmental-response trends rather than to claim plant-specific predictive accuracy. These assumptions enable large-scale cycle-resolved simulations while retaining the dominant thermodynamic and kinetic drivers governing adsorption performance. Only condensed parameter tables are shown in the main text. Full parameter definitions, coefficient sets, default values, and numerical safeguards are provided in Supplementary Material A. Hourly temperature, relative humidity, electricity carbon intensity, and electricity price for 2025 were collected for two representative locations, Atoka, Texas and Madison, Wyoming (Gagnon et al., 2024; Buster, 2023; An et al., 2024). The two representative locations are Atoka, Texas and Madison, Wyoming, using representative site coordinates of 31.99° N, 99.67° W and 44.64° N, 110.87° W, respectively, for weather, grid, and regional input extraction. Two locations capture contrasting temperature-humidity regimes and electricity carbon intensity and cost profiles within the United States. The full hourly input profiles are provided in Supplementary Figures S1-S3. For model evaluation and dataset generation, prescribed driver profiles are primarily used, while adaptive cycle-level control is introduced in the surrogate-based optimization framework.

2.2 Dynamic state formulation

To represent the dynamic behavior of the adsorption cycle, the reduced-order model tracks a compact set of thermodynamic and gas-phase states. These variables describe the instantaneous condition of the sorbent bed and evolve as the TVSA cycle proceeds through its operational stages. The system state is represented by the state vector given in Equation 1:

x = q CO 2 , f , q CO 2 , s , q H 2 O , f , q H 2 O , b , y CO 2 , g , T bed (1)

This six-state representation was chosen to balance physical interpretability and computational efficiency for long-term ambient condition-resolved simulations. The first two states describe CO₂ loading on the sorbent. q C O 2 , f represents a fast adsorption pool associated with rapidly accessible adsorption sites, while q C O 2 , s represents a slower kinetic pool associated with less accessible or more strongly retained CO₂ (mol CO₂ kg^-1 sorbent). This two-pool formulation captures multi-timescale sorption behavior while maintaining computational efficiency (Young et al., 2021; Sendi et al., 2022; Low et al., 2023). The next two states describe water uptake. q H 2 O , f represents weakly bound or mobile water, while q H 2 O , b represents more strongly bound water associated with the sorbent structure (mol H₂O kg^-1 sorbent). Separating water uptake into free and bound components enables the model to represent humidity-dependent sorption behavior and the influence of water on CO₂ adsorption equilibria, which has been widely reported for amine sorbents (Young et al., 2021; Sendi et al., 2022). A two-pool representation provides a compact approximation of experimentally observed multi-timescale adsorption behavior while maintaining computational efficiency for large-scale simulations.

The gas-phase condition inside the collector is represented by y C O 2 , g, the CO₂ mole fraction in the collector void space. This variable evolves dynamically as adsorption removes CO₂ from the inlet air and desorption releases CO₂ during regeneration. Finally, T b e d represents the average temperature of the sorbent bed, which reflects the combined effects of adsorption heat release, regeneration heating, and heat exchange with the surrounding process environment.

At each simulation time step, the model updates these coupled adsorption, gas-phase, and thermal states according to the active stage of the TVSA cycle. The updated state vector is then propagated to the next time step and carried forward between cycles. This state propagation allows the model to capture path-dependent behavior, where the current adsorption performance depends on the thermal and sorption history of previous cycle steps. For clarity, the main manuscript presents only the compact state formulation. The complete numerical update procedure, including the time integration scheme and stepwise update equations, is provided in Supplementary Material B, D. Definitions of all variables, parameters, and units are summarized in Supplementary Material Appendix A.

2.3 Adsorption equilibrium, kinetics, and gas-phase coupling

This section defines the governing relationships linking adsorption equilibrium, kinetics, and gas-phase dynamics within the ROM. Only the compact governing equations are presented here to clarify the physical structure of the model. Detailed mechanistic formulations, coefficient definitions, and numerical implementation procedures are provided in Supplementary Material B. The equilibrium CO₂ loading depends on the thermodynamic state of the sorbent bed and the surrounding gas phase. Equilibrium loading varies with bed temperature, CO₂ partial pressure, and water loading on the sorbent due to co-adsorption interactions (Young et al., 2021; Sendi et al., 2022). The equilibrium CO₂ loading is given by Equation 2:

q CO 2 , eq = f T bed , p CO 2 , q H 2 O (2)

q C O 2 , eq: equilibrium CO₂ loading on the sorbent (mol CO₂kg^-1 sorbent).

T bed: average sorbent bed temperature (K).

p C O 2: CO₂ partial pressure in the gas phase (Pa).

q H 2 O: water loading on the sorbent (mol H₂O kg^-1 sorbent).

The explicit functional form describing the co-adsorption interaction between CO₂ and H₂O, including blocking effects and temperature-dependent coefficients, is provided in Supplementary Material B, (Young et al., 2021).

Water uptake on the sorbent is represented using the Guggenheim-Anderson-de Boer (GAB) isotherm (Young et al., 2021), which relates equilibrium water loading to water activity and temperature. The GAB model is widely used to describe multilayer water adsorption on porous materials and has been applied extensively to amine-based sorbents. The equilibrium water loading and water activity are given by Equations 3, 4, respectively: q H 2 O , eq: equilibrium water loading (mol H₂O kg^-1 sorbent).

q H 2 O , eq = q m C GAB K GAB a w 1 − K GAB a w 1 + C GAB − 1 K GAB a w (3)

a w = p H 2 O p sat T bed (4)

q m: monolayer water capacity (mol kg^-1 sorbent).

C GAB: GAB adsorption constant associated with the heat of sorption (−)

K GAB: multilayer interaction constant (−)

a w: water activity (−)

p H 2 O: water vapor partial pressure in the gas phase (Pa).

p sat T bed: saturation vapor pressure at bed temperature (Pa).

The temperature-dependent parameterization of the GAB constants and the numerical inversion used during dynamic simulation are described in Supplementary Material B.

The transient approach to equilibrium is represented using the linear driving force (LDF) approximation (Young et al., 2021). The LDF approximation provides a computationally efficient representation of mass-transfer limitations and is commonly used in reduced-order adsorption models (Sendi et al., 2022). In this formulation, the adsorption rate is proportional to the difference between the instantaneous and equilibrium loadings, as given by Equation 5, while the temperature-dependent kinetic constant is given by Equation 6:

d q i dt = k q i , eq − q i (5)

k = k ref ⁡ exp − E a R 1 T − 1 T r e f (6)

q i: instantaneous loading of species i (mol kg^-1 sorbent).

q i , eq: equilibrium loading of species i (mol kg^-1 sorbent).

k: kinetic rate constant (s^-1).

k r e f: reference kinetic constant at temperature T r e f (s^-1)

E a: activation energy (J mol^-1).

R: universal gas constant (J mol^-1 K⁻¹).

T: temperature (K).

The temperature dependence of the kinetic constant follows an Arrhenius-type relationship (Low et al., 2023). Additional kinetic modifiers used in the numerical implementation, including fast/slow pool partitioning, stage-dependent multipliers, and humidity-dependent modifiers, are described in Supplementary Material B, F.

Gas-phase CO₂ dynamics within the collector are represented using a dry-gas inventory formulation (Schellevis et al., 2024; Rezo et al., 2026). The dry-gas inventory is defined in Equation 7:

n d r y = P d r y V g R T b e d (7)

n d r y: dry gas inventory (mol).

P d r y: dry gas pressure (Pa).

V g: gas-phase void volume in the collector (m³).

R: universal gas constant (J mol^-1 K⁻¹).

T b e d: bed temperature (K).

The inversion of inlet and outlet flows from the prescribed pressure trajectory, along with smoothing operations and numerical rate limits, is described in Supplementary Material C, D. All remaining variables, parameter values, and default constants are summarized in Supplementary Material A.

2.4 Energy formulation and performance metrics

This section summarizes the compact energy formulation and performance metrics used in the reduced-order model. Only the core energy relationships are presented to illustrate the thermodynamic structure of the model. Detailed component-level formulations for fan power, vacuum pumping, regeneration heating, CO₂ compression, efficiency corrections, and auxiliary adjustments are provided in Supplementary Material E.

The temperature evolution of the sorbent bed is represented by the first-order thermal response in Equation 8, which relaxes toward a stage-dependent command temperature. Stage-specific effective thermal time constants are used for vacuum, heating, desorption, cooling, pressurization, and adsorption (Wurzbacher et al., 2016). This formulation captures the dominant thermal lag associated with heating and cooling of the bed.

d T b e d d t = T c m d − T b e d τ T (8)

T b e d: average sorbent bed temperature (K).

T c m d: imposed temperature trajectory representing the process heating or cooling condition (K).

τ T: stage-specific effective thermal time constant representing the lumped heat-transfer response of the bed (s).

The instantaneous sensible heat requirement, given by Equation 9, accounts for heating of the solid sorbent and the gas contained in the bed void space (Stampi-Bombelli et al., 2020).

Q ˙ s e n s = m s C p , s + ε V bed c CO 2 C p , CO 2 , g + c H 2 O C p , H 2 O , g + c air C p , air , g dT b ed dt (9)

m s: sorbent mass (kg).

C p , s: specific heat capacity of the sorbent (J kg^-1 K⁻¹).

ε: bed void fraction (−)

V bed: total bed volume (m³).

c i: molar concentration of gas species i (mol m^-3).

C p , i , g: molar heat capacity of gas species i (J mol^-1 K⁻¹).

Adsorption and desorption involve reaction heat associated with CO₂ and H₂O uptake on the sorbent (Young et al., 2021; Sendi et al., 2022). The adsorption heat rate is given by Equation 10. Positive heat release corresponds to exothermic adsorption, while desorption results in heat consumption. The fast and slow pools represent effective kinetic timescales rather than independently parameterized chemical species. A single effective adsorption enthalpy is therefore applied to the total loading change.

Q ˙ ads = − m s Δ H CO 2 dq CO 2 dt + Δ H H 2 O dq H 2 O dt (10)

Δ H C O 2: heat of CO₂ adsorption (J mol^-1).

Δ H H 2 O: heat of water adsorption (J mol^-1).

q C O 2: CO₂ loading on the sorbent (mol kg^-1).

q H 2 O: water loading on the sorbent (mol kg^-1).

Bed and steam-related thermal duties are first converted to electrical-equivalent heating demand using the modeled coefficient of performance (COP). Total electricity demand is then calculated using Equation 11 by combining this heating demand with vacuum, CO₂ compression, fan, and auxiliary electricity (Sendi et al., 2022; Rezo et al., 2026; Fasihi et al., 2019).

E tot = E heat + E vac + E comp + E fan + E par (11)

E heat: electricity or equivalent energy used for regeneration heating (MWh).

E vac: vacuum pumping electricity (MWh).

E comp: CO₂ compression electricity (MWh).

E fan: air fan electricity for moving ambient air through the collector (MWh).

E par: parasitic or auxiliary electricity consumption (MWh).

A nominal efficiency of 0.60 is used for both vacuum evacuation and product CO₂ compression in the reference calculation. These two contributions are calculated and reported separately in Supplementary Material E.

Two performance metrics are used throughout the model evaluation and results sections: annual productivity and specific energy consumption (SEC), defined by Equations 12, 13, respectively:

P rod = m CO 2 , cyc N cyc , yr 1000 (12)

SEC = E tot m CO 2 , cyc MWh tCO 2 − 1 (13)

Prod: annual CO₂ captured per collector (tCO₂ yr^-1).

m C O 2 , cyc: mass of CO₂ captured per cycle (kg).

N cyc , yr: number of cycles completed per year (−)

SEC: specific energy consumption (MWh tCO₂ ⁻¹).

E tot: total energy consumption per cycle (MWh).

To maintain computational efficiency while preserving realistic cycle behavior, the adsorption and desorption stages are terminated using the criteria in Equations 14, 15, respectively.

q CO 2 t − q CO 2 , 0 q CO 2 , eq T , p CO 2 , 0 , q H 2 O − q CO 2 , 0 ≥ η ads (14)

q CO 2 t ≤ q target (15)

q C O 2 , 0: initial CO₂ loading at the start of adsorption (mol kg^-1).

η ads: adsorption equilibrium-approach threshold (−)

The nonzero residual loading represents incomplete regeneration caused by strongly bound species and practical temperature-vacuum limitations.

The sorbent loading states q C O 2 and q H 2 O act as memory variables that determine the working capacity available in the next cycle, the humidity-dependent adsorption behavior, and the regeneration energy requirement. Because of this cycle-to-cycle coupling, system performance under variable environmental conditions cannot be represented using independent steady-state calculations. Instead, sequential ambient condition-driven simulations are required to capture path-dependent behavior (Young et al., 2021; Sendi et al., 2022). For large-scale analysis, the reduced-order model is used to generate a dataset spanning environmental conditions, operating controls, and system design parameters. After applying physical feasibility screening, approximately 120,000 cycle-resolved simulations are produced for subsequent surrogate-model development. All remaining variables, parameter values, and default constants are summarized in Supplementary Material A. A bounded dry-condition adjustment is applied only to vacuum electricity when the minimum regeneration water activity falls below 0.25. Other optional RH-dependent and H₂O energy corrections were disabled for all simulations reported in this study. The total thermal duty is computed by combining sensible and adsorption heat contributions and integrating over the cycle duration.

2.5 Surrogate modeling and optimization

To enable computationally efficient optimization of cyclic operation, a control-oriented surrogate model is constructed to approximate the cycle-resolved dynamics of the TVSA system. While the physics-informed ROM captures the underlying thermodynamic and kinetic behavior, its direct use within optimization frameworks remains computationally expensive for long-horizon or real-time applications.

To address this limitation, a data-driven surrogate model is developed using a feedforward neural network (NN). The surrogate is trained on the cycle-resolved dataset generated by the ROM, where each sample represents a single cycle mapping between system inputs and outputs. The NN architecture consists of a single hidden layer with 39 neurons and sigmoid activation functions. The model takes 13 normalized input variables, including environmental conditions, control variables, and selected state descriptors, and predicts 5 outputs representing system performance and state evolution (see Supplementary Material Appendix F).

The dataset was divided once using a fixed random seed into 80% training data and 20% held-out validation data. The validation samples were excluded from neural-network weight updates. Training was performed using the Adam optimizer with a learning rate of 1 × 10⁻⁴ over 5,000 epochs, with min–max normalization applied consistently to the two subsets. Surrogate fidelity was evaluated on the held-out validation subset using ROM–NN parity plots and R ², RMSE, and MAPE for CO₂ productivity and specific energy consumption. The training behavior and held-out validation results are provided in Supplementary Figure S6 and Supplementary Table S7. A compact architecture comparison is provided in Supplementary Table S8.

With the surrogate model established, the cyclic operation of the system is optimized using a model predictive control (MPC) framework (Rawlings et al., 2017). At each cycle index k, the controller predicts system evolution over a finite horizon (N_h) and computes optimal control actions. The MPC optimization problem is formulated in Equations 16–20:subject to: where:

min u , y ∑ k = 0 N h − 1 J k (16)

s . t . x k + 1 , y k = f T V S A x k , u k , d k ∀ k ∈ 0 , … , N h (17)

x 0 = x ^ (18)

u _ ≤ u k ≤ u ¯ ∀ k ∈ 0 , … , N h (19)

y _ ≤ y k ≤ y ¯ ∀ k ∈ 0 , … , N h (20)

x k: system states at cycle k

u k: control variables (e.g., regeneration temperature, vacuum pressure, timing).

d k: environmental disturbances (temperature, humidity, CO2 concentration).

y k: system outputs (energy consumption, productivity, etc.)

The system dynamics f T V S A are approximated using the NN surrogate model.

The stage-wise objective function is defined in Equation 21:where:

J k = c e k y e k + i e k y e k − y p k (21)

c e k: electricity price at cycle k ($ MWh^-1).

i e k: electricity carbon intensity at cycle k (t-CO₂ MWh^-1).

y e k: energy consumption per cycle (MWh).

y p k: CO₂ productivity per cycle (tCO₂).

The first term represents operational energy cost, while the second term captures net carbon performance by accounting for emissions associated with energy consumption and CO₂ captured per cycle. This formulation allows the controller to balance economic and performance objectives under time-varying conditions.

The optimization problem is implemented using the Pyomo optimization framework (Bynum, 2020), with the NN surrogate embedded through the OMLT toolkit (Ceccon et al., 2022). At each cycle, the optimal control action u k is computed, only the first action is applied, and the system state is updated before solving the optimization problem again at the next cycle. By replacing the physics-informed ROM with the surrogate model, the computational cost of repeated evaluations is significantly reduced. This enables efficient exploration of optimal operating strategies under dynamic environmental conditions while preserving the path-dependent behavior of the system.

2.6 Cycle-resolved dataset structure and physics-to-surrogate workflow

Beyond simulating adsorption system performance, the ROM is also used as a data-generation engine for large parametric studies. The overall modeling workflow linking physics-informed simulation and surrogate modeling is illustrated in Figure 1. Instead of storing only annual summary results, the workflow records cycle-resolved information for each simulation. For every simulated cycle, the dataset stores key model inputs and outputs, including:

• Ambient conditions (e.g., ambient temperature, relative humidity)
• Controllable operating variables (e.g., regeneration temperature, vacuum pressure, and stage durations)
• Internal state descriptors (e.g., sorbent loading states and bed temperature)
• Resulting performance metrics (e.g., productivity and electricity consumption)

Recording these variables at the cycle level preserves the dynamic coupling between environmental conditions, system states, and system performance. As a result, the dataset can be used to train surrogate models that approximate the behavior of the ROM. The surrogate takes environmental conditions, operating variables, and relevant state descriptors as inputs and predicts quantities such as CO₂ productivity and energy demand. In this study, carbon-aware performance is reported using the positive-form metric, N e t p r o d u c t i v i t y = P C O 2 − E e l e c × C I e l e c, so larger values indicate better net CO₂ removal performance. Once trained, this surrogate layer enables rapid evaluation of system performance and supports large-scale optimization under time-varying environmental conditions.

In the present study, the ROM formulation, model evaluation, large-scale dataset generation, surrogate training, and proof-of-concept optimization are integrated within a single physics-to-surrogate workflow. This cycle-resolved representation is essential for capturing path-dependent behavior that cannot be represented by steady-state or cycle-averaged models.

3 Model evaluation and benchmark assessment

The ROM is evaluated for its intended role as a computationally efficient, control-oriented state-transition model. The evaluation includes: (i) physical consistency of cycle trajectories, (ii) comparison of productivity and electricity demand with literature benchmarks, and (iii) consistency of environmental-response trends. These assessments evaluate fitness for comparative screening, dataset generation, surrogate development, and cyclic optimization rather than equipment-level prediction of a specific DAC plant.

3.1 Cycle-level physical consistency

Figure 2 presents representative TVSA cycles simulated using the reduced-order model under three ambient conditions: Figure 2a shows Scenario A, cold/dry conditions (4 °C, 20% RH); Figure 2b shows Scenario B, moderate conditions (20 °C, 50% RH); and Figure 2c shows Scenario C, hot/humid conditions (40 °C, 90% RH). The figure tracks the temporal evolution of CO₂ loading, water loading, bed temperature, water activity, and system pressure across the cycle stages. The temperature and humidity trajectories are prescribed through actuator-level setpoints (T_cmd, RH_cmd), which define feasible operating envelopes without explicitly resolving equipment-level dynamics. The actual bed temperature (T_bed) follows the imposed trajectory with finite thermal inertia. Gas-phase humidity is represented through water activity, where a_w,gas reflects vapor-phase conditions and a_w,eff governs sorbent equilibrium.

Across all ambient conditions, consistent stage-wise behavior is observed. During the heating stage, the bed temperature increases toward the regeneration setpoint under low pressure, initiating desorption and causing rapid decreases in both CO₂ and water loading. In the desorption stage, loading continues to decline toward a non-zero residual level associated with strongly bound species (Young et al., 2021; Sendi et al., 2022). During cooling, externally imposed heat rejection drives the commanded bed temperature toward ambient conditions while pressure remains near the vacuum level. Cooling-equipment and coolant-flow dynamics are not explicitly resolved, and adsorption remains limited because of the low CO₂ partial pressure. Pressurization restores ambient pressure and humidity, and in the adsorption stage, both CO₂ and water uptake increase as the sorbent approaches equilibrium with the incoming air stream.

While Figure 2 can be interpreted as representative cycle profiles, their role in this study is more fundamental. Each cycle should be viewed as a state-transition trajectory, where the system evolves from an initial state to a new post-cycle state. Importantly, the CO₂ loading at the end of adsorption does not exactly match the loading at the start of the subsequent vacuum stage (t = 0), indicating that the system does not fully return to an identical state after a single cycle. This mismatch reflects incomplete regeneration and residual loading, which propagate cycle-to-cycle memory and directly influence the working capacity available in the subsequent cycle.

In this framework, system behavior is not defined by a single steady cycle, but by the evolution of states across successive cycles. The start and post-cycle states form the basis of the dataset used in this work, where each simulation maps an initial condition (q_CO2,start, q_H2O,start) and operating conditions to a new state (q_CO2,post, q_H2O,post).

Furthermore, the cycle trajectories are not fixed. They vary systematically with operating parameters such as adsorption time, desorption time, regeneration temperature, and vacuum pressure. This dependence is explicitly demonstrated in Supplementary Figure S4 through controlled perturbations in adsorption time and regeneration severity. Shortened adsorption reduces CO₂ uptake, while milder regeneration conditions increase residual loading, both of which directly affect cycle-to-cycle state evolution.

As a result, system performance emerges from the interaction between intra-cycle dynamics and inter-cycle state evolution. Building on this formulation, the reduced-order model is used to simulate cyclic operation over extended time horizons (e.g., year-long operation under varying environmental conditions), enabling cycle-by-cycle optimization and capturing the cumulative effects of dynamic operating strategies.

3.2 Benchmark comparison with literature data

To provide a quantitative benchmark, the ROM predictions are compared with the 12-point productivity and electricity dataset reported by Sendi et al. Because one bounded energy correction was informed by this dataset, the electricity comparison is treated as a calibration-consistency assessment rather than a fully independent validation. The benchmark spans combinations of temperature and relative humidity that represent typical environmental conditions for atmospheric CO₂ adsorption systems. The full comparison table is provided in Supplementary Table S3. For each case, the pointwise percentage error was calculated as the absolute difference between the ROM prediction and the reference value, divided by the absolute reference value and multiplied by 100. The mean absolute percentage error was then calculated as the average of the 12 pointwise percentage errors. Across the 12 benchmark cases, the ROM reproduces reference productivity and electricity consumption with a mean absolute percentage error of 10.2% for productivity and 11.4% for electricity demand. These errors are evaluated together with cycle-level physical consistency and trend agreement to assess fitness for comparative screening and optimization; they are not treated as a universal acceptance threshold or plant-level validation.

The largest deviations occur under extreme environmental conditions. The maximum productivity deviation (17.7%) appears at high temperature and low humidity, while the largest electricity deviation (19.3%) occurs under very low temperature conditions. These boundary-condition deviations likely reflect the combined effects of nonlinear adsorption behavior and the ROM’s lumped 0-D, prescribed-driver, and simplified energy assumptions. Given the reduced-order nature of the model, perfect numerical agreement is not expected. Instead, the goal is to reproduce the magnitude, direction, and relative sensitivity of key performance metrics.

Beyond pointwise accuracy, the benchmark assessment also evaluates whether the ROM preserves the directional response of system performance to environmental changes. Trend agreement metrics (TrendG) and sign-consistency analysis (SignAgree) show that the model captures the dominant response of productivity and electricity consumption to variations in temperature and humidity across the benchmark space. This trend fidelity is particularly important for reduced-order models used in large parametric studies, where capturing the correct sensitivity to environmental drivers is often more critical than exact numerical agreement. Overall, the benchmark comparison indicates that the ROM provides a physically consistent representation of cyclic adsorption performance across the evaluated temperature and humidity conditions.

3.3 Model accuracy and limitations

Although the ROM reproduces the overall trends of the benchmark dataset, moderate deviations appear under some extreme temperature–humidity conditions. This framework enables computationally efficient simulation of large numbers of cycles under time-varying environmental conditions. As discussed in Section 3.2, the largest discrepancies occur under high-temperature-low-humidity and very low-temperature conditions. These deviations arise primarily from simplifications in the reduced-order formulation. The model represents the sorbent bed using a lumped zero-dimensional structure, applies prescribed temperature and pressure trajectories for the cycle stages, and uses simplified gas-phase and energy formulations. While these assumptions significantly reduce computational cost, they can introduce error when adsorption behavior becomes strongly nonlinear, particularly under extreme environmental conditions. Such tradeoffs are expected for reduced-order adsorption models designed for large parametric studies. In the present framework, the ROM prioritizes physical consistency and computational efficiency rather than exact pointwise agreement with detailed models. This design enables long-term ambient condition-driven simulations and the generation of large cycle-resolved datasets for subsequent surrogate-model development.

The use of lumped assumptions, prescribed driver trajectories, and simplified gas-phase dynamics introduces known limitations, particularly under strongly nonlinear conditions. The literature benchmark provides an aggregate assessment of these reduced-order errors, but the isolated contribution of neglected axial gradients cannot be quantified without comparison to a spatially distributed model. Accordingly, the surrogate assessment quantifies fidelity to the ROM outputs rather than eliminating the structural uncertainty associated with the 0-D formulation. However, the model includes the dominant dependencies of adsorption performance on temperature, humidity, and operating conditions, making it suitable for dataset generation and surrogate modeling workflows. These limitations reflect the intended trade-off between computational efficiency and physical detail. Because the surrogate is trained on ROM-generated data, ROM uncertainty also propagates into the optimization results. Accordingly, the reported improvements should be interpreted as relative, model-based estimates under a common set of assumptions rather than as plant-level performance guarantees. Vacuum electricity is also sensitive to the assumed pump efficiency; however, the representative-condition sensitivity analysis indicates that this assumption changes the magnitude of total SEC without altering the qualitative ranking across ambient conditions.

4 Results

4.1 Sensitivity analysis

A one-at-a-time (OAT) sensitivity analysis was conducted to assess the local influence of environmental conditions, operating controls, and system-design parameters on CO₂ productivity and specific electricity consumption. Each parameter was varied individually between the low and high values summarized in Supplementary Table S4 and annotated in Figure 3, while all other model inputs and initial sorbent states were held fixed at their reference values. This OAT analysis was conducted independently of the surrogate-training dataset.

For CO₂ productivity (Figure 3a), regeneration temperature is the dominant parameter, with an increase from 70 °C to 130 °C leading to a reduction in productivity of approximately 10 tCO₂ m^-3 bed yr^-1. This behavior reflects the strong trade-off between regeneration completeness and cycle time, where higher temperatures enhance desorption but reduce overall cycle efficiency due to increased thermal loading and diminished adsorption capacity in subsequent cycles. Among the operational parameters, desorption time exhibits a strong positive influence, with an increase from 14,000 s to 26,000 s improving productivity by approximately 4–5 tCO₂ m^-3 bed yr^-1. This highlights the importance of regeneration completeness in systems with non-negligible residual loading. In contrast, adsorption time has a more moderate effect, indicating that cycle performance is more strongly limited by incomplete desorption than by adsorption kinetics under the examined conditions.

Environmental variables also play a significant role. Increasing ambient temperature from 4 °C to 45 °C improves productivity by approximately 3–4 tCO₂ m^-3 bed yr^-1, primarily due to reduced adsorption affinity penalties at low temperatures. Similarly, higher inlet CO₂ mole fraction enhances productivity by increasing the driving force for adsorption. In contrast, increasing altitude reduces productivity due to lower total pressure and reduced CO₂ partial pressure.

Electricity consumption (Figure 3b) exhibits a distinct sensitivity structure. Regeneration temperature remains the most influential parameter, with higher temperatures significantly increasing electricity demand (on the order of 0.7–0.8 MWh tCO₂ ⁻¹ across the examined range). Ambient humidity is the second most important factor, where increasing RH from 0.1 to 0.9 raises electricity consumption by approximately 0.5 MWh tCO₂ ⁻¹. This behavior is directly associated with water co-adsorption, which increases the thermal energy required for regeneration.

Additional contributions arise from CO₂ mole fraction, altitude, and adsorption time. Higher CO₂ concentration reduces electricity intensity by improving capture efficiency per cycle, while higher altitude increases electricity demand due to reduced gas density and increased volumetric processing requirements. Operating parameters such as feed velocity and vacuum pressure also influence energy consumption, though their impact is secondary compared to thermal effects.

The sensitivity analysis reveals a strongly asymmetric and nonlinear response of system performance to parameter variation. Productivity is primarily governed by regeneration completeness and cycle throughput, while electricity consumption is dominated by thermal requirements and water co-adsorption effects. Because the OAT analysis varies one parameter at a time, it provides a local screening of individual effects and does not quantify parameter interactions or global variable importance. Combined variations among environmental, operating, design, and state variables are considered separately through the Latin-hypercube dataset and surrogate-based optimization framework. A targeted one-factor sensitivity analysis was also conducted for vacuum-pump efficiency over a range of 0.30–0.80. At the reference efficiency of 0.60, vacuum electricity represented approximately 12%–20% of total SEC across the three representative ambient conditions. Reducing the efficiency to 0.30 increased total SEC by 12.1%–19.7%, whereas increasing it to 0.80 reduced total SEC by 3.0%–4.9%. The ambient-condition ranking remained unchanged, with the hot/humid case retaining the highest total electricity demand. Detailed results are provided in Supplementary Table S6 and Supplementary Figure S5.

4.2 Ambient condition-driven performance variation

System performance varies significantly with environmental conditions, as demonstrated by the hourly input profiles shown in Supplementary Figures S1–S3. These figures illustrate the temporal variability of ambient temperature, relative humidity, electricity carbon intensity, and electricity price for the two representative locations, Atoka, Texas and Madison, Wyoming.

Supplementary Figure S1 shows that Atoka, Texas experiences higher average temperatures and consistently higher humidity compared to Madison, Wyoming, which exhibits lower temperatures and generally drier conditions. These differences directly affect adsorption thermodynamics. Higher humidity in Atoka leads to increased water co-adsorption, reducing effective CO₂ capacity and increasing regeneration energy demand. In contrast, the drier conditions in Madison favor CO₂ adsorption capacity but require longer or more energy-intensive regeneration to overcome lower baseline temperatures.

In addition to temperature and humidity differences, Supplementary Figures S2, S3 show strong temporal variability in electricity carbon intensity and price. Both locations exhibit significant intra-day and seasonal fluctuations, with periods of low carbon intensity and low electricity price interspersed with higher-intensity intervals. As a result, the same control policy does not remain optimal throughout the year. Periods with lower electricity carbon intensity are more favorable for regeneration, whereas periods with lower humidity are more favorable for maintaining higher adsorption capacity.

The combined variability in temperature, humidity, and electricity characteristics leads to substantial cycle-to-cycle variation in system performance. As a result, adsorption capacity, regeneration energy, and overall productivity evolve dynamically over time rather than remaining fixed. These results confirm that system behavior cannot be represented by a single operating point or steady-state assumption. Instead, performance emerges from the interaction between environmental forcing and cycle-resolved state evolution. Importantly, the temporal structure observed in Supplementary Figures S1–S3 provides the underlying driver for the optimization results presented in Section 4.4. By aligning cycle-level control actions with favorable environmental and grid conditions, the system can exploit this variability to improve both energy efficiency and carbon-aware performance. These results support the use of ambient condition-resolved, cycle-level modeling for evaluating adsorption-system operation.

4.3 Large-scale dataset generation

The ROM was used to generate a dataset of 120,000 cycle-resolved simulations spanning environmental conditions, operating controls, system design parameters, and initial sorbent states. The dataset was generated using Latin hypercube sampling of 30,000 environmental, operating, and design combinations, each paired with four feasible initial CO₂ and H₂O loading states. Physically infeasible cases were removed using thermodynamic and operational screening before surrogate training. This sampling strategy provides broad coverage of the modeled operating domain while preserving cycle-to-cycle state variability.

The dataset reveals a well-defined structure in system performance, even prior to surrogate training. CO₂ productivity and electricity consumption exhibit a consistent trade-off across operating conditions, indicating that system performance is governed by a multidimensional response surface rather than a single optimal operating point. Because the surrogate model is trained on the ROM-generated dataset, the present study focuses not only on dataset generation but also on demonstrating how the dataset supports control-oriented optimization. The resulting dataset provides the basis for surrogate-model training, control-oriented prediction, and the optimization analysis presented in Section 4.4.

4.4 Cycle-resolved surrogate-based optimization

Before comparing annual summary metrics, the cycle-resolved cumulative behavior of the system was examined under baseline and optimized operation. Figure 4 shows the cumulative evolution of electricity demand, CO₂ productivity, and operational emissions during year-long operation for Atoka, Texas and Madison, Wyoming under dynamically varying ambient and electricity conditions. The baseline strategy uses fixed cycle conditions and completes approximately 1,094 cycles in both locations. In contrast, the optimized strategy adjusts cycle-level operation in response to changing temperature, humidity, electricity price, and electricity carbon intensity, increasing the annual cycle count to approximately 1,449 cycles in Atoka, Texas and 1,379 cycles in Madison, Wyoming. The resulting trajectories show how cycle-level operational changes accumulate over time into meaningful differences in energy use, CO₂ productivity, and operational emissions. The divergence between baseline and optimized trajectories becomes progressively larger over the annual horizon, describing how cycle-level control adjustments accumulate into substantial system-level performance differences. These cumulative trends provide the dynamic basis for the annual performance comparisons summarized in Figure 5.

The cycle-resolved dataset generated by the ROM was used to train a surrogate model and evaluate dynamic optimization across representative locations. Figure 5 compares baseline and optimized operation of the TVSA system across representative locations. In the baseline case, all control variables are fixed throughout operation, independent of environmental conditions. In contrast, the optimized strategy dynamically adjusts cycle-level operating conditions in response to time-varying inputs, including temperature, humidity, electricity price, and carbon intensity of electricity.

To quantify carbon-aware performance, the net productivity metric is defined in Equation 22:where P C O 2 is CO₂ productivity, E e l e c is electricity demand, and C I e l e c is the carbon intensity of electricity. This metric represents the net CO₂ removal after accounting for emissions associated with energy use. As shown in Figure 5, optimization leads to clear improvements in electricity demand and CO2 productivity. For Atoka, Texas, electricity demand decreases from 48 to 39 MWh, corresponding to an 18.8% reduction, while productivity increases from 55 to 69 tCO₂, corresponding to a 25.5% increase. For Madison, Wyoming, electricity demand decreases from 56 to 51 MWh, corresponding to an 8.9% reduction, while productivity increases from 58 to 71 tCO₂, corresponding to a 22.4% increase.

N e t P r o d u c t i v i t y = P C O 2 − E e l e c × C I e l e c (22)

The carbon-adjusted net productivity also improves substantially under the optimized strategy. For Atoka, Texas, net productivity increases from 21 to 41 tCO₂, corresponding to an absolute gain of 20 tCO₂ and a relative increase of 95.2%. For Madison, Wyoming, net productivity increases from 18 to 35 tCO₂, corresponding to an absolute gain of 17 tCO₂ and a relative increase of 94.4%. Electricity cost trends further illustrate the trade-off between economic and carbon-aware objectives. For Atoka, Texas, total electricity cost decreases from $1,442 to $1,420, corresponding to a 1.5% reduction. In contrast, for Madison, Wyoming, electricity cost increases from $1,697 to $1759, corresponding to a 3.7% increase. This behavior indicates that the optimization framework does not uniformly minimize all objectives, but instead balances electricity use, carbon intensity, and operating performance under time-varying conditions.

These results show that cycle-resolved optimization can shift annual performance relative to fixed operation. By coupling dynamic state evolution with time-varying external conditions, the system can achieve simultaneous improvements in productivity, electricity efficiency, and carbon-aware performance outcomes that are not accessible under fixed operating strategies. Additional surrogate-based Pareto analyses are provided in Supplementary Material Appendix G.

5 Conclusion

This study developed a physics-informed dynamic ROM for cyclic atmospheric CO₂ separation under time-varying environmental conditions and showed that system behavior is governed not only by operating conditions, but also by path-dependent state evolution across cycles. The reference cycle analysis highlights that adsorption and desorption do not return the system to an identical state, leading to persistent memory effects that influence subsequent cycle behavior. Comparison with literature benchmarks indicates that the ROM captures the dominant trends and approximate magnitudes of productivity and electricity demand across the evaluated conditions, supporting its intended use for large-scale parametric analysis and surrogate-based optimization.

Using the ROM, approximately 120,000 cycle-resolved simulations were generated across environmental, operational, and design spaces. The resulting dataset reveals a structured performance landscape characterized by strong trade-offs between productivity, electricity demand, and environmental conditions. Sensitivity analysis shows that regeneration temperature, humidity, and cycle timing parameters dominate system behavior, while geometric parameters play a secondary role in the sensitivity analysis.

Cycle-resolved optimization under time-varying conditions leads to substantial improvements in operational performance. Electricity demand is reduced by up to 18.8%, while CO₂ productivity increases by up to 25.5% across the representative locations. These gains translate into substantial increases in carbon-adjusted net productivity, with improvements of 95.2% in Atoka, Texas and 94.4% in Madison, Wyoming. However, the results also show that economic and carbon-aware objectives do not always align, as electricity cost may decrease or increase depending on local grid conditions, highlighting the importance of explicitly accounting for temporal variability in electricity price and carbon intensity. Importantly, these improvements are not achieved solely by reducing total energy consumption, but by shifting energy-intensive steps to periods with more favorable environmental and grid conditions. This demonstrates that system performance depends not only on how much energy is used, but also on its temporal alignment with ambient and grid conditions.

Overall, this work shows a scalable physics-to-surrogate framework that integrates adsorption physics, dynamic simulation, large-scale dataset generation, and optimization. By enabling efficient exploration of control strategies under realistic, time-varying conditions, the proposed approach provides a computational framework for evaluating adsorption systems where environmental variability and grid interactions are significant.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fenve.2026.1881554/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Cristina Gutiérrez, Iberian Centre for Research in Energy Storage, Spain

Reviewed by: Miriam Navarrete, Universidad Autónoma del Estado de Morelos, Mexico

Udit Gupta, Siemens, Germany