1 DAC Thermodynamic Simulation: MATLAB Simscape↩︎

Figure 1 illustrates a detailed thermal-fluid management model constructed in MATLAB/Simulink using the Simscape Fluids and Simscape Thermal libraries. The system is centered on a solar-thermal source, represented here by a “Solar Heat-Up Deck” that delivers heat to a dedicated thermal storage reservoir. In the parameterized simulation model connected to optimization model, the solar thermal flux data is automatically generated and fed to the simulation as input. From this reservoir, heat flow can be directed through a thermal switch that toggles between heating and cooling pathways, depending on DAC system temperature demands. The heating path circulates hot fluid (using forced convection steam heat transfer coefficient) to supply energy for processe: direct air capture (DAC) regeneration, while the cooling path leverages a thermal cooling subsystem to dissipate heat when required (cooling to ambient temperature for initiating adsorption again). A series of pumps, and flow/thermal controllers (using variable thermal resistor) regulate fluid/heat circulation, allowing for dynamic control of heat transfer. Temperature sensors and feedback control loops are integrated throughout the model to monitor the chamber environment (i.e., the DAC chamber which contains DAC thermal unit as a thermal mass), adjust heat transfer rates, and maintain user-defined setpoints. This multi-domain approach—combining thermodynamics, thermal energy storage, and control algorithms—enables simulation of both steady-state and transient behavior under variable conrtol signal or data input conditions. By capturing the interactions among heat sources, storage elements, and demand-side processes, the model facilitates comprehensive performance evaluation and design optimization of solar-based thermal systems and detailed DAC unit thermodynamic behavior, ultimately guiding system design parameters and verification of the simplified optimization model.

2 Optimization Methodology: Extended↩︎

In this section, a more detailed version of the methodology for DAC operation optimization is presented, including all constraints for the mixed-integer linear programming (MILP) optimization model written in Julia, and the custom strategic bidding algorithm algorithm written in MATLAB for cyclic DAC strategic operation which greatly improves the computational speed.

2.1 MILP for Cyclic DACs↩︎

The mixed-integer linear programming (MILP) is a rigorous optimization formulation for linearizing and solving inherently nonlinear and nonconvex process models. While MILP provides accurate profit-maximization results with optimality-bound guarantees, it is computationally expensive and practically unsolvable for technologies with long cycling periods or long optimization horizons.

Indexes:

• \(t\): Time period index \(t \in \{1,2,...,T\}\), a total of T time periods in optimization horizon.

Binary decision variables:

• \(u_t\): 1 if DAC is during the absorption phase at time period \(t\), otherwise zero.

• \(v_t\): 1 if DAC is during the desorption phase at time period \(t\), otherwise zero.

• \(z_t\): 1 if DAC is switched to a new cycle at time period \(t\), otherwise zero

• \(k_t\): introduce new sign-variable to determine the change of status at time period \(t\) to facilitate calculation of \(z_t\)

Continuous decision variables:

• \(X_t\): state-of-saturation capacity of DAC system at time period \(t\)

• \(a_t\): absorption amount of DAC system at time period \(t\)

• \(d_t\): desorption amount of DAC system at time period \(t\)

• \(h_t\): available thermal energy at time period \(t\)

System input parameters:

• \(\lambda_t\): electricity price at time period \(t\), can subjected to modification of \(CO_2\) (see \(\lambda e_t\))

• \(s_t\): solar thermal charging at time period \(t\), given be data

• \(e_t\): electricity \(CO_2\)-intensity at time period \(t\)

• \(\rho_e\): \(CO_2\) value of carbon for electricity, 0 if for wholesale electricity price case, \(\pi\) if for carbon-tax adjusted electricity price case.

• \(\lambda e_t\): electricity price without \(CO_2\)-intensity correction (\(\lambda_t = \lambda e_t + \rho_e e_t\))

• \(\eta_{t}\): adsorption rate correction factor at time period \(t\), which is a function of relative humidity and temperature.

• \(C_{t}\): energy consumption correction factor at time period \(t\), which is a function of relative humidity and temperature.

• \(\pi\): \(CO_2\) value constant including selling price, subsidies, carbon-tax, etc.

DAC input parameters:

• \(P^a\): electricity consumption for absorption phase per unit time period

• \(P^d\): electricity consumption for desorption phase per unit time period

• \(\bar{X}\): max available DAC capacity

• \(\bar{h}\): max available thermal energy storage capacity

• \(\eta\): thermal energy loss efficiency, \(0 \leq \eta \leq 1\)

• \(S\): switching cycle cost for consumption of sorbent

• \(H\): heating thermal energy load during desorption phasematerials

Piecewise linear approximation for absorption/desorption rate using quadratic coefficients:

• \(\beta^{a}_{1}\): first-order coefficient for absorption

• \(\beta^{a}_{2}\): second-order coefficient for absorption

• \(\beta^{d}_{1}\): first-order coefficient for desorption

• \(\beta^{d}_{2}\): second-order coefficient for desorption

The objective function maximizes the total profit of the DAC operation.

\[max \sum_t \; \pi d_t - \lambda_t C_{t} (P^a u_t + P^d v_t)-Sz_t\] where \(\pi d_t\) is the total revenue by captured (desorbed) \(CO_2\), minus the cost of power consumption corrected by ambient conditions at each time step \(\lambda_t C_{t} (P^a u_t + P^d v_t)\), minus the cost of material consumption for switching cycles \(Sz_t\).
Binary constraints for absorption/desorption status:

\[u_t + v_t \leq 1\] where the DAC system can only be adsorption/desorption at one time period, it can be neither absorbing nor desorbing.
Absorption and desorption rate (using inequality here to avoid contradiction with binary constraints, which is relaxation of the constraints), be mindful that only adsorption (capture) process is corrected by ambient condition of each time step, desorption is a controlled process which is not affected by ambient condition: \[a_t \leq \eta^{a}_{t} (\beta^{a}_{1} + \beta^{a}_{2} X_t)\] \[d_t \leq \beta^{d}_{1} + \beta^{d}_{2} X_t\] additionally, the absorption rate and desorption rate shall be bounded by the binary variable for each as well. \[a_t \leq M u_t\] \[d_t \leq M v_t\] where \(M\) is a sufficiently large number which does not bind the absorption and desorption rate if \(u_t\) and \(v_t\) are 1.

Be careful that although the below constraints are not required to be added explicitly, they shall be automatically satisfied with the above absorption and desorption rate constraints: \[0 \leq a_t \leq a_{max}\] \[0 \leq d_t \leq d_{max}\] where absorption and desorption rates are always bounded by 0 and its maximum designed capacity
State-of-saturation capacity of DAC system updates with absorption/desorption rates: \[X_t-X_{t-1} = a_t - d_t\] where the change of state-of-saturation between time periods equals: adding absorption; subtracting desorption. Both absorption and desorption are corrected by the binary decision variables.
\[0 \leq X_t \leq \bar{X}\] where the state-of-saturation is always lower bounded by 0, and upper bounded by its maximum capacity \(\bar{X}\)
Switching cycle constraints: \[k_0 = 0\] where the initial state of the sign-variable equals zero. \[-M(1-k_t) \leq k_{t-1} + (u_t - v_t) - 0.5\] \[M k_t \geq k_{t-1} + (u_t - v_t) - 0.5\] where the sign function is defined here with a sufficiently large \(M\). The comparison is subtracted by 0.5 to avoid the possible case that \(k_{t-1} + (u_t - v_t) = 0\). The solution matrix for \(k_t\) for all possible combinations of sign-function is given in below table:

The above two equations together define the sign-variable \(k_t\) which is used to determine the binary cycle counting variable \(z_t\): \[z_t \geq k_t - k_{t-1}\] where \(z_t\) will be minimized in the objective function that only when \(k_t = 1\) and \(k_{t-1}=0\), \(z_t = 1\).

Eventually, the thermal storage and energy consumption behaviors. Thermal energy bounding constraint: \[0 \leq h_t \leq \bar{h}\]

Thermal energy charging/discharging constraint: \[h_t \leq \eta h_{t-1} + s_t - v_t H\]

These two sets of thermal behavior constraints represents a significant simplification of the comlicated thermodynamic behavior that’s been simulated in the previous mentioned MATLAB Simscape model. The key differences are: (1) completed ignoring the temperature profile of DAC unit itself (see Figure 2), only tracking the thermal energy required for operating the system; (2) all thermal losses to ambient environment in various processes (e.g., storage, thermal switch, DAC unit) are combined into one single energy loss assign to the solar input or as correction of the thermal energy consumption; (3) the heat transfer during the regeneration process linearize the highly non-linear heat profile, which is very high during the heating up stage and moderate in maintaining the required temperature for enough time. These simplifications introduces only trivial differences in DAC operation that the optimization code and Simscape simulation achieve \(>\)​99% matching accuracy.

The above formulation summarizes the optimization framework of the DAC system using the temperature-swing DAC technology (all 3 DAC technologies tested in this paper). In practice, the DAC input parameters including the piecewise linear approximation for absorption and desorption rate using quadratic coefficients will be determined by different DAC specifications. The actual optimization programs use a look-ahead framework (similar concept of MPC control) which looks longer optimization horizon but applies only the first few steps for action.

2.2 Custom Algorithm: Strategic Operation↩︎

A threshold-based formulation is rooted in a custom strategic bidding MATLAB algorithm, specifically designed to handle nonlinear processes. By participating in electricity markets through the bidding process, DAC operators can strategically operate during the low-price periods: activate operation if prices are lower than the threshold and idle if higher, hence mimicking market clearing of flexible demand bids, potentially generating more profit at market price valleys. The optimization problem is therefore greatly simplified to optimizing only 1 variable, the price threshold for each optimization horizon, increasing the computational efficiency.

The operation of the DAC follows exactly the same formulation as MILP, with additional key terms in this formulation including:

• \(\lambda_{\text{opt}}\): The price threshold, or the electricity price at which DAC operators are willing to operate.

• \(c\): A data “chunk" for processing. Each chunk is optimized individually.

A typical chunk size \(c\) is 24 hours, equivalent to 288 time steps (5-min temporal resolution), reflecting daily market price behaviors. However, the algorithm is designed with flexibility, allowing for adjustments in chunk size to cater to different market dynamics or operational needs for different DAC cycle times.

The DAC system’s operation is determined by the threshold, \(\lambda_{\text{opt}}\). The decision to either accept or reject is defined as: \[\begin{cases} \text{Activate} & \text{if } \lambda_t(t)\leq \lambda_{\text{opt}} \\ \text{Idle} & \text{otherwise} \end{cases}\]

For each chunk \(c\), the algorithm employs an iterative process to find the optimal \(\lambda_{\text{opt}}\) that maximizes profit. It utilizes MATLAB’s fminsearch function, which optimizes 1 variable to find the minimum of the objective function. Specifically, fminsearch is tasked to find the value of \(\lambda_{\text{opt}}\) that minimizes the negative profit objective function, effectively maximizing profit.

To circumvent the issue of local minima and to ensure identification of the global minimum, the algorithm iterates over a series of initial guesses \(\lambda_{\text{series}}\) for \(\lambda_{\text{opt}}\) in the range from -10 to 500 USD. One may increase the granularity of the initial guesses to get a marginally better objective with a computation time trade-off. For each initial guess in the \(\lambda_{\text{series}}\), fminsearch conducts an optimization to locate a local maximum. Upon the completion of all such iterations, the algorithm chooses the highest profit from these local maxima as the global maximum. This approach reduces the risk of the optimization process settling for a suboptimal local maximum.

The final determination of \(\lambda_{\text{opt}}^*\) involves selecting the optimal value that corresponds to the maximum profit achieved across all the local optimizations:

\[\lambda_{\text{opt}}^* = \max_{\lambda_{\text{series}}} \text{fminsearch}(-\text{Objective}(c, \lambda_{\text{guess}})).\]

The \(\max\) function in the equation is a representation of the selection process that identifies the highest value from all the local maxima obtained by fminsearch, starting from different initial guesses within the series. This comprehensive exploration across a diverse set of starting points helps in ensuring that the resulting \(\lambda_{\text{opt}}^*\) is the most favorable for maximizing profit.

2.2.1 Look-Ahead Mechanism:↩︎

• \(L\): Look-ahead parameter representing the number of future data points considered during optimization.

• \(t\): Starting time of the given chunk \(c\).

For a given chunk \(c\) starting at time \(t\), the algorithm factors in data from \(t\) to \(t+L\) to find the optimal \(\lambda_{\text{opt}}\). The optimization process employs the MATLAB function fminsearch:

\[\lambda_{\text{opt}}^* = \text{fminsearch}(-\text{Objective}(c, \lambda_{\text{opt}}, L))\]

Where: \[\text{Objective}(c, \lambda_{\text{opt}} , L)\] is the profit obtained for chunk \(c\) when using threshold \(\lambda_{\text{opt}}\)and considering \(L\) future data points.

2.2.2 Boost Mechanism:↩︎

• X_remain: State-of-saturation at the end of the processed chunk. This represents the amount of CO\(_2\) credit still in the system that hasn’t been cleared yet.

• Profit_chunk: Total profit from the processed chunk.

• CO2_chunk: Total amount of CO\(_2\) desorbed during the processed chunk

The boost mechanism is introduced to account for the potential value of the remaining CO\(_2\) in the system at the end of the DAC loop to avoid selling all CO2 at the end of each chunk to maximize profit, which will force the DAC operation to terminate at the end of each optimization horizon. The boost is calculated using the following formula:

\[\text{boost} = \left( \frac{\text{Profit\_chunk}}{\text{CO2\_chunk}} \right) \times \text{X\_remain}\]

Where:

• \(\frac{\text{Profit\_chunk}}{\text{CO2\_chunk}}\) represents the average profit per unit of CO\(_2\) sold during the processed chunk. It indicates the expected value or profit obtained from selling one unit of CO\(_2\).

After calculating the boost, it’s added to the profit for that chunk. This accounts for both the profit from the CO\(_2\) that was sold and the potential value of the CO\(_2\) that remains in the system:

\[\text{Profit\_total} = \text{Profit\_chunk} + \text{boost}\]

However, it’s important to note that this inflated profit is not used when calculating the overall profits. It is solely used for the process of finding the optimal lambda. This ensures that the optimization process considers both actual and potential profits, valuing scenarios where there’s a significant amount of CO\(_2\) left in the system at the end of the loop.

2.3 Ambient correction calculations↩︎

Assuming all DAC nominal capacity and power consumption is measured under standard lab environment with baseline temperature of 20 \(^{\circ}\)C and relative humidity of 50%. We applied the ambient sensitivity from two studies: [1] for cyclic DAC operation based on steam-assisted vacuum-pressure temperature swing adsorption with amine-functionalized solid sorbent; and [2] for continuous liquid solvent KOH DAC.

For cyclic DAC operation, energy consumption prefers mild relative humidity, where relative humidity either too high or low will increase energy consumption with very high sensitivity. Energy consumption is not very sensitive to ambient temperature, slightly in favor of lower temperature. From capture rate or abatement productivity perspective, it strongly in favor of low temperature and mild relative humidity.

The electricity consumption correction after numerical fitting is given by the following, using quadratic equation for temperature \(T\) fitting and exponential function for skewed symmetric behavior of relative humidity \(RH\):

\[C_t = [1.9 + 0.01(T-20)](RH-0.4)^2 e^{RH-0.4}+[1.5+0.003(T-20)^2]\]

The capture rate or abatement correction similarly:

\[\eta_t = 65 - 0.01 T^2 - (T+20)(RH-0.4)^2\]

For continuous KOH liquid solvent DAC, capture rate or abatement productivity prefers high temperature and high relative humidity. Little information about energy consumption’s sensitivity on ambient conditions. It can be explained by the fact that majoroty of power consumption is used for driving moving equipments and thermal energy for regeneration of solvent at high temperature of 800 \(^{\circ}\)C, both insensitive of ambient conditions.

The capture rate or abatement correction after numerical fitting is given by:

\[\eta_t = 74 + 8(RH-0.5) + (T-20)\]

2.4 Solar Thermal Modeling↩︎

The solar heating collector efficiency is a function of both input data (solar DNI) and design parameters.

\[\begin{align} \eta_{c_t} (DNI_t, cr, T) &= 0.78 - \eta_{loss} \\ &= 0.78 - \alpha T^2 * \frac{\beta}{DNI_t + m} * \frac{\gamma}{cr}\\ \end{align}\] the function states that the collector efficiency is baseline (0.78) - loss. The loss is growing with \(T^2\) corrected by \(\alpha\). The loss will reduce when solar input (direct normal irradiation) \(DNI_t\) and solar concentration ratio \(cr\) is larger. For simplicity, the DNI is fitted using solar capacity factor instead of original \(W/m^2\). \(m\) is a small number to avoid calculation error when \(DNI_t\) is 0. After fitting the experimental data in literature, the coefficients are calculated to be.

• \(\alpha = 8.8*10^{-7}\)

• \(\beta = 1.1\)

• \(m = 0.1\)

• \(\gamma = 1\)

So the solar heating collector efficiency is eventually given by:

\[\eta_{c_t} (DNI_t, cr, T) = 0.78 - 8.8*10^{-7} *T^2 * \frac{1.1}{DNI_t + 0.1} * \frac{1}{cr}\]

The available heat is given by the following:

Without thermal energy storage: \[s_t = DNI_t * cp * \eta_{c_t}\]

With thermal energy storage: \[s_t = DNI_t * cp * \eta_{c_t} * \eta_{th}(T)\] where \(\eta_{th}(T)\) is the charging efficiency of the thermal energy storage system, which is a function of target temperature T.

The results of above semi-physics fitting compared to experimental results in literature is shown:

The solar Heating + Storage System CAPEX is given by function of the design parameters:

\[\begin{align} CAPEX_s (cp, cr, \bar{h}) &= CAPEX_{sh} (cp, cr) + CAPEX_{th} (T, \bar{h}) \\ &= CAPEX_{sh}^{unit}*cp*cr*(1-0.15)^{log_2(cp*cr)} + CAPEX_{th}^{unit}*\bar{h}\\ \end{align}\]

Sand thermal energy storage as a special type of storage. Sand is chemically very stable and the energy storage capacity is a function of both operation temperature and sand mass (size of the storage tank). Under the current formulation, the sand mass (size of the storage tank) is determined by the investment on the storage size, while the target temperature is determined by solar collector design.

\[\bar{h}_{effect} = \bar{h} *\frac{T - T_{min}}{T_{unit}}\]

Here the \(T_{unit}\) is the designed temperature increment for energy storage capacity \(\bar{h}\), and the effective energy storage capacity will be changing proportionally with the actual designed temperature increment (\(T - T_{min}\)).

3 Data and Processing↩︎

NY, CA, and TX data collection and processing are presented, and the processed data input for the model can be accessed from Github Repo.

3.1 NY↩︎

5-minute resolution NY pricing data for the “MHK VL” zone was downloaded from NYISO’s operational data website. This zone was chosen for its high number of industrial facilities, resulting in price fluctuations primarily based on demand.

Emission data was sourced from processing NYISO’s Real-Time Fuel Mix data archive, available at a 5-minute resolution for the year 2022. To calculate the Total Adjusted Emission Rate, the Total Demand (MW) was determined by summing up the energy generated from various fuel sources:

\[\text{Total Demand} = \sum_{i} \text{Demand}_{i}\]

Where \(i\) represents each fuel source.

Using standard emission rates for each fuel type, the Total Emission Rate (TER) for New York was determined by taking the dot product of the energy demand values and the respective emission values, resulting in a weighted sum of the emissions.

\[\text{Total Adjusted Emission Rate} = \frac{\text{TER}}{\text{Total Demand}}\]

The Total Adjusted Emission Rate was then calculated using equation (18).

The NY case results are not shown since low solar capacity factor and very large gap to TX and CA cases. As shown in the solar map filter (see below), the NY region is not within the practical areas for solar-thermal DAC systems.

3.2 CA↩︎

Pricing data was downloaded from CAISO OASIS, capturing 5-minute resolution Locational Marginal Price (LMP) data for the year 2022. The node selected for this study was NEWHALL_1_N001, located in Santa Clarita, California. This area was chosen due to the pronounced representation of the solar “duck curve".

5-min resolution CA emission data (measured in ton-CO₂/MWh) was obtained from CAISO’s “Today’s Outlook”. The Total Emission Rate (TER) was then divided by the total demand in MW, sourced from the LMP data, to derive the Total Adjusted Emission Rate.

3.3 TX↩︎

Texas data from ERCOT using the real-time price (RTP) data and fuel mix data. The TER calculation follows the same formula as NY and CA. This paper uses the price data from “HB WEST" of the ERCOT settlement points in west Texas, this is where the largest ongoing DAC project, STRATOS, is located.

ERCOT has a 15-minute resolution for market clearing. To be consistent with the NY and CA 5-minute resolution comparison, this study reconstructed the 15-minute resolution data to 5-minute resolution by repeating each data point three times, where three 5-minute resolution time steps have the same data to create the 15-minute market clearing interval. Any missing data from sources are linearly extrapolated from the nearest available data.

3.4 Temperature and Relative Humidity Data↩︎

3.5 Grid-connection: Site-specific Data↩︎

Ambient environmental data of hourly resolution temperature and relative humidity time series is downloaded from Climate Data Store Climate Data Store “ERA5 pressure levels” based on 1000hPa atmospheric pressure in year 2022 [3]. Details of data description about specific spot data retrieval is in the following table Table 1.

3.6 Global Impact Data↩︎

The global study data is obtained from the exact same data source but expanded to global inclusion. To minimize the computation load, several masks on the original data is applied to eliminate point on water surface and low solar capacity regions, leading to the presented feasible study areas, seeFigure 5.

4 DAC Power System Model Integration↩︎

To better understand the impact of DAC load to power system, both the power price and electricity carbon intensity, an integrated DAC + CAISO power system simulation is added here. The results show that the DAC’s scale explored in this study has very limited impact on the overall power system. Even scaled up by \(\leq\) 100 times, the change of annual power consumption cost is \(\leq\) 5%.

The power market model is using the open-sourced model from [4], presenting a Western Electricity Coordinating Council (WECC) power market simulation tool with two-stage market clearing of both day-ahead unit commitment at hourly resolution and real-time price from economic dispatch with 5-min resolution, both consistent with CAISO power market.

To better understand the impact of unpredictable load change due to flexible DAC operation, the DAC load is added to real-time load forecast error instead of base-load that’s within the prediction. The DAC load is added to Region 2, where solar duck-curve behavior is most prominent and average power price is similar to historical data. Meanwhile, considering the duck-curve driven DAC operation recovers the solar renewable profile, two different scenarios are generated: (1) DAC scaling without new PV capacity installation; (2) DAC scaling with new PV installation that matches DAC capacity.

The results of DAC power system integration model is presented in Table 2. Due to load added to real-time load forecast error, the day-ahead unit commitment would not see the DAC load and would dispatch the commited generators to satisfy the DAC load, which could be either curtailed solar PV generation or fossil generators. Therefore, adding PV with DAC load don’t always results in lower costs or lower emissions. This problem can be resolved if the DAC load is added to day-ahead load, meaning predictable load. It can be seen that unless DAC is scaled up by 1000 times, meaning 0.8 GW power when turned on, it’s impact on it’s local locational marginal price and average power carbon intensity is very small, limited within 5%. This proves that using price-taker and carbon-taker assumptions is well defended and very accurate, as the 5% cost gap is smaller than optimization profitability gap from custom algorithm.

On the other hand, the integrated simulation with power market model is significantly more computationally expensive. A full year simulation using price-taker and carbon-taker assumptions with custom strategic algorithm takes seconds to solve, while the power market integrated model takes 6-8 hours, mostly repetitively solving unit commitment and economic dispatch optimization. Consider the wide ranges of scenarios and geography coverage in this study include thousands of simulations to run, the power market integrated model is practically impossible. Here we show the results that defend the price-taker and carbon-taker assumptions and all results presented in the manuscript is generated using the assumptions.

5 DAC Technology Models↩︎

Three different DAC technology samples are extracted from the literature and tested in this study. The details of the technology and parameters for model inputs are summarized in Table 3. As the short cycle MOF technology with about 1 hour cycle time most ideal for solar-thermal DAC considering reducing the solar curtailment, only MOF results is presented in the main text.

In this study, two amine-functionalized sorbents, SI-AEATPMS and APDES-NFC-FD, are compared against the Nobel MOF sorbent. Leonzio et al. [5] juxtapose these amine-functionalized sorbents with three Metal-Organic Frameworks (MOFs) - MIL-101, MOF-177, and MOF-5, investigating their efficacy in capturing CO2 from dry air via a temperature swing adsorption system. To get more detailed non-linear adsorption/desorption rates behavior for the MOF sorbent technology, this paper also combined the results from [[6]][7] to calculate the \(\beta\)-values.

SI-AEATPMS, which stands for [N-(2-aminoethyl)-3-aminopropyl]trimethoxysilane grafted on silica gel, is created by drying silica gel beads and subsequently loading them with AEATPMS. This technology is characterized by a long, 89.6 hour cycle time, adsorbing for \(3.06 \times 10^5\) seconds, and desorbing for \(1.67 \times 10^4\) seconds.

Conversely, APDES-NFC-FD, which is 3-aminopropylmethyldiethoxysilane on nanofibrillated cellulose, is made by adding APDES to an NFC hydrogel and then undergoing a freeze-drying process. This sorbent has a shorter, 9.6 hour cycle time, adsorbing for \(3.26 \times 10^4\)seconds, and desorbing for \(1.85 \times 10^4\) seconds.

It’s worth noting that Climeworks, a leading company in the Direct Air Capture (DAC) field, has endorsed these amine-based sorbents. In fact, APDES-NFC-FD is currently being used in their facility in Switzerland.

5.1 Parameterization: Translating Source Data to Code Inputs↩︎

5.1.1 Calculation of Cycle Switching Cost↩︎

\(S\) represents the total cost per cycle, which is the sum of various operational expenses. It is computed as:

\[S = S_{\text{sorbent material}} + S_{\text{thermal}}\]

5.1.2 Calculation of Power Consumption Rates↩︎

The power consumption parameters are primarily focused on electrical energy. It’s assumed that the regeneration energy is predominantly thermal, which can be sourced at a lower cost from waste heat. The most significant cost component is the electrical energy which is primarily used for fans.

Adsorption Power Consumption: \[P_{\text{a}} = \frac{\text{Total Electricity Consumption (MWh/ton-CO}_2\text{)}}{\text{Number of 5-minute time steps for adsorption}}\]

Desorption Power Consumption: \[P_{\text{d}} = \frac{\text{Total Electricity Consumption (MWh/ton-CO}_2\text{)}}{\text{Number of 5-minute time steps for desorption}}\]

5.1.3 Calculation of Beta Coefficients↩︎

The beta coefficients, which are used in the piecewise linear approximation for both absorption and desorption, were derived from the figures presented by Leonzio et al.[5] Values were chosen to recover the number of time steps for the desired cycle length. Be minded that to keep consistent across different technologies, these \(\beta\)-values are calculated using unit capacity where \(\bar{X}\) = 1.

Absorption Coefficients:

For both amine-functionalized sorbents, SI-AEATPMS and APDES-NFC-FD, the adsorption curve figure displays a long flat region before the ratio of outlet to inlet \(CO_2\) concentration reaches 1, indicating system saturation. This flat region preceding the breakthrough suggests a linear behavior, which allows for the extrapolation of a linear relationship. Because of this, in the case of SI-AEATPMS and APDES-NFC-FD, the secondary term for the beta values is set to zero:

• \(\beta^{a}_{1}\): Represents the first-order coefficient for absorption.

• \(\beta^{a}_{2}\): Represents the second-order coefficient for absorption.

For the calculation of \(\beta^{a}_{1}\), a line is fit using two distinct points from the figure showing the loading for chemisorbents SI-AEATPMS and APDES-NFC-FD, for both the adsorption rates are effective linear and the second order coefficient is 0.

Desorption Coefficients:

For the desorption curve, the behavior is non-linear, so three distinct points from the figure (figure 8) showing CO2 concentration during the desorption step were selected and fit to the equation \(X \sim (1-\beta^{d}_{2})^{x}\), where \(X\) represents the state of saturation, ranging from 0 (completely unsaturated) to 1 (fully saturated). The derived relationship provides the value for \(\beta^{d}_{2}\).

• \(\beta^{d}_{1}\): Represents the first-order coefficient for desorption, which is consistently set to zero unless explicitly specified otherwise.

• \(\beta^{d}_{2}\): Represents the second-order coefficient for desorption derived from the exponential decay fit of the desorption curve.

5.1.4 Calculation of \(X_{\text{hat}}\)↩︎

\(X_{\text{hat}}\) represents the maximum capture per one absorption cycle. It is derived from the ratio of the static plant capacity to the number of cycles per year.

The number of cycles per year can be computed as: \[\text{Number of cycles per year} = \frac{8760}{\text{cycle time (hours)}}\]

Subsequently, \(X_{\text{hat}}\) is calculated with: \[\bar{X} = \frac{\text{100\% Plant Capacity}}{\text{Number of cycles per year}}\]

6 Supplementary Results and Discussions↩︎

6.1 Temporal behavior discussion with grid connection↩︎

Power market pricing is highly volatile, driven by fluctuations in supply and demand that vary by time of day and season. Prices tend to be high when net demand is high, while low prices can result from either low overall demand (e.g., midnight) or an abundance of renewable energy generation. In regions like California, the “duck curve" phenomenon highlights how solar power production peaks during midday, significantly reducing the price. Steep price peaks occur during sunset as the decline in solar power generation significantly increases net demand. Although the average daily power price typically ranges from $20 to $100, prices at specific hour or days can be far more volatile, dropping to zero or even negative during periods of renewable energy curtailment and soaring above $1000 during price spikes.

Solar-thermal DAC system performs better in locations with solar-dominated electricity generation. As shown in Figure 6, when connected to the solar-dominated California power market, solar-thermal DAC system demonstrate robust design optimization in thermal storage sizing and higher profits without sacrificing net-CO₂ removal volume. When incentive selling prices of CO₂ increase, California grid-connection firstly recovers the solar profile which have lower electricity prices without need of thermal storage. The “solar synchronization” also suggest DAC integration to solar-dominated power market can help absorb curtailed solar generation, leading to more cost efficient and sustainable operation (California has 7% higher CO₂ capture efficiency than Texas). California case performs slightly better in both CO₂ abatement volume and profitability when incentive is sufficiently high despite the fact that the annual average power price in California ($70/MWh) is higher than Texas ($57/MWh). At low incentive levels (\(<\$150/ton\)), the solar-dominated California power market outperforms Texas by an order of magnitude in profitability and investment payback period, highlighting its long-term potential as the technology matures and incentives gradually decline. Additionally, deeper solar penetration is expected to intensify the "duck curve" phenomenon in the California grid.

While the main text and figures present results as annual averages or as representative daily averages over 365 days, it is important to note that significant seasonal and daily temporal variations exist. These variations necessitate high temporal resolution analyses. For instance, in the TX case, a noticeable discrepancy is observed: the electricity price peaks in July while CO₂ removal reaches a valley in August. This counterintuitive result arises because aggregating daily data into monthly averages can mask the impact of outlier events. In July, although the average price is high due to several extremely expensive days, there are still eight days with average prices below $60/MWh, which renders DAC operation marginally profitable and sustains a moderate capacity factor. Conversely, in August, despite a lower overall average price—attributable to the absence of extreme outliers—all 31 days exhibit prices exceeding $60/MWh, rendering DAC operations unprofitable except during isolated, exceptional hours, and thus leading to a significantly reduced capacity factor. This mismatch underscores that aggregating higher-resolution data into lower-resolution averages can obscure critical dynamics that directly affect profitability. Consequently, employing high temporal resolution data—particularly the 5-minute resolution wholesale electricity prices—is essential for accurately capturing market volatility and optimizing operation scheduling for DAC systems. In contrast, relying on daily or annual averages can lead to substantial uncertainties in the analysis. Notably, while hourly-resolution solar data may suffice for modeling solar-driven processes such as thermal energy storage and DAC regeneration heat scheduling, the high variability of electricity prices with 5-min temporal resolution plays a non-trivial role in determining DAC profitability and therefore shall be carefully modeled.

6.2 Global impact analysis↩︎

It can be seen clearly that annual average solar capacity factor and capacity factor standard deviation across 365 daily average are most important predictors for solar-DAC performance. For each annual solar average capacity factor, there exist a lower bound for the LCCO₂ of stand-alone solar-DAC systems, and this lower bound can be reached when the standard deviation across 365 daily average capacity factor is minimized.

6.3 MATLAB Simscape Model Details↩︎

Figures 8 to Figures 14 showing details about MATLAB Simscape model structure.

References↩︎