# Direct Air Capture, Part 1: the Entropy Penalty

Direct air capture, or DAC, is what climate industry professionals call any process that takes atmospheric air and removes (some) carbon dioxide. DAC gets people, myself included, very excited. Wouldn’t it be great if we could undo CO$_2$ emissions?

The idea is not, strictly speaking, that new. Various working technologies already exist: submarines have had “CO$_2$ scrubbers” that heat and cool monoehtanolamine to absorb CO$_2$ from the air and release it outside since the mid-20th century. Despite this, several large, well-funded DAC projects since 2010 have failed to deliver. IEA.

DAC today has two problems: 1), nobody will pay you to take CO$_2$ out of the air, and even if someone did, 2), it would cost too much in electricity per CO$_2$ removed.

You can solve part of the second problem with better technology. But there is
also a **fundamental physical limit on the energy required for any process that
pulls one gas out of a mix of gases**. Soherman, Jones, and Dauenhauer
amusingly call this the ‘entropy penalty’ of DAC, because it is due to the
entropy of mixing.

I’d like to do a series of posts about DAC. I’ll start with what I’m most qualified to discuss, which also happens to be the least solvable problem.

Here is a simple sketch of the entropy penalty using physics found in most thermodynamics textbooks.

## Ideal mixing and separation; maximum efficiency

Physically, the difference between DAC and capture at, e.g., a powerplant flue is the composition of the initial mixture, and in particular the initial concentration of CO$_2$. I will treat both.

Under conditions of atmospheric temperature and pressure, air can be treated as an ideal gas with molar mass 28.97 g$\cdot$mol$^{−1}$, and with 44.01 g$\cdot$mol$^{−1}$.

Call air with all CO$_2$ removed *A*, and pure CO$_2$ *B*. Because the initial
concentration of *B* is low compared to the major constituents of air (O$_2$
at 21% and at N$_2$ at 78%), I can neglect the effect of removal of *B* on the
change in ideal gas properties of atmospheric air versus *A*. In addition, when
*B* is removed from air, there is a negligible change in internal energy.
Although air with *B* removed is not a single gas, summing over the partial
pressures of the constituents of air, and neglecting the change when CO$_2$ is
removed, the air mixture can be treated as a binary mixture.

The entropy of a binary mixture of two ideal gases is higher than the sum of the separate entropies. Assuming constant U and V, with $x = \frac{N_B}{N_A + N_B}$:

$\Delta S_{\text{mixing}} = -R \left[ x \ln x + \left( 1 - x \right) \ln \left(1 - x \right) \right]$

$\text{and given } G = -TS + U + PV$,

$\Delta G _{\text{mixing}} = -T\Delta S _{\text{mixing}}$ (Schroeder 187).

At constant pressure, each additional molecule of $B$ replaces a molecule of *A*, so
we have $x = \frac{N_A}{N_{init} }$, where $N_{init}$ is
constant and given by the ideal gas law. Therefore
$\Delta G_{\text{mixing}}$ is a function of $N_B$ only.

At the emitter, CO$2$ mixes into the atmosphere, and it is completely mixed by the time atmosphere touches the separator. The change in Gibbs free energy due to the entropy of mixing is therefore not recoverable.

Assuming that other (e.g. mechanical) processes in the separator are reversible,
$\Delta G_{\text{mixing}}$ is then the minimum energy that an ideal
separator must supply to unmix the gases at constant pressure and capture
$N_B$ molecules of CO$_2$. A real separator will use some higher amount of
energy *E* for the same result, and will have an efficiency $r = \frac{\Delta G_{mixing}}{E} < 1$.

Under this model, the minumum energy required to separate from atmospheric air is $\Delta G_{\text{mixing}}$= $RT(x\ln x +(1− x)\ln (1− x))$

# Results

The goal is to capture some fixed $N_B$ of CO$_2$ . Estimate the energy required to capture 1 ton, or $N_B = 2.3 \cdot 10^4 mol$, depending on the intial concentration, and compare values for capture at the plant and directly from air.

*As a function of input concentration*. Figure 1 gives the energy to
extract 1 ton as a function of input concentration. When concentration
approaches 0, as expected, the work approaches infinity (Schroeder).
However, in the range of interest (atmospheric to tailpipe emissions)
the energy varies by less than 1 order of magnitude.

*Direct air capture*. Atmospheric air consists of 415 (molar) ppm
CO$_2$, or 0.0415% (NOAA). Using this value,$x = \frac{N_B}{ N_{init}} = 0.0415$. Under approximate
atmospheric temperature and pressure (300 K, 100 kPa), the energy required for direct air capture is 0.498 MJ$\cdot$t$^{−1}$.

It is interesting to note that carbon capture is subject to a negative feedback loop: as the average atmospheric temperature increases, (i.e. as global warming occurs) the energy required to extract also increases, but since average temperature increases are only of a few degrees the effect is small, on the order of $\frac{1}{300}$.

*Power plant exhaust*. Typical concentrations at a power plant exhaust, in
contrast, are approximately 15%. The corresponding value for $\Delta G$ is 0.160 MJ $\cdot$t$^{−1}$, or approximately 3 times lower than direct air capture.

*Underground storage*. As an aside, Figure 2 gives the work required to compress
by atmospheric volume into a theoretical 1 million cubic meter depeted oil
field, over a volume range from 1 to approximately 100 tons of air. The energy
for storage alone is close to the separation step.

Thus, the theoretical energy requirement of unmixing depends on the initial concentration but does not vary by orders of magnitude accross the range. In the extremes: for CCS, the energy of separation due to $\Delta S_{\text{mixing}}$ is lower than for DAC by a factor of 3. Underground storage of high volumes of CO$_2$, however, comes at an even higher energy cost.

## Current technology

The state of the art for separation technology is in chemical methods. The most
efficient of those, however, only approach 2.5 GJ t^{−1}, which is $r =
\frac{0.2}{1000} = \frac{1}{5000}$ .

# Conclusion

The energy requirements for a carbon capture system are not limited to the change in Gibbs: they also include pumping air accross the separator. Here, there is a tradeoff between running the separator at saturation and the energy cost of the airflow.

Finally, with respect to storage, the formation of stable chemical compounds is a more encouraging avenue than pumped storage .

I estimated the energy requirements of a perfectly efficient carbon capture and storage system. Even in the ideal case, the entropy of mixing at the point of emissions results in unrecoverable energy. This is the so-called entropy penalty. It places an unavoidable thermodynamic limit on the efficiency of a DAC process, which current technologies do not approach.

Next, if I can learn enough chemistry, I’d like to take a look at recent progress in the solvable part of the efficiency problem (planned references: Nohra, et al., Zhu, et al.). Later, I’d like to model the economic incentives that would be required for a sustainable DAC project assuming perfect technology.

**Questions? Comments?** Email me at blog@silasbailey.com.

# References

- Nohra, Woo, Aalavi, and Ripmeester. https://www.sciencedirect.com/science/article/abs/pii/S0021961411003016
- Zhu, et al. https://www.nature.com/articles/s41586-023-06060-1
- IEA. https://www.iea.org/reports/direct-air-capture-2022/executive-summary
- Soeherman, Jones, and Dauenhauer. https://pubs.acs.org/doi/full/10.1021/acsengineeringau.2c00043