# Physical and thermodynamic constraints

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While any individual air capture technology will face a host of engineering challenges, there are two fundamental factors that make air capture more difficult than conventional post combustion CO2 capture processes: first, the energy and materials cost of moving air through an absorber, and second the thermodynamic barrier due to the low concentration of CO2 in air. In this section we describe the physics that constrains an idealized air capture system with respect to these two factors.

There is no lower bound to the energy required to move air through an absorber. The energy required to move air can be made arbitrarily low if the speed of air through the absorber approaches zero. However, there is a strong trade-off between this energy cost and the capital cost of the absorber. As the flow velocity approaches zero, the rate of capture per unit absorber surface will also approach zero; so while energy costs approach zero, the amortized cost of capital will approach infinity. This is because each unit of absorber structure captures CO2 at a rate that approaches zero yet still has a finite cost for amortizing the capital required for its construction along with the cost of maintenance to keep it operational. Any practical design must balance the cost of energy required to drive air through the absorber against the cost of capital and maintenance.

If we make a reasonable - though not universally applicable - assumption that the flow through the CO2-absorbing structure is laminar, then the energy required to drive air through the absorber and the capture rate are linked by the fact that - for laminar flow - the transport of both CO2 and momentum are diffusive processes that occur in a fixed ratio.

Neglecting factors of order unity that depend on the specific geometry, we can compute the energy required as compression work to capture a unit of CO2 in a laminar-flow absorber made from a substance that is sufficiently absorptive that the uptake rate is limited by CO2 transport in air (air-side limited). Under these assumptions, the specific energy per unit CO2, E, is

rpco2 D

where air has density p, velocity V and kinematic viscosity v; and CO2 has diffusion constant D, density pCO2 and mole fraction r. For air at standard conditions with 400 ppm CO2 at an air velocity of 10 m/s, the minimum energy is 0.15 GJ/tCO2.

This result can be obtained in a physically intuitive way by thinking about a parcel of air that moves through a structure coated with a perfect CO2 absorbing material. As the parcel moves, gas molecules contact the surface by diffusion, transferring momentum to the surface and losing CO2. When the parcel has travelled far enough that most of the CO2 has been absorbed it will also have transferred most of its initial momentum to the surface, which would be sufficient to bring the parcel to a stop if there were no pressure gradient sustaining the flow. The minimum pressure drop needed to capture a minor constituent gas in a perfect laminar absorber is therefore, to order unity, the stagnation pressure at the flow velocity, that is, Apmin = 1/2pV2 scaled by the ratio of diffusion rates which itself is the ratio of relative concentrations of momentum or CO2 and their transport coefficients.

The thermodynamic minimum energy required to separate CO2 from the air is given by the free energy of mixing,

where p is the final pressure of pure CO2, p0 is the initial partial pressure, R is the ideal gas constant and T is the working temperature. (Note that this formula ignores the change of free energy of the air when CO2 is extracted, a ~ 1% correction.) At a 20 °C operating temperature the minimum energy required to capture CO2 from the atmosphere at 400 ppm and deliver it at one atmosphere is 0.43 GJ/tCO2.

Now, consider the comparison between capturing CO2 from the air and from the exhaust stream of a coal-fired power plant assuming that the CO2 is to be delivered in compressed form suitable for pipeline transport at a pressure of 150 bar. A process which captures CO2 from the air can, in principle, trade off the cost of scrubbing a larger fraction of CO2 from the air against the cost of moving additional air if the fraction captured is smaller. In practice, the fraction captured will depend on a complex optimization between capital and operating costs. Suppose that half of the CO2 is captured and thus that CO2 must be removed from air at an average concentration of about 300 ppm and, for the last molecules captured, a final concentration of 200 ppm. Assuming the worst-case 200 ppm requirement for all the capture, the minimum energy to go from 200 ppm to 1 bar is 0.47 GJ/tCO2 and the minimum energy cost of compressing CO2 from 1 to 150 bar is 0.28 GJ/tCO2 for a total of 0.75 GJ/tCO2.

Most designs for post-combustion capture from power plants assume that at least 90% of the CO2 must be scrubbed from the exhaust gases with a representative concentration of 15% CO2. Again, what matters is the minimum energy to capture the final bit of CO2 from exhaust gases at 1.5% or 15 000 ppm for which the minimum energy is 0.23 GJ/tCO2 and, counting the same compression cost to 1 bar, the total minimum energy to deliver it at 150 bar is 0.51 GJ/tCO2.

Comparing CO2 capture from air and power plant exhausts, the intrinsic thermo-dynamic penalty due to the lower starting concentration of CO2 in air is therefore about a factor of 2 if the product is a 1 bar pure CO2 stream and a factor of about 1.5 if the product is pipeline pressure CO2. These ratios move towards a factor of 1.0 as the percentage capture of CO2 increases; however, the energy cost per ton of CO2 increases simultaneously. The primary reason to move in this direction is the possibility of building and operating smaller air-contacting equipment because not as much air must be treated to capture the target tonnage of CO2.

In practice, proposed designs for both air capture and post-combustion capture are a long way from thermodynamic efficiency limits. Aqueous amines, the most commonly considered method for post-combustion capture require about 23 GJ/tCO2 of regeneration heat (IPCC, 2005; Rao et al., 2006) and the NaOH solutions which we are exploring for air capture have a thermodynamic minimum regeneration energy of 2.4 GJ/tCO2.

The physical limits are, nevertheless, an important guide to the development of energy technologies (Keith et al., 2005):

These thermodynamic arguments do not, of course, prove that practical air capture systems can be realized, nor is the performance of air capture technologies likely to approach thermo-dynamic limits in the near future. The ultimate thermodynamic limits are nevertheless an important basis for suggesting that air capture can be achieved at comparatively low cost. From the liberation of pure metals from their oxides to the performance of internal combustion engines, electric motors and heat pumps, the historical record strongly supports the view that thermodynamic and other physical limits serve as an important guide to the long run performance of energy technologies.