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Energy Considerations in Direct Air CaptureOf the many schemes for capturing and storing carbon dioxide from the air, DACCS (scheme 11 in Table 1) is the closest in terms of technology to a wellknown and researched method of carbon management, namely CCS fr om flue gas. However, the deployment of FGC has been slow, for various reasons, including concern about its energy requirement, which must of course be paid for. Similarly, DACCS has been criticised as energy intensive, and therefore expensive, particularly compared to schemes utilising photosynthesis, where the energy for capture appears to be free. However, DACCS remains a potentially important weapon to tackle atmospheric carbon, so we devote this section to discussing its energy requirement in detail. Comparison is made with flue gas capture, and the approach is extended to estimating likely costs. Supporting calculations can be found in the Appendix to the chapter. The calculation of energy required for FGC and DAC involves a number of steps, so it is usefitl to follow a clear calculation strategy. This is shown in Figure 2. We start with (i) the calculation of the (minimum) shaft work requirement ir_{m} for an idealised reversible separation process that involves no energy losses. Then calculation (ii) using the Carnot efficiency yields the (minimum) reversible heat requirement. The heat required by an actual process plant 0,_{ep} is greater than the reversible amount due to losses, and this is calculated last (iii). We then calculate and compare values of Q_{sep}/w_{m} for actual process plant data. Figure 2. Strategy for calculating heat requirement of separation process. For many separations with regenerable sorbents, process efficiency is about 'A to 'A. Note: In this section quantities of thermal energy are designated MJ(th), and quantities of electrical energy or work are denoted MJ(e) when this distinction might otherwise be unclear Reversible work of separationIn Direct Air Capture, ah' is separated into a gas enriched in carbon dioxide and a second “reject” stream of inerts which may also contain some CO„ as shown in Figure 3. The inerts are components which are not differentiated from each other in this particular separation. We analyse an arrangement in which all streams are at the same pressure, and at the same temperature T_{g}. The mole fraction of CO, in the feed is у and in the CO,enriched stream is у . The separation can be characterised by two further parameters, and we choose these to be a, the fraction of inlet CO, that is recovered in the CO,enriched stream, and Figure 3. Separation of air into CO,ennched stream and reject stream. Basis of calculation: 1 mole of feed with mole fraction y_{f} of carbon dioxide. ft, the fraction of the inerts in the feed that slip through into the CO,enriched stream. The parameter ft is found by mass balance to be given by a.y_{c}(l y_{c}J/{y_{c}/l  у ft). Shaft work is supplied in order to bring about the separation, and, by the first law, an equal quantity of waste heat is produced because the system is isothermal. The minimum quantity of work needed is that required by a perfectly reversible process, and this can be calculated from the change in Gibbs free energy between inlet and outlet streams. For this calculation, we need no information about the separation process used in Figure 3, beyond the constraint that it is thermodynamically reversible. This means that all gradients of temperature, pressure, velocity and concentration in the processing equipment are infinitesimally small, so that all internal transfer processes proceed infinitely slowly and there are no internal losses of energy. The advantage of considering such an idealised process is that its performance can be calculated unambiguously. It represents a clearly defined standard with which real processes can be compared. No separation process working with real equipment will be able to affect the same separation, at the same temperature and pressur e, with less expenditure of work. The reversible work,^{1} which is, thus, the minimum required for the separation, w per mole of CO, recovered in the CO,enriched stream, is given by
The four terms on the righthand side of equation (11) are the Gibbs free energy change of, respectively, (1) CO, in the CO,enriched stream, (2) inerts in the CO,enriched stream, (3) CO, in the reject stream, and (4) inerts in the reject stream. When concentrating dilute streams of carbon dioxide, the reversible work requirement is dominated by the first term. Allowing some inert to slip into the enriched stream reduces the work requirement, but the effect is initially slight, as shown in Figure 4. For air at 293 К and a CO, concentration of 400 ppm, the reversible work required for a complete separation (a = 1 and ft = 0) is 21.49 MJ/kmol CO,, and only falls to half this value when the enriched stream purity drops to around 3% CO,. Somewhat less work is required if the recovery is reduced, but again the effect is slight. Even restricting recovery (a) to 10%, also shown in Figure 4, only reduces the reversible work requirement by ~ 10% in the range of interest (purities > 96%). A low' recovery fraction means that a lot more air must be treated per kmol CO, captured, incurring greater capital and operating expense. This derivation assumes the mixtures are ideal. For the treatment of nonideal mixtures see, e.g., King (1980) Chapter 13. Figure 4. Reversible work of separating CO, from air. Mole fraction in air 0.0004. Calculated for T_{0} = 293 K. a. is the fraction of inlet CO, recovered in the CO,ennched stream. Energy requirements in DACprocess plantThe standard enthalpy of formation of CO, is ()393.5 MJ/kmol, some 18 times the reversible work required for a complete separation of400 ppm CO, from air. from equation (11). So it might seem that the energy cost of capturing an amount of CO, from the air would be modest compared to the heat that was obtained when it was formed duiing combustion. However w is a work, not a heat, requirement. The sorbents used for DAC require regeneration, a step involving heat input, so the overall sorption process functions as a “separation engine”. Heat is supplied to the engine and it produces work of separation as well as waste heat that must be removed. The Camot efficiency of this separation engine, >)_{Crn}, can be estimated. For example for an amine treater with regeneration using low pressure steam at T_{H}~ 403 K, and removing heat in the overhead condenser and trim cooler at a mean temperature of T_{c} ~ 343 K, t]_{Car} (= 1  T_{C}/T_{H}) is 0.17 or ~ 1/6. Furthermore, the Camot efficiency refers to a reversible process. In practice, when materials are processed at a commercial rate and scale, energy is dissipated by friction and in the diffusion of heat and mass across finite gradients of temperature and concentration, respectively. The efficiency of a real process might be one third to one half of the (reversible) Camot value, though this will depend on the process and the design choices made. The heat requirement of an actual DAC plant producing pure CO, would then be about (2 to 3) x (l//;_{Co},,) or 1218 times u'_{TO}, that is some 250400 MJ/ kmol CO,. Thus, the energy needed for DAC might approach, and even exceed, the energy obtained by burning the carbon in the first place. From published process data, it is possible to estimate the factor by which the energy required for carbon capture, 0,_{ep}. exceeds the minimum work of separation, n^{}m. The calculations are described in the Appendix, and summarised in Table 2. The data refer to four FGC plants (Metz et al., 2005) at the limit of their effectiveness, and the DAC process developed by Carbon Engineering (CE) (Keith et al., 2018). Our initial supposition that the heat actually used to drive a capture process for carbon dioxide might be some 1218 times the minimum reversible work is seen from Table 2 to be approximately true, though the spread of values is greater, about 524. Part of this spread is due to assumptions and simplification in the calculations and the degree of optimality of the process design. In particular, it is likely that some of the variation in 0._{ep}/w_{m} represents different approaches to the tradeoff between capital expenditure and energy efficiency, a wellknown optimisation problem in process design (Smith. 2016, p460). Energy efficiency can be improved in general by a greater degree of process integration, and by reducing temperature differences in exchange of heat between hot and cold streams. This, in turn, requires more heat transfer area, and thus, an increase in capital expenditure. Losses in mass transfer equipment can also be reduced by operating closer to equilibrium, but analogously with heat transfer, Table 2. Reversible work and actual heat requirement for CO, capture processes, MJ/kmol CO, captured. Data and calculations for pulverised coal (PC) and natural gas combined cycle (NGCC) flue gas capture: for DAC by CE process. T„ = 21 °C.
this also requires more transfer area and, thus, a larger plant. Also, a more highly integrated design may har e less flexibility with regard to accommodating summer/winter conditions, changes in flowrate or composition and so on. Selection of the optimum design will then depend on a wide range of factors, such as the cost of capital, the cost of energy and consumed chemicals and the possibilities of integration with neighbouring process plant. It seems reasonable to assume that currently optimised processes for both FGC and DAC should be able to operate at a ratio of approximately 0,_{ep}Av_{m}, ~ 14. This ratio is suggested as appropriate for preliminary design and estimating purposes. It is not, however, a ratio that should be considered as permanently fixed. It obviously varies somewhat from design to design, and further optimisation and improvements in process and equipment will reduce energy requirements further. And importantly, new process concepts could lead to step changes, just as the introduction of combined cycle power generation schemes led to significant improvements in power plant efficiency that were once thought impossible. Such innovation might result, for example, from electrochemical schemes, where chemical energy is converted to electrical energy (work) without the need for heat generation which involves a Carnot efficiency. The requirement for shaft work or electricity within the process might be met by renewables (solar, wind, hydro), reducing the need to bum fossil fuel. Innovations in material science might lead to feasible membrane separation processes powered by renewable energy. Similarly, use of new adsorbents tailored to the specific FGC and DAC requirements might prove more efficient than existing schemes. It seems likely that such innovations could be applied both to FGC and DAC. 
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