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A thermodynamic model, borrowing ideas from psychrometric principles, of a cryogenic direct-air CO2-capture system utilizing a precooler is used to estimate the optimal CO2 removal fraction to minimize energy input per tonne of CO2. Energy costs to operate the system scale almost linearly with the temperature drop between the ingested air and the cryogenic desublimation temperature of CO2, driving siting to the coldest accessible locations. System performance in three Arctic/Antarctic regions where the proposed system can potentially be located is analyzed. Colder ambient temperatures provide colder system input air temperature yielding lower CO2 removal energy requirements. A case is also presented using direct-sky radiative cooling to feed colder-than-ambient air into the system. Removing greater fractions of the ingested CO2 lowers the CO2 desublimation temperature, thereby demanding greater energy input for air cooling. It therefore is disadvantageous to remove all CO2 from the processed air, and the optimal mass fraction of CO2 desublimated under this scheme is found to be ~0.8-0.9. In addition, a variety of precooler effectiveness (ε) values are evaluated. Increasing effectiveness reduces the required system power input. However, beyond ε = 0.7, at certain higher values of desublimated CO2 mass fraction, the CO2 begins to solidify inside the precooler before reaching the cryocooler. This phenomenon fouls the precooler, negating its effectiveness. Further system efficiencies can be realized via a precooler designed to capture solidified CO2 and eliminate fouling. [ABSTRACT FROM AUTHOR]