Direct air capture is a carbon-negative technology that absorbs dilute carbon dioxide from the air directly, which requires a significant amount of energy and the construction of large-scale equipment. The main challenge for direct air carbon dioxide capture is the development of non-fossil energy sources for carbon dioxide compression and purification. In this work, a multi-stage adsorption system based on temperature swing adsorption with adsorption materials that can be driven by thermal energy is designed for direct air capture. The model of adsorption process is built on thermodynamic principles and conservation of mass, and the common tangent plane approach of adsorption is adopted at the end of each step calculation to validate the selected solution. Analysis of the process shows that, the CO2 purity in the storage bed can exceed 90% and the pressure can exceed 14 bar with zeolite 13X at a regeneration temperature of 368 K. Exploration of thermal performance has been investigated regarding specific energy and exergy efficiency of various regeneration temperatures. The influence of mass of adsorbent on the purity and specific energy has also been evaluated. Results show that the specific energy declines from 11.92 MJth/molCO2 to 8.73 MJth/molCO2, and the optimal exergy efficiency increases from 0.205 % to 0.238 % as the regeneration temperature grows from 368 K to 373 K. The values of adsorbent mass in the second and third beds for optimal purity and specific energy are calculated. It means that a thermally-powered carbon separation device can not only meet the requirements of direct air capture but also get the optimal purity of carbon dioxide or specific energy by altering the operation parameters.