Provided herein are methods and apparatus for capturing or otherwise decreasing the amount of CO2 in an exhaust stream. The separation process for removing CO2 from a waste stream preferably operates substantially at atmospheric pressure and at a preselected temperature, and without the need of thermal swing regeneration. This novel elimination of the heat up and cool down periods allows the inventive cycle herein to be run faster, and more efficiently than in previously known systems. Further, in some examples herein, the heat of adsorption is used to provide the heat of regeneration.