Cryogenic carbon capture removes CO2 and other pollutants from flue and waste stream gases produced from the combustion of fossil fuels such as coal, natural gas, and oil and the production of cement. A transient, 1-dimensional numerical model was developed to study the temperature profile within a counter-current surface CO2 desublimation-falling liquid or solid heat exchanger. Effects of desublimation heat and mass transfer as well as convective and conductive heat transfer relationships were taken into account. Experiments show that CO2 can be captured on a falling spherical particle when appropriate column operating conditions are met.