Carbon dioxide (CO$2$) is the main contributor to greenhouse gases (GHG). Cryogenic carbon capture is a potential alternative for systems with higher CO$2$ concentration. These methods do not involve the use of any chemicals (like solvents, adsorbents etc.) or supports (like membrane), so that the costs of raw materials and/or their regeneration are done away with. Desublimation is one of the cryogenic carbon capture methods involving the conversion of gaseous CO$2$ to solid CO$2$ /dry ice by cooling the feed gas mixture. Before the desublimation of CO$2$ , all the condensable components (like water, particulate matter etc.) are removed. The primary unit of a desublimation-based carbon capture system is the desublimator that should ensure not only efficient cooling of the CO$2$ -laden gas but also effective removal of desublimated CO$2$ (solid) from the system. Nitrogen vapor, liquid nitrogen, liquid methane, hydrocarbon based refrigerant blend etc. are some possible coolants. A few numerical studies on the desublimation-based carbon capture have been reported in the literature. However, all the studies are based on feed gas with low CO$2$ concentration (up to 20 mol%) and carried out at lab scale only. In this work, a one dimensional numerical model of carbon capture process is developed for better understanding of CO$2$ desublimation process, with feed gas with higher CO$2$ concentration (>20 mol%). The model contains a tube-in-tube parallel-flow heat exchanger in which the CO$2$ -laden feed gas and the coolant are passed through the inner-and outer-tube of the heat exchanger respectively. The presented model shows the effects of the various process variables such as temperature, velocity etc. on the efficacy of carbon capture by desublimation. A parametric study is carried out to know the significance of each process variables.