The Cryogenic Carbon Capture process can decrease the energy demand for CO2 separation and pressurisation to 150 bar significantly, compared to other conventional technologies. The CCC process is a post-combustion technology that utilises cryogenic conditions to separate CO2 from the other contaminants in the cement plant off-gas. The energy duty for the CCC process is estimated both with the use of an empirical off-gas composition and a modelled off-gas composition. The modelled calcination and combustion process is based on a methane fired combustion and a calcination process with a pure CaCO3 supply. Furthermore, the CCC process is benchmarked to the theoretical minimum energy duty for CO2 separation to consider the potential for further improvements. It appears that the CCC technology's high potential for heat integration is crucial in order to achieve a competitive energy duty. Furthermore, based on the sensitivity study, it appears that the most dominating parameter affecting the energy duty is the CO2 concentration in the off-gas. Moreover, the synergies between the calcination and combustion process, the CCC process, and a Power to X facility are investigated to explore the possibility of converting the captured CO2 into methane, which can be utilised to force the calcination reaction and make a self-supplying process.