The use of natural calcium carbonates as regenerable CO2 sorbents in industrial processes is limited by the rapid decay of the carbonation conversion with the number of cycles carbonation/calcination. However, new processes are emerging to capture CO2 using these cycles, that can take advantage of the intrinsic benefits of high temperature separations in energy systems. This work presents an analysis of a general carbonation/calcination cycle to capture CO2, incorporating a fresh feed of sorbent to compensate for the decay in activity during sorbent re-cycling. A general design equation for the maximum CO2 capture efficiency is obtained by incorporating to the cycle mass balances a simple but realistic equation to estimate the decay in sorbent activity with the number of cycles.