For the flexible operation of mono-ethanol-amine-based post-combustion carbon capture processes, recent studies concentrate on model-based protocols which require underline model parameters of carbon capture processes for controller design. In this paper, a novel application of the model-free adaptive control algorithm is proposed that only uses measured input-output data for carbon capture processes. Compared with proportional-integral control, the stability of the closed-loop system can be easily guaranteed by increasing a stabilizing parameter. By updating the pseudo-partial derivative vector to estimate a dynamic model of the controlled plant on-line, this new protocol is robust to plant uncertainties. Compared with model predictive control, tuning tests of the protocol can be conducted on-line without non-trivial repetitive off-line sensitivity or identification tests. Performances of the model-free adaptive control are demonstrated within a neural network carbon capture plant model, identified and validated with data generated by a first-principle carbon capture model.