Recently, a new composite "LiCl inside Multi-Wall Carbon NanoTubes" (LiCl/MWCNT) has been suggested as water sorbent for Adsorption Thermal Energy Storage (ATES), because it has a large thermal storage capacity of 1.7 kJ/g for a daily heat storage cycle. This work addresses the results of the study of water sorption dynamics on the novel composite loaded into representative small scale fragments of a common finned flat-tube HEx. The study consists of four parts: (1) shaping the LiCl/MWCNT composite as grains using polyvinyl alcohol (PVA) as a binder; (2) analysis of sorption equilibrium for the pair "LiCl/ MWCNT/PVA - water"; (3) measuring the thermal storage capacity of the granulated LiCl/ MWCNT/PVA composite; (4) investigation of water sorption dynamics on the LiCl/MWCNT/PVA under typical conditions of the daily storage cycle. It is shown that the thermal storage capacity of the LiCl/MWCNT/PVA composite equals 1.5-1.6 kJ/g. The specific power reaches 4.2 and 9.8 kW/kg of the heat release and thermal storage stages, respectively. The results obtained show that the working pair "LiCl/MWCNT/PVA - water" appears to be advantageous for ATES.