Transient natural convection combined heat and mass transfer in enclosures filled with humid air, including evaporation from or condensation to the walls, which are subjected to time varying prescribed temperatures, is studied numerically. Emphasis is given to the two-dimensional enclosures of circular cross-section, emulating horizontal containers or ducts filled with humid air. Starting from uniform temperature and concentration distributions, wall temperature decrease with time leads to some water condensation at the walls, and wall temperature increase with time leads to evaporation of some liquid water from the walls if it exists there. During the final period of both situations temperature ceases changing, and a final steady-state regime is reached. Situation of prescribed time-periodic wall temperature is also considered, and a periodic solution is obtained after some few periods of operation. Results reveal the flow structure and also the temperature and concentration time evolutions in the enclosure. Special attention is given to the time evolution of the overall Nusselt and Sherwood numbers over the walls, and also to the time and space evolutions of the condensate layer over the walls. (C) 2012 Elsevier Ltd. All rights reserved.