A plume of hot fluid issues from a source of finite size with finite momentum located in the floor of a confined region. Small openings in either the floor or ceiling exhaust the mass flux. The plume rises to the ceiling entraining cooler fluid. There it spreads laterally with steadily decreasing velocity and so enters the environment. It supplies the downward flow which is induced by the entrainment. The equations for conservation of mass, momentum and energy in both plume and environment define the temperature field. Two limiting cases have been solved. First is the initial front which is a step change moving down from the ceiling with decreasing velocity. Second is the asymptotic state in which temperature at all points increases linearly with time. Results for both cases have been verified by laboratory experiments using fresh and salt water.