We analyse the density evolution of fluid within a confined ventilated space resulting from the action of a dense turbulent plume originating at the top of the space with finite source volume flux, $Q_0$, and initial source buoyancy flux, $B_0$. The space is ventilated through upper and lower openings of areas $A_u$ and $A_l$ respectively, which are separated by a vertical distance $H$. We show that if $Q_0^3\,{<}\, 2 B_0 H c_l^2 A_l^2$ (where $c_l$ is an empirically determined discharge coefficient) then a two-layer steady stratification becomes established in the room, with outflow through the lower opening and inflow through the upper opening. The interface location depends not only on the geometry of the openings, but also the source conditions. We show that as $Q_0$ increases for fixed $B_0$, the height of the interface, which equals the depth of the lower layer of relatively dense fluid, increases. Eventually, when the source volume flux has a value greater than $Q_m\,{=}\,(c_l A_l)^{2/3}(2B_0 H)^{1/3}$, the natural exchange flow becomes blocked and a steady outflow through both of the openings develops. As a result, the density of the fluid throughout the room gradually evolves towards the density of the incoming dense fluid. We compare our theoretical predictions with a series of laboratory experiments, and discuss the implications of our model for the design of ventilation systems.