Plume-fed doorway ventilation ($z_i^5=(8/27){\gamma }^3{W}^2(D-z_i{)}^3$)

For a pure plume with $g_p^{\prime }=\gamma B^{2/3}{z}^{-5/3}$, a steady sharp interface at $z_i$ has $Q_p=B^{1/3}{z}_i^{5/3}/\gamma$ and warm-layer reduced gravity $g^{\prime }=B/Q_p$. A freely discharging upper doorway of height $D$ and width $W$ has available head $D-z_i$. Equating plume supply to the broad-crested weir flux gives the displayed implicit height relation, with $0<z_i<D$. Warm air above the doorway stores heat but adds no available head at its crest in this uniform-layer model.