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The room's nearly motionless upper reservoir has interface $z_i=H-h$. A doorway roof at height $D$ has available light-layer head $\delta =D-z_i=h-(H-D)>0$. Evaluate the Bernoulli equation in the room and at a freely controlled doorway: $E=-g^{\prime }{z}_i$ upstream and $E=3g^{\prime }{h}_c/2-g^{\prime }D$ at its critical section. Therefore $h_c=2\delta /3$ andThis is the ideal broad-crested weir law, not a small-orifice law with $C_d$. A nonideal doorway would require its own measured loss correction.

When the floor heat source is on, volume conservation and buoyancy flux conservation still give $Q=B_0^{1/3}{z}_i^{5/3}/\gamma$ and $g^{\prime }=\gamma B_0^{2/3}{z}_i^{-5/3}$. Combine these with the doorway relation to obtain the plume-fed doorway ventilation equilibrium:There is one interface-height root in the indicated interval. The result is physically appropriate only if its upper outflow is thin enough for the one-layer doorway model to remain valid.

If $D<H$, air in the volume above the doorway is part of the upper thermal reservoir and can store heat and alter transients. But its extra depth $H-D$ does not add available head at the doorway: only $D-z_i$ enters the critical condition. In the steady ideal well-mixed model it shares the same density as the outflow. Treating the whole $h$ as the doorway head would overestimate discharge.