= Solution
Let $q_b$ be floor inflow and $q_t$ roof outflow. Use effective-area discharge $A\sqrt{\delta p/\rho}$, matching the factors in the supplied relations. With negligible source volume, both equal $q$. The lower region has exterior <mass density> while the warm upper layer has <reduced gravity> $g'$ and depth $H-h$. <hydrostatic pressure> therefore supplies a total opening <pressure> head $g'(H-h)$. Equal openings sharing equal discharge take half each, giving
$$
q=A\sqrt{\frac{g'(H-h)}2}.
$$
Steady lower-layer volume conservation sets this ventilation rate equal to the entrained <turbulent plume> flux at the interface, $q=\lambda B^{1/3}h^{5/3}$. Upper-layer buoyancy conservation gives $B=qg'$. These are the required balances for <displacement ventilation>:
$$
\boxed{A\sqrt{\frac{g'(H-h)}2}=\lambda B^{1/3}h^{5/3},\qquad
B=Ag'\sqrt{\frac{g'(H-h)}2}}.
$$
Eliminating $g'$ and $q$ gives the <displacement-ventilation interface height> equation
$$
\boxed{2\lambda^3h^5=A^2(H-h)},\qquad
\boxed{\frac{\zeta^5}{1-\zeta}=\frac{A^2}{2\lambda^3H^4},\quad\zeta=\frac hH\in(0,1)}.
$$
The left side is strictly increasing from zero to infinity, so there is one interface height. It depends on opening geometry and plume entrainment, not on $B$; the resulting $g'=B^{2/3}/(\lambda h^{5/3})$ and throughflow do depend on $B$.
For finite source volume $Q$, conservation instead gives $q_t=q_b+Q$. The two <pressure> drops are no longer equal:
$$
\frac{q_b^2+q_t^2}{A^2}=g'(H-h),\qquad B=q_tg'.
$$
Let $q_p(h;Q)$ be a physically entraining source plume: $q_p(0;Q)=Q$ and $q_p(h;Q)>Q$ for $h>0$. Its interface balance is $q_p=q_t=Q+q_b$. Therefore, as $q_b$ reaches zero, a positive-depth lower layer has no replenishment and cannot remain steady: this is <no steady displacement layer without ambient supply>. The limiting interface is $h=0$. At the threshold $q_t=Q$ and $g'=B/Q$, and the entire hydrostatic head drives the roof discharge. Hence
$$
\boxed{Q_c=A\sqrt{\frac{BH}{Q_c}},\qquad Q_c=(A^2BH)^{1/3}}.
$$
This is <source-volume blocking of displacement ventilation>. The half-head factor of the negligible-source state must not be kept after the floor inflow vanishes. In the usual physical-area convention $q=A_{\rm phys}C_d\sqrt{2\delta p/\rho}$, the equivalent formula is $Q_c=(2C_d^2A_{\rm phys}^2BH)^{1/3}$.
The quoted pure-plume law cannot be used unmodified down to the origin for a source with nonzero volume: it would incorrectly give zero flux there. A compatible finite-source example uses a virtual origin,
$$
q_p(h;Q)=\lambda B^{1/3}(h+h_0)^{5/3},\qquad
h_0=\left(\frac{Q}{\lambda B^{1/3}}\right)^{3/5}.
$$
It recovers the supplied pure-plume limit when $Q\to0$, and gives the same blocking threshold. The complete finite-$Q$ interface trajectory depends on that plume model, but the threshold needs only source-volume conservation and positive entrainment. For $Q>Q_c$, the assumed steady regime with a lower inflow no longer exists.
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