= Solution
Write $z_i=H-h$ for the height of the warm-layer interface above the heat source. A negligible-volume point heat source supplies a <pure plume> with conserved <buoyancy flux> $B_0$. The given plume buoyancy implies its <volume flux> at the interface is
$$
Q_p(z_i)=\frac{B_0}{g'_p(z_i)}=\frac{B_0^{1/3}}\gamma z_i^{5/3}.
$$
At steady <displacement ventilation>, the plume feeds the warm layer at the same rate it empties through the upper vent: $q_v=Q_p(z_i)$. The upper-layer buoyancy balance gives $q_vg'=B_0$, so
$$
g'=\gamma B_0^{2/3}z_i^{-5/3},\qquad q_v=C_dA\sqrt{g'h}.
$$
Eliminate $q_v$ and $g'$ to get
$$
\boxed{(H-h)^5=\gamma^3C_d^2A^2h,\qquad
\Delta\rho=\frac{\rho_0}{g}\gamma B_0^{2/3}(H-h)^{-5/3}.}
$$
The height equation has one solution with $0<h<H$. Its height is independent of source strength in this ideal pure-plume model, while its warm-layer density contrast increases as $B_0^{2/3}$. The sketch shows cool replacement air below, an entraining plume through that region, and the upper mixed layer feeding the ceiling vent.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-43-ventilation.png]
{title=Plume-fed displacement ventilation through two vents and through a hydraulically controlled upper doorway}
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