= Solution
Assume <Boussinesq approximation>, no wind, a uniform exterior, negligible source volume, sharp well-mixed internal layers, point-source <pure plume> entrainment and thin openings represented by discharge coefficients $C_b,C_t$. Let $g_1'=g\Delta\rho_1/\rho$ be the upper-layer reduced gravity. Steady volume conservation makes the magnitudes of inlet and outlet flow equal to $Q$.
If the opening pressure drops are $\Delta p_b$ and $\Delta p_t$, the orifice laws and hydrostatic pressure difference give
$$
Q=C_bA_b\sqrt{2\Delta p_b/\rho}=C_tA_t\sqrt{2\Delta p_t/\rho},
\qquad \Delta p_b+\Delta p_t=\rho g_1'(H-h_1).
$$
Define the reduced area by
$$
\boxed{A^*=\left[\frac{2}{(C_bA_b)^{-2}+(C_tA_t)^{-2}}\right]^{1/2}.}
$$
Then the throughflow in this <displacement ventilation> model is
$$
\boxed{Q_b=Q_t=Q=A^*\sqrt{g_1'(H-h_1)}.}
$$
Lower-layer conservation requires $Q=Q_p(h_1)=CB_1^{1/3}h_1^{5/3}$. Global buoyancy conservation gives $g_1'Q=B_1$. Eliminating $Q,g_1'$ yields
$$
Q^3=(A^*)^2B_1(H-h_1),\qquad
\boxed{\frac{A^*}{H^2C^{3/2}}=\left[\frac{(h_1/H)^5}{1-h_1/H}\right]^{1/2}.}
$$
This uniquely determines an interface height between zero and $H$ for a given positive area parameter. Opening area controls the nondimensional height; source strength controls the density contrast and flow magnitude. Factors of two depend on the stated effective-area convention, which has been fixed explicitly here.
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