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Assume both openings have positive area, no imposed wind pressure, a light buoyant plume, and a Boussinesq approximation. Write $S=\pi Q_s$ and $B_s=\pi F_s$ for physical source volume and buoyancy flux. Include discharge coefficients $C_H,C_0$; if ideal openings are intended, set these to one. In a steady two-layer displacement ventilation state with interface height $z_{\ast }$, let the floor inflow be $q_0$ and roof outflow $q_H$. Volume and buoyancy balances giveThe hydrostatic pressure head across the warm upper layer must equal the two opening losses:A positive unprocessed lower layer needs $q_0>0$. At the boundary where this layer disappears, $z_{\ast }=0$, the plume can entrain no lower ambient fluid, so $q_0=0$, $q_H=S$ and $g_u^{\prime }=B_s/S$. The critical condition is therefore $S^3=2C_H^2{A}_H^2{B}_{s}H$.

To determine the inequality, consider a fully processed room. Its steady reduced gravity is $g^{\prime }=B_s/S$. If $p_0$ is the interior-to-exterior pressure difference at the floor, source-volume blocking of fresh floor inflow requires $p_0\ge 0$. Both openings then exhaust, withThe right-hand side is increasing in $p_0$ and has its minimum at zero. Consequently a fully processed state exists exactly whenThis is source-volume blocking of displacement ventilation. At equality floor inflow vanishes; above it the floor can also exhaust and no fresh lower-layer supply is needed. Below it, the floor pressure must be negative, admitting fresh fluid and supporting a positive lower layer. In fact the steady interface equation has a unique positive root: its loss side increases with $z_{\ast }$ because $Q(z_{\ast })$ increases, while its buoyancy-head side decreases.

The floor-opening area affects the positive-interface state and its approach, but drops out of the blocking threshold because floor flux is zero at the threshold. Equal splitting of pressure head, or using a source-free effective opening area at this transition, would miss this distinction. If the floor opening is sealed, the fresh-inflow argument no longer applies: a finite source with only roof exhaust instead cycles the room asymptotically as in the ceiling-opening case of part (c).

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 $g^{\prime }$ and depth $H-h$. hydrostatic pressure therefore supplies a total opening pressure head $g^{\prime }(H-h)$. Equal openings sharing equal discharge take half each, givingSteady 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 $B=q{g}^{\prime }$. These are the required balances for displacement ventilation:Eliminating $g^{\prime }$ and $q$ gives the displacement-ventilation interface height equationThe 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 $g^{\prime }=B^{2/3}/(\lambda h^{5/3})$ 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: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 $g^{\prime }=B/Q$, and the entire hydrostatic head drives the roof discharge. HenceThis 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 $q=A_{\mathrm{phys}}{C}_d\sqrt{2\delta p/\rho }$, the equivalent formula is $Q_c=(2C_d^2{A}_{\mathrm{phys}}^{2}BH{)}^{1/3}$.

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,It recovers the supplied pure-plume limit when $Q\to 0$, 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.