Two-plume three-layer displacement ventilation ($g_1^{\prime }/g_2^{\prime }=(\psi +{\psi }^{2/3})/(1+\psi )$)

Two independent point-source pure plumes with $\psi =B_1/B_2<1$ remove lower-layer volume $Q=C(B_1^{1/3}+B_2^{1/3})h_1^{5/3}$. If plume 1 feeds the middle layer, its reduced gravity is $g_1^{\prime }=B_1/[C{B}_1^{1/3}{h}_1^{5/3}]$. The upper outflow carries both sources, so $g_2^{\prime }=(B_1+B_2)/Q$, yielding the displayed ratio. Opening flow is driven by the integrated hydrostatic head $g_1^{\prime }(h_2-h_1)+g_2^{\prime }(H-h_2)$. A plume crossing the lower interface has altered buoyancy relative to its new ambient and must be continued with nonzero incoming mass and momentum fluxes rather than reset as a zero-flux pure plume.