= Multiple-plume displacement ventilation
{title2=$h_*,b_*$}
Suppose $n$ independent <axisymmetric pure plumes> share total <buoyancy flux> $B$ equally, have negligible source <volume flux>, and are supplied with a separate ambient ventilation flux $F$ at floor level. Their combined flux is $P(h)=\lambda n^{2/3}B^{1/3}h^{5/3}$. <Volume conservation> at a steady <displacement-ventilation interface height> and <buoyancy flux> conservation give
$$
h_* = \left(\frac{F}{\lambda n^{2/3}B^{1/3}}\right)^{3/5},\qquad b_* = \frac BF.
$$
A positive two-layer state needs $0<h_*<H$, sources small relative to $h_*$, and plumes sufficiently separated to remain independent. Splitting the sources increases total <fluid entrainment> and reduces the interface height by $n^{-2/5}$. A finite source flux cannot simultaneously be neglected and supply all the replacement air; <no steady displacement layer without ambient supply> explains this restriction.
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