= Solution
Let $z_f(t)$ be the <filling-box first front>, with $z_f(0)=H$. Below it the original fluid has not yet been through the plume. In the <filling box model>, the <turbulent plume> adjusts rapidly and occupies a negligible fraction of the room's area. A point source with negligible source volume flux supplies $Q(z)=CF_s^{1/3}z^{5/3}$. All its volume crossing the front grows the processed region, so <volume conservation> gives
$$
A_c\dot z_f=-\pi CF_s^{1/3}z_f^{5/3}.
$$
Put $a_p=\pi CF_s^{1/3}/A_c$. Integrating $d(z_f^{-2/3})/dt=2a_p/3$ yields
$$
\boxed{z_f(t)=\left(H^{-2/3}+\frac23a_pt\right)^{-3/2}
=H\left[1+\frac{2\pi CF_s^{1/3}H^{2/3}}{3A_c}t\right]^{-3/2}.}
$$
The first front approaches the floor only as $t\to\infty$. A nonzero source volume flux would require an equal exhaust to keep the room volume fixed; it is treated in the following parts.
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