 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-43/5/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Write

z_{i} = H - h

for the height of the warm-layer interface above the heat source. A negligible-volume point heat source supplies a pure plume with conserved buoyancy flux

B_{0}

. The given plume buoyancy implies its volume flux at the interface is

Q_{p} (z_{i}) = \frac{B _{0}}{g _{p}^{'} ( z _{i} )} = \frac{B _{0}^{1/3}}{γ} z_{i}^{5/3} .

At steady displacement ventilation, the plume feeds the warm layer at the same rate it empties through the upper vent:

q_{v} = Q_{p} (z_{i})

. The upper-layer buoyancy balance gives

q_{v} g^{'} = B_{0}

, so

g^{'} = γ B_{0}^{2/3} z_{i}^{- 5/3}, q_{v} = C_{d} A g^{'} h .

Eliminate

q_{v}

and

g^{'}

to get

(H - h)^{5} = γ^{3} C_{d}^{2} A^{2} h, Δ ρ = \frac{ρ _{0}}{g} γ B_{0}^{2/3} (H - h)^{- 5/3} .

The height equation has one solution with

0 < h < H

. Its height is independent of source strength in this ideal pure-plume model, while its warm-layer density contrast increases as

B_{0}^{2/3}

. The sketch shows cool replacement air below, an entraining plume through that region, and the upper mixed layer feeding the ceiling vent.

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