Rigid-column screening — the Michaud downsurge
When the pump dies, the column keeps moving and the static head Hs is what decelerates it; the pump-side pressure falls by the inertia term L·V₀/(g·tr) — the same Michaud relation as gradual valve closure, with the rundown time tr in place of tc. Anchor it: L = 1000 m, V₀ = 1.5 m/s, tr = 2 s → ΔH = 76.5 m; with Hs = 20 m the pump-side head dives to −56.5 m gauge — far past the vapor line. Stretch the rundown to 10 s and the same line stays at +4.7 m: no separation. The static head that stops the column acts on the time \(t_{rev}=V_0 L/(g\,H_s)\) — 7.6 s for the same line.
Vapor-pressure truncation — why the line cannot go below −10.3 m
Liquid water cannot sustain less than its vapor-pressure head: at the moment the computed head line would fall below −10.3 m gauge (full vacuum minus the vapor pressure), the column tears open and a vapor cavity occupies the gap instead. The computed trace is truncated there — the missing head reappears as a growing cavity, and when the reversed column returns the cavity collapses: the two water bodies meet at relative velocity and the rejoin spike reaches \(\Delta H_{up}=a\,\Delta V_{rev}/g\) above the vapor line — for a = 1200 m/s and ΔVrev = 0.75 m/s that is +92 m on top of −10.3 m. This model is a schematic: rigid-column screening with vapor truncation, for ranking risk and protection need — not a substitute for a full elastic-column (MOC) transient study.
The five phases drawn by the animation
| Phase | What you see | Model event |
|---|---|---|
| ① Rundown | Pump slows, head at the pump slides down the HGL | H(t) = H_s − L·V₀/(g·tr)·t/tr |
| ② Vapor line reached | Head line touches the red dashed line; cavity cracks open | Hmin would cross −10.3 m |
| ③ Cavity growth | Gap between pump and column widens, then closes as flow reverses | H pinned at vapor head; duration ≈ 0.6·trev |
| ④ Rejoining collision | 💥 — columns slam; spike on the H(t) strip | H jumps to −10.3 + a·ΔVrev/g |
| ⑤ Settling | Damped surging toward steady Hs | Exponential decay envelope |
Decision criteria
| Minimum gauge head at pump | Verdict | Action |
|---|---|---|
| < −10.3 m (−34 ft) | 🔴 Column separation | Protection mandatory — air chamber, vacuum-breaking air valve or one-way surge tank; then re-run a full transient study. |
| −10.3 m … −7 m | 🟡 Thin margin | Less than 3 m above the vapor line — protection review recommended; check warmer-water and longer-line scenarios. |
| > −7 m | 🟢 Column holds | No separation indicated; keep ≥ 2 m margin and re-check after any layout change. |
Symbol table
| Symbol | Meaning | Units |
|---|---|---|
| L | pipeline length (pump to reservoir) | m / ft |
| V₀ | steady flow velocity before the trip | m/s |
| tr | pump rundown time (to zero speed) | s |
| Hs | static head (pump to reservoir level) | m / ft |
| a | pressure-wave speed | m/s |
| trev | time for the static head to stop and reverse the column, V₀L/(gHs) | s |
Worked anchor (self-check)
Defaults — L = 1000 m, tr = 2 s, Hs = 20 m: ΔHdown = 1000·1.5/(9.807·2) = 76.5 m, minimum head 20 − 76.5 = −56.5 m gauge → 🔴 separation. The head line reaches −10.3 m at tsep = 2·(20+10.3)/76.5 = 0.79 s into the trip, the cavity lives about 0.6·trev = 4.6 s, and the rejoin spike lands at −10.3 + 1200·0.75/9.807 ≈ +81 m gauge. Set tr = 10 s and every number relaxes: ΔH = 15.3 m, Hmin = +4.7 m, no cavity, 🟢.
Engineering criteria applied
- Separation screen — the −10.3 m (−34 ft) vapor-line criterion with a 3 m amber approach band and a 2 m green keep-out.
- Rejoin severity — Joukowsky ceiling on the reversed velocity, a·ΔVrev/g above the vapor line, shown on the H(t) strip with the 💥 marker.
- Model class — rigid-column (Michaud) with vapor-pressure truncation, explicitly labelled schematic on the canvas and readouts; final design belongs to an elastic-column transient study.