Tools · Piping & flow
Acoustic resonance & dead legs
A closed side branch (dead leg, relief valve stub, PSV inlet) is a quarter-wave resonator. Flow across its mouth sheds vortices, and at the wrong velocity they lock onto the acoustic mode and pump up large pulsations.
Gas defaults are roughly a light natural gas. For a dead leg, L is measured from the main pipe wall to the closed end; adding an end correction of about 0.4·d gives a slightly better estimate.
| Mode | Hz | Wavelength (m) |
|---|
The formulas
c = √( γ · Z · Ru · T / Mw ) (Ru = 8314.46 J/kmol·K)
closed–open: fn = (2n − 1) · c / 4L closed–closed or open–open: fn = n · c / 2L
shedding at the branch mouth: fs = St · V / d
Lab studies on closed side branches show the acoustic modes being excited over a Strouhal range of roughly 0.27 to 0.55 (based on branch diameter and main-pipe velocity). The calculator turns that band into a range of main-line velocities that can excite the first mode, so you can see whether normal operation, turndown or start-up passes through it.
Typical fixes: shorten the branch to push its frequency above the excitation, round or chamfer the branch entrance, or relocate it away from flow disturbances. Pulsation from this mechanism is a classic cause of PSV chatter and small-bore connection fatigue.
Screening estimate only. Branch-to-main diameter ratio, the neighbouring branches and the system's acoustic response all change the onset. Source for the Strouhal band: experimental study of flow-excited acoustic resonance in ducts with closed side branches, FIV 2008 conference proceedings, paper 291.
Method & sources
What it calculates
- Speed of sound c = √(γ·Z·Ru·T/Mw), the ideal-gas expression with a compressibility correction.
- Plane-wave acoustic modes: closed–open (quarter-wave) fn = (2n − 1)c/4L; closed–closed and open–open fn = n·c/2L.
- Excitation band for a closed side branch: shedding fs = St·V/d with St from 0.27 to 0.55 (experimental range from the FIV 2008 study).
Assumptions & limits
- The γ·Z form is an approximation; for dense gases near critical conditions use the real-gas isentropic exponent from a property package.
- Plane waves only: valid below the first cross-mode, about 1.84c/(πD).
- End correction at the branch mouth (about 0.4d) is not added automatically.
Verified: γ = 1.3, Z = 0.9, Mw = 19 at 40 °C gives c = 400.4 m/s; a 1.2 m dead leg gives 83.42 Hz, and 50 mm branch gives an excitation band of 7.58–15.45 m/s, all matching Python.
Sources
- Flow-excited acoustic resonance in ducts with closed side branches. Proceedings of the 9th International Conference on Flow-Induced Vibration (FIV 2008), paper 291 ↗
- Blevins, R. D. (1979, reprinted 2001). Formulas for natural frequency and mode shape. Krieger
- API Standard 618, 5th edition. Reciprocating compressors for petroleum, chemical and gas industry services ↗
- Energy Institute (2008). Guidelines for the avoidance of vibration induced fatigue failure in process pipework, 2nd edition ↗