Library / Hydraulics, open channels and fluid power / HYD-47
Hydraulics, open channels and fluid power
Wave speed in an elastic thin-walled pipe
Speed of a pressure wave in a liquid-filled thin-walled elastic pipe, from the liquid density ρ and bulk modulus B, and the pipe diameter D, wall Young’s modulus E and wall thickness tw.
- Ref
- HYD-47
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- a [m/s]
A pressure wave in a liquid-filled pipe travels slower than in free liquid, because the wall stretches. With a thin elastic wall of thickness tw, the wave speed is a = [ρ(1/B + D/(E tw))]^(−1/2). The first term in the bracket is the liquid’s compressibility and the second is the wall flexibility. A soft wall (plastic) gives a much lower speed than steel. Axial restraint of the pipe changes the wall term slightly.
Speed of a pressure wave in a liquid-filled thin-walled elastic pipe, from the liquid density ρ and bulk modulus B, and the pipe diameter D, wall Young’s modulus E and wall thickness tw. A wall coefficient of one is assumed.
Calculator
- Wave speed in free liquid, √(B/ρ)1,414 m/s
- Compliance factor, 1 + B D/(E tw)2
- Reduction compared with free liquid29.29 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The wave moves at 1,000 m/s in this pipe, 29.3 % below the 1,414 m/s of free liquid.
- A rigid wall (very large E) gives the free-liquid value, and a thin plastic wall can drop the wave speed to a few hundred m/s, which lowers surge pressures.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| E Wall Young’s modulus | 1e9 to 4e11 Pa | 99.75 | 1,155 | 1,055 | +10% | +2.35 % |
| D Internal pipe diameter | 0.01 to 5 m | 426.4 | 1,400 | 973.9 | +10% | −2.41 % |
| tw Wall thickness | 5e-4 to 0.1 m | 426.4 | 1,380 | 953.7 | +10% | +2.35 % |
| B Liquid bulk modulus | 1e8 to 5e9 Pa | 308.6 | 1,195 | 886.6 | +10% | +2.35 % |
| ρ Liquid density | 500 to 2,000 kg/m³ | 707.1 | 1,414 | 707.1 | +10% | −4.65 % |
Sweep of Wall thickness
| Wall thickness (m) | Wave speed (m/s) |
|---|---|
| 5e-4 | 426.4 |
| 0.01294 | 1,201 |
| 0.02538 | 1,293 |
| 0.03781 | 1,329 |
| 0.05025 | 1,349 |
| 0.06269 | 1,361 |
| 0.07512 | 1,369 |
| 0.08756 | 1,375 |
| 0.1 | 1,380 |
Sweep one variable
Variables and how each one contributes
Wave speed
What the equation gives you. Everything else feeds it.
Liquid density
The liquid density, in kg/m³. A denser liquid lowers the wave speed. Raising it by 10 % (to 1,100) moves the result by −4.65 %, from 1,000 to 953.5 m/s.
Liquid bulk modulus
The liquid bulk modulus, in Pa. A stiffer liquid raises the wave speed. Raising it by 10 % (to 2.2×109) moves the result by +2.35 %, from 1,000 to 1,024 m/s.
Internal pipe diameter
The pipe internal diameter, in m. A wider pipe is more flexible for its wall thickness, so the wave speed falls. Raising it by 10 % (to 0.55) moves the result by −2.41 %, from 1,000 to 975.9 m/s.
Wall Young’s modulus
The wall Young’s modulus, in Pa. A stiffer wall raises the wave speed. Raising it by 10 % (to 2.2×1011) moves the result by +2.35 %, from 1,000 to 1,024 m/s.
Wall thickness
The wall thickness, in m. A thicker wall raises the wave speed. Raising it by 10 % (to 0.0055) moves the result by +2.35 %, from 1,000 to 1,024 m/s.
Worked example with the values above
- Compressibility term: 1/B = 5×10−10 1/Pa.
- Wall term: D/(E tw) = 0.5 / (2×1011 × 0.005) = 5×10−10 1/Pa.
- a = [1,000 × 1×10−9]^(−1/2) = 1,000 m/s.
Assumptions and limits
- A single-phase liquid in a uniform line, with a linear elastic wall and the wave travelling without loss.
- Rapid closure, with the valve moving in less than the wave return time.
Common mistakes
- Using the speed of sound in free liquid.
- Using the pipe outer diameter in place of the inner diameter.
- Ignoring trapped air, which lowers the wave speed sharply.
Statistics and uncertainty
Wave speed is sensitive to trapped air and pipe support, so surge predictions can be off by large factors. The first Joukowsky peak is usually the largest, but reflections and column separation can create bigger ones.
Where it is used
- Checking pipelines and pump stations for water hammer.
- Sizing surge tanks and gas-charged accumulators.
Maths you need
- Water hammer: a pressure wave from a sudden velocity change
- Bulk modulus: B, the stiffness of the liquid
- Polytropic process: p Vⁿ = constant