---
title: "Pressure rise: rigid versus elastic pipe"
subject: "Hydraulics"
chapter: 4
key: "semester_4/hydraulics:rigid-vs-elastic-pipe-pressure-rise-5km"
source: https://www.bctengineer.com/q/semester_4/hydraulics/rigid-vs-elastic-pipe-pressure-rise-5km
site: BCT Engineering
---

# Pressure rise: rigid versus elastic pipe

Hydraulics · Chapter 4: Unsteady flow in pipes

- Asked in: 2070 Bhadra, 5 marks
- Video: [Unit-3:Unsteady Flow in pipe|Water Hammer, Pressure rise in elastic pipe|Hydraulics PU PoU TU KU BE](https://www.youtube.com/watch?v=t4lMu8Ws1JM) — Civil Engineering Solution Channel
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/rigid-vs-elastic-pipe-pressure-rise-5km
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-4

## Question

A 300 mm diameter pipe of mild steel having 6 mm thickness carries water at the rate of 200 l/s. What will be the rise in pressure if the valve at the downstream end is closed instantaneously? Compare results assuming the pipe to be rigid as well as elastic. What should be the maximum closing time for the computed results to be valid? Take pipe length as 5.0 km, Modulus of elasticity of pipe material as $2.25\times10^{11}$ N/m², Bulk modulus of elasticity of water as $2.0\times10^9$ N/m².

## Short answer

For elastic pipe: wave speed c=1176.7 m/s, pressure rise Δp=3.33 MPa (339 m of water). For rigid pipe: c=1414.2 m/s, Δp=4.00 MPa (408 m). Results require closure time ≤ 2L/c: 8.50 s (elastic) or 7.07 s (rigid).

## Answer

For instantaneous closure the pressure rise is $\Delta p=\rho cV$. The wave celerity $c$ depends on whether the pipe wall is treated as rigid (only the water is compressible) or elastic (the wall also stretches).

#### Given data
$D=0.3$ m, $t=6$ mm, $Q=0.2$ m³/s, $L=5000$ m, $E=2.25\times10^{11}$ N/m², $K=2.0\times10^9$ N/m².
$$
V=\frac{Q}{A}=\frac{0.2}{\frac{\pi}{4}(0.3)^2}=2.829\ \text{m/s}
$$

#### (a) Rigid pipe
$$
c_r=\sqrt{\frac{K}{\rho}}=\sqrt{\frac{2\times10^9}{1000}}=1414.2\ \text{m/s}
$$
$$
\Delta p=\rho c_rV=1000\times1414.2\times2.829=4.001\ \text{MPa}\ (407.9\ \text{m of water})
$$

#### (b) Elastic pipe
$$
c_e=\sqrt{\frac{K/\rho}{1+\dfrac{KD}{Et}}}=\sqrt{\frac{2\times10^6}{1+\dfrac{2\times10^9\times0.3}{2.25\times10^{11}\times0.006}}}=1176.7\ \text{m/s}
$$
$$
\Delta p=\rho c_eV=1000\times1176.7\times2.829=3.329\ \text{MPa}\ (339.4\ \text{m})
$$

#### Comparison
| Pipe | $c$ (m/s) | $\Delta p$ (MPa) | Head (m) |
|---|---|---|---|
| Rigid | 1414.2 | 4.001 | 407.9 |
| Elastic | 1176.7 | 3.329 | 339.4 |

The rigid-pipe assumption over-estimates the pressure by about 20%, because the elastic wall expands and absorbs some energy.

#### Maximum closing time for the result to be valid
The result holds for rapid closure, i.e. $t_c\le 2L/c$:
- Elastic: $t_{max}=\dfrac{2\times5000}{1176.7}=8.50$ s.
- Rigid: $t_{max}=\dfrac{2\times5000}{1414.2}=7.07$ s.

**Answer: elastic pipe: $\Delta p=3.33$ MPa (339 m); rigid pipe: $\Delta p=4.00$ MPa (408 m); closing time must not exceed 8.50 s (elastic) / 7.07 s (rigid).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For a rigid pipe, the wave speed is
   A. sqrt(K/rho)
   B. K/rho
   C. sqrt(K D / E t)
   D. sqrt(E/rho)
   Answer: A. Only water compressibility matters, c = sqrt(2e9/1000) = 1414 m/s. (Key point: Rigid pipe celerity)

2. Why is the pressure rise lower for the elastic pipe?
   A. Friction increases
   B. The velocity is lower
   C. The wall stretches and absorbs energy, lowering c
   D. The water is lighter
   Answer: C. The elastic wall expands, giving c = 1177 m/s and dp = 3.33 MPa against 4.00 MPa. (Key point: Elastic vs rigid)

3. The pressure rise for instantaneous closure is
   A. K V^2
   B. rho g V
   C. rho c V
   D. rho L V / t
   Answer: C. dp = rho c V (Joukowsky). (Key point: Instantaneous closure formula)

4. The results are valid if closing time is not more than
   A. 4L/c
   B. L/2c
   C. 2L/c
   D. L/c
   Answer: C. Closure must finish before the reflected wave returns, t_c <= 2L/c (8.50 s elastic). (Key point: Validity of result)

## Related in Hydraulics

- [Water hammer in 1.2 m steel pipe, wall thickness](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-steel-pipe-thickness-1200mm) — Hydraulics, chapter 4
- [Pressure rise for instantaneous closure, elastic pipe](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aelastic-pipe-pressure-rise-derivation) — Hydraulics, chapter 4
- [Water hammer in 20 m steel pipeline with valve](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-20m-pipeline-pressure-time-curves) — Hydraulics, chapter 4
- [Pipe wall stress for 3 s valve closure](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Avalve-closure-3s-stress-in-pipe-wall) — Hydraulics, chapter 4
- [Diameter to thickness ratio for stress limit](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Apipe-diameter-thickness-ratio-hoop-stress) — Hydraulics, chapter 4
- [Penstock pressure rise and pressure-time sketch](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asurge-tank-penstock-pressure-rise) — Hydraulics, chapter 4

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- [Water pipe thickness from pressure head](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Awater-pipe-thickness-pressure-head) — Strength of Materials, chapter 5
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