---
title: "Pressure rise for gradual and sudden closure"
subject: "Hydro Power"
chapter: 1
key: "semester_6/hydro_power:water-hammer-valve-closure"
source: https://www.bctengineer.com/q/semester_6/hydro_power/water-hammer-valve-closure
site: BCT Engineering
---

# Pressure rise for gradual and sudden closure

Hydro Power · Chapter 1: Introduction to Basic Fluid Mechanics and Hydraulics

- Asked in: 2081 Chaitra, 3+3 marks
- Video: [Sudden closure of valve, when pipe is rigid](https://www.youtube.com/watch?v=aU-7L2Vnz2k) — Pavika fluid says Tawi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/water-hammer-valve-closure
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-1

## Question

In a pipe of 500 mm diameter and 2500 m length, provided with a valve at its end, water is flowing with a velocity of 1.5 m/s. Assuming velocity of pressure wave = 1460 m/s, find (i) the rise in pressure if the valve is closed in 25 seconds and (ii) the rise in pressure if the valve is closed in 2 seconds. Assume the pipe to be rigid one and take bulk modulus of water as 2.132 GN/m².

## Short answer

Critical time 2L/c = 3.425 s. For 25 s closure (gradual): Δp = ρLV/T = 0.15 MN/m². For 2 s closure (sudden): Δp = ρcV = 2.19 MN/m².

## Answer

First compare the closure time with the critical time $2L/c$ to decide whether the closure is gradual or sudden.

#### Critical time
$$
t_c = \frac{2L}{c} = \frac{2 \times 2500}{1460} = 3.425\ \text{s}
$$

#### (i) Closure in 25 s
$25\ \text{s} > 3.425\ \text{s}$, so it is **gradual closure**. The water column decelerates uniformly:
$$
\begin{aligned}
p &= \frac{\rho L V}{T} = \frac{1000 \times 2500 \times 1.5}{25} = 150\,000\ \text{N/m}^2 = 0.15\ \text{MN/m}^2 \\
H &= \frac{p}{\rho g} = \frac{150000}{9810} = 15.29\ \text{m}
\end{aligned}
$$

#### (ii) Closure in 2 s
$2\ \text{s} < 3.425\ \text{s}$, so it is **sudden closure**. For a rigid pipe:
$$
\begin{aligned}
p &= V\sqrt{K\rho} = 1.5\sqrt{2.132 \times 10^9 \times 1000} \\
&= 1.5 \times 1.460 \times 10^6 = 2.19 \times 10^6\ \text{N/m}^2 = 2.19\ \text{MN/m}^2
\end{aligned}
$$
(Check: $\rho c V = 1000 \times 1460 \times 1.5 = 2.19$ MN/m², same, since $c = \sqrt{K/\rho} = 1460$ m/s.) This equals about 223.3 m of water.

**Answer:** (i) 0.15 MN/m² (150 kN/m²) for 25 s closure; (ii) 2.19 MN/m² for 2 s closure.

## Check yourself

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

1. How is closure classified as gradual or sudden?
   A. Compare closing time with 2L/c
   B. Compare closing time with L/V
   C. Compare velocity with wave speed
   D. Compare diameter with length
   Answer: A. If closing time exceeds 2L/c it is gradual; if less it is sudden. Here 2L/c = 3.425 s. (Key point: Critical time)

2. For the 25 s closure, which formula applies and what is the result?
   A. p = rho c V = 2.19 MPa
   B. p = rho L V / T = 0.15 MN/m^2
   C. p = rho V / T = 60 N/m^2
   D. p = V sqrt(K rho) = 2.19 MPa
   Answer: B. 25 s > 3.425 s so it is gradual: p = 1000 x 2500 x 1.5 / 25 = 150 kN/m^2. (Key point: Gradual closure)

3. For the 2 s closure of a rigid pipe, what gives the pressure rise?
   A. rho L V / T with T = 2
   B. V sqrt(K rho) = rho c V = 2.19 MN/m^2
   C. K x V
   D. Half of rho c V
   Answer: B. 2 s < 3.425 s so closure is sudden; p = V sqrt(K rho) = 2.19 MN/m^2, equal to rho c V since c = sqrt(K/rho). (Key point: Sudden closure)

## Related in Hydro Power

- [Water hammer effects and gradual closure](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Awater-hammer-gradual-closure) — Hydro Power, chapter 1
- [Water hammer in elastic cast-iron pipe](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Awater-hammer-cast-iron-pipe) — Hydro Power, chapter 1
- [Water hammer in 200 mm pipe](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Awater-hammer-200mm-pipe) — Hydro Power, chapter 1
- [Head loss at sudden enlargement](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Asudden-enlargement-loss) — Hydro Power, chapter 1
- [Flow between reservoirs (15 cm and 25 cm pipe)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Areservoirs-pipeline-15-25cm) — Hydro Power, chapter 1
- [Head loss with enlarged central pipe length](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Apipe-replaced-central-length) — Hydro Power, chapter 1

## Related in other subjects

- [Pressure variation after sudden valve closure](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Awater-hammer-pressure-time-valve-and-midpoint) — Hydraulics, chapter 4
- [Pressure rise: rigid versus elastic pipe](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Arigid-vs-elastic-pipe-pressure-rise-5km) — Hydraulics, chapter 4
- [Gradual valve closure with retardation law](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Agradual-valve-closure-head-at-valve) — 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