---
title: "Head loss at sudden enlargement"
subject: "Hydro Power"
chapter: 1
key: "semester_6/hydro_power:sudden-enlargement-loss"
source: https://www.bctengineer.com/q/semester_6/hydro_power/sudden-enlargement-loss
site: BCT Engineering
---

# Head loss at sudden enlargement

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

- Asked in: 2080 Asoj, 6 marks
- Video: [Loss of Head Due to Sudden Enlargement | Head loss Due to Sudden Enlargement | Flow Through Pipe](https://www.youtube.com/watch?v=dmRW-H4TkKI) — Professor Nidhi Trivedi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/sudden-enlargement-loss
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-1

## Question

A horizontal pipe 150 mm in diameter is joined by sudden enlargement to a 225 mm diameter pipe. Water is flowing through it at the rate of 0.05 m³/s. Find (i) loss of head due to sudden expansion, (ii) pressure difference in the two pipes, and (iii) change in pressure if the change of section is gradual without any loss.

## Short answer

Head loss h_e = (V₁−V₂)²/(2g) = 0.126 m. Pressure difference with loss is p₂−p₁ ≈1.98 kN/m²; with gradual loss-free enlargement it would be 3.21 kN/m².

## Answer

**Given:** $D_1 = 0.15$ m, $D_2 = 0.225$ m, $Q = 0.05$ m³/s, horizontal pipe.

#### Velocities
$$
\begin{aligned}
A_1 &= \frac{\pi}{4}(0.15)^2 = 0.017671\ \text{m}^2, \quad V_1 = \frac{0.05}{0.017671} = 2.829\ \text{m/s} \\
A_2 &= \frac{\pi}{4}(0.225)^2 = 0.039761\ \text{m}^2, \quad V_2 = \frac{0.05}{0.039761} = 1.258\ \text{m/s}
\end{aligned}
$$

#### (i) Loss of head due to sudden expansion
$$
h_e = \frac{(V_1 - V_2)^2}{2g} = \frac{(2.829 - 1.258)^2}{2 \times 9.81} = 0.1259\ \text{m}
$$

#### (ii) Pressure difference (Bernoulli with loss, $z_1 = z_2$)
$$
\frac{p_1}{\rho g} + \frac{V_1^2}{2g} = \frac{p_2}{\rho g} + \frac{V_2^2}{2g} + h_e
$$
$$
\begin{aligned}
\frac{p_2 - p_1}{\rho g} &= \frac{V_1^2 - V_2^2}{2g} - h_e = \frac{2.829^2 - 1.258^2}{19.62} - 0.1259 \\
&= 0.3274 - 0.1259 = 0.2015\ \text{m} \\
p_2 - p_1 &= 9810 \times 0.2015 = 1977\ \text{N/m}^2
\end{aligned}
$$

#### (iii) Gradual enlargement without loss ($h_e = 0$)
$$
p_2 - p_1 = \rho\frac{V_1^2 - V_2^2}{2} = 9810 \times 0.3274 = 3212\ \text{N/m}^2
$$

**Answer:** (i) $h_e = 0.126$ m; (ii) pressure rises by $p_2 - p_1 \approx 1.98$ kN/m²; (iii) with a gradual, loss-free enlargement the rise would be $\approx 3.21$ kN/m².

## Check yourself

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

1. Which formula gives head loss in a sudden enlargement?
   A. (V1 - V2)^2/2g
   B. (V1^2 - V2^2)/2g
   C. 0.5 V1^2/2g
   D. V2^2/2g
   Answer: A. Borda-Carnot loss: 0.126 m. (Key point: Loss formula)

2. Why is the pressure rise smaller than for a loss-free change?
   A. Friction adds pressure
   B. Area is smaller
   C. The shock loss h_e is subtracted from the velocity head recovery
   D. Velocity increases
   Answer: C. p2-p1 over rho g = (V1^2 - V2^2)/2g - h_e = 0.3274 - 0.1259 = 0.2015 m. (Key point: Pressure rise with loss)

3. What is the pressure rise for a loss-free gradual enlargement?
   A. About 0.126 kN/m^2
   B. About 1.98 kN/m^2
   C. Zero
   D. About 3.21 kN/m^2
   Answer: D. With h_e = 0 the full velocity head difference converts: 9810 x 0.3274 = 3212 N/m^2. (Key point: Gradual enlargement)

4. Which are the velocities V1 and V2?
   A. 5.0 and 2.2 m/s
   B. 2.829 and 1.258 m/s
   C. 2.829 and 2.829 m/s
   D. 1.258 and 2.829 m/s
   Answer: B. V = Q/A: 0.05/0.017671 and 0.05/0.039761. (Key point: Velocities)

## Related in Hydro Power

- [Discharge and head loss in horizontal pipe](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Apipe-discharge-head-loss) — 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
- [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
- [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
- [Pressure rise for gradual and sudden closure](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Awater-hammer-valve-closure) — Hydro Power, chapter 1
- [Head loss and flow direction in tapering oil pipe](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Atapering-pipe-oil-flow) — Hydro Power, chapter 1

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