---
title: "Contraction width that just gives critical flow"
subject: "Hydraulics"
chapter: 7
key: "semester_4/hydraulics:contraction-width-for-critical-flow-3m-channel"
source: https://www.bctengineer.com/q/semester_4/hydraulics/contraction-width-for-critical-flow-3m-channel
site: BCT Engineering
---

# Contraction width that just gives critical flow

Hydraulics · Chapter 7: Energy and Momentum Principles in Open channel flow

- Asked in: 2072 Magh, 3 marks
- Video: [Hydraulics || Unit-6 || Critical flow in rectangular channel || old question solution || Pou,Pu,TU ](https://www.youtube.com/watch?v=zPQyNf9U-FY) — Study with Jyoti
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/contraction-width-for-critical-flow-3m-channel
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-7

## Question

Water flows at a depth of 1.8 m and velocity of 1.5 m/s in a 3 m wide rectangular channel. Find the width at contraction which just causes critical flow without a change in the upstream depth.

## Short answer

Set E1 = 1.915 m as the critical energy in the contraction. Then yc = 1.277 m, q2 = √(g yc³) = 4.517 m²/s, and the required width b2 = Q/q2 = 1.79 m. A narrower width would choke the flow.

## Answer

#### Given data
$b_1=3$ m, $y_1=1.8$ m, $V_1=1.5$ m/s, no change in bed level.
$$
Q=3\times1.8\times1.5=8.10\ \text{m}^3/\text{s},\qquad q_1=2.7\ \text{m}^2/\text{s}
$$
$$
E_1=1.8+\frac{1.5^2}{19.62}=1.9147\ \text{m}
$$

#### Condition
The upstream depth does not change, so $E_1$ is just enough for critical flow in the contraction: $E_{min2}=E_1$.
$$
y_c=\frac23E_1=1.2765\ \text{m}
$$
$$
q_2=\sqrt{gy_c^3}=\sqrt{9.81\times1.2765^3}=4.5169\ \text{m}^2/\text{s}
$$
$$
b_2=\frac{Q}{q_2}=\frac{8.10}{4.5169}=1.793\ \text{m}
$$
**Answer: contracted width $=1.79$ m** (critical depth $1.28$ m). A narrower width would choke the flow and raise the upstream depth.

## Check yourself

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

1. Upstream depth is to stay unchanged while the contraction just produces critical flow. What condition is used?
   A. The minimum specific energy at the contraction equals the upstream specific energy E1
   B. The upstream Froude number equals 1
   C. The critical depth equals the upstream depth 1.8 m
   D. The discharge per unit width stays at 2.7 m^2/s
   Answer: A. With no bed change and no upstream depth change, E1 is only just enough, so E_min at the throat = 1.5 yc = E1. (Key point: Condition for just-critical contraction)

2. With E1 = 1.9147 m, what is the critical depth in the contraction and why?
   A. 1.9147 m, since yc = E1
   B. 1.8 m, since the depth is unchanged
   C. 1.2765 m, since yc = (2/3)E1 for a rectangular section
   D. 0.957 m, since yc = E1/2
   Answer: C. For a rectangular channel Emin = 1.5 yc, so yc = (2/3)(1.9147) = 1.2765 m. (Key point: Critical depth from energy)

3. After finding yc, how is the contracted width b2 obtained?
   A. b2 = q1 / yc
   B. b2 = Q / q2, where q2 = sqrt(g yc^3)
   C. b2 = b1 x yc / y1
   D. b2 = Q / (V1 x yc)
   Answer: B. At critical flow q2 = sqrt(g yc^3) = 4.517 m^2/s, so b2 = 8.10/4.517 = 1.79 m. (Key point: Finding contracted width)

4. What happens if the contraction is made narrower than the computed 1.79 m?
   A. Flow becomes supercritical upstream
   B. Nothing changes since depth is fixed
   C. The upstream depth falls
   D. The flow chokes and the upstream depth rises
   Answer: D. A narrower throat needs more energy than E1 is available, so the flow chokes and upstream depth rises. (Key point: Effect of narrower width)

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