---
title: "Economical trapezoid for given velocity"
subject: "Hydraulics"
chapter: 6
key: "semester_4/hydraulics:economical-trapezoidal-velocity-0-6-bed-slope"
source: https://www.bctengineer.com/q/semester_4/hydraulics/economical-trapezoidal-velocity-0-6-bed-slope
site: BCT Engineering
---

# Economical trapezoid for given velocity

Hydraulics · Chapter 6: Uniform flow in open channel

- Asked in: 2073 Bhadra, 6 marks
- Video: [most economical trapezoidal channel section || fluid mechanics || h&hm](https://www.youtube.com/watch?v=stWSGCsfgEE) — e Tution
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/economical-trapezoidal-velocity-0-6-bed-slope
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-6

## Question

Design an economical trapezoidal channel with a velocity of 0.6 m/s. The side slope Z of channel is 1.5 and conveys a discharge of 3 m³/s. Take manning's coefficient as 0.003. Also find the required bed slope.

## Short answer

With Q=3 m³/s, V=0.6 m/s, z=1.5, area 5 m² gives economical depth 1.54 m, bed width 0.93 m. Manning's equation with n=0.003 yields bed slope 4.59×10⁻⁶.

## Answer

#### Given data
$V=0.6$ m/s, $z=1.5$, $Q=3$ m³/s, $n=0.003$ (as given in the question).

#### Area
$$
A=\frac QV=\frac{3}{0.6}=5.0\ \text{m}^2
$$

#### Economical section
$$
A=y^2\left(2\sqrt{1+z^2}-z\right)=y^2\left(2\sqrt{3.25}-1.5\right)=2.1056\,y^2
$$
$$
y=\sqrt{\frac{5}{2.1056}}=1.541\ \text{m}
$$
$$
b=2y\left(\sqrt{1+z^2}-z\right)=2\times1.541\times(1.8028-1.5)=0.933\ \text{m}
$$
$$
P=b+2y\sqrt{1+z^2}=6.489\ \text{m},\qquad R=\frac AP=\frac y2=0.770\ \text{m}
$$
Check: $(b+zy)y=(0.933+1.5\times1.541)\times1.541=5.00$ m².

#### Bed slope from Manning's equation
$$
V=\frac1nR^{2/3}S_0^{1/2}\;\Rightarrow\;S_0=\left(\frac{nV}{R^{2/3}}\right)^2=\left(\frac{0.003\times0.6}{0.770^{2/3}}\right)^2=4.587\times10^{-6}
$$
i.e. a slope of about 1 in 218 000.

**Answer: depth $y=1.54$ m, bed width $b=0.93$ m, side slope 1.5H:1V, bed slope $S_0=4.59\times10^{-6}$.**

Note: $n=0.003$ is unusually small. If the intended value were $n=0.03$ (earth channel), the slope would be $4.59\times10^{-4}$ (1 in 2180).

## Check yourself

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

1. How is the required flow area obtained?
   A. A = Q x V = 1.8 m2
   B. A = Q/V = 5 m2
   C. A = V/Q = 0.2 m2
   D. A = Q/V^2 = 8.3 m2
   Answer: B. By continuity A = Q/V = 3/0.6 = 5 m2. (Key point: Area from continuity)

2. For z = 1.5 and A = 5 m2, what gives the depth and bed width of the economical section?
   A. b = y and A = 2.5 y^2
   B. A = y^2(2 sqrt(1+z^2) - z) = 2.1056 y^2 so y = 1.54 m, and b = 2y(sqrt(1+z^2) - z) = 0.93 m
   C. b = 2y and y = 1.12 m
   D. y = A/b with b = 3 m
   Answer: B. The economical-trapezoid relations give y = 1.541 m and b = 0.933 m. (Key point: Economical section dimensions)

3. Which step gives the bed slope?
   A. Use R = y (not y/2) in Manning's equation
   B. S0 = n V / R^(2/3) without squaring
   C. Rearrange Manning's equation: S0 = (n V / R^(2/3))^2 with R = y/2 = 0.77 m
   D. Take S0 = 1/z
   Answer: C. From V = (1/n) R^(2/3) S0^(1/2), S0 = (nV/R^(2/3))^2 = 4.59e-6. (Key point: Bed slope)

4. The given n = 0.003 produces a very tiny slope. If n were 0.03, the slope would be:
   A. 10 times smaller
   B. 100 times larger, 4.59e-4
   C. Unchanged
   D. 10 times larger
   Answer: B. S0 is proportional to n^2, so a tenfold larger n gives a 100 times larger slope. (Key point: Sensible roughness)

## Related in Hydraulics

- [Economical trapezoid and power saving](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aeconomical-trapezoid-saving-in-power-vs-rectangular) — Hydraulics, chapter 6
- [Most economical trapezoid of area 40 m²](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aeconomical-trapezoidal-section-area-40) — Hydraulics, chapter 6
- [Half-hexagon channel dimensions and Manning n](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ahalf-hexagon-channel-chezy-manning-n) — Hydraulics, chapter 6
- [Efficient trapezoid with unequal side slopes](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aefficient-trapezoidal-two-side-slopes-critical-depth) — Hydraulics, chapter 6
- [Chezy's and Manning's coefficient for Seti River](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aseti-river-chezy-manning-coefficient) — Hydraulics, chapter 6
- [Economical channel: unlined versus concrete lined](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aeconomical-trapezoidal-unlined-vs-lined) — Hydraulics, chapter 6, asked 2×

## Related in other subjects

- [Economical rectangular condition and trapezoidal bed slope](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Atrapezoidal-channel-bed-slope) — Hydro Power, chapter 1
- [Economical trapezoidal channel side length](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aeconomical-trapezoidal-side-length) — Hydro Power, chapter 1
- [Most economical trapezoidal channel of area 50 m²](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aeconomical-trapezoidal-area-50) — Hydro Power, chapter 1
- [Economical lined channel: Q 20](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Alined-canal-q20-slope-30cm-km) — Irrigation and Drainage Engineering, chapter 4