---
title: "Economical channel: unlined versus concrete lined"
subject: "Hydraulics"
chapter: 6
key: "semester_4/hydraulics:economical-trapezoidal-unlined-vs-lined"
source: https://www.bctengineer.com/q/semester_4/hydraulics/economical-trapezoidal-unlined-vs-lined
site: BCT Engineering
---

# Economical channel: unlined versus concrete lined

Hydraulics · Chapter 6: Uniform flow in open channel

- Asked in: 2073 Magh, 6 marks; 2071 Magh, 8 marks (asked 2 times)
- Video: [Design of lined channel](https://www.youtube.com/watch?v=hv6dTDBEugM) — Epsilon One
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/economical-trapezoidal-unlined-vs-lined
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-6

## Question

A trapezoidal channel having side slope of 1:1 has to carry a flow of 15 m³/s. The bed slope is 1 in 1000. Chezy's C is 45 if the channel is unlined and 70 if the channel is lined with concrete. The cost per m³ of excavation is 3 times cost per m² of lining. Find which arrangement is economical.

## Short answer

Using most-efficient trapezoid (b=0.828y, R=y/2) and Chezy’s formula, unlined depth is 2.315m, area 9.80m², cost 29.39 units/m; lined depth is 1.94m, area 6.88m², cost 27.74 units/m. The concrete-lined channel is more economical.

## Answer

For a given area and side slope, the most efficient (least wetted perimeter) trapezoid is used for each case. Then the cost per metre of channel is compared in terms of the unit cost of lining.

#### Given data
$Q=15$ m³/s, $z=1$ (1H:1V), $S_0=1/1000$, $C=45$ (unlined) or $70$ (lined). Let cost of lining per m² $=1$ unit; excavation cost per m³ $=3$ units.

#### Most efficient trapezoidal section ($z=1$)
$$
b=2y(\sqrt{1+z^2}-z)=2y(\sqrt2-1)=0.8284\,y,\qquad R=\frac y2
$$
$$
A=(b+zy)y=1.8284\,y^2,\qquad P=b+2y\sqrt{1+z^2}=3.6569\,y
$$

#### Chezy's equation $Q=CA\sqrt{RS_0}$
$$
15=C\,(1.8284\,y^2)\sqrt{\frac y2\times0.001}
$$
| Item | Unlined ($C=45$) | Lined ($C=70$) |
|---|---|---|
| Depth $y$ (m) | 2.315 | 1.940 |
| Bed width $b=0.8284\,y$ (m) | 1.918 | 1.607 |
| Area $A$ (m²) | 9.798 | 6.881 |
| Wetted perimeter $P$ (m) | 8.465 | 7.094 |

#### Cost per metre length (assuming the excavated section equals the flow section)
- Unlined: excavation only $=3\times A=3\times9.798=29.39$ units.
- Lined: excavation $=3\times6.881=20.64$ units, plus lining $=1\times P=7.09$ units; total $=27.74$ units.

**Answer: lined channel costs 27.74 units/m against 29.39 units/m for the unlined channel, so the concrete-lined channel (y = 1.94 m, b = 1.61 m) is more economical (cheaper by about 5.6%).**

## Check yourself

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

1. For each case, which trapezoidal section is used for the comparison?
   A. Most efficient section (least wetted perimeter) with z = 1
   B. Rectangular section
   C. Section with b = y
   D. Triangular section
   Answer: A. Each case uses the most efficient trapezoid, b = 2y(sqrt(1+z^2) - z) and R = y/2. (Key point: Section choice)

2. Why is the lined channel smaller than the unlined one?
   A. Its bed slope is steeper
   B. Its Chezy C is larger (70 vs 45), so less area carries 15 m^3/s
   C. It has a higher discharge
   D. Its side slope is flatter
   Answer: B. A higher C gives higher velocity, so a smaller area carries the same discharge. (Key point: Effect of C)

3. How is the cost per metre of the lined channel calculated?
   A. 3 x P only
   B. 1 x A only
   C. 3 x P + 1 x A
   D. 3 x A (excavation) + 1 x P (lining)
   Answer: D. Excavation costs 3 units per m^3 (times area) and lining 1 unit per m^2 (times wetted perimeter). (Key point: Cost model)

4. What is the conclusion?
   A. Unlined is cheaper at 27.74 units/m
   B. Both cost the same
   C. Lined costs 27.74 units/m against 29.39 for unlined, so lining is more economical
   D. Lined costs 20.64 units/m in total
   Answer: C. Lined: 20.64 + 7.09 = 27.74 units/m, less than unlined at 29.39. (Key point: Result)

## Related in Hydraulics

- [Trapezoidal to rectangular transition with bed lowering](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Atrapezoid-to-rectangular-transition-bed-lowering) — Hydraulics, chapter 7
- [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
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- [Contraction and bed rise, 3.5 m channel (1.2 m depth)](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acontraction-bed-rise-3-5m-wide-1-2m-depth) — Hydraulics, chapter 7
- [Bed slope for critical normal depth](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Abed-slope-for-critical-flow-at-normal-depth) — Hydraulics, chapter 7
- [Bed motion check and non-moving gravel size](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asediment-bed-motion-and-minimum-gravel-size) — Hydraulics, chapter 10, asked 2×

## Related in other subjects

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