---
title: "Hydraulically best trapezoidal section"
subject: "Hydro Power"
chapter: 1
key: "semester_6/hydro_power:best-trapezoidal-section-proof"
source: https://www.bctengineer.com/q/semester_6/hydro_power/best-trapezoidal-section-proof
site: BCT Engineering
---

# Hydraulically best trapezoidal section

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

- Asked in: 2070 Bhadra, 2+6 marks
- Video: [Most economical trapezoid channel section //open channel flow //](https://www.youtube.com/watch?v=9PJhN517foA) — Civil Engineer FBH
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/best-trapezoidal-section-proof
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-1

## Question

What do you mean by a channel of hydraulically best section? Show that the hydraulic mean depth for a trapezoidal channel is half the depth of flow.

## Short answer

Hydraulically best section maximizes discharge for given area, slope, roughness by minimizing wetted perimeter. For trapezoidal channel, minimizing P yields hydraulic mean depth R = y/2.

## Answer

#### Hydraulically best section
A channel section is **hydraulically best** (most economical or most efficient) when, for a given area, bed slope and roughness, it carries the **maximum discharge**. Since $Q = AC\sqrt{RS}$ (or Manning), Q is maximum when the wetted perimeter $P$ is minimum, i.e. hydraulic radius $R = A/P$ is maximum. Such a section also needs the least lining and excavation.

#### Proof: $R = y/2$ for the best trapezoidal section
Let bottom width $b$, depth $y$, side slope $n$ H : 1 V.

```text
   |<-------- b + 2ny -------->|
   \                          /
    \  y                     /  side slope n:1
     \_______________________/
          |<---- b ---->|
```

$$
\begin{aligned}
A &= (b + ny)y \;\Rightarrow\; b = \frac{A}{y} - ny \\
P &= b + 2y\sqrt{1+n^2} = \frac{A}{y} - ny + 2y\sqrt{1+n^2}
\end{aligned}
$$

For maximum discharge, $P$ must be minimum with $A$ constant:
$$
\begin{aligned}
\frac{dP}{dy} &= -\frac{A}{y^2} - n + 2\sqrt{1+n^2} = 0 \\
\frac{A}{y^2} &= 2\sqrt{1+n^2} - n \\
A &= y^2\left(2\sqrt{1+n^2} - n\right)
\end{aligned}
$$
Using $A = (b+ny)y$: $b + ny = 2y\sqrt{1+n^2} - ny$, i.e.
$$
\frac{b + 2ny}{2} = y\sqrt{1+n^2}
$$
(half the top width equals one sloping side).

Now the perimeter:
$$
\begin{aligned}
P &= \frac{A}{y} - ny + 2y\sqrt{1+n^2} \\
&= y(2\sqrt{1+n^2} - n) - ny + 2y\sqrt{1+n^2} \\
&= 2y(2\sqrt{1+n^2} - n) = \frac{2A}{y}
\end{aligned}
$$
Hence
$$
R = \frac{A}{P} = \frac{A}{2A/y} = \frac{y}{2}
$$

So for the most economical trapezoidal channel, the **hydraulic mean depth equals half the depth of flow**. (It is also shown that the three sides are tangent to a semicircle with centre on the water surface.)

## Check yourself

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

1. What does a hydraulically best section give for fixed area, slope and roughness?
   A. Minimum discharge
   B. Maximum discharge
   C. Minimum velocity
   D. Maximum wetted perimeter
   Answer: B. It carries the maximum discharge, which occurs when the wetted perimeter is minimum. (Key point: Best section meaning)

2. Q is maximum when which quantity is maximum?
   A. Wetted perimeter P
   B. Top width
   C. Hydraulic radius R = A/P
   D. Bed width b
   Answer: C. With A fixed, minimum P means maximum R, which maximises Q = AC sqrt(RS). (Key point: Condition)

3. How is the condition for minimum perimeter obtained?
   A. Put dP/dy = 0 with A constant
   B. Put dA/dy = 0 with P constant
   C. Put dQ/dS = 0
   D. Put b = y
   Answer: A. With b = A/y - ny, setting dP/dy = 0 gives A = y^2(2 sqrt(1+n^2) - n). (Key point: Method of proof)

4. What does the proof show for R?
   A. R = y
   B. R = y/2
   C. R = 2y
   D. R = y/sqrt(2)
   Answer: B. P = 2A/y at the optimum, so R = A/P = y/2. (Key point: Final result)

## Related in Hydro Power

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- [Definition: best hydraulic section](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Abest-hydraulic-section-definition) — Hydro Power, chapter 1
- [Efficient rectangular section and triangular channel depth](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Atriangular-channel-normal-depth-rect) — Hydro Power, chapter 1, asked 2×
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- [Economical trapezoidal channel side length](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Aeconomical-trapezoidal-side-length) — Hydro Power, chapter 1

## Related in other subjects

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