---
title: "Hydraulic exponents for rectangular and triangular"
subject: "Hydraulics"
chapter: 6
key: "semester_4/hydraulics:hydraulic-exponents-rect-triangular-numerical"
source: https://www.bctengineer.com/q/semester_4/hydraulics/hydraulic-exponents-rect-triangular-numerical
site: BCT Engineering
---

# Hydraulic exponents for rectangular and triangular

Hydraulics · Chapter 6: Uniform flow in open channel

- Asked in: 2078 Chaitra, 4 marks
- Video: [Hydraulic Engineering and Machines I Lecture-16 I Hydraulic Exponents](https://www.youtube.com/watch?v=flKASPwdboM) — Effective Learning Center
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/hydraulic-exponents-rect-triangular-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-6

## Question

Calculate first hydraulic exponent for critical flow and second hydraulic exponent for uniform flow for wide rectangular channel and triangular channel.

## Short answer

From A³/T ∝ y^M: wide rectangular M=3, triangular M=5. From K² ∝ y^N: wide rectangular N=10/3=3.33, triangular N=16/3=5.33.

## Answer

#### Definitions
- **First hydraulic exponent $M$** (for critical flow): $Z^2=\dfrac{A^3}{T}\propto y^{M}$. It is used in the critical flow condition $\dfrac{Q^2}{g}=\dfrac{A^3}{T}$.
- **Second hydraulic exponent $N$** (for uniform flow): the conveyance $K=\dfrac1nAR^{2/3}\propto y^{N/2}$, i.e. $K^2\propto y^N$.

#### Wide rectangular channel (width $b$, depth $y$)
- $A=by$, $T=b$, and $R\approx y$ since $b\gg y$.
$$
\frac{A^3}{T}=\frac{b^3y^3}{b}=b^2y^3\;\Rightarrow\;\boxed{M=3}
$$
$$
K=\frac{1}{n}(by)\,y^{2/3}=\frac bn\,y^{5/3}\;\Rightarrow\;K^2\propto y^{10/3}\;\Rightarrow\;\boxed{N=\frac{10}{3}=3.33}
$$

#### Triangular channel (side slope $z$)
- $A=zy^2$, $T=2zy$, $P=2y\sqrt{1+z^2}$, so $R=\dfrac{zy}{2\sqrt{1+z^2}}=c\,y$, where $c=\dfrac{z}{2\sqrt{1+z^2}}$ is a constant.
$$
\frac{A^3}{T}=\frac{z^3y^6}{2zy}=\frac{z^2}{2}y^5\;\Rightarrow\;\boxed{M=5}
$$
$$
K=\frac1n(zy^2)\,(c\,y)^{2/3}\propto y^{2+2/3}=y^{8/3}\;\Rightarrow\;K^2\propto y^{16/3}\;\Rightarrow\;\boxed{N=\frac{16}{3}=5.33}
$$

| Channel | $M$ (first exponent) | $N$ (second exponent) |
|---|---|---|
| Wide rectangular | 3 | 10/3 = 3.33 |
| Triangular | 5 | 16/3 = 5.33 |

## Check yourself

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

1. The first hydraulic exponent M is defined by:
   A. K^2 proportional to y^M
   B. A^3/T proportional to y^M
   C. A proportional to y^M
   D. R proportional to y^M
   Answer: B. Z^2 = A^3/T varies as y^M and is used in the critical flow condition. (Key point: Definition of M)

2. What is M for a wide rectangular channel?
   A. 3
   B. 5
   C. 2
   D. 10/3
   Answer: A. A^3/T = b^3 y^3 / b = b^2 y^3, so M = 3. (Key point: Rectangular M)

3. What is N for a wide rectangular channel?
   A. 3
   B. 5/3
   C. 10/3
   D. 16/3
   Answer: C. K is proportional to y^(5/3), so K^2 is proportional to y^(10/3) and N = 10/3. (Key point: Rectangular N)

4. What are M and N for a triangular channel?
   A. M = 3, N = 10/3
   B. M = 5, N = 16/3
   C. M = 5, N = 10/3
   D. M = 3, N = 16/3
   Answer: B. A^3/T is proportional to y^5, and K to y^(8/3), so K^2 to y^(16/3). (Key point: Triangular exponents)

## Related in Hydraulics

- [Hydraulic exponent for triangular section 16/3](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ahydraulic-exponent-triangular-16-3) — Hydraulics, chapter 6, asked 2×
- [Hydraulic exponent for rectangular section 10/3](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ahydraulic-exponent-rectangular-10-3) — Hydraulics, chapter 6, asked 2×
- [Hydraulic jump in 90° triangular channel](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Ajump-90-degree-triangular-channel-flow-rate) — Hydraulics, chapter 9
- [Program for uniform depth in rectangular channel](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aalgorithm-uniform-depth-rectangular) — Hydraulics, chapter 6
- [Conditions for uniform flow in open channel](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aconditions-of-uniform-flow) — Hydraulics, chapter 6, asked 2×
- [Define non-uniform open channel flow with examples](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Adefine-non-uniform-open-channel-flow) — Hydraulics, chapter 5

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