---
title: "Energy coefficient for compound channel"
subject: "Hydraulics"
chapter: 6
key: "semester_4/hydraulics:compound-channel-energy-coefficient-alpha"
source: https://www.bctengineer.com/q/semester_4/hydraulics/compound-channel-energy-coefficient-alpha
site: BCT Engineering
---

# Energy coefficient for compound channel

Hydraulics · Chapter 6: Uniform flow in open channel

- Asked in: 2070 Magh, 4 marks
- Video: [Lec14- Open Channel Flow-Composite Sections & Compound Channels](https://www.youtube.com/watch?v=aYY-HmcbkAY) — learning studio for water resources
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/compound-channel-energy-coefficient-alpha
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-6

## Question

Prove that for compound open channel, velocity distribution coefficient (Energy correction factor) $\alpha = \frac{\sum\left(\frac{K_i^3}{A_i^2}\right)\left(\sum A_i\right)^2}{\left(\sum K_i\right)^3}$, where $K_i$ = Conveyance factor of $i^{th}$ section, $A_i$ = Cross section area of $i^{th}$ section.

## Short answer

Kinetic energy correction factor α is defined as ∑ V_i³A_i / (V³A). Substituting V_i = K_i√S_f / A_i and V = Q/A leads to α = [∑(K_i³/A_i²) (∑A_i)²] / (∑K_i)³.

## Answer

#### Definition
The energy (kinetic energy) correction factor for a section of area $A$ and mean velocity $V=Q/A$ is
$$
\alpha=\frac{\int v^3\,dA}{V^3A}
$$
For a compound channel with subsections of area $A_i$ and mean velocity $V_i$:
$$
\alpha=\frac{\sum V_i^3A_i}{V^3A}
$$

#### Proof
1. $Q=\sum Q_i$, $A=\sum A_i$, $V=Q/A$.
2. The conveyance of each part is $K_i$, with $Q_i=K_i\sqrt{S_f}$ (all subsections share the same energy slope $S_f$). Then
$$
Q=\sqrt{S_f}\sum K_i,\qquad V_i=\frac{Q_i}{A_i}=\frac{K_i\sqrt{S_f}}{A_i}
$$
3. Numerator:
$$
\sum V_i^3A_i=\sum\frac{K_i^3S_f^{3/2}}{A_i^3}A_i=S_f^{3/2}\sum\frac{K_i^3}{A_i^2}
$$
4. Denominator:
$$
V^3A=\frac{Q^3}{A^2}=\frac{S_f^{3/2}\left(\sum K_i\right)^3}{\left(\sum A_i\right)^2}
$$
5. Dividing,
$$
\boxed{\alpha=\frac{\sum\left(\dfrac{K_i^3}{A_i^2}\right)\left(\sum A_i\right)^2}{\left(\sum K_i\right)^3}}
$$
Hence proved. For one uniform section $\alpha=1$; for a compound section $\alpha>1$ (typically 1.1 to 2 or more), and it must be used in $E=y+\alpha V^2/2g$.

## Check yourself

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

1. The energy correction factor for subsections of velocity V_i is:
   A. alpha = sum(V_i^2 A_i)/(V^2 A)
   B. alpha = sum(V_i A_i)/(V A)
   C. alpha = sum(V_i^3 A_i)/(V^3 A)
   D. alpha = sum(V_i)/V
   Answer: C. Kinetic energy flux involves V^3, so alpha = sum V_i^3 A_i/(V^3 A). (Key point: Definition of alpha)

2. With Q_i = K_i sqrt(S_f) and a common S_f, the velocity in part i is:
   A. V_i = K_i sqrt(S_f)
   B. V_i = K_i/A_i
   C. V_i = K_i sqrt(S_f)/A_i
   D. V_i = A_i/(K_i sqrt(S_f))
   Answer: C. V_i = Q_i/A_i = K_i sqrt(S_f)/A_i. (Key point: Discharge and velocity of a part)

3. After substitution, sum V_i^3 A_i and V^3 A become:
   A. S_f sum(K_i^2/A_i) and S_f (sum K_i)^2/sum A_i
   B. S_f^(3/2) sum(K_i^3/A_i^2) and S_f^(3/2)(sum K_i)^3/(sum A_i)^2
   C. sum(K_i^3/A_i) and (sum K_i)^3/sum A_i
   D. S_f^(3/2) sum(K_i^3/A_i^3) and S_f^(3/2)(sum K_i)^3/(sum A_i)^3
   Answer: B. V_i^3 A_i = K_i^3 S_f^(3/2)/A_i^2 and V^3 A = Q^3/A^2, and S_f^(3/2) cancels. (Key point: Numerator and denominator)

4. For a compound section, alpha is:
   A. Equal to 1
   B. Always less than beta
   C. Greater than 1 and used in E = y + alpha V^2/2g
   D. Negative
   Answer: C. Unequal velocities make alpha > 1, and it corrects the kinetic energy term in specific energy. (Key point: Value and use of alpha)

## Close variants set in other exams

- [Momentum coefficient for compound channel](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Acompound-channel-momentum-coefficient-beta) — Hydraulics, chapter 6, asked 2×

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- [Correction factors for given velocity profile](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Aenergy-momentum-correction-factors-linear-profile) — Hydraulics, chapter 7

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