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

# Momentum coefficient for compound channel

Hydraulics · Chapter 6: Uniform flow in open channel

- Asked in: 2071 Bhadra, 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-momentum-coefficient-beta
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-6

## Question

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

## Short answer

The momentum correction factor β = ∑ V_i²A_i / (V²A). With V_i = K_i√S_f / A_i and V = Q/A, and Q = √S_f ∑K_i, substitution yields β = [∑(K_i²/A_i) ∑A_i] / (∑K_i)².

## Answer

#### Definition
The momentum correction factor for a section of total area $A$, discharge $Q$ and mean velocity $V=Q/A$ is
$$
\beta=\frac{\int v^2\,dA}{V^2A}
$$
For a compound channel divided into subsections, each with mean velocity $V_i$, area $A_i$ (velocity taken uniform in each subsection):
$$
\beta=\frac{\sum V_i^2A_i}{V^2A}
$$

#### Proof
1. Total discharge $Q=\sum Q_i$, $A=\sum A_i$ and $V=\dfrac{Q}{A}$.
2. For each subsection, conveyance $K_i$ is defined by $Q_i=K_i\sqrt{S_f}$. All subsections have the same energy slope $S_f$ (same water surface and energy line). Therefore
$$
Q=\sqrt{S_f}\sum K_i,\qquad V_i=\frac{Q_i}{A_i}=\frac{K_i\sqrt{S_f}}{A_i}
$$
3. Substituting,
$$
\sum V_i^2A_i=\sum\frac{K_i^2S_f}{A_i^2}A_i=S_f\sum\frac{K_i^2}{A_i}
$$
$$
V^2A=\frac{Q^2}{A}=\frac{S_f\left(\sum K_i\right)^2}{\sum A_i}
$$
4. Dividing,
$$
\beta=\frac{S_f\sum\dfrac{K_i^2}{A_i}}{\dfrac{S_f\left(\sum K_i\right)^2}{\sum A_i}}
$$
$$
\boxed{\beta=\frac{\sum\left(\dfrac{K_i^2}{A_i}\right)\left(\sum A_i\right)}{\left(\sum K_i\right)^2}}
$$
Hence proved. For a single uniform section, $\beta=1$; for compound sections $\beta>1$ because the velocities in the main channel and flood plains differ greatly.

## Check yourself

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

1. For subsections with uniform velocity V_i, the momentum correction factor is:
   A. beta = sum(V_i A_i)/(V A)
   B. beta = sum(V_i^3 A_i)/(V^3 A)
   C. beta = sum(V_i^2 A_i)/(V^2 A)
   D. beta = V/sum(V_i)
   Answer: C. beta is the ratio of true momentum flux to that of the mean velocity: sum V_i^2 A_i/(V^2 A). (Key point: Definition of beta)

2. Why can Q_i = K_i sqrt(S_f) be used with the same S_f for every subsection?
   A. All subsections have the same roughness
   B. All subsections have the same area
   C. All subsections have the same hydraulic radius
   D. They share the same water surface and energy line, hence the same energy slope
   Answer: D. The sections are parts of one cross-section, so they share the same energy slope. (Key point: Common slope)

3. Using V_i = K_i sqrt(S_f)/A_i, the numerator sum V_i^2 A_i becomes:
   A. S_f sum(K_i^2/A_i)
   B. sqrt(S_f) sum(K_i/A_i)
   C. S_f sum(K_i A_i)
   D. S_f^(3/2) sum(K_i^3/A_i^2)
   Answer: A. V_i^2 A_i = K_i^2 S_f/A_i, and the sum factors out S_f. (Key point: Substitution)

4. What is beta for a single uniform section and for a compound section?
   A. Less than 1 for compound sections
   B. Exactly 1 for a single section; greater than 1 for compound sections because velocities differ widely
   C. Always greater than 2
   D. Always exactly 1
   Answer: B. A uniform velocity gives beta = 1; unequal velocities in main channel and flood plains push beta above 1. (Key point: Meaning of beta)

## Close variants set in other exams

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

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