---
title: "Velocity and discharge, 1.8 m sewer"
subject: "Sanitary Engineering"
chapter: 3
key: "semester_6/sanitary_engineering:sewer-velocity-discharge-1p8m"
source: https://www.bctengineer.com/q/semester_6/sanitary_engineering/sewer-velocity-discharge-1p8m
site: BCT Engineering
---

# Velocity and discharge, 1.8 m sewer

Sanitary Engineering · Chapter 3: Design and Construction of Sewers

- Asked in: 2076 Bhadra, 8 marks
- Video: [Sewer Design Example | Waste Water Engineering](https://www.youtube.com/watch?v=2me10Fk1oEM) — APSEd
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/sanitary_engineering/sewer-velocity-discharge-1p8m
- Whole chapter: https://www.bctengineer.com/md/semester_6/sanitary_engineering/chapter-3

## Question

Calculate the velocity and discharge in a sewer of circular section having diameter of 1.8 m laid at a gradient of 1 in 600. The sewer runs partially full at 0.75 depth. Use Manning's formula taking n = 0.013.

## Short answer

Full-flow velocity = 1.844 m/s, discharge = 4.693 m³/s. At 0.75 depth, apply proportional ratios: velocity = 1.844×1.1335 = 2.09 m/s, discharge = 4.693×0.9119 = 4.28 m³/s.

## Answer

#### Given
$D=1.8$ m, $S=1/600$, $n=0.013$, depth of flow $d=0.75D$.

#### Full-flow values
$$
\begin{aligned}
v_f&=\frac{0.3969}{n}D^{2/3}S^{1/2}=\frac{0.3969}{0.013}(1.8)^{2/3}\left(\frac{1}{600}\right)^{1/2}=1.844\ \text{m/s}\\
Q_f&=v_f\times\frac{\pi}{4}D^2=1.844\times2.545=4.693\ \text{m}^3/\text{s}
\end{aligned}
$$

#### Proportionate values at $d/D=0.75$
For $d/D=0.75$: $\theta = 2\cos^{-1}(1-2d/D)=240.00^\circ$, $A/A_f=\dfrac{\theta-\sin\theta}{2\pi}=0.8045$, $R/R_f=1-\dfrac{\sin\theta}{\theta}=1.2067$, so $v/v_f=(R/R_f)^{2/3}=1.1335$ and $Q/Q_f=(A/A_f)(v/v_f)=0.9119$ (n taken constant).

$$
\begin{aligned}
v&=1.1335\times1.844=2.09\ \text{m/s}\\
Q&=0.9119\times4.693=4.28\ \text{m}^3/\text{s}
\end{aligned}
$$

**Answer: Velocity = 2.09 m/s; discharge = 4.28 m³/s.**

## Check yourself

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

1. Which formula gives velocity of a circular sewer running full by Manning?
   A. v_f = (0.3969/n) D^(2/3) S^(1/2)
   B. v_f = (0.3117/n) D^(8/3) S^(1/2)
   C. v_f = (1/n) D^(2/3) S
   D. v_f = (0.3969/n) D^(8/3) S^(1/2)
   Answer: A. With R = D/4, v = (1/n) R^(2/3) S^(1/2) becomes 0.3969/n x D^(2/3) S^(1/2); here 1.844 m/s. (Key point: Full-flow velocity)

2. For d/D = 0.75 (constant n), v/v_f and Q/Q_f are about
   A. 0.75 and 0.75
   B. 0.8045 and 1.1335
   C. 1.1335 and 0.9119
   D. 1.2067 and 0.8045
   Answer: C. v/v_f = (R/R_f)^(2/3) = 1.1335 and Q/Q_f = 0.8045 x 1.1335 = 0.9119. (Key point: Proportionate values at 0.75 depth)

3. The velocity and discharge at 0.75 depth are obtained by
   A. Using v_f and Q_f unchanged
   B. Multiplying v_f and Q_f by the proportionate ratios: v = 2.09 m/s, Q = 4.28 m3/s
   C. Multiplying both by 0.75
   D. Dividing v_f and Q_f by the ratios
   Answer: B. Partial flow values = ratio x full-flow values. (Key point: Partial-flow results)

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