---
title: "Stable alluvial canal, Q 50, d 0.5 mm"
subject: "Irrigation and Drainage Engineering"
chapter: 4
key: "semester_6/irrigation_and_drainage_engineering:alluvial-canal-q50-sketch"
source: https://www.bctengineer.com/q/semester_6/irrigation_and_drainage_engineering/alluvial-canal-q50-sketch
site: BCT Engineering
---

# Stable alluvial canal, Q 50, d 0.5 mm

Irrigation and Drainage Engineering · Chapter 4: Design of Canals

- Asked in: 2072 Asoj, 5 marks
- Video: [Lacey's sum | Type 1 when Silt factor is given & Side slope is 1/2 : 1 | Lacey's  Design of canal](https://www.youtube.com/watch?v=U4SwFYekkLg) — Professor Nidhi Trivedi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/irrigation_and_drainage_engineering/alluvial-canal-q50-sketch
- Whole chapter: https://www.bctengineer.com/md/semester_6/irrigation_and_drainage_engineering/chapter-4

## Question

Design a stable irrigation canal carrying a discharge of 50 m³/s, which passes through alluvium (d_mean = 0.50 mm). Draw a sketch of the designed section.

## Short answer

For Q=50 m³/s in alluvium with d=0.5 mm, using Lacey's theory, silt factor f=1.245 gives bed width 29.5 m, depth 1.81 m, side slope 0.5H:1V, bed slope 1 in 4452, velocity 0.91 m/s.

## Answer

Alluvium with $d_{mean} = 0.5$ mm carrying silt is designed on the regime principle (Lacey). Side slope 0.5H:1V is assumed.

#### Silt factor

$$
f = 1.76\sqrt{d_{mm}} = 1.76\sqrt{0.5} = 1.245
$$

#### Lacey's regime values

Given: $Q = 50\ \text{m}^3/\text{s}$, $f = 1.245$, side slope 0.5H:1V.

$$
\begin{aligned}
V &= \left(\frac{Q f^2}{140}\right)^{1/6} = \left(\frac{50\times 1.245^2}{140}\right)^{1/6} = 0.906\ \text{m/s} \\
A &= \frac{Q}{V} = \frac{50}{0.906} = 55.19\ \text{m}^2 \\
R &= \frac{5V^2}{2f} = \frac{5\times 0.906^2}{2\times 1.245} = 1.649\ \text{m} \\
P &= 4.75\sqrt{Q} = 4.75\sqrt{50} = 33.59\ \text{m} \\
S &= \frac{f^{5/3}}{3340\,Q^{1/6}} = \frac{1.245^{5/3}}{3340\times 50^{1/6}} = 0.000225 = \frac{1}{4452}
\end{aligned}
$$

#### Trapezoidal section

For side slope $z = 0.5$: $A = (B + zD)D$ and $P = B + 2D\sqrt{1+z^2} = B + 2.2361D$.

Eliminating $B$:

$$
\begin{aligned}
1.7361\,D^2 - 33.59\,D + 55.19 &= 0 \\
D &= \frac{33.59 - \sqrt{33.59^2 - 4\times 1.7361\times 55.19}}{2\times 1.7361} = 1.813\ \text{m} \\
B &= P - 2.2361D = 33.59 - 2.2361\times 1.813 = 29.53\ \text{m}
\end{aligned}
$$

(The other root of the quadratic gives a very deep, narrow section, so the smaller root is taken.)

Check: $A = (29.53 + 0.5\times 1.813)\times 1.813 = 55.19\ \text{m}^2$ against required 55.19 $\text{m}^2$.

**Answer:** Bed width $B \approx 29.5$ m, depth of water $D \approx 1.81$ m, side slope 0.5H:1V, bed slope $S = 1$ in 4452 (0.000225), velocity $V = 0.91$ m/s, hydraulic mean radius $R = 1.65$ m.

#### Sketch of designed section

```text
      FSL
  ____|~~~~~~~~~~~~~~~~~~~~~~~|____
     /<--------- T --------->\
    /  D = 1.81 m            \  1V : 0.5H
   /___________________________\
        B = 29.5 m
```
Top width $T = B + 2zD = 31.35$ m (freeboard to be added above FSL).

## Check yourself

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

1. How is f found from the mean particle size?
   A. f = 1.76 d = 0.88
   B. f = 0.55 d^0.64
   C. f = 4.75 sqrt(d)
   D. f = 1.76 sqrt(d in mm) = 1.245
   Answer: D. Lacey's silt factor is 1.76 sqrt(d_mm); for 0.5 mm it is 1.245. (Key point: Silt factor)

2. Which set gives Lacey's V, R and P?
   A. V = (Q f^2/140)^(1/6), R = 5V^2/2f, P = 4.75 sqrt(Q)
   B. V = 0.55 m D^0.64, R = D/2, P = B + 2D
   C. V = Q/f, R = V^2, P = 2 sqrt(Q)
   D. V = (Q/140)^(1/2), R = f V, P = Q/V
   Answer: A. These regime equations give V = 0.906 m/s, R = 1.649 m, P = 33.59 m. (Key point: Lacey set)

3. How are B and D found?
   A. Set B = D and scale to Q
   B. Use A = (B + 0.5D)D and P = B + 2.2361D, eliminate B, and take the smaller root of the quadratic
   C. Use Kennedy's trial and error
   D. Set D = R and B = P
   Answer: B. This gives D = 1.81 m and B = 29.5 m. (Key point: Section from A and P)

4. How is the top width T in the sketch found?
   A. T = B - 2zD
   B. T = P
   C. T = B + 2zD = 29.5 + 2 x 0.5 x 1.81 = 31.35 m, with freeboard added above FSL
   D. T = B + D
   Answer: C. Each side slope adds zD of horizontal width. (Key point: Top width)

## Related in Irrigation and Drainage Engineering

- [Canal in alluvium, Q 64, d 0.5 mm](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Aalluvial-canal-q64-d05mm) — Irrigation and Drainage Engineering, chapter 4
- [Canal in alluvium, Q 40, d 0.5 mm](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Aalluvial-canal-q40-d05mm) — Irrigation and Drainage Engineering, chapter 4
- [Stable canal: Q 32, d 30 mm](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Astable-canal-q32-d30mm-phi37) — Irrigation and Drainage Engineering, chapter 4, asked 2×
- [Stable depth: Q 50, slope 1:1000](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Astable-depth-q50-d50mm-slope1000) — Irrigation and Drainage Engineering, chapter 4, asked 2×
- [Stable depth: Q 20, d 40 mm](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Astable-depth-q20-d40mm) — Irrigation and Drainage Engineering, chapter 4, asked 2×
- [Stable canal: Q 40, d 5 cm](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Astable-canal-q40-d5cm-phi37) — Irrigation and Drainage Engineering, chapter 4, asked 2×

## Related in other subjects

- [Design stable channel in 8 mm gravel](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Astable-channel-8mm-gravel-design) — Hydraulics, chapter 10
- [Checking whether root locus passes through point P](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arl-point-p-dominant) — Control Systems, chapter 5
- [Tractive force design in coarse alluvium, 20 m³/s](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Atractive-force-trapezoid-20m3s-d75-2cm) — Hydraulics, chapter 10
- [Irrigation canal earthwork with vertical drop](https://www.bctengineer.com/question.md?key=semester_7%2Festimating_and_costing%3Acanal-a-drop-0-400) — Estimating and Costing, chapter 6