---
title: "Lacey: max discharge at slope 1/6000"
subject: "Irrigation and Drainage Engineering"
chapter: 4
key: "semester_6/irrigation_and_drainage_engineering:lacey-max-discharge-slope-16000"
source: https://www.bctengineer.com/q/semester_6/irrigation_and_drainage_engineering/lacey-max-discharge-slope-16000
site: BCT Engineering
---

# Lacey: max discharge at slope 1/6000

Irrigation and Drainage Engineering · Chapter 4: Design of Canals

- Asked in: 2071 Bhadra, 5 marks
- 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/lacey-max-discharge-slope-16000
- Whole chapter: https://www.bctengineer.com/md/semester_6/irrigation_and_drainage_engineering/chapter-4

## Question

The slope of a channel in alluvium is 1/6000. Find the channel section and the maximum discharge which can be allowed to flow in it. Take f = 1.0.

## Short answer

Maximum regime discharge 33.6 m³/s. Channel section: bed width 23.6 m, depth 1.74 m, side slope 0.5H:1V, bed slope 1/6000, velocity 0.79 m/s.

## Answer

Given: $S = 1/6000$, $f = 1.0$, side slope 0.5H:1V (assumed where not stated). The maximum discharge is the one for which the channel is in regime at this slope.

#### Discharge from Lacey's slope equation

$$
S = \frac{f^{5/3}}{3340\,Q^{1/6}} \Rightarrow Q = \left(\frac{f^{5/3}}{3340\,S}\right)^6 = \left(\frac{1.0^{5/3}}{3340/6000}\right)^6 = 33.607\ \text{m}^3/\text{s}
$$

**Maximum discharge $Q \approx 33.61$ m³/s** (used for the section below).

Section for this discharge:

#### Lacey's regime values

Given: $Q = 33.61\ \text{m}^3/\text{s}$, $f = 1.000$, side slope 0.5H:1V.

$$
\begin{aligned}
V &= \left(\frac{Q f^2}{140}\right)^{1/6} = \left(\frac{33.61\times 1.000^2}{140}\right)^{1/6} = 0.788\ \text{m/s} \\
A &= \frac{Q}{V} = \frac{33.61}{0.788} = 42.63\ \text{m}^2 \\
R &= \frac{5V^2}{2f} = \frac{5\times 0.788^2}{2\times 1.000} = 1.554\ \text{m} \\
P &= 4.75\sqrt{Q} = 4.75\sqrt{33.61} = 27.54\ \text{m} \\
S &= \frac{f^{5/3}}{3340\,Q^{1/6}} = \frac{1.000^{5/3}}{3340\times 33.61^{1/6}} = 0.000167 = \frac{1}{6000}
\end{aligned}
$$

#### Trapezoidal section

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

Eliminating $B$:

$$
\begin{aligned}
1.7361\,D^2 - 27.54\,D + 42.63 &= 0 \\
D &= \frac{27.54 - \sqrt{27.54^2 - 4\times 1.7361\times 42.63}}{2\times 1.7361} = 1.739\ \text{m} \\
B &= P - 2.2361D = 27.54 - 2.2361\times 1.739 = 23.65\ \text{m}
\end{aligned}
$$

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

Check: $A = (23.65 + 0.5\times 1.739)\times 1.739 = 42.63\ \text{m}^2$ against required 42.63 $\text{m}^2$.

**Answer:** Bed width $B \approx 23.6$ m, depth of water $D \approx 1.74$ m, side slope 0.5H:1V, bed slope $S = 1$ in 6000 (0.000167), velocity $V = 0.79$ m/s, hydraulic mean radius $R = 1.55$ m.

## Check yourself

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

1. How is the maximum discharge found when only the slope S = 1/6000 and f = 1.0 are given?
   A. Use V = (Q f^2/140)^(1/6) with Q = A x V directly
   B. Rearrange Lacey's slope equation S = f^(5/3)/(3340 Q^(1/6)) to solve for Q
   C. Use P = 4.75 sqrt(Q) and set P equal to the slope
   D. Use Manning's formula with an assumed n and depth
   Answer: B. Lacey's slope equation links S, f and Q, so solving it for Q gives the discharge at which the channel is in regime on that slope. (Key point: Finding Q from slope)

2. Once Q is known, which Lacey relation gives the wetted perimeter?
   A. P = 2.5 sqrt(Q)
   B. P = 5V^2/(2f)
   C. P = 4.75 sqrt(Q)
   D. P = Q/V
   Answer: C. Lacey's perimeter is P = 4.75 sqrt(Q); for Q = 33.61 this gives 27.54 m. R = 5V^2/2f is the hydraulic mean radius and Q/V is the area. (Key point: Lacey regime values)

## Related in Irrigation and Drainage Engineering

- [Lacey: max discharge at slope 1/4000](https://www.bctengineer.com/question.md?key=semester_6%2Firrigation_and_drainage_engineering%3Alacey-max-discharge-slope-14000) — Irrigation and Drainage Engineering, chapter 4
- [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

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