Problem Statement
A flow of water of 150 litres per second flows down in a rectangular flume of width 70 cm and having adjustable bottom slope. If Chezy's constant C is 60, find the bottom slope necessary for uniform flow with a depth of flow of 40 cm. Also find the conveyance K of the flume.
Given Data & Constants
- Discharge, \(Q = 150 \, \text{L/s} = 0.15 \, \text{m}^3/\text{s}\)
- Width of flume, \(B = 70 \, \text{cm} = 0.7 \, \text{m}\)
- Depth of flow, \(d = 40 \, \text{cm} = 0.4 \, \text{m}\)
- Chezy's constant, \(C = 60\)
Solution
1. Calculate Geometric Properties
$$ \text{Area of flow, } A = B \times d = 0.7 \times 0.4 = 0.28 \, \text{m}^2 $$
$$ \text{Wetted Perimeter, } P = B + 2d = 0.7 + 2 \times 0.4 = 1.5 \, \text{m} $$
$$ \text{Hydraulic Mean Depth, } m = \frac{A}{P} = \frac{0.28}{1.5} \approx 0.1867 \, \text{m} $$
2. Calculate the Necessary Bottom Slope (\(i\))
First, find the velocity of flow. Then, rearrange Chezy's formula to solve for the slope.
$$ \text{Velocity, } V = \frac{Q}{A} = \frac{0.15 \, \text{m}^3/\text{s}}{0.28 \, \text{m}^2} \approx 0.5357 \, \text{m/s} $$
$$ V = C \sqrt{m \cdot i} \implies i = \frac{V^2}{C^2 \cdot m} $$
$$ i = \frac{(0.5357)^2}{60^2 \times 0.1867} = \frac{0.287}{3600 \times 0.1867} = \frac{0.287}{672.12} \approx 0.000427 $$
$$ \text{Slope} = 1 \text{ in } \frac{1}{0.000427} \approx 1 \text{ in } 2342 $$
3. Find the Conveyance (K) of the Flume
The conveyance of a channel section is a measure of its carrying capacity and is defined by the equation \(Q = K \sqrt{i}\).
$$ K = \frac{Q}{\sqrt{i}} = \frac{0.15}{\sqrt{0.000427}} \approx \frac{0.15}{0.02066} \approx 7.26 $$
$$ \text{Alternatively: } K = A \cdot C \sqrt{m} = 0.28 \times 60 \times \sqrt{0.1867} \approx 7.26 $$
Final Results:
Bottom slope necessary: \( \approx 0.000427 \) or 1 in 2342
Conveyance of the flume: \( K \approx 7.26 \, \text{m}^3/\text{s} \)
Explanation of Concepts
- Uniform Flow: This is a condition where the depth and velocity of the flow remain constant along the length of the channel. For this to happen, the driving force (gravity, represented by the bed slope) must exactly balance the resisting force (friction, represented by Chezy's constant and the channel geometry).
- Conveyance (K): This is a useful property of a channel section that combines all the geometric and roughness factors (\(A, C, m\)) into a single value. It represents the discharge capacity of the channel per unit root of slope. Once calculated, you can easily find the discharge for any given slope, or the required slope for any given discharge, without re-calculating all the geometric properties.
