Problem Statement
Find the slope of the free water surface in a rectangular channel of width 15 m, having depth of flow 4 m. The discharge through the channel is 40 m³/s. The bed of the channel is having a slope of 1 in 4000. Take the value of Chezy's constant, C = 50.
Given Data & Constants
- Channel width, \(B = 15 \, \text{m}\)
- Depth of flow, \(d = 4 \, \text{m}\)
- Discharge, \(Q = 40 \, \text{m}^3/\text{s}\)
- Bed slope, \(S_0 = 1 \text{ in } 4000 = 0.00025\)
- Chezy's constant, \(C = 50\)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)
Solution
1. Calculate Geometric and Flow Properties
$$ \text{Area of flow, } A = B \times d = 15 \times 4 = 60 \, \text{m}^2 $$
$$ \text{Wetted Perimeter, } P = B + 2d = 15 + 2 \times 4 = 23 \, \text{m} $$
$$ \text{Hydraulic Mean Depth, } m = \frac{A}{P} = \frac{60}{23} \approx 2.6087 \, \text{m} $$
$$ \text{Velocity, } V = \frac{Q}{A} = \frac{40}{60} \approx 0.6667 \, \text{m/s} $$
2. Calculate the Energy Line Slope (\(S_f\))
The energy line slope (or friction slope) is found by rearranging Chezy's formula.
$$ V = C \sqrt{m \cdot S_f} \implies S_f = \frac{V^2}{C^2 \cdot m} $$
$$ S_f = \frac{(0.6667)^2}{50^2 \times 2.6087} = \frac{0.4445}{2500 \times 2.6087} \approx 0.00006816 $$
3. Calculate the Froude Number (\(Fr\))
$$ Fr = \frac{V}{\sqrt{g d}} = \frac{0.6667}{\sqrt{9.81 \times 4}} \approx 0.1066 $$
4. Calculate the Slope of the Free Water Surface
The slope of the water surface relative to the horizontal (\(S_w\)) is found using the gradually varied flow equation.
$$ \frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2} = \frac{0.00025 - 0.00006816}{1 - (0.1066)^2} \approx 0.0001839 $$
$$ S_w = S_0 - \frac{dy}{dx} = 0.00025 - 0.0001839 = 0.0000661 $$
Final Result:
The slope of the free water surface is approximately 0.000184 (relative to the channel bed).
(The slope relative to the horizontal is approximately 0.0000661, or 1 in 15128).
Explanation of the Slopes
In Gradually Varied Flow, it's important to distinguish between three different slopes:
- Bed Slope (\(S_0\)): The physical slope of the channel bottom.
- Energy Line Slope (\(S_f\)): The rate at which energy is lost due to friction. In uniform flow, \(S_f = S_0\).
- Water Surface Slope (\(S_w\)): The slope of the free water surface relative to the horizontal. This is what you would visually observe. It is related to the rate of change of depth (\(dy/dx\)) by the equation \(S_w = S_0 - dy/dx\). The term \(dy/dx\) represents the slope of the water surface *relative to the channel bed*. In many contexts, this is the value referred to as the "slope of the free water surface."
