Problem Statement
Find the rate of flow of water through a V-shaped channel having total angle between the sides as 60°. Take the value of C = 50 and slope of the bed 1 in 1500. The depth of flow is 6 m.
Given Data & Constants
- Total angle of V-channel, \(2\theta = 60^\circ\), so angle with vertical, \(\theta = 30^\circ\)
- Depth of flow, \(d = 6 \, \text{m}\)
- Bed slope, \(i = 1 \text{ in } 1500 = \frac{1}{1500}\)
- Chezy's constant, \(C = 50\)
Solution
1. Calculate Geometric Properties
For a V-shaped channel, the area and wetted perimeter are calculated based on the depth and the angle with the vertical.
$$ \text{Area of flow, } A = d^2 \tan(\theta) = (6)^2 \times \tan(30^\circ) $$
$$ A = 36 \times 0.57735 \approx 20.785 \, \text{m}^2 $$
$$ \text{Wetted Perimeter, } P = 2 \times \frac{d}{\cos(\theta)} = 2 \times \frac{6}{\cos(30^\circ)} $$
$$ P = \frac{12}{0.866} \approx 13.856 \, \text{m} $$
$$ \text{Hydraulic Mean Depth, } m = \frac{A}{P} = \frac{20.785}{13.856} \approx 1.5 \, \text{m} $$
2. Calculate Velocity and Discharge
We use Chezy's formula to find the velocity, and then the discharge.
$$ V = C \sqrt{m \cdot i} $$
$$ V = 50 \times \sqrt{1.5 \times \frac{1}{1500}} = 50 \times \sqrt{0.001} $$
$$ V = 50 \times 0.03162 \approx 1.581 \, \text{m/s} $$
$$ Q = A \times V = 20.785 \, \text{m}^2 \times 1.581 \, \text{m/s} \approx 32.86 \, \text{m}^3/\text{s} $$
Final Result:
The rate of flow (discharge) through the V-shaped channel is approximately \(32.86 \, \text{m}^3/\text{s}\).
Explanation of V-Shaped Channel Calculation
Calculating the flow in a V-shaped (or triangular) channel follows the same principles as a rectangular or trapezoidal channel, but with different formulas for the geometric properties.
- Angle (\(\theta\)): It is crucial to note that the angle used in the trigonometric formulas is the angle one side makes with the vertical centerline, which is half of the total included angle of the 'V'.
- Geometric Properties: The area (\(A\)) and wetted perimeter (\(P\)) are calculated using standard trigonometry for a triangle. These are then used to find the hydraulic mean depth (\(m\)), which is a key parameter in Chezy's formula.
- Chezy's Formula: Once the hydraulic mean depth is known, Chezy's formula is applied in the standard way to find the velocity, which is then multiplied by the area to find the total discharge.
