Problem Statement
A trapezoidal channel has side slopes of 1 horizontal to 2 vertical and the slope of the bed is 1 in 2000. The area of the section is 42 m². Find the dimensions of the section if it is most economical. Determine the discharge of the most economical section if C = 60.
Given Data & Constants
- Side slope = 1 Horizontal to 2 Vertical. Let \(n\) be the horizontal part for a 1-unit vertical drop, so \(n = 1/2 = 0.5\).
- Bed slope, \(i = 1 \text{ in } 2000 = \frac{1}{2000}\)
- Area of section, \(A = 42 \, \text{m}^2\)
- Chezy's constant, \(C = 60\)
Solution
1. Conditions for a Most Economical Trapezoidal Section
For a trapezoidal channel to be most economical, two conditions must be met:
- Half of the top width is equal to the length of one sloping side: \( \frac{B + 2nd}{2} = d\sqrt{1+n^2} \)
- The hydraulic mean depth is half the depth of flow: \(m = d/2\)
We will use these conditions to find the dimensions.
2. Find the Dimensions of the Section (B and d)
First, use the first condition to find a relationship between the bed width (B) and the depth (d).
Now substitute this relationship into the formula for the area of a trapezoid.
3. Determine the Discharge (Q)
Using the second condition for an economical section, we find the hydraulic mean depth.
Now we use Chezy's formula to find the velocity and then the discharge.
The most economical dimensions are: Bed Width (B) \( \approx 6.08 \, \text{m} \), Depth (d) \( \approx 4.92 \, \text{m} \)
The discharge of the section is: \( Q \approx 88.37 \, \text{m}^3/\text{s} \)
