Design (Find y₁, B, T and Fb) a concrete-lined drainage channel (n = 0.015) to carry 150 m³/s with 1.5H:1V side slopes, B/y = 3, and a longitudinal slope of 0.0005. Compute the velocity and Froude Number at design capacity. Note: rectangular channel ss-0 triangular channel B-0 Freeboard (a) Trapezoidal channel A-By+ssy² P=B+2√√1+²
Design (Find y₁, B, T and Fb) a concrete-lined drainage channel (n = 0.015) to carry 150 m³/s with 1.5H:1V side slopes, B/y = 3, and a longitudinal slope of 0.0005. Compute the velocity and Froude Number at design capacity. Note: rectangular channel ss-0 triangular channel B-0 Freeboard (a) Trapezoidal channel A-By+ssy² P=B+2√√1+²
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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![### Design of a Concrete-Lined Drainage Channel
**Problem Statement:**
Design (Find \(y_n\), \(B\), \(T\), and \(F_b\)) a concrete-lined drainage channel (\(n = 0.015\)) to carry 150 m³/s with 1.5H:1V side slopes, \(B/y_n = 3\), and a longitudinal slope of 0.0005. Compute the velocity and Froude Number at design capacity.
**Given:**
- Discharge, \(Q = 150 \, \text{m}^3/\text{s}\)
- Manning's Roughness Coefficient, \(n = 0.015\)
- Side slopes, \(1.5H:1V\)
- Ratio of bottom width to normal depth, \(B/y_n = 3\)
- Longitudinal slope, \(S = 0.0005\)
**Diagram Analysis:**
The diagram provided is of a trapezoidal channel. Here is a breakdown of the key components:
- **\(y_n\)**: Normal Depth
- **\(B\)**: Bottom Width of the channel
- **\(T\)**: Top Width of the water surface level at normal depth
- **\(F_b\)**: Freeboard, which is the additional height above the normal depth to ensure safety against overflow.
### Understanding the Channel Geometry:
A trapezoidal channel is defined by its side slopes, bottom width, and normal depth. The area (\(A\)) and the wetted perimeter (\(P\)) are calculated using the following equations:
\[ A = B y_n + s y_n^2 \]
\[ P = B + 2y_n\sqrt{1 + s^2} \]
Where,
- \(s\) is the side slope (horizontal component), here \(s = 1.5\)
### Computations:
#### Step 1: Determining Normal Depth (\(y_n\)):
Given \(B/y_n = 3\),
\[ B = 3 y_n \]
Using Manning's Equation:
\[ Q = \frac{1}{n} A R^{2/3} S^{1/2} \]
Where,
\[ R = \frac{A}{P} \] (Hydraulic Radius)
Substitute \(A\), \(P\), and rearrange the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ea503b5-9839-4a05-b482-50f8a035eb54%2F3a7a3835-c3e1-426f-ac02-abce3befdd90%2Fu4xyl0e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Design of a Concrete-Lined Drainage Channel
**Problem Statement:**
Design (Find \(y_n\), \(B\), \(T\), and \(F_b\)) a concrete-lined drainage channel (\(n = 0.015\)) to carry 150 m³/s with 1.5H:1V side slopes, \(B/y_n = 3\), and a longitudinal slope of 0.0005. Compute the velocity and Froude Number at design capacity.
**Given:**
- Discharge, \(Q = 150 \, \text{m}^3/\text{s}\)
- Manning's Roughness Coefficient, \(n = 0.015\)
- Side slopes, \(1.5H:1V\)
- Ratio of bottom width to normal depth, \(B/y_n = 3\)
- Longitudinal slope, \(S = 0.0005\)
**Diagram Analysis:**
The diagram provided is of a trapezoidal channel. Here is a breakdown of the key components:
- **\(y_n\)**: Normal Depth
- **\(B\)**: Bottom Width of the channel
- **\(T\)**: Top Width of the water surface level at normal depth
- **\(F_b\)**: Freeboard, which is the additional height above the normal depth to ensure safety against overflow.
### Understanding the Channel Geometry:
A trapezoidal channel is defined by its side slopes, bottom width, and normal depth. The area (\(A\)) and the wetted perimeter (\(P\)) are calculated using the following equations:
\[ A = B y_n + s y_n^2 \]
\[ P = B + 2y_n\sqrt{1 + s^2} \]
Where,
- \(s\) is the side slope (horizontal component), here \(s = 1.5\)
### Computations:
#### Step 1: Determining Normal Depth (\(y_n\)):
Given \(B/y_n = 3\),
\[ B = 3 y_n \]
Using Manning's Equation:
\[ Q = \frac{1}{n} A R^{2/3} S^{1/2} \]
Where,
\[ R = \frac{A}{P} \] (Hydraulic Radius)
Substitute \(A\), \(P\), and rearrange the
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