Problem Statement
Find the velocity of flow and rate of flow of water through a rectangular channel of 5 m wide and 2 m deep, when it is running full. The channel is having bed slope of 1 in 3000. Take Chezy's constant C = 50.
Given Data & Constants
- Width of channel, \(B = 5 \, \text{m}\)
- Depth of channel, \(d = 2 \, \text{m}\)
- Bed slope, \(i = 1 \text{ in } 3000 = \frac{1}{3000}\)
- Chezy's constant, \(C = 50\)
Solution
1. Calculate Area (A) and Wetted Perimeter (P)
$$ \text{Area of flow, } A = B \times d = 5 \times 2 = 10 \, \text{m}^2 $$
$$ \text{Wetted Perimeter, } P = B + 2d = 5 + 2 \times 2 = 9 \, \text{m} $$
2. Calculate Hydraulic Mean Depth (m)
The hydraulic mean depth (or hydraulic radius) is the ratio of the flow area to the wetted perimeter.
$$ m = \frac{A}{P} = \frac{10}{9} \approx 1.111 \, \text{m} $$
3. Calculate Velocity of Flow (V)
We use Chezy's formula to find the velocity.
$$ V = C \sqrt{m \cdot i} $$
$$ V = 50 \times \sqrt{1.111 \times \frac{1}{3000}} = 50 \times \sqrt{0.0003703} $$
$$ V = 50 \times 0.01924 \approx 0.962 \, \text{m/s} $$
4. Calculate Rate of Flow (Discharge, Q)
The rate of flow is the area multiplied by the velocity.
$$ Q = A \times V $$
$$ Q = 10 \, \text{m}^2 \times 0.962 \, \text{m/s} = 9.62 \, \text{m}^3/\text{s} $$
Final Results:
Velocity of flow: \( \approx 0.962 \, \text{m/s} \)
Rate of flow (Discharge): \( \approx 9.62 \, \text{m}^3/\text{s} \)
Explanation of Chezy's Formula
Chezy's formula is an empirical equation used in fluid dynamics to calculate the mean velocity of uniform flow in an open channel. The formula, \(V = C \sqrt{m \cdot i}\), relates the velocity to three key factors:
- Chezy's Constant (C): An empirical value that accounts for the roughness of the channel bed and sides. A higher value of C means a smoother channel and thus a higher velocity.
- Hydraulic Mean Depth (m): Also known as the hydraulic radius, this is the ratio of the cross-sectional area of the flow to the wetted perimeter. It represents the efficiency of the channel's shape in conveying water.
- Bed Slope (i): The gradient or slope of the channel bed. This is the primary driving force for the flow, as it is the component of gravity that acts in the direction of motion.
