Problem Statement
The discharge of water through a rectangular channel of width 6 m, is 18 m³/s when depth of flow of water is 2 m. Calculate : (i) specific energy of the flowing water, (ii) critical depth and critical velocity and (iii) value of minimum specific energy.
Given Data & Constants
- Width of channel, \(B = 6 \, \text{m}\)
- Discharge, \(Q = 18 \, \text{m}^3/\text{s}\)
- Depth of flow, \(d = 2 \, \text{m}\)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)
Solution
(i) Specific Energy of the Flowing Water (\(E\))
First, we calculate the area of flow and the velocity.
Now, we calculate the specific energy, which is the sum of the depth head and the velocity head.
(ii) Critical Depth (\(d_c\)) and Critical Velocity (\(V_c\))
First, we find the discharge per unit width (\(q\)).
Now, we can calculate the critical depth and critical velocity.
(iii) Value of Minimum Specific Energy (\(E_{min}\))
The minimum specific energy occurs at the critical depth.
(i) Specific energy: \( \approx 2.115 \, \text{m} \)
(ii) Critical depth: \( \approx 0.972 \, \text{m} \), Critical velocity: \( \approx 3.086 \, \text{m/s} \)
(iii) Minimum specific energy: \( \approx 1.458 \, \text{m} \)
Explanation of Concepts
- Specific Energy (E): This is the total energy of the flow per unit weight of water, relative to the channel bed. It's the sum of the potential energy (depth, \(d\)) and the kinetic energy (velocity head, \(V^2/2g\)).
- Critical Depth (\(d_c\)): For a given discharge, this is the depth at which the specific energy is at its absolute minimum. It represents a transition point between subcritical (slow, deep) flow and supercritical (fast, shallow) flow.
- Critical Velocity (\(V_c\)): This is the velocity of the flow when it is at the critical depth.
- Minimum Specific Energy (\(E_{min}\)): This is the value of the specific energy when the flow is at the critical depth. It's the lowest possible energy state for a given discharge.
