Maximum velocity condition in a flow-through circular channel section is:
H = 0.81d; Area of flow = R2/2(2θ - sin 2θ)
Understanding Maximum Velocity in Circular Channels
In open channel flow, the velocity of water varies depending on the depth of flow, especially in non-rectangular sections like a circular channel. For a given discharge, the maximum velocity does not necessarily occur when the channel is full. It occurs at a specific partial depth of flow.
Condition for Maximum Velocity
The velocity in an open channel is often estimated using empirical formulas like Manning's equation or Chezy's equation. These equations relate velocity (\(V\)) to the hydraulic radius (\(R_h\)) and the slope (\(S\)).
Manning's Equation: \(V = \frac{1}{n} R_h^{2/3} S^{1/2}\)
Where:
- \(V\) is the velocity
- \(n\) is Manning's roughness coefficient
- \(R_h\) is the hydraulic radius (\(A/P\), Area of flow divided by Wetted Perimeter)
- \(S\) is the bed slope
For a circular channel with a fixed slope (\(S\)) and roughness (\(n\)), the velocity \(V\) is maximum when the hydraulic radius \(R_h\) is maximum. The hydraulic radius \(R_h = A/P\) changes with the depth of flow.
Depth of Flow for Maximum Velocity
Calculations show that the hydraulic radius, and thus the velocity, is maximized in a circular channel when the depth of flow (\(H\)) is approximately 0.81 times the diameter (\(d\)) of the channel.
This critical depth for maximum velocity is given by:
\(H \approx 0.81d\)
Or, in terms of radius (\(R = d/2\)):
\(H \approx 0.81 \times 2R = 1.62R\)
Area of Flow Calculation for a Circular Section
For a partially filled circular channel, the area of flow is the area of the circular segment below the water surface. If \(R\) is the radius of the pipe and \(2\theta\) is the angle (in radians) subtended by the water surface at the center of the circle, the area of flow (\(A\)) is given by:
\(A = \text{Area of sector} - \text{Area of triangle}\)
\(A = \frac{1}{2}R^2 (2\theta) - \frac{1}{2}R^2 \sin(2\theta)\)
\(A = R^2 \theta - \frac{1}{2}R^2 \sin(2\theta)\)
\(A = \frac{R^2}{2}(2\theta - \sin 2\theta)\)
This formula for the area of flow is standard for a circular segment where \(2\theta\) is the central angle subtended by the water surface chord.
Connecting Depth and Area for Maximum Velocity
The condition for maximum velocity requires the depth of flow to be \(H \approx 0.81d\). The formula for the area of flow for a partial circle is \(A = \frac{R^2}{2}(2\theta - \sin 2\theta)\).
Comparing this with the given options, the condition that matches the established depth for maximum velocity and the standard area formula is:
- Depth \(H = 0.81d\)
- Area of flow \(A = \frac{R^2}{2}(2\theta - \sin 2\theta)\)
Therefore, the maximum velocity condition in a flow-through circular channel section is characterized by these two parameters.
| Parameter | Condition for Maximum Velocity |
|---|---|
| Depth of Flow (H) | \(\approx 0.81d\) (where \(d\) is the diameter) |
| Area of Flow (A) | \(\frac{R^2}{2}(2\theta - \sin 2\theta)\) (where \(R\) is radius, \(2\theta\) is central angle) |
Conclusion
The condition for maximum velocity in a circular channel occurs at a specific depth, which maximizes the hydraulic radius. This depth is approximately 0.81 times the channel diameter, and the area of flow at this depth is calculated using the formula for a circular segment based on the central angle subtended by the water surface.
Revision Table: Circular Channel Flow Conditions
| Condition | Depth (H) | Area (A) | Wetted Perimeter (P) | Notes |
|---|---|---|---|---|
| Maximum Velocity | \(\approx 0.81d\) | \(\frac{R^2}{2}(2\theta - \sin 2\theta)\) | \(R(2\theta)\) | Maximizes Hydraulic Radius (\(R_h = A/P\)) |
| Maximum Discharge | \(\approx 0.95d\) | \(R^2(\theta - \frac{1}{2}\sin 2\theta)\) at this depth | \(R(2\theta)\) at this depth | Maximizes \(A \cdot R_h^{2/3}\) |
| Full Pipe | \(d\) | \(\pi R^2\) | \(\pi d\) or \(2\pi R\) | Not max velocity or max discharge |
Additional Information on Open Channel Hydraulics
Open channel flow refers to the flow of liquid in a conduit with a free surface exposed to the atmosphere. Circular channels, such as storm sewers and culverts, are common examples.
- Hydraulic Radius: A key parameter in open channel flow calculations, defined as the ratio of the flow area to the wetted perimeter (\(R_h = A/P\)).
- Wetted Perimeter: The length of the boundary of the flow section that is in contact with the fluid.
- Manning's Roughness Coefficient (n): Represents the resistance to flow caused by the channel surface roughness.
- Maximum Discharge vs. Maximum Velocity: It is important to note that the depth for maximum velocity (\(\approx 0.81d\)) is different from the depth for maximum discharge (\(\approx 0.95d\)). Maximum discharge occurs when the product of the area and a power of the hydraulic radius (e.g., \(A \cdot R_h^{2/3}\) in Manning's equation) is maximized.
Understanding these conditions is crucial for the efficient design and analysis of circular drainage systems.
Important Questions from Design of Sewer
Which of the following statements is NOT correct?
The average daily flow of wastewater generated from residential, commercial, and industrial sources, specifically excluding any contribution from rainfall or stormwater, is commonly referred to as:
As per NBO, the gradient required to generate self-cleansing velocity for a 100 mm Φ sewer is _____.
Which of the following problems does proper sewer ventilation help to mitigate?
Maximum discharge through a circular channel takes place when depth of flow is equal to
