An open channel of trapezoidal section, \(2.5 \mathrm{~m}\) wide at the base and having sides inclined at \(60^{\circ}\) to the horizontal, has a bed slope of 1 in \(500 .\) It is found that when the rate of flow, is \(1.24 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\) the depth of water in the channel is \(350 \mathrm{~mm}\). Assuming the validity of Manning's formula, calculate the rate of flow when the depth is \(500 \mathrm{~mm}\).

Manning's formula is a crucial tool in open channel flow calculations. It is used to determine the velocity or flow rate of water through a channel based on the channel's physical characteristics and surface roughness. The formula is: `Q = (1/n) * A * R^(2/3) * S^(1/2)` Here:

• `Q` is the flow rate (m³/s)

• `A` is the cross-sectional area of flow (m²)

• `R` is the hydraulic radius (m)

• `S` is the slope of the channel bed (dimensionless)

• `n` is the Manning's roughness coefficient

Manning's formula is widely used due to its simplicity and effectiveness, especially for natural channels where flow conditions can be irregular.

The hydraulic radius is an important concept in open channel flow as it relates to the efficiency of the channel in conveying water. It is defined as: `R = A / P` Where:

• `R` is the hydraulic radius

• `A` is the cross-sectional area of the flow

• `P` is the wetted perimeter

The hydraulic radius helps to account for the shape of the channel by considering both the area of flow and the perimeter in contact with water. A higher hydraulic radius generally means a more efficient channel. For a trapezoidal channel, calculating the hydraulic radius is a bit more complex due to the shape, but it follows the same general principle.

A trapezoidal channel is a type of open channel that has a flat bottom and sides that slope outward. This shape is common in man-made channels because it is relatively easy to construct and maintain. The key parameters for a trapezoidal channel include:

• Base width: The width of the bottom section.

• Side slopes: The angle at which the sides are inclined from the base.

Calculating Areas and Perimeters

To calculate the cross-sectional area of a trapezoidal channel, the formula used is: `A = 1/2 (base width + top width) * depth` The top width can be found by considering the depth and side slopes. The wetted perimeter (P), which is the contact length between water and the channel, is calculated as: `P = base width + 2 * side length` Side length itself can be found using the trigonometric relationship involving the depth and the cosine of the slope angle.

Flow rate calculation in open channel flow involves using Manning's formula and requires accurate determination of parameters such as area, hydraulic radius, and slope. Here’s a step-by-step approach:

Determine Cross-Sectional Area (A)

Find the cross-sectional area by considering the channel shape and depth

• For a trapezoidal channel, use `A = 1/2 (base width + top width) * depth`

Compute Wetted Perimeter (P)

Calculate the wetted perimeter with:

• `P = base width + 2 * side length`

Calculate Hydraulic Radius (R)

You can find the hydraulic radius using `R = A / P`

• Ensure you have the correct hydraulic radius as it impacts the final flow rate.

Apply Manning's Formula

Finally, use the Manning's formula to compute `Q` (flow rate):

• `Q = (1/n) * A * R^(2/3) * S^(1/2)`

This requires the Manning's roughness coefficient, which is often based on the channel material and condition.