In a long rectangular channel \(3 \mathrm{~m}\) wide the specific energy is \(1.8 \mathrm{~m}\) and the rate of flow is \(12 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\). Calculate two possible depths of flow and the corresponding Froude numbers. If Manning's \(n=0.014\) what is the critical slope for this discharge?

In open channel flow, **specific energy** is an important concept. It refers to the total energy per unit weight of water at a particular section of a channel relative to the channel bed. It combines both the potential energy (represented by the depth of flow) and the kinetic energy (represented by the velocity of flow).

The formula for specific energy (\textbf{E}) is:

\[ E = y + \frac{V^2}{2g} \]

Here, \textbf{y} is the flow depth, \textbf{V} is the flow velocity, and \textbf{g} is the acceleration due to gravity. This equation gives us crucial insights into the conditions of the flow, helping us determine the different depths of flow and other related properties.

The **Froude number** (\textbf{Fr}) is a dimensionless number that helps characterize the type of flow in open channels. It's defined as the ratio of inertial forces to gravitational forces. This number is crucial in distinguishing between different flow regimes:

• Subcritical flow (\textbf{Fr} < 1)
• Critical flow (\textbf{Fr} = 1)
• Supercritical flow (\textbf{Fr} > 1)

The Froude number can be calculated using the formula:

\[ Fr = \frac{V}{\root g y} \]

Where \textbf{V} is the flow velocity, \textbf{g} is the acceleration due to gravity, and \textbf{y} is the flow depth. Understanding the Froude number helps in designing channels and predicting flow behaviors, ensuring the efficient and safe transport of water.

**Manning's equation** is an empirical formula used to estimate the average velocity of water flowing in an open channel. The equation is given by:

\[ V = \frac{1}{n} R^{2/3} S^{1/2} \]

Where:

• \textbf{V} is the flow velocity
• \textbf{n} is Manning's roughness coefficient
• \textbf{R} is the hydraulic radius (\textbf{R} = \textbf{A} / \textbf{P})
• \textbf{S} is the slope of the energy grade line

The hydraulic radius (\textbf{R}) is defined as the ratio of the cross-sectional area of the flow (\textbf{A}) to the wetted perimeter (\textbf{P}). For a rectangular channel, \textbf{R} simplifies to:

\[ R = \frac{y}{2y + b} \]

Manning's equation is extensively used in hydraulic engineering to design channels and estimate flow rates, considering the channel bed's roughness and other flow properties.

The **critical slope** (\textbf{S\textsubscript{c}}) in open channel flow indicates the slope at which the flow transitions from subcritical to supercritical. To find this slope, we can use Manning's equation and reorganize it for critical conditions.

The critical slope can be calculated using:

\[ S_c = \frac{(n \times Q)^{10/3}}{(A \times R^{2/3})} \]

For our example, where the width of the channel is known, we first determine the cross-sectional area (\textbf{A}) and hydraulic radius (\textbf{R}) for the depths of flow. Substituting these values into the equation will give us the critical slope. Identifying the critical slope is essential for ensuring proper channel design, as it indicates how steep the slope can be while maintaining desired flow conditions.