Kennedy's Theory for Critical Velocity in Irrigation Channels

Understanding the critical velocity in irrigation channels is fundamental for designing stable and efficient canal systems. Kennedy's theory is an empirical method used to determine this critical velocity, which represents the mean velocity of flow that prevents both silting (deposition of sediment) and scouring (erosion of the bed and banks) in an unlined channel.

Calculating Critical Velocity with Kennedy's Theory

According to Kennedy's theory, the critical velocity (\(V_0\)) for a channel is given by the formula:

\[V_0 = 0.55 \times m \times D^{0.64}\]

Where:

• \(V_0\) is the critical velocity in meters per second (m/sec).
• \(m\) is the critical velocity ratio (C.V.R.), which is a factor that depends on the type and fineness of the silt. For standard silt, \(m\) is typically 1.0. For finer silt, \(m\) is less than 1, and for coarser silt, \(m\) is greater than 1.
• \(D\) is the depth of flow in meters (m).

Given Parameters for the Irrigation Channel

The problem provides us with the following data for the irrigation channel:

• Depth of flow, \(D = 1.75 \text{ m}\)
• Value of critical velocity ratio, \(m = 1.4\)

Step-by-Step Determination of Critical Velocity

To determine the critical velocity of the channel, we will substitute the given values into Kennedy's formula:

1. Substitute the provided values into the formula:

   \[V_0 = 0.55 \times 1.4 \times (1.75)^{0.64}\]

2. Perform the multiplication of the constant term and the critical velocity ratio:

   First, multiply \(0.55\) by the critical velocity ratio \(m = 1.4\):

   \[0.55 \times 1.4 = 0.77\]

3. Formulate the final expression for the critical velocity:

   Now, substitute this calculated value back into the formula to get the final expression for \(V_0\):

   \[V_0 = 0.77 \times (1.75)^{0.64}\]

Comparing the Result with Options

Let's compare our derived expression for the critical velocity with the given options to find the matching answer:

The calculated critical velocity expression, \(0.77 \times 1.75^{0.64}\), perfectly matches the expression provided in Option 4. This demonstrates the application of Kennedy's theory for determining the critical velocity in an irrigation channel considering the critical velocity ratio and depth of flow.