Question

A rectangular venturi of width \(10 \, \text{m}\) is designed to carry excess runoff. Compute the specific energy of water when discharge is \(30 \, \text{m}^3/\text{sec}\) and depth of water is \(1.5 \, \text{m}\).

• A. \(1.70 \, \text{m}\)
• B. \(2.00 \, \text{m}\)
• C. \(0.85 \, \text{m}\)
• D. \(4.80 \, \text{m}\)

Hint:

Specific energy equation: \[ \boxed{ E = y + \frac{V^2}{2g} } \] For rectangular channels: \[ \boxed{ V = \frac{Q}{by} } \]

Answer 1

The Correct Option is A

Concept: Specific energy in open channel flow is defined as: \[ E = y + \frac{V^2}{2g} \] where:
• \(E\) = specific energy
• \(y\) = depth of flow
• \(V\) = velocity of flow
• \(g\) = acceleration due to gravity The first term represents potential energy and the second term represents kinetic energy.

Step 1: Writing the given data. Width of rectangular channel: \[ b = 10 \, \text{m} \] Discharge: \[ Q = 30 \, \text{m}^3/\text{sec} \] Depth of flow: \[ y = 1.5 \, \text{m} \]

Step 2: Calculating flow area. For rectangular channel: \[ A = by \] Substituting values: \[ A = 10 \times 1.5 \] \[ A = 15 \, \text{m}^2 \]

Step 3: Calculating velocity of flow. Using discharge relation: \[ Q = AV \] Thus: \[ V = \frac{Q}{A} \] Substituting values: \[ V = \frac{30}{15} \] \[ V = 2 \, \text{m/sec} \]

Step 4: Calculating velocity head. Velocity head: \[ \frac{V^2}{2g} \] Substituting values: \[ \frac{(2)^2}{2 \times 9.81} \] \[ = \frac{4}{19.62} \] \[ = 0.204 \, \text{m} \]

Step 5: Calculating specific energy. \[ E = y + \frac{V^2}{2g} \] Substituting values: \[ E = 1.5 + 0.204 \] \[ E = 1.704 \, \text{m} \] Approximately: \[ \boxed{ E \approx 1.70 \, \text{m} } \]

Step 6: Checking all options carefully. Option (A): Matches calculated value. \[ \boxed{ \text{Option (A) is correct} } \] Option (B): Incorrect. \[ \boxed{ \text{Option (B) is incorrect} } \] Option (C): Too small. \[ \boxed{ \text{Option (C) is incorrect} } \] Option (D): Too large. \[ \boxed{ \text{Option (D) is incorrect} } \] Final Conclusion: Specific energy of water is: \[ \boxed{ 1.70 \, \text{m} } \] Hence the correct answer is: \[ \boxed{ (A) } \]