Question

Water is flowing FULL through a rectangular tunnel of size 3 m (width) × 2 m (height). The average velocity of flow is 1 m/s. The frictional head loss is observed to be 1 m per km. Consider acceleration due to gravity (g) as 10 \(m/s^2\). The correct statement(s) is/are

• A. Hydraulic radius is 0.6 m
• B. Darcy-Weisbach friction factor is 0.048
• C. Hydraulic radius is 2 m
• D. Darcy-Weisbach friction factor is 0.024

Answer 1

The Correct Option is A, B

Step 1: Compute the hydraulic radius. The hydraulic radius (\( R_h \)) is given by: \[ R_h = \frac{\text{Area of flow}}{\text{Wetted perimeter}} \] For a rectangular cross-section of width \( b = 3 \, \text{m} \) and height \( h = 2 \, \text{m} \): \[ \text{Area of flow}, A = b \cdot h = 3 \cdot 2 = 6 \, \text{m}^2 \] \[ \text{Wetted perimeter}, P = 2(b + h) = 2(3 + 2) = 10 \, \text{m} \] Substitute these into the formula for \( R_h \): \[ R_h = \frac{A}{P} = \frac{6}{10} = 0.6 \, \text{m} \] Step 2: Compute the Darcy–Weisbach friction factor. The Darcy–Weisbach equation for head loss (\( h_f \)) is: \[ h_f = f \cdot \frac{L}{D_h} \cdot \frac{v^2}{2g} \] Rearrange to solve for \( f \) (friction factor): \[ f = \frac{h_f \cdot 2g \cdot D_h}{L \cdot v^2} \] Substitute these values: \[ f = \frac{1 \cdot 2 \cdot 10 \cdot 2.4}{1000 \cdot 1^2} = \frac{48}{1000} = 0.048 \] Step 3: Verify the options. 1. Hydraulic radius is \( 0.6 \, \text{m} \):
This is correct, as calculated in Step 1. 2. Darcy–Weisbach friction factor is \( 0.048 \):
This is correct, as calculated in Step 2. 3. Hydraulic radius is \( 2 \, \text{m} \):
This is incorrect, as \( R_h = 0.6 \, \text{m} \). 4. Darcy–Weisbach friction factor is \( 0.024 \):
This is incorrect, as \( f = 0.048 \).