Question

Liquid is flowing through two pipes having the same dimensions and same material. If the ratio of velocities are 2:3, with all other factors remaining same, the ratio of loss of head due to friction is

• A. \( 4:9 \)
• B. \( 9:4 \)
• C. \( 2:3 \)
• D. \( 8:27 \)

Hint:

The Darcy-Weisbach equation \( h_f = f \frac{L}{D} \frac{V^2}{2g} \) is fundamental for calculating head loss due to friction. When comparing head losses between pipes with same dimensions and material, and the flow is typically turbulent, assume the friction factor \( f \) is constant. In this case, head loss is directly proportional to the square of the velocity (\( h_f \propto V^2 \)).

Answer 1

The Correct Option is A

Given:

• Two pipes with identical dimensions (same diameter \(D\) and length \(L\)) and material
• Velocity ratio \(v_1 : v_2 = 2 : 3\)
• All other factors remain the same

Step 1: Darcy-Weisbach Equation
The head loss due to friction in pipe flow is given by: \[ h_f = f \frac{L}{D} \frac{v^2}{2g} \] where:

• \(h_f\) = head loss due to friction
• \(f\) = Darcy friction factor (same for both pipes)
• \(L\) = pipe length (same)
• \(D\) = pipe diameter (same)
• \(v\) = flow velocity
• \(g\) = gravitational acceleration (constant)

Step 2: Proportionality
Since \(f\), \(L\), \(D\), and \(g\) are identical for both pipes: \[ h_f \propto v^2 \] Therefore, the ratio of head losses is: \[ \frac{h_{f1}}{h_{f2}} = \left(\frac{v_1}{v_2}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]
Step 3: Conclusion
The ratio of head loss due to friction is \(4:9\). \framebox[1.1\width]{Answer: (1) \(4:9\)}