Question

A liquid flows with same velocity through pipes 1 & 2 having same diameter. If the length of the second pipe is twice that of first pipe, what should be the ratio of head loss in two pipes?

• A. 1:2
• B. 2:1
• C. 1:4
• D. 4:1

Hint:

When all other factors remain constant, the head loss in a pipe system is linearly proportional to the length. Doubling the length always doubles the frictional loss.

Answer 1

The Correct Option is A

Concept: The Darcy-Weisbach equation is used to calculate the head loss due to friction in a pipe.

• Formula: \( h_f = \frac{f \cdot L \cdot v^2}{2 \cdot g \cdot D} \)

• \( h_f \): Head loss, \( f \): Friction factor, \( L \): Length.

• \( v \): Velocity, \( g \): Gravity, \( D \): Diameter.

Step 1: Setting up the proportionality.
Given: Velocity (\(v\)), Diameter (\(D\)), and Friction factor (\(f\)) are identical for both pipes. From the formula, we see that \( h_f \propto L \).

Step 2: Applying the given conditions.
Let the length of Pipe 1 be \( L_1 \) and Pipe 2 be \( L_2 \). We are told that \( L_2 = 2 \cdot L_1 \). \[ \frac{h_{f1}}{h_{f2}} = \frac{L_1}{L_2} \]

Step 3: Calculating the ratio.
Substitute the relationship into the ratio: \[ \frac{h_{f1}}{h_{f2}} = \frac{L_1}{2 \cdot L_1} = \frac{1}{2} \] The ratio of head loss in pipe 1 to pipe 2 is 1:2. Final Answer: (A)