Question

For a long pipe of diameter D with a given discharge, fiction factor and length of pipe, the loss of head due to friction ($h_f$) is

• A. inversely proportional to D
• B. directly proportional to D
• C. inversely proportional to $D^3$
• D. inversely proportional to $D^5$

Hint:

Be careful to distinguish between problems with constant velocity and constant discharge.
- If velocity ($v$) is constant: $h_f \propto 1/D$.
- If discharge ($Q$) is constant: $h_f \propto 1/D^5$.
The second case is much more common in practical pipe flow problems.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Constant Q, f, L. How h_f varies with D?

Step 2: Key Derivation (Alternate):
h_f = fLv²/(2gD). v = Q/A = 4Q/(πD²). Substitute: h_f ∝ 1/D⁵.

Step 3: Detailed Explanation:
h_f ∝ v²/D ∝ (1/D²)²/D = 1/D⁴×1/D = 1/D⁵. Doubling diameter reduces head loss by factor 32. Very sensitive to diameter.

Step 4: Final Answer:
Inversely proportional to D⁵.