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:
The question asks for the relationship between the head loss due to friction ($h_f$) and the pipe diameter ($D$) when the discharge ($Q$) is held constant.

Step 2: Key Formula or Approach:
The head loss due to friction is given by the Darcy-Weisbach equation:
\[ h_f = \frac{f L v^2}{2 g D} \] where $f$ is the friction factor, $L$ is the pipe length, $v$ is the average velocity, $g$ is gravity, and $D$ is the pipe diameter.
The discharge ($Q$) is related to velocity and diameter by the continuity equation:
\[ Q = A v = \left(\frac{\pi D^2}{4}\right) v \] This means the velocity can be expressed in terms of the constant discharge:
\[ v = \frac{4Q}{\pi D^2} \]

Step 3: Detailed Explanation:
Now, substitute the expression for velocity ($v$) into the Darcy-Weisbach equation to see how head loss depends on diameter for a constant discharge.
\[ h_f = \frac{f L}{2 g D} \left( \frac{4Q}{\pi D^2} \right)^2 \] \[ h_f = \frac{f L}{2 g D} \left( \frac{16 Q^2}{\pi^2 D^4} \right) \] \[ h_f = \left( \frac{16 f L Q^2}{2 g \pi^2} \right) \frac{1}{D \cdot D^4} \] \[ h_f = \left( \frac{8 f L Q^2}{g \pi^2} \right) \frac{1}{D^5} \] In this problem, the discharge ($Q$), friction factor ($f$), and length ($L$) are all given as constants. Therefore, the entire term in the parenthesis is a constant.
This leaves the relationship:
\[ h_f \propto \frac{1}{D^5} \] The head loss due to friction is inversely proportional to the fifth power of the diameter.

Step 4: Final Answer:
The loss of head due to friction is inversely proportional to $D^5$.